Multiple Choice Questions in Mathematics by Jimmy Ocampo

MULTIPLE CHOICE QUESTIONS IN

MATHEMATICS Fausto Uy and Jimmy Ocampo

1. A sequence of numbers such that the same quotient is obtained by dividing a term by the preceding term is called A. arithmetic progression B. harmonic progression C. infinite progression D. geometric progression 2.If x is an irrational number not equal to zero and x2 = N, then which of the following best describes N? A. N is a natural number. B. N is any rational number. C. N is a positive rational number. D. N is a positive integral number. 3. In the expression an , the number n is referred to as the A. power B. exponent C. degree D. index 4. The polynomial 2x3 y + 8xyz4 – 3x2y3 has a degree of A. 6 B. 3 C. 4 D. 8 5. The equations x + y = 2 and 2x + 2y = 8 are examples of equations which are A. dependent B. independent C. conditional D. inconsistent 6. A non-terminating and non-periodic decimal is A. rational B. prime C. irrational D. imaginary

7. The probability that an event is certain to occur is A. greater than one B. less than one C. equal to one D. equal to zero 8. Radicals can be added to form a single radical if they have the same radicand and the same A. power B. exponent C. index D. coefficient 9. A set of elements that is taken without regard to the order in which the elements are arranged is called a A. sequence B. permutation C. combination D. progression 10. If b = 0, then the number a + bi is A. complex B. real C. imaginary D. irrational 11. How many prime numbers are there between 200 and 210? A. one B. three C. none D. two 12. In the expression A. power B. exponent C. index D. radicand

, the number n is called the

13. A harmonic progression is a sequence of numbers such that the reciprocals of the numbers will form a A. geometric progression B. arithmetic progression C. infinite progression D. finite progression 14. If a, b and c are rational numbers and if b2 – 4ac is positive but not perfect square, then the roots of the quadratic equation ax2 + bx + c = 0 are A. real, irrational and unequal B. real, rational and unequal C. real, rational and equal D. real, irrational and unequal 15. The equation xy = 0 implies that A. x = 0 and y = 0 B. x = 0 or y = 0 C. x = 0 and y is not equal to zero D. x = 0 or y is not equal to zero 16. Which of the following events are mutually exclusive? A. event “Ace” and event “black card” B. event “Queen” and event “heart” C. event “Ten” and event “Spade” D. event “diamond” and event “club” 17.Which of the following best describes (-3)1/2? A. irrational number B. pure imaginary number C. natural number D. complex number 18. It is a sequence of numbers such that the successive terms differ by a constant. A. geometric progression B. arithmetic progression C. harmonic progression D. infinite progression

19. If the discriminant of a quadratic equation is greater than zero, the roots of the equation are A. real and equal B. real and distinct C. complex and unequal D. imaginary and distinct 20. Which of the following terms is not rational in x? A. 6x2 B. -4x C. x4 D. 21. In the theory of sets, the relation (A ᴗ B)’ = A’ ᴖ B’ expresses which of the following laws on set operations? A. De Morgan’s Law B. Involution Law C. Complement Law D. Identity Law 22. The set of odd integers is closed under the operation of A. addition B. subtraction C. multiplication D. division 23. Which of the following law states that the factors of a product may be grouped in any manner without affecting the result? A. commutative law B. associative law C. distributive law D. inverse law 24. Which of the following terms has a degree of 4? A. x4 y B. xy4 C. 4xy D. xy3

25. The product of two conjugate complex numbers is A. a real number B. an imaginary number C. zero D. an irrational number 26. The statement “The examinees are not more than 30 years old.” implies that the examinees are A. less than 30 years old B. at least 30 years old C. 30 years old or less D. 30 years old or more 27. The closure property of numbers is not satisfied by the set of all integers under the operation of A. addition B. multiplication C. subtraction D. division 28. The conditional probability of B given A is denoted symbolically by P(B/A). If P(B/A) = P(B), then the events A and B are A. dependent B. independent C. mutually inclusive D. disjoint 29. What is the value of k that will make x2 – 28x + k a perfect square trinomial? A. 100 B. 121 C. 144 D. 196 30. The roots of 6x2 + 7x + 34 = 0 are A. real and equal B. real and unequal C. complex and unequal D. pure imaginary

31. What is the conjugate of -6 A. 6 B. -6 C. 6i D. -6i 32. Which of the following is true? A. B. (a + b)2 = a2 + b2 C. a / (b – c) = a/b – a/c D. 33. Which of the following cannot be a probability value? A. (0.99)4 B. 88/100 C. D. (0.5)-1 34. How many subsets has the set {c, u, t, e}? A. 12 B. 14 C. 16 D. 18 35. Using the remainder theorem, find the remainder when x6 – x + 1 is divided by x – 2. A. 61 B. 62 C. 63 D. 64 36. What is the sum of the numerical coefficient of (2x – y)20? A. zero B. one C. greater than one D. less than one

37. How many subsets of one or more elements can be formed from a set containing 12 elements? A. 4,096 B. 4,095 C. 4,094 D. 4,093 38. What is the product of

and

?

A. 6i B. -6i C. 6 D. -6 39. Which of the following is an irrational number? A. (16)3/4 B. 0.0075 C. 1.36363636... D. 3(5)1/2 40. Two prime numbers which differ by 2 are called prime twins. Which of the following pairs of numbers are prime twins? A. 1 and 3 B. 7 and 9 C. 17 and 19 D. 13 and 15 41. If A ᴖ B ᴖ C is not equal to zero, then which of the following notations refers to the set of elements found in A and B but not in C? A. A ᴖ B ᴗ C’ B. A ᴗ B ᴖ C’ C. A ᴖ B ᴖ C’ D. A ᴗ B ᴗ C’ 42. Which of the following sequence is a geometric progression? A. 16, 12, 8, ... B. 16, 8, 2, ... C. 16, 12, 9, ... D. 16, 14, 12, ...

43. The point (x, y) where x = 2 and y = -x is in what quadrant? A. first B. second C. third D. fourth 44. Experiment: A die is tossed. Event: A prime number results. Which of the following is not an outcome of the event? A. 1 B. 2 C. 3 D. 5 45. The logarithmic equation equivalent to 1/a = bc is A. logc(1/a) = b B. logb(1/a) = c C. logc b = 1/a D. logb c = 1/a 46. If P(A) = 0.78 and P(B) = 0.35, what is P(A’) + P(B’) ? A. 0.83 B. 0.85 C. 0.87 D. 0.89 47. Which of the following is a polynomial in x ? A. x -2 + x + 4 B. + 3x + 5 C. x3 + 2x + 3 D. 4/x + 3x + 1 48. If a set A has 1,024 subsets, how many elements does A contain? A. 8 B. 9 C. 10 D. 11

49. Which of the terms in the expansion (y3 + y -1)10 will involve y2 ? A. 6th term B. 7th term C. 8th term D. 9th term 50. P(A) = 0.60 and P(B’) = 0.30 while P(AᴖB) = 0.15, find P(AᴖB’) by using Venn Diagram. A. 0.90 B. 0.30 C. 0.45 D. 0.75 51. Evaluate (i – 1)8. A. 16 B. -16 C. 16i D. -16i 52. If 16 is 4 more than 3x, then 2x – 5 = A. 2 B. 3 C. 4 D. 5 53. In the series 2, -4, 8, -16, x, -64, ..., what is x? A. -24 B. -32 C. 24 D. 32 54. If a, b, 2b, -a, ... is an arithmetic progression, find the next term. A. 2b – 3a B. 3b – 2a C. 2b + 3a D. 2b + a

55. In how many ways can 6 boys be seated in a row? A. 520 B. 620 C. 720 D. 820 56.

is true only if

A. x > 2y B. x = 2y C. x = 2y 57. Find the fourth proportional to 3, 5 and 21. A. 27 B. 56 C. 65 D. 35 58. Give the value of –(-1/27)-2/3 A. 9 B. -9 C. 1/9 D. -1/9 59. Simplify (a -1 + b -1) / (ab) -1 A. ab B. b + a C. 1/ab D. a/b 60. If a die is thrown once, what is the probability of getting a prime number? A. 1/3 B. ¼ C. ½ D. 1/6

61. Which of the following are similar radicals A. and B.

and

C.

and

D.

and

62. Evaluate x = log 2 8 A. 4 B. 3 C. 2 D. 1 63. What is the greatest common factor (GCF) of 48 and 72 ? A. 12 B. 24 C. 36 D. 42 64. If x, y and 5x are three consecutive terms of an arithmetic progression whose sum is 81, find x. A. 9 B. 10 C. 11 D. 12 65. If f(x) = 2x3 – 3x + 1, then f(1) = A. 0 B. 1 C. 2 D. 3 66. Find the sum to infinity of 3 -1, 3 -3, 3 -5, ... A. 1/8 B. 3/8 C. 7/8 D. 5/8

67. Find the value of x if

3 2 1 x

= 10

A. 3 B. 4 C. 5 D. 6 68. Find the value of x in the series 1, 8, 27, x, 125, ... A. 100 B. 81 C. 30 D. 64 69. Find the least common multiple (LCM) of 72x3y2, 108x2 y3 and 9x2 y A. 108x3y3 B. 648x3 y3 C. 972x3 y3 D. 216x3y3 70. Evaluate (-1/27)-2/3 + (-1/32)-2/5 A. 6.25 B. 3.25 C. 9.25 D. 7.25 71. For what values of x will (x+3) < 2(2x+1)? A. x=3 B. x>1/3 C. x0) and the x-axis is equal to 8/3, find k. A. -1 B. 1 C. 2 D. -3 487. Evaluate A. [(4x2+1)3/2]/20 + C B. [(4x2+1)3/2]/8 + C C. [(4x2+1)5/2]/20 + C D. [(4x2+1)5/2]/8 + C

.

488. The length of the arc of the curve y = ln sec x from x = 0 to x = pi/3 is A. 1.4170 B. 1.3170 C. 1.2170 D. 1.1170 489. If

and

evaluate A. 3 B. 7 C. 6 D. 5 490. Find the area bounded by y=x2-1 and y=3. A. 31/3 B. 32/3 C. 35/3 D. 37/3

,

491. Integrate A. B. C. D. 492. Find the moment of inertia with respect to the x-axis of the area bounded by y2 = 4x, y = 4 and x = 0. A. 21.2 B. 31.2 C. 41.2 D. 51.2 493. Find the y-coordinate (ŷ) of the centroid of the first-quadrant area under the curve y = ex between x = 0 and x = 1. A. 0.91 B. 0.93 C. 0.95 D. 0.97 494. Evaluate A. 1.7726 B. 1.7627 C. 1.6772 D. 1.6727 495. Find the integral of

from x = 0 to x = 1.

A. pi/6 B. pi/7 C. pi/8 D. pi.9 496. Find the area bounded by y2 = 1 – x, y = x -2, y=1 and y=-1. A. 7/3 B. 8/3 C. 10/3 D. 11/3

497. If the second-degree equation Ax2 + Bxy + Cy2 +Dx + Ey + F = 0 represents a real conic and B2 – 4AC is positive, then it is a. ellipse b. circle c. parabola d. hyperbola 498. If the slopes of two lines are equal and their y-intercepts are different, then the lines are a. intersecting b. parallel c. coincident d. perpendicular 499. A line with inclination between 0° and 90° has a. zero slope b. no slope c. positive slope d. negative slope 500. The parabola x2 – 4x + 2y – 6 = 0 opens a. downward b. upward c. to the right d. to the left 501. The locus of a point on a circle which rolls without slipping on a straight line is called a. strophoid b. trochoid c. astroid d. cycloid 502. If b2 – 4ac < 0, then the graph of y = ax2 + bx + c a. crosses the x-axis once b. crosses the x-axis twice c. does not cross the x-axis d. touches the x-axis once 503. The point (4,y) where y < 0 lies in quadrant a. I b. II c. III d. IV

504. The slope of a vertical line is a. zero b. one c. 90° d. undefined 505. The graph of y2 – 1 = 0 is a. a pair of parallel lines b. a pair of intersecting lines c. a parabola d. a point 506. The curve y = x3 is symmetric with respect to a. the z-axis b. the y-axis c. the origin d. both axes 507. The polar equation of the line parallel to the polar axis and 4 units above it is a. r = 4cscθ b. r = 4secθ c. r = 4sinθ d. r = 4cosθ 508. The equation y2 + 12y + 36 = 0 represents a. two parallel lines b. two intersecting lines c. a point d. a straight line 509. If C = 0, then the graph of the line Ax + By + C = 0 a. is parallel to the x-axis b. is parallel to the y-axis c. crosses the positive x-axis d. passes through the origin 510. If the inclination θ of a line is an obtuse angle, then the tangent of θ is a. positive b. negative c. zero d. infinity

511. Which of the following as no graph? a. x2 + y2 – 9 = 0 b. x2 + y2 + 9 = 0 c. x2 – y2 – 9 = 0 d. x2 – y2 + 9 = 0 512. The ellipse is symmetric with respect to a. the x-axis only b. the y-axis only c. the origin only d. both axes and the origin 513. The circle x2 + y2 = 100 has a radius of a. 25 b. 30 c. 10 d. 50 514. If the eccentricity of a conic is 3/5, then it is a. an ellipse b. a circle c. a parabola d. a hyperbola 515. The graph of the polar equation r(2 + 4sinθ) = 3 is a. a circle b. a hyperbola c. a parabola d. an ellipse 516. if a line slants downward to the right, then it has a. negative slope b. positive slope c. no slope d. zero slope 517. the equation of the directrix of the parabola x2 =16y is a. x + 4 = 0 b. x – 4 = 0 c. y – 4 = 0 d. y + 4 = 0

518. the locus of a point such that its radius vector is proportional to its vectorial angle is called the a. Conchoid of Nicomedes b. Spiral of Archimedes c. Cissoid of Diocles d. Folium of Descartes 519. If A = 0 and B∙C ≠ 0, then the line Ax + By + C = 0 is a. parallel to the x-axis b. parallel to the y-axis c. perpendicular to the x-axis d. coincident with the y-axis 520. The graph of the equation 4y2 = 8 – x2 is a. a circle b. an ellipse c. a parabola d. a hyperbola 521. If the directed distance from a point to the line is negative, then which of the following is true? a. The point and the origin are not on the side of the line. b. The point and the origin are on the opposite sides of the line. c. The point is below the line. d. The point is above the line. 522. It is the locus of a point which moves in a plane so that the sum of its distance from two fixed points is constant. a. a circle b. a parabola c. an ellipse d. a hyperbola 523. If M is a point that is 1/3 of the distance from point A to point B, then M divides the line segment AB in the ratio a. 1:3 b. 1:2 c. 2:3 d. 1:4 524. Which of the following is the polar equation of a limacon? a. r = 1 + sinθ b. r = 2(1 – sinθ) c. r = 2 – sinθ d. r = 2sinθ

525. A line will have a positive slope under which of the following conditions? a. positive x-intercept and positive y-intercept b. negative x-intercept and positive y-intercept c. negative x-intercept and negative y-intercept d. both b and c 526. If two lines with slopes m1 and m2 are perpendicular to each other, then which of the following relations is true? a. m1 = m2 b. m1m2 = -1 c. m1/m2 = -1 d. m1 – m2 = 1 527. The graph of y2 + 4x = 0 has symmetry with respect to the a. x-axis only b. y-axis only c. origin only d. all of a, b and c 528. If the eccentricity of a conic is greater than one, then it is a a. an ellipse b. a circle c. a parabola d. a hyperbola 529. The graph of Ax2 + Cy2 + Dx +Ey +F = 0 where A and C are not both zero is a parabola if a. AC = 0 b. AC > 0 c. AC < 0 d. AC ≠ 0 530. Which of the following curves is symmetric with respect to the x-axis? a. y2 = 2x3 b. y = 2x3 c. xy = 2 d. y = 3x2 531. the graph of a limacon r = a + bcosθ has an inner loop if a. a = b b. 0 < a/b < 0 c. ab = 1 d. 0 < b/a < 1

532. Which of the following is the equation of a pair of parallel lines? a. y2 – x2 = 0 b. x2 + y2 +7 = 0 c. y2 + 4y = 0 d. x2 – 6x + 9 = 0 533. Which of the following is an equation of a pair of semicubical parabola? a. y = x3/2 b. y = x1/2 c. y = x4 d. y = 1/x 534. The graph of 3x2 – y = y2 + 6x is a. a parabola b. an ellipse c. a circle d. a hyperbola 535. The equation Ax2 + Cy2 + Dx + Ey +F = 0 is an ellipse if a. both A and C are not zero, A = C and they have the same sign b. neither A nor C is zero, A ≠ C and they have the same sign c. both A and C are not zero, A = C and they have opposite signs d. neither A nor C is zero, A ≠ C and they have opposite signs 536. The distance between the foci of an ellipse 6x2 + 2y2 = 12 a. 4 b. 5 c. 6 d. 7 537. The distance between the directrices of an ellipse in problem 40 is a. 5 b. 6 c. 7 d. 8 538. What is the polar equation of the line passing through (3, 0°) and perpendicular to the polar axis? a. r = 3cscθ b. r = 3secθ c. r = 3cosθ d. r = 3sinθ

539. Find the equation of the radical axis of the following circles: C1: x2 + y2 – 5x +3y -2 = 0 C2: x2 + y2 + 4x – y – 7 = 0 a. 9x + 4y – 5 = 0 b. 9x – 4y + 5 = 0 c. 9x – 4y – 5 = 0 d. 9x + 4y + 5 = 0 540. Find the distance between the points A(-3,0) and B(-4,7). a. b. c. d. 541. If the slope of the line determined by the points (x,5) and (1,8) is -3, find x. a. 2 b. 1 c. 0 d. 3 542. The focus of the parabola y2 = 4x is at a. (4,0) b. (0,4) c. (1,0) d. (0,1) 543. The inclination of the line determined by the points (2,5) and (1,8) is a. 106.41° b. 107.42° c. 108.43° d. 109.44° 544. The length of the latus rectum of 27x2 + 36y2 = 972 is a. 8 b. 9 c. 10 d. 11 545. The slope of the line through the points (-4,-5) and (2,7) is a. 2 b. -2 c. 3 d. -3

546. The equivalent of x2 + y2 – y = 0 in polar form is a. r = 2cosθ b. r = 2sinθ c. r2 = 2sinθ d. r2 = 2cosθ 547. The area of the ellipse x2/64 + y2/16 = 1 is a. 30π b. 31π c. 32π d. 33π 548. Find the equation of the ellipse which has the line 2x – 3y = 0 as one of its asymptotes. a. 2x2 – 3y2 = 6 b. 3y2 – 2y2 = 6 c. 4x2 – 9y2 = 36 d. 9y2 – 4x2 = 36 549. The transverse axis of the hyperbola 36x2 – 25y2 = 900 is a. 13 b. 12 c. 11 d. 10 550. The parabola y = 3x2 – 6x + 5 has its vertex at a. (0,5) b. (1,2) c. (-1,14) d. (2,5) 551. The line 4x – 6y + 14 = 0 is coincident with the line a. 2x = 3y – 7 b. 2x = 3y + 7 c. 4x = 6y + 14 d. 4x = 14 – 6y 552. Determine the axis of symmetry of the parabola (y + 5) 2 = 24x a. y = 5 b. y = -5 c. x = 5 d. x = -5

553. Find the area of the triangle which the line 2x – 3y + 6 = 0 forms with the coordinate axes. a. 2 b. 3 c. 4 d. 5 554. The directrix of the parabola is y = 5 and its focus is at (4,-3). What is the latus rectum? a. 14 b. 15 c. 16 d. 17 555. Find the equation of the circle containing the point (1,-4) and center at the origin. a. x2 + y2 = 16 b. x2 + y2 = 17 c. x2 + y2 = 18 d. x2 + y2 = 19 556. Find the equation of the line containing the point (2,-3) and is parallel to the line 3x + y – 5 = 0. a. 3x + y – 1 = 0 b. 3x + y – 4 = 0 c. 3x + y – 2 = 0 d. 3x + y – 3 = 0 557. The distance from the point (2,1) to the line 4x – 3y + 5 = 0 is a. -2 b. 2 c. -3 d. 4 558. If the slope of the line (k + 1)x + ky – 3 = 0 is -2, find k. a. 2 b. 1 c. -3 d. -2 559. Write the equation of the line with x-intercept -6 and y-intercept 3. a. x + 2y – 6 = 0 b. x – 2y – 6 = 0 c. x – 2y + 6 = 0 d. x + 2y + 6 = 0

560. Write the equation of the tangent line to the circle x2 + y2 = 80 at the point in the first quadrant where x = 4. a. x – 2y – 20 = 0 b. x – 2y + 20 = 0 c. x + 2y – 20 = 0 d. x + 2y + 20 = 0 561. If the distance between (8,7) and (3,y) is 13, what is the value of y? a. -5 or 19 b. 5 or 19 c. 5 or -19 d. -5 or -19 562. If the major axis of an ellipse is twice its minor axis, find its eccentricity. a. 0.965 b. 0.866 c. 0.767 d. 0.668 563. The center of the circle x2 + y2 – 18x +10y +25 = 0 is a. (9,5) b. (-9,5) c. (-5,9) d. (9,-5) 564. Compute the area of the polygon with vertices at (6,1), (3,-10), (-3,-5) and (-2,0). a. 60 b. 50 c. 40 d. 30 565. A line with the equation y = mx + k passes through the points (-1/3,-6) and (2,1). Find m. a. 2 b. 3 c. 4 d. 5

566. Find the tangential distance from the point (8,5) to the circle (x – 2)2 + (y – 1)2 = 16. a. 7 b. 8 c. 9 d. 6 567. Find the equation of the line through (-1,3) and is perpendicular to the line 5x – 2y + 3 = 0. a. 2x + 5y – 13 = 0 b. 2x + 5y – 12 = 0 c. 2x + 5y – 11 = 0 d. 2x + 5y – 10 = 0 568. Find the distance between the two lines represented by the two linear equations 4x – 3y – 12 = 0 and 4x – 3y + 8 = 0. a. 8 b. 6 c. 5 d. 4 569. The distance between the points (sinθ,cosθ) and (cosθ,-sinθ) is a. 1 b. 2 c. d. 570. Find the equation of the line parallel to 3x + 4y + 2 = 0 and -3 units from it. a. 3x + 4y + 13 = 0 b. 3x + 4y – 13 = 0 c. 3x + 4y + 17 = 0 d. 3x + 4y – 17 = 0 571. If the circle has its center (-3,1) and passes through (5,7), then its radius is a. 7 b. 8 c. 9 d. 10

572. Find the area of the triangle whose vertices lie at A, B and C whose coordinates are (4,1), (6,2) and (2,-5), respectively. a. 4 b. 5 c. 6 d. 7 573. Express y3 = 4x2 in polar form a. r = 4cot2θcscθ b. r = 4cotθcsc2θ c. r = 4cot2θcsc2θ d. r = 4cotθcscθ 574. If the slopes of the lines L1 and L2 are 3 and -1 respectively, find the angle between them measured counterclockwise from L1 to L2. a. 64.33° b. 36.43° c. 63.43° d. 43.36° 575. What is the length of the latus rectum of a hyperbola with foci at (-3,15) and (-3,-5) and a transverse axis equal to 12? a. 44/3 b. 54/3 c. 64/3 d. 74/3 576. If the line through (-1,3) and (-3,-2) is perpendicular to the line through (-7,4) and (x,2), find x if x is positive. a. 3 b. 2 c. 4 d. 1 577. Determine k so that the line y = kx – 3 will be parallel to the line 4x + 12y = 12. a. 1/2 b. 1/3 c. -1/3 d. -3

578. Find the equation of the parabola with focus at (0,8) and directrix y + 8 = 0. a. x2 = -32y b. x2 = 32y c. y2 = -32x d. y2 = 32x 579. find the tangent of the angle from the line through (-2,-3) and (4,3) to the line through (1,6) and (3,-2) a. 3 b. 4 c. 2 d. 1 580. The second-degree equation 19x2 + 6xy 11y2 + 20x – 60y +80 = 0 represents a conic. To remove the xy-term, we rotate the coordinate axes through an angle of a. 16.40° b. 17.41° c. 18.43° d. 19.45° 581. Find the value of k given that the slope of the line joining (3,1) and (5,k) is 2. a. 2 b. 3 c. 4 d. 5 582. If the focus of a parabola is at (-6,0) and its vertex is at (0,0), the equation of its directrix is a. x + 6 = 0 b. x – 6 = 0 c. x + 3 = 0 d. x – 3 = 0 583. For what value of k is the line 6y + (2k – 1 )x = 12 perpendicular to the line 3y – 2x = 6? a. 5 b. 4 c. 3 d. 2

584. The circumference of the circle x2 + y2 – 8x +2y + 8 = 0 is a. 18.85 b. 17.85 c. 16.85 d. 15.85 585. If the perpendicular distance from the line kx – 3y + 15 = 0 to the point (2,1) is -4, find k. a. -4 b. -3 c. -2 d. -1 586. The eccentricity of the hyperbola 16(y – 6)2 – 9(x – 7)2 = 144 is equal to a. 4/3 b. 5/3 c. 7/3 d. 9/4 587. If the tangent of the angle from the line through (6,y) and (-4,2) to the line through (6,6) and (3,0) is 8/9, find the value of y if y is positive. a. 4 b. 5 c. 6 d. 7 588. Find the equation of the line which passes through the point (8,3) and forms with the coordinate axes a triangle of area 54. a. 4x + 3y – 41 = 0 b. 2x + 4y – 28 = 0 c. 5x + 2y – 46 = 0 d. 3x + 4y – 36 = 0 589. If P0(x0,y0) is such that P1P0/P0P2 = 7/6 where P1(2,5) and P2(5 ,-1), find x0. a. 45/13 b. 46/13 c. 47/13 d. 18/13 590. Find the polar equation of the line perpendicular to θ = 20° and passing through the point (6,20°). a. r = 6sec(θ + 20°) b. r = 6sec(θ – 20°) c. r = -6sec(θ + 20°) d. r = -6sec(θ – 20°)

591. Determine b so that x2 + y2 + 2x – 3y – 5 = 0 and x2 + y2 + 4x + by + 2 = 0 are orthogonal. a. 10/3 b. 11/3 c. 13/3 d. 14/3 592. If the value of the invariant B2 – 4AC is negative, then the second-degree equation Ax2 + Bxy +Cy2 + Dx + Ey + F = 0 represents either an ellipse or a. a pair of parallel lines b. two intersecting lines c. a point d. a line 593. Find the distance between the points (4,40°) and (4,220°). a. 7 b. 8 c. 10 d. 9 594. Identify the locus of the curve whose parametric equations are x = 3sinθ, y = 2cosθ. a. a circle b. a parabola c. an ellipse d. a hyperbola 595. Find the equation of the line through the midpoint of AB where A(-3,1), B(2,-1) and is perpendicular to AB. a. 10x + 4y + 5 = 0 b. 10x + 4y – 5 = 0 c. 10x – 4y + 5 = 0 d. 10x – 4y – 5 = 0 596. Find the length of the tangent line from the point P(4,-7) to the circle x2 + y2 – 10x – 4y + 25 = 0. a. b. c. d. 597. Find the equation of the circle with center at the midpoint of A(4,2), B(-1,-2) and having a radius 3. a. 4x2 + 4y2 + 12x + 27 = 0 b. 4x2 + 4y2 – 12x – 27 = 0 c. 4x2 + 4y2 + 12x – 27 = 0 d. 4x2 + 4y2 – 12x + 27 = 0

598. Write the polar equation of the circle with center (-5,π) and radius 5. a. r = 5cosθ b. r = -5cosθ c. r = 10cosθ d. r = -10cosθ 599. Give the Cartesian equation of the line whose parametric equations are x = 2t – 1, y = 3t + 5 where t is the parameter. a. 3x – 2y + 13 = 0 b. 3x + 2y – 13 = 0 c. 3x – 2y – 13 = 0 d. 3x + 2y + 13 = 0 600. Find the equation of the line through (6,-3) and parallel to the line through (2,8) and (5,1). a. 3x + y + 15 = 0 b. 3x – y – 15 = 0 c. 3x – y – 15 = 0 d. 3x – y + 15 = 0 601. The vertices of a triangle are A(4,6), B(2,-4) and C(-4,2). Find the length of the median of the triangle from the vertex C to the side AB. a. b. c. d. 602. Find the equation of the circle containing (1,-4) and center at the origin. a. x2 + y2 = 14 b. x2 + y2 = 15 c. x2 + y2 = 16 d. x2 + y2 = 17 603. If AB is perpendicular to CD and A(-1,0), B(2,5), C(3,-1), D(-3,a), find the value of a. a. 13/4 b. 13/5 c. 13/6 d. 13/7 604. Find the equation of the line through (4,0) and is parallel to the altitude from A to BC of the triangle A(1,3), B(2,-6) and C(-3,0). a. 5x + 6y + 20 = 0 b. 5x – 6y – 20 = 0 c. 5x + 6y – 20 = 0 d. 5x – 6y + 20 = 0

605. Find the equation of the circle which has the line joining (4,7) and (2,-3) as diameter. a. (x – 2)2 + (y – 3)2 = 26 b. (x – 2)2 + (y – 3)2 = 27 c. (x – 2)2 + (y – 3)2 = 28 d. (x – 2)2 + (y – 3)2 = 29 606. Write the equation of the line with x-intercept -6 and y-intercept 3. a. x – 2y – 6 = 0 b. x + 2y + 6 = 0 c. x – 2y + 6 = 0 d. x + 2y – 6 = 0 607. Find the abscissa of the point P0 which divides P 1P2 in the ratio P1P0/P0P2 = r1/r2 were P1(2,5), P2(6,-3), r1 = 3, r2 = 4. a. 25/7 b. 26/7 c. 27/7 d. 28/7 608. Find the equation of the conic with eccentricity 7/4 and foci at (7,0) and (-7,0). a. x2/33 + y2/16 = 1 b. x2/16 + y2/33 = 1 c. x2/33 – y2/16 = 1 d. x2/16 – y2/33 = 1 609. Find the equation of the line passing trough (2,-3) and is parallel to the line 3x – y = 5. a. 3x + y – 2 = 0 b. 3x + y – 3 = 0 c. 3x + y – 4 = 0 d. 3x + y – 5 = 0 610. If the slope of the line (k + 1)x + ky – 3 = 0 is arctan(-2), find the value of k. a. 1 b. 2 c. 3 d. 4 611. Find the equation of the line parallel to 5y – 5x + 12 = 0 and contains the point (0,-3). a. x – y + 3 = 0 b. x + y – 3 = 0 c. x – y – 3 = 0 d. x + y + 3 = 0

612. Find k so that the circle x2 + y2 + 2kx + 4y – 5 = 0 will pass through the point (5,1). a. -3/2 b. -5/2 c. -7/2 d. -9/2 613. Find the equation of the line through the points (-7,-3) and (-1,9). a. 2x – y + 11 = 0 b. 2x + y – 11 = 0 c. 2x + y + 11 = 0 d. 2x – y – 11 = 0 614. The equation of the parabola with vertex (-1,2) and directrix at x = -3 is a. (y – 2)2 = 8(x + 1) b. (y + 2)2 = 8(x + 1) c. (x + 1)2 = 8(y + 2) d. (x – 1)2 = -8(y + 2) 615. Find the length of the latus rectum of a parabola with focus at (-2,-6) and directrix x – 2 = 0. a. 6 b. 4 c. 8 d. 10 616. Write the equation of the line tangent to the circle x2 + y2 + 14x + 18 y – 39 = 0 at the point in the second quadrant where x = -2. a. 5x + 12y + 26 = 0 b. 5x – 12y – 26 = 0 c. 5x + 12y – 26 = 0 d. 5x – 12y + 26 = 0 617. The two points on the line 2x + 3y + 4 = 0 which are at a distance 2 from the line 3x + 4y – 6 = 0 are a. (7,-6) and (-11,6) b. (-88,-8) and (-16,-16) c. (64,-44) and (4,-4) d. (-44,64) and (10,-10)

618. Find the equation of the line which forms with the axes in the first quadrant a triangle of area 2 and whose intercepts differ by 3. a. x + 4y – 4 = 0 b. x – 4y + 4 = 0 c. x + 4y + 4 = 0 d. x – 4y – 4 = 0 619. What is the locus of a point which moves so that its distance from the line x = 8 is twice its distance from the point (2,8)? a. a circle b. an ellipse c. a parabola d. a hyperbola 620. Write the polar equation of a line which passes through the points (2,π/2) and (-1,0). a. r(2cosθ + sinθ) – 2 = 0 b. r(2cosθ – sinθ) – 2 = 0 c. r(2cosθ + sinθ) + 2 = 0 d. r(2cosθ – sinθ) + 2 = 0 621. The line segment with end points A(-1,-6) and B(3,0) is extended beyond point A to a point C so that C is 4 times as far from B as from A. find the abscissa of point C. a. -5/3 b. -7/3 c. -8/3 d. -4/3 622. A semi-elliptic arch is 20-ft high at the center and as a span of 50-ft. find the height of the arch at a point 10-ft from one end of the base. a. 14 ft b. 15 ft c. 16 ft d. 17 ft 623. If the slope of a line 3x + y – 5 + k(x + 2y – 3) = 0 is 11/3, find k. a. -4/5 b. -3/5 c. -2/5 d. -1/5

624. The equation of the ellipse with vertices at (-3,-2) and (1,-2) and which passes through (2,-1) is a. x2 + 3y2 + 2x + 12y + 9 = 0 b. 3x2 + y2 + 2x + 12y – 9 = 0 c. x2 + 3y2 – 2x + 12y + 9 = 0 d. 3x2 + y2 – 2x + 12y – 9 = 0 625. Find the diameter of the ellipse 9x2 + 16y2 = 144 defined by the system of parallel chords of slope 2. a. 9x – 32y = 0 b. 9x + 32y = 0 c. 32x – 9y = 0 d. 32x + 9y = 0 626. The locus of 4x2 + 4xy + y2 + 2x + y – 2 = 0 is a pair of parallel lines. What is the slope of each line? a. -1 b. -2 c. 1 d. 2 627. Find the area of a triangle with one vertex at the pole and the two others are (5,60°) and (4,-30°). a. 13 b. 12 c. 11 d. 10 628. Given A(3,7), B(-6,4), C(-2,8) and D(-7,0). Find the tangent of the angle measured counterclockwise from AB to CD. a. 17/23 b. 18/23 c. 19/23 d. 20/23

629. Find the equation of the hyperbola with vertices at (4,0) and (-4,0) and asymptotes y = 2x and y = -2x. a. x2/64 – y2/16 = 1 b. x2/16 – y2/64 = 1 c. y2/64 – x2/16 = 1 d. y2/64 – x2/64 = 1 630. The equation of the perpendicular bisector of the line segment joining the points (2,6) and (-4,3) is a. x + 2y – 8 = 0 b. 4x + 2y – 5 = 0 c. x – 2y + 10 = 0 d. 4x + 2y – 13 = 0 631. Assume that power cables hang in a parabolic arc between two pole 100-ft apart. If the poles are 40-ft high and if the lowest point on the suspended cable is 35-ft above the ground, find the height of the cable at a point 20-ft from the pole. a. 34.8 ft b. 35.8 ft c. 36.8 ft d. 37.8 ft 632. Transform the rectangular equation (x2 + y2)3 = 4x2 y2 into polar coordinates. a. r = 2sinθ b. r = sin2θ c. r = 2cosθ d. r = cos2θ 633. What is the eccentricity of an equilateral hyperbola? a. b. c. 1.5 d. 2 634. Find the equation of the locus of a point which moves so that its distance from (4,0) is equal to two thirds of its distance from the line x = 9. a. 9x2 – 5y2 = 180 b. 5x2 – 9y2 = 180 c. 9x2 + 5y2 = 180 d. 5x2 + 9y2 = 180

635. Find the equation of the line through the point which divides A(-1,-1/2), B(6,3) in the ratio AP/PB = 3/4 and through the point Q which is equidistant from C(1,-1), D(-3,1) and E(-1,3). a. x – 8y – 6 = 0 b. x – 8y + 6 = 0 c. x + 8y + 6 = 0 d. x + 8y – 6 = 0 636. Find the equation of the line tangent to the hyperbola 9x2 – 2y2 = 18 at the point (-2,3). a. 3x + y + 3 = 0 b. 3x – y + 3 = 0 c. 3x + y – 3 = 0 d. 3x – y – 3 = 0 637. For the conic 2x2 – xy + x + y – 5 = 0, find the equation of the diameter defined by the cords of slope ½. a. 7x + 2y – 3 = 0 b. 7x + 2y + 3 = 0 c. 7x – 2y + 3 = 0 d. 7x – 2y – 3 = 0 638. The equation of the hyperbola with foci at (0,9) and (0,-9) and conjugate axis 10 units is a. x2/56 – y2/25 = 1 b. x2/25 – y2/56 = 1 c. y2/56 – x2/25 = 1 d. y2/25 – x2/56 = 1 639.

An arch is in the form of a semi-ellipse with major axis as the span. If the span is 24.4 m and the maximum eight is 9.2 m, find the height of the arch at a point 4.6 m from the semi-minor axis. a. 6.9 m b. 5.9 m c. 8.9 m d. 7.9 m

640. If the area of the quadrilateral with vertices at (-5,-1), (x,2), (10,-4) and (-2,7) is 78.5, find x if x is positive. a. 5 b. 6 c. 7 d. 8 641. Find the value of k so that the radius of the circle x2 + y2 – kx + 6y – 3 = 0 is equal to 4. a. 3 b. 4 c. 5 d. 6 642. A parabolic segment is 32 dm high and its base is 16 dm. What is the focal distance? a. 0.5 dm b. 0.4 dm c. 0.6 dm d. 0.3 dm 643. Write the equation of the hyperbola conjugate to the hyperbola 4x2 – 3y2 + 32x + 18y + 25 = 0. a. 4x2 – 3y2 + 32x + 18y – 49 = 0 b. 4x2 – 3y2 + 32x + 18y – 36 = 0 c. 4x2 – 3y2 + 32x + 18y – 16 = 0 d. 4x2 – 3y2 + 32x + 18y – 64 = 0 644. Find the abscissa of the point P on the line segment AP for A(-8,4) and B(-13,6) if AP:PB = 3:2. a. -10 b. -11 c. -9 d. -12 645. Find the point on the parabola x2 = 16y at which there is a tangent with a slope ½. a. (8,4) b. (-8,4) c. (4,1) d. (-4,1)

646. What is the equation of the line tangent to the hyperbola

if the slope

of the line is 2? a. 2x + y + 23 = 0 b. 2x + y – 23 = 0 c. 2x – y + 23 = 0 d. 2x – y – 23 = 0 647. Find the eccentricity of an ellipse whose latus rectum is 2/3 of the major axis. a. 0.58 b. 0.68 c. 0.78 d. 0.88 648. The vertices of a triangle are (2,4), (x,-6) and (-3,5). If x is negative and the area of the triangle is 28.5, find x. a. -5 b. -6 c. -4 d. -7 649. A parabolic arch has a span of 20 m and a maximum height of 15 m. how high is the arch 4 m from the center of the span? a. 10.6 m b. 11.6 m c. 12.6 m d. 13.6 m 650. Determine the value of k so the following circles are orthogonal: C1: x2 + y2 + 2x – 3y – 5 = 0 C2: x2 + y2 + 4x + ky + 2 = 0 a. 11/2 b. 13/3 c. 14/3 d. 10/3 651. An ellipse has its foci at (0,c) and (0,-c) and its eccentricity is ½. Find the length of the latus rectum. a. 2c b. 3c c. 4c d. 5c

652. The earth’s orbit is an ellipse with eccentricity 1/60. If the semi-major axis of the orbit is 93M miles and the sun is at one of the foci, what is the shortest distance between the earth and the sun? a. 89.43M mi b. 90.44M mi c. 91.45M mi d. 92.46M mi 653. If the length of the latus rectum of an ellipse is ¾ of the length of its minor axis, then its eccentricity is a. 0.46 b. 0.56 c. 0.66 d. 0.76 654. If the point P(9,2) divides the line segment from A(6,8) to B(x,y) such that AP:AB = 3:10, find y. a. -11 b. -10 c. -9 d. -12 655. Find the rectangular equation for the curve whose parametric equations are x = 2cosθ, y = cos2θ. a. x2 = 2(y + 1) b. x2 = 2(y – 1) c. y2 = 2(x + 1) d. y2 = 2(x – 1) 656. A parabolic arch spans 200-ft wide. How high must the arch be above the stream to give a minimum clearance of 40-ft over a tunnel in the center which is 120-ft wide? a. 60.5 ft b. 61.5 ft c. 62.5 ft d. 63.5 ft

657. In the parabola x2 = 4y, an equilateral triangle is inscribed with one vertex at the origin. Find the length of each side of the triangle. a. 13.86 b. 12.85 c. 11.84 d. 10.83 658. The foci of a hyperbola are (4,3) and (4,-9) and the length of the conjugate axis is

.

Find its eccentricity. a. 1.3 b. 1.5 c. 1.7 d. 1.9 659. Find the length of the common chord of the curves whose equations are x2 + y2 = 48 and x2 + 8y = 0. a. b. c. d. 660. The point (8,5) bisects a chord of the circle whose equation is x2 + y2 – 4x + 8y = 110. Find the equation of the cord. a. 3x + 2y = 0 b. 3x – 2y = 14 c. 2x + 3y = 31 d. 2x – 3y = 1 661. Find the length of the latus rectum of the parabola with focus at (-2,-6) and directrix x – 2 = 0. a. 8 b. 7 c. 6 d. 4

662. Find the distance between (1,2,-5) and (-1,-1,4). a. b. c. d. 663. What is the distance from the origin to the point (4,-3,2)? a. b. c. d. 664. Find the direction numbers of the line through (4,-1,-3) and (0,1,4). a. 4,-2,-7 b. -4,2,-7 c. -4,-2,7 d. -4,2,-7 665. The direction numbers of two lines are 2,-1,4 and -3,y,2 respectively. Find y if the lines are perpendicular to each other. a. -1 b. 3 c. -2 d. 2 666. Transform p = 6θ to spherical coordinates. a. r2 – z2 = 36θ2 b. r2 – z2 = 6θ c. r2 + z2 = 36θ2 d. r2 + z2 = 6θ 667. the surface described by the equation 4x2 + y2 + 26z = 100 is an a. elliptic hyperboloid b. elliptic paraboloid c. ellipsoid d. elliptic cone

668. Find the Cartesian coordinates of the point having the cylindrical coordinates (3,π/2,5). a. (5,0,3) b. (3,0,5) c. (0,5,3) d. (0,3,5) 669. Find the cylindrical coordinates of the point having the rectangular coordinates (4,4,-2). a. ( b. ( c. ( d. ( 670. The distance of the point (-4,5,2) from the x-axis is a. b. c. d. 671. The equivalent of (3,4,5) in the cylindrical coordinate system is a. (5,31.53°,5) b. (5,51.33°,5) c. (5,53.13°,5) d. (5,35.31°,5) 672. If one end of a line is (-2,4,8) and its midpoint is (1,-2,5), find the x-coordinate of the other end. a. 4 b. 3 c. 5 d. 6 673. Find the value of k such that the plane x + ky – 2z – 9 = 0 shall pass through the point (5,-4,-6). a. 2 b. 1 c. 3 d. 4

674. The locus of 9x2 – 4z2 – 36y = 0 is a/an a. elliptic cone b. hyperbolic paraboloid c. parabolic cylinder d. ellipsoid 675. The trace of x2 + 4z2 – 8y = 0 on the xy-plane is a. a hyperbola b. an ellipse c. a parabola d. a point 676. The locus of y2 + z2 – 4x = 0 has symmetry with respect to a. xz-plane only b. yz- and xy-planes c. z-axis d. xz- and xy-planes 677. If the plane curve b2x2 + a2y2 = a2b2 is revolved about the x-axis, the surface generated is a/an a. ellipsoid of revolution b. hyperbolic paraboloid c. paraboloid of revolution d. parabolic cylinder 678. The rectangular coordinates for the point whose cylindrical coordinates are (6,120°,-2) are a. (3,3 ,-2) b. (2,3

,-3)

c. (-3,3

,-2)

d. (-2,3

,-3)

679. Which of the following has a locus that is a hyperbolic paraboloid? a. x2 + y2 – 2z = 0 b. x2 + 5z2 – 6y = 0 c. z2 – 2y2 + 4x = 0 d. 4x2 + y2 – 4z = 0

680. Find the z-coordinate of the midpoint of the segment whose end points are (4,5,6) and (3,1,2). a. 3 b. 4 c. 5 d. 6 681. The traces of the surface a. b. c. d.

on the coordinate planes are

circles ellipses parabolas hyperbolas

682. Transform the equation θ = tanφ to cylindrical coordinates. a. r = zθ b. z = rθ c. θ = rz d. r = zφ 683. Which of the following is a quadric cone? a. x2 – y2 – 4z2 = 0 b. x2 – y2 – 4z = 0 c. x2 + y2 – 4z2 = 0 d. x2 + y2 – 4z = 0 684. Transform z 2r = 1 to spherical coordinates. a. pcosφ – 2sinφ = 1 b. p(sinφ – 2cosφ) = 1 c. cosφ – 2psinφ = 1 d. p(cosφ – 2sinφ) = 1 685. If z = 0 in the equation 2y2 + 3z2 – x2 = 0, then the trace of the surface on the xy-plane is a a. pair of parallel lines b. pair of intersecting lines c. line d. point

686. Find the cylindrical coordinates for the point (6,3,2). a. ( b. ( c. ( d. ( 687. A line makes an angle of 45 degrees with the x-axis and an angle of 60 degrees with the y-axis. What angle does it make with the z-axis? a. 30° b. 45° c. 60° d. 55° 688. Two directions cosines of a line are 1/3 and -2/3. What is the third? a. 2/3 b. 4/3 c. 5/3 d. 7/3 689. A line makes equal angles with the coordinate axes. Find the angle. a. 44.64° b. 54.74° c. 64.84° d. 74.94° 690. Find the distance of the point (6,2,3) from the x-axis. a. b. c. d. 691. What a. b. c. d.

is the locus of any equation of the form x2 + y2 = f(z)? hyperboloid of revolution ellipsoid of revolution paraboloid of revolution cylinder of revolution

692. The radius of the sphere x2 + y2 + z2 – 6x + 4z – 3 = 0 is a. 2 b. 3 c. 5 d. 4 693. The direction numbers of two lines are 2,-1,k and -3,2,2 respectively. Find k if the lines are perpendicular. a. 4 b. 2 c. 5 d. 3 694. Find the equation of the locus of a point which moves so that it is 4 units in front of the xz-plane. a. y +4 = 0 b. z – 4 = 0 c. x + 4 = 0 d. y – 4 = 0 695. The equation x2 + z2 = 5y is a paraboloid of revolution that is symmetric with respect to a. x-axis b. y-axis c. z-axis d. origin 696. The equation of the plane through the point (-1,2,4) and parallel to the plane 2x – 3y – 5z + 6 = 0. a. 2x – 3y – 5z + 27 = 0 b. 2x – 3y – 5z + 26 = 0 c. 2x – 3y – 5z + 28 = 0 d. 2x – 3y – 5z + 29 = 0 697. Find the distance of the point (6,2,3) from the z-axis. a. b. c. d. 7

698. A line drawn from the origin to the point (-6,2,3). Find the angle which the line makes with the z-axis. a. 147° b. 149° c. 151° d. 150° 699. Find the length of the line segment whose end points are (3,5,-4) and (-1,1,2). a. b. c. d. 700. Find the locus of a point whose distance from the point (-3,2,1) is 4. a. x2 + y2 + z2 + 6x – 4y – 2z + 3 = 0 b. x2 + y2 + z2 + 6x – 4y – 2z – 4 = 0 c. x2 + y2 + z2 + 6x – 4y – 2z + 1 = 0 d. x2 + y2 + z2 + 6x – 4y – 2z – 2 = 0 701. Find the center of the sphere x2 + y2 + z2 – 6x + 4y – 8z = 7. a. C(3,2) b. C(-3,2) c. C(3,-2) d. C(-3,-2) 702. Find the rectangular coordinates for the point (4,210°,30°). a. ( b. ( c. ( d. ( 703. The vertices of a triangle are A(2,-3,1), B(-6,5,3) and C(8,7,-7). Find the length of the median drawn from A to BC. a. b. c. d.

704. Find the angle between the line L1 with direction numbers 3,4,1 and the line L2 with direction numbers 5,3,-6. a. 55.41° b. 60.51° c. 65.61° d. 70.71° 705. Find spherical coordinates for the point (-2,2,-1). a. (3,315°,109.5°) b. (3,240°,107.5°) c. (3,300°,110°) d. (3,215°,100°) 706. Find the distance from the plane 2x + 7y + 4z – 3 = 0 to the point (2,3,3). a. b. c. d. 707. Transform psinφsinθtanθ = 5 to rectangular coordinates. a. x2 = 5y b. y2 = 5x2 c. y2 = 5x d. y = 5x2 708. Two direction angles of a line are 45 degrees and 60 degrees. Find the third direction angle. a. 30° b. 35° c. 40° d. 45° 709. Find m so that the plane 5x – 6y – 7z = 0 and the plane 3x + 2y – mz + 1 = 0 are parallel. a. -5/3 b. -7/3 c. -4/3 d. -2/3

710. Transform y2 = 4ax to cylindrical coordinates. a. rcosθtanθ = 4a b. rcosθcotθ = 4a c. rsinθtanθ = 4a d. rsinθcotθ = 4a 711. The triangle with vertices (3,5,-4),(-1,1,2) and (-5,-5,-2) is a. equilateral b. isosceles c. right d. equiangular 712.

The sphere x2 + y2 + z2 – 2x + 6y +2z – 14 = 0 has a radius a. 2 b. 4 c. 5 d. 3

713. Find the x-coordinate of a point which is 10 units from the origin and has direction cosines cosβ = 1/3 and cosγ = -2/3. a. 19/3 b. 20/3 c. 17/3 d. 22/3 714. Give the equivalent spherical coordinates of (3,4,6). a. ( b. c. d. ( 715. If the line L1 has direction numbers x,-2x3 and line L2 has direction numbers -2,x,4 and if L1 is perpendicular to L2, find x. a. 5 b. 4 c. 3 d. 2

716. Find the cosine of the angle between the line directed from (3,2,5) to (8,6,2) and the line directed from (-4,5,3) to (-3,4,3). a. 1/12 b. 1/10 c. 1/11 d. 1/13 717. Find the angle between the planes 3x – y + z – 5 = 0 and x + 2y + 2z + 2 = 0. a. 69.42° b. 70.43° c. 71.44° d. 72.45° 718. Find the coordinates of the point P(x,y,z) which divides the line segment P 1P2 where P1(2,5,-3) and P2(-4,0,1) in the ratio 2:3. a. (2/5,-3,-7/5) b. (-2/5,3,7/5) c. (-2/5,3,-7/5) d. (-2/5,-3,-7/5) 719. Find the Cartesian coordinates of the point having the spherical coordinates (4, . a. ( b. ( c. ( d. ( 720. Find the equations of the line through (2,-1,3) and parallel to the x-axis. a. y + 1 = 0, z – 3 = 0 b. y – 1 = 0, z + 3 = 0 c. y – 1 = 0, z – 3 = 0 d. y + 1 = 0, z + 3 = 0 721.

Give the polar coordinates for the point (1,-2,2). a. (3,48.2°,131.8°,70.5°) b. (3,70.5°,131.8°,48.2°) c. (3,48.2°,70.5°,131.8°) d. (3,131.8°,70.5°,48.2°)

722. Transform the equation cosγ = p(cos2α – cos2β) to rectangular coordinates. a. y = x2 – z2 b. x = y2 – z2 c. z = x2 – y2 d. z = x2 + y2 723. A point P(x,y,z) moves so that its distance from the z-axis is 4 times its distance from the x-axis. Find the equation of the locus. a. 15y2 + 16z2 – x2 = 0 b. 15y2 – 16z2 + x2 = 0 c. 15y2 – 16z2 – x2 = 0 d. 15y2 + 16z2 + x2 = 0 724. Write the equation in rectangular coordinates of p = 5acosφ. a. x2 – y2 + z2 = 5az b. x2 + y2 – z2 = 5az c. x2 – y2 – z2 = 5az d. x2 + y2 + z2 = 5az 725. The rectangular coordinates for the point (2,90°,30°,60°) is a. (0, b. (0, c. (1, d. (1, 726. Find the equations of the line through (1,-1,6) with direction numbers 2,-1,1. a. x = 2z + 11, y = z – 5 b. x = 2z – 11, y = z + 5 c. x = 2z – 11, y = 5 – z d. x = 2z + 11, y = 5 – z 727. If the angle between two lines with direction numbers 1,4,-8 and x,3x-6 respectively is arccos(62/63),find x. a. 4 b. 5 c. 2 d. 3

728. Find the polar coordinates of the point (0,-2,-2) a. (2 b. (2 c. ( d. ( 729. Find the point where the line through the points (3,-1,0) and (1,3,4) pierces the xz-plane. a. (1,0,1) b. (1.5,0,1) c. (2,0,1) d. (2.5,0,1) 730. Find the equation of the plane such that the foot of the perpendicular from the origin to the plane is (-6,3,6). a. 2x + y + 2z – 27 = 0 b. 2x – y – 2z + 27 = 0 c. 2x – y + 2z + 27 = 0 d. 2x + y – 2z – 27 = 0 731. Find angle A of the triangle whose vertices are A(4,6,1), B(6,4,0) and C(-2,3,3). a. 112.39° b. 111.38° c. 110.37° d. 109.36° 732. Find the equation of the plane that passes through (3,-2,1), (2,4,-2) and (-1,3,2). a. 21x + 13y + 19z – 56 = 0 b. 21x + 13y – 19z – 56 = 0 c. 21x + 13y + 19z + 56 = 0 d. 21x – 13y – 19z – 56 = 0 733. Find the acute angle between the lines x + y + z + 1 = 0, x – y + z + 1 = 0 and x – y – z – 1 = 0, x + y = 0. a. 71.20° b. 72.21° c. 73.22° d. 74.23°

734. Find the equation of the plane through the point (-1,2,3) and perpendicular to the line for which cosα = 2/3, cosβ = -1/3, cosγ = 2/3. a. 2x – y + 2z – 2 = 0 b. 2x – y – 2z + 2 = 0 c. 2x + y – 2z – 2 = 0 d. 2x + y + 2z – 2 = 0 735. Find the area of the triangle with vertices (1,3,3), (0,1,0) and (4,-1,0). a. b. c. d. 736. If the acute angle between the planes 2x – y + z – 7 = 0 and x + y + kz – 11 = 0 is 60°, find k. a. 4 b. 3 c. 1 d. 2 737. Transform the cylindrical coordinates (8,120°,6) to spherical coordinates. a. (10,120°,53.13°) b. (11,120°,53.13°) c. (12,120°,53.31°) d. (10,120°,51.33°) 738. Find the locus of the point equidistant from the plane y = 7 and the point (0,5,0). a. x2 – z2 + 4y – 24 = 0 b. x2 – z2 – 4y + 24 = 0 c. x2 + z2 + 4y – 24 = 0 d. x2 + z2 – 4y + 24 = 0 739. Find the direction numbers of the line 2x – y + 3z + 4 = 0, 3x + 2y – z + 7 = 0. a. 5,-11,7 b. -5,11,7 c. -5,7,11 d. 5,-7,11

740. Find the equation of the plane perpendicular to the line joining (2,5,-3) and (4,-1,0) and which passes through the point (1,4,-7). a. 2x – 6y – 3z + 43 = 0 b. 2x + 6y – 3z + 43 = 0 c. 2x – 6y + 3z + 43 = 0 d. 2x + 6y + 3z + 43 = 0 741. Find the equation of the line which passes through (-1,-3,6) and which is parallel to the plane 4x – 9y + 7z + 2 = 0. a. 4x – 9y + 7z – 65 = 0 b. 4x + 9y + 7z – 65 = 0 c. 4x – 9y – 7z + 65 = 0 d. 4x + 9y – 7z + 65 = 0 742. Find the value of m so that the line passing through (-m,-1,2) and (0,2,4) be perpendicular to the line through (1,m,1) and (m+1,0,2). a. 1 or 5 b. 1 or 4 c. 1 or 3 d. 1 or 2 743. Find the acute angle between the line

and the line

.

a. b. c. d. 744. If the angle between the planes 2x – 3y + 6z = 18 and 2x – y + kz = 12 is arccos(19/21), find k. a. 4 b. 3 c. 2 d. 1

745.

A plane contains the point P1(4,-4,2) and is perpendicular to the line segment from P1 to P2(0,6,6). Find the equation of the plane. a. 2x + 5y + 2z – 24 = 0 b. 2x + 5y – 2z + 24 = 0 c. 2x – 5y + 2z + 24 = 0 d. 2x – 5y – 2z – 24 = 0

746. A

line

whose

parametric

equations

are

is

perpendicular to the plane 2x + ky + 12z = 3. Find the value of k. a. -3 b. -4 c. -5 d. -6 747. Write the equations of the line through (-2, 2, -3) and (2, -2, 3). a. x – y = 0, 3y + 2z = 0 b. x + y = 0, 3y + 2z = 0 c. x – y = 0, 3y – 2z = 0 d. x + y = 0, 3y – 2z = 0 748. Find the equation of the paraboloid with vertex at (0, 0, 0), axis along the y-axis and passing through (1, 1, 1) and (3/2, 7/12, 1/2). a. x2 + 5z2 = 6y b. x2 + 6z2 = 5y c. 5x2 + z2 = 6y d. 6z2 + z2 = 5y 749. Find the equation of the plane determined by the points (6,-4,1), (0,1,-3) and (2,2,-7). a. x + 2y – z + 1 = 0 b. x – 2y + z – 1 = 0 c. x + 2y + z + 1 = 0 d. x – 2y – z – 1 = 0

750. What is the locus of the moving point, the difference of whose distance from (0,0,3) and (0,0,-3) is 4? a. b. c. d. 751. Find the piercing point in the xy-plane of the line x + y – z – 3 = 0, x + 2y + z – 4 = 0. a. (1,2,0) b. (1,0,2) c. (2,0,1) d. (2,1,0) 752. Find the acute angle between the line 0. a. b. c. d.

and the plane 2x – 2y + z – 3 =

25.3° 26.4° 27.5° 28.6°

753. Find the equation of the plane through (1,-2,3) and perpendicular to the line of intersections of the plane 3x + 2y – 2z = 12 and x + 2y + 2z = 0. a. 2x – 2y – z – 9 = 0 b. 2x – 2y + z – 9 = 0 c. 2x + 2y – z + 9 = 0 d. 2x + 2y + z + 9 = 0 754. A plane contains the points (3,1,7) and (-3,-2,3) and as an x-intercept equal to three times its z-intercepts. Find the equation of the plane. a. x + 6y – 3z + 18 = 0 b. x – 6y – 3z + 18 = 0 c. x – 6y + 3z – 18 = 0 d. x + 6y – 3z – 18 = 0

755. Find the acute angle between the lines through the points (-2,3,1) and (4,6,7) and the plane x + 4y + z – 10 = 0. a. 35.64° b. 36.74° c. 37.84° d. 38.94° 756. Find the equation of the plane which contains the line x – 2y + z = 1, 2x = y – z and is perpendicular to the plane 3x + 2y – 3z = 0. a. 9x – 6y + 5z – 1 = 0 b. 9x + 6y – 5z + 1 = 0 c. 9x + 6y – 5z – 1 = 0 d. 9x + 6y + 5z + 1 = 0 757. Find the equation of the plane which is perpendicular to the xy-plane and which passes through (2,-1,0) and (3,0,5). a. x + y + 3 = 0 b. x – y – 3 = 0 c. x + y – 3 = 0 d. x – y + 3 = 0 758. Find the acute angle between the lines

and 2x + 2y + z – 4 = 0, x – 3y +

2z = 0. a. 46°24’ b. 47°25’ c. 48°26’ d. 49°27’ 759. Find the equations of the line through (2,-3,4) and perpendicular to the plane 3x – y + 2z = 4. a. x = 3y – 7, z = 2y – 2 b. x = 3y + 7, z = 2y + 2 c. x = -3y – 7, z = -2y – 2 d. x = -3y + 7, z = -2y + 2

760. Find the point of intersection of the plane 3x + 2y + z = 1 and the line a. b. c. d.

.

(1,0,1) (1,1,0) (-1,1,0) (1,-1,0)

761. Transform 3x2 – 3y2 = 8z to spherical coordinates. a. 2psin2φcos2θ = 8pcosφ b. 2psin2φcos2θ = 8pcosφ c. 2p2sin2φcos2θ = 8pcosφ d. 2p2sin2φcos2θ = 8pcosφ 762. Find the equation of the sphere whose center is (2,1,-1) and which is tangent to the plane x – 2y + z + 7 = 0. a. x2 + y2 – 4z – 2y + 2z = 0 b. x2 + y2 – 4z + 2y + 2z = 0 c. x2 + y2 + 4z – 2y – 2z = 0 d. x2 + y2 + 4z + 2y – 2z = 0 763. If the line k. a. b. c. d.

is parallel to the plane 6x + ky – 5z – 8 = 0, find the value of

2 3 -2 -3

764. Find the equation of the plane that is perpendicular to the yz-plane and having 5 and -2 as its y- and z-intercepts respectively. a. 2y + 5z – 10 = 0 b. 2y – 5z – 10 = 0 c. 2y + 5z + 10 = 0 d. 2y – 5z + 10 = 0 765. Find the angle between the line with direction numbers 1,-1,-1 and the plane 3x – 4y + 2z – 5 = 0. a. 32.42° b. 34.22° c. 42.32° d. 43.22°

766. Find the equation of the locus of a point whose distance from the xy-plane is equal to its distance from (-1,2,-3). a. x2 + y2 – 2x + 4y – 6z – 14 = 0 b. x2 + y2 – 2x – 4y + 6z + 14 = 0 c. x2 + y2 + 2x + 4y – 6z – 14 = 0 d. x2 + y2 + 2x – 4y + 6z + 14 = 0 767. Given the points A(k,1,-1), B(2k,0,2) and C(2+2k,k,1). Find k so that the line segment AB shall be perpendicular to the line segment BC. a. 3 b. 1 c. 2 d. 4 768. The angle between two lines with direction numbers 4,3,5 and x,-1,2 respectively is 45 degrees. Find x. a. 4 b. 5 c. 2 d. 3 769. At the minimum point, the slope of the tangent line to a curve is a. positive b. negative c. zero d. infinity 770. A curve y = f(x) is concave downward if the value of y’’ is a. negative b. positive c. unity d. zero 771. The point where the concavity of a curve changes is called the a. maximum point b. minimum point c. inflection point d. tangent point

772. If the 1st derivative of a function is a constant, then its graph is a. a point b. a line c. a parabola d. a circle 773. At the minimum point of y = f(x), the value of d2 y/dx2 is a. zero b. undefined c. positive d. negative 774. If at x = a, f’’(a) is positive, then f’(x) increases as x a. increases b. decreases c. becomes infinite d. becomes zero 775. If the first derivative of a function is a constant, then the function is a. sinusoidal b. exponential c. linear d. quadratic 776. A function f(x) is said to be an even function if its graph is symmetric with respect to a. the x-axis b. the y-axis c. the origin d. both axes 777. Which of the following is an odd function? a. f(x) = xcosx b. f(x) = xsinx c. f(x) = ecosx d. f(x) = sin2x

778. The notation f’(x) was invented by a. Leibniz b. Newton c. Wallis d. Lagrange 779. At the inflection point of y = f(x) where x = a, a. f”(a) < 0 b. f”(a) = 0 c. f”(a) > 0 d. f”(a) = ∞ 780. If a function f(x) is concave downward on the interval (1,10), then f(8) and f(3) a. may be true b. cannot be true c. must be true d. is never true 781. If a tangent to a curve y = f(x) is horizontal at x = a, then f’(a) is a. positive b. negative c. zero d. infinity 782. For a function y = f(x), if f”(x) = -f(x), then the function is a. logarithmic b. exponential c. transcendental d. sinusoidal 783. Which of the following notations is an open interval? a. (-3,4) b. [-3,4] c. [-3,∞) d. (-∞,4)

784. The graph of y = x5 – x will cross the x-axis a. twice b. 3 times c. 4 times d. 5 times 785. The derivative of an increasing function f(x) must be a. strictly positive b. always positive c. nonnegative d. negative 786. If the function f(x) increases at x = a, then which of the following is definitely true? a. f'(a) = 0 or f’(a) > 0 b. f’(a) = 0 or f’(a) < 0 c. f’(a) ≠ 0 or f’(a) > 0 d. f’(a) ≠ 0 or f’(a) < 0 787. At the maximum point, the value of the 2nd derivative of a function is a. positive b. negative c. zero d. infinite 788. At the inflection point, the value of y” is a. zero b. positive c. negative d. unity 789. Which of the following functions will have an inflection point? a. y = x4 b. y = x3 c. y = x2 d. y = x

790. The function y = f(x) has a maximum value of x = 2 if f’(2) = 0 and f”(2) is a. equal to zero b. less than zero c. greater than zero d. unity 791. At the maximum point, the tangent line is a. slanting upward b. oblique c. horizontal d. vertical 792. Which of the following is true? a. ∞ – ∞ = 0 b. ∞ + ∞ = ∞ c. ∞/∞ = ∞ d. both a and b 793. Which of the following functions is neither even nor odd? a. h(x) = x2 b. g(x) = x3 c. f(x) = x2 + x d. t(x) = x3 + x 794. Find the rate of change of the volume of a cube with respect to its side when the side is 6 cm. a. 108 cm3/cm b. 107 cm3/cm c. 106 cm3/cm d. 105 cm3/cm 795. If f(x) = e –x+1, then f’(1) is equal to a. 0 b. 1 c. -1 d. ∞

796. If f(x) = Aekx, f(0) = 5 and f(3) = 10, find k. a. 0.1184 b. 0.1285 c. 0.1386 d. 0.1487 797. The function a. b. c. d.

is discontinuous at x =

1 or -3 1 or -2 -1 or 2 -1 or 3

798. Find the slope of the line tangent to y = 4/x at x = 2. a. 1 b. -1 c. 2 d. -2 799. If y = cos24x, find dy/dx. a. 2cos4x b. 2sin4x c. -4sin8x d. -8sin4x 800. Evaluate the limit of ln(1 – x)/x as x approaches zero. a. 0 b. -1 c. 1 d. ∞ 801. Evaluate a. b. c. d.

∞ 0 ½ 2

.

802. The rate of change of the area of a circle with respect to its radius when the diameter is 6cm is a. 4π cm2/cm b. 5π cm2/cm c. 6π cm2/cm d. 7π cm2/cm 803. At what point of the curve y = x3 + 3x are the values of y’ and y” equal? a. (0,0) b. (-1,-4) c. (2,14) d. (1,4) 804. If f(x) = ln x and g(x) = log x and if g(x) = kf(x), find k. a. 0.4433 b. 0.3434 c. 0.3344 d. 0.4343 805. If N(x) = sin x – sin θ and D(x) = x – θ, find the limit of N(x)/D(x) as x approaches θ. a. sinθ b. cosθ c. zero d. no limit 806. Given z2 + x2 + y2 = 0, find a. b. c. d. 807. What a. b. c. d.

x/z –x/z z/x –z/x is the 50th derivative of y = cosx sinx –sinx cosx –cosx

808. Which of the following has no horizontal asymptote? a. b. c. d. 809. If f(x) = a. b. c. d.

if f(x) = x – 2 and g(x) = x2 – 1.

∞ 0 ½ ¼

811. Evaluate a. b. c. d.

.

infinity unity zero undefined

810. Evaluate a. b. c. d.

, find

.

0 ∞ 1 e

812. If z = xy2 + yx3, find zxyx. a. 6yx b. 6x c. 3xy d. 3x2 813. If y = x2, find ∆y – dy when x = 2 and dx = 0.01. a. 0.0001 b. 0.001 c. 0.0002 d. 0.002

814. If f(x) = x3 + 2x, find f”(2). a. 10 b. 11 c. 12 d. 13 815. The motion of a particle along the x-axis is given by the equation x = 2t 3 – 3t2. Find the velocity of the particle when t = 2. a. 10 b. 9 c. 11 d. 12 816. Find x for which the line tangent to the parabola y = 4x – x2 is horizontal. a. 4 b. -4 c. 2 d. -2 817. The slope of the tangent to y = 2 – x2 at the point (1,1) is a. -2 b. -1 c. 0 d. -4 818. If y = sin2x, the derivative dy/dx is equal to a. cos2x b. sin2x c. 2cosx d. 2sinx 819. If y = x3 – 2x2 + 3x – 1, then d2 y/dx2 is equal to a. 6x b. 6x + 4 c. 6x – 4 d. 3x – 4

820. If y = x2 – 2x and x changes from 2 to 2.01, find ∆y. a. 0.0102 b. 0.0210 c. 0.0120 d. 0.0201 821. The radius R of a circle is increasing at the rate of 1cm per sec. how fast is the area changing when R = 4cm? a. 8π cm2/s b. 10π cm2/s c. 6π cm2/s d. 12π cm2/s 822. Find the slope of y = 1 – x3 at the point where y = 9. a. -11 b. -12 c. -10 d. -13 823. If an error of 1 percent is made in measuring the edge of a cube, what is the percentage error in the computed volume? a. 3% b. 2% c. 4% d. 5% 824. Find the derivative of y with respect to x of y = xlnx – x. a. 1 b. x c. lnx d. lnx – 1 825. For what value of x will the curve y = x3 – 3x2 + 4 be concave upward? a. 1 b. 2 c. 3 d. 4

826. How fast does the diagonal of a cube increase if each edge of the cube increases at a constant rate of 5cm/s? a. 6.7 cm/s b. 7.7 cm/s c. 8.7 cm/s d. 9.7 cm/s 827. If f(x) = tanx – x and g(x) = x3, evaluate the limit of f(x)/g(x) as x approaches zero. a. 0 b. ∞ c. 3 d. 1/3 828. Find the 3rd derivative of y = xlnx. a. -1/x b. -1/x2 c. -1/x3 d. -1 829. Evaluate a. b. c. d.

.

∞ 1 e 1/e

830. If xy3 + x3y = 2, find dy/dx at the point (1,1). a. 1 b. -1 c. 2 d. -2 831. The tangent line to the curve y = x3 at the point (1,1) will intersect the x-axis at x = a. 2/3 b. 4/3 c. 1/3 d. 5/3

832. If y = ex + xe + xx, find y’ at x = 1. a. e +1 b. e – 1 c. 2e + 1 d. 2e – 1 833. Evaluate a. b. c. d.

.

0 ∞ ½ 1

834. Find the value of x for which y = x3 – 3x2 has a minimum value. a. 1 b. 2 c. 0 d. -2 835. Find the angle of intersection between the curve y = x2 and x = y2. a. b. c. d. 836. If z = xy2, and x changes from 1 to 1.0, and y changes from 2 to 1.98, find the approximate change in z. a. -0.0202 b. -0.0303 c. -0.0404 d. -0.0505 837. A ball is thrown vertically upward from a roof 112-ft above the ground. The height s of the ball above the roof is given by the equation s = 96t -16t2 where s is measured in ft and the time t in sec. calculate its velocity wen it strikes the ground. a. -130 fps b. -128 fps c. -126 fps d. -124 fps

838.

If y = ln(tanhx), find dy/dx. a. 2sech2x b. 2sech2x c. 2csch2x d. 2coth2x

839.

Find the approximate surface area of a sphere of radius 5.02 cm. a. 317 sq. cm b. 315 sq. cm c. 313 sq. cm d. 311 sq. cm

840.

Find the value of x for which y = x5 – 5x3 – 20x – 2 will have a maximum point. a. -1 b. -2 c. 1 d. 2

841.

A man is walking at a rate of 1.5 m/s toward a street light which is 5 m above the level ground. At what rate is the tip of his shadow moving if the man is 2 m tall? a. -1.5 m/s b. -2.5 m/s c. -3.5 m/s d. -5 m/s

842.

If y = ln(x2ex), find y”. a. -1/x2 b. -2/x2 c. -1/x d. -2/x

843.

Find the radius of curvature of y = x3 at the point (1,1). a. 3.25 b. 4.26 c. 5.27 d. 6.25

844.

A particle moves along the circumference of a circle of radius 10-ft in such a manner that its distance measured along the circumference from a fixed point at the end of t sec is given by the equation s = t2. Find the angular velocity at the end of 3 seconds. a. 0.40 rad/s b. 0.50 rad/s c. 0.60 rad/s d. 0.70 rad/s

845.

Find the point on the curve y = x3 – 3xfor which the tangent line is parallel to the x-axis. a. (-1,2) b. (2,2) c. (1,2) d. (0,0)

846.

If y = 1/2tan2x + ln(cosx), find y’. a. tan3x b. tanx – sinx c. tanxsec2 x d. 0

847.

If S = 4πR2, find ∆S – dS when R = 2 and ∆R = 0.01. a. 0.0021 b. 0.0102 c. 0.0210 d. 0.0012

848.

Find two numbers whose sum is 8 if the product of one number and the cube of the other is a maximum. a. 3 and 5 b. 4 and 4 c. 2 and 6 d. 1 and 7

849.

Find the approximate height of the curve y = x3 – 2x2 + 7 at the point where x = 2.98. a. 14.8 b. 15.7 c. 16.6 d. 17.5

850.

If y =

, find x for which dy/dx = 0.

a. b. c. d. 851.

Te volume of a cube is increasing at the rate of 6 cm3/min. How fast is the surface increasing when the length of each edge is 12 cm? a. 3 cm2/min b. 4 cm2/min c. 2 cm2/min d. 5 cm2/min

852.

If u =

, find the approximate change in u as x changes from 10 to 10.02 and

y changes from 4 to 4.01. a. -0.00170 b. -0.00701 c. -0.00107 d. -0.00017 853.

Find the equation of the line tangent to y = x2 – 3x – 5 and parallel to the line y = 3x – 2. a. y = 3x – 14 b. y = 3x – 13 c. y = 3x – 12 d. y = 3x – 11

854.

A garden is in the form of an ellipse with semi-major axis 4 and semi-minor axis 3. If the axes are increased by 0.18 unit each, find the approximate increase in the area. a. 3.92 b. 3.94 c. 3.96 d. 3.98

855.

Find the relative error in the computed area of an equilateral triangle due to an error of 3 percent in measuring the edge of the triangle. a. 0.05 b. 0.06 c. 0.07 d. 0.08

856.

A body is thrown vertically upward from the ground. After 2 seconds, its velocity is 10 ft/sec. Find its initial velocity. a. 54 fps b. 64 fps c. 74 fps d. 84 fps

857.

In problem 345, find the rate at which the length of the shadow of the man is shortening. a. -1 cm/s b. -1.5 cm/s c. -2 cm/s d. -2.5 cm/s

858.

A rectangular field is fenced off, an existing wall being used as one side. If the area of the field is 7,200 sq. ft, find the least amount of fencing needed. a. 250 ft b. 240 ft c. 230 ft d. 220 ft

859.

The side of an equilateral triangle is increasing at the rate of 0.50 cm/s. Find the rate at which its altitude is increasing. a. 0.334 cm/s b. 0.443 cm/s c. 0.433 cm/s d. 0.343 cm/s

860.

Find C co that the line y = 4x + 3 is tangent to the curve y = x2 + C. a. 3 b. 4 c. 5 d. 6

861.

At what acute angle does the curve y = 1 – 1/2x2 cut the x-axis? a. 34.54° b. 44.64° c. 54.74° d. 64.84°

862.

The angle θ, made by a swinging pendulum with the vertical direction, is given at time t by the equation θ = asin(bt + c), where a, b and c are constants. Find the angular acceleration at time t. a. –a2θ b. –b2θ c. –aθ d. –bθ

863.

If y = a. b. c. d.

find y’ at x = 5. 1/13 1/14 1/15 1/16

864.

Find the equation of the line with slope -1/2 and tangent to the ellipse x2 + y2 = 8. a. x + 2y – 4 = 0 b. x – 2y + 4 = 0 c. x + 2y + 4 = 0 d. x – 2y – 4 = 0

865.

Find the second derivative (y”) of 4x2 + 9y2 = 36 by implicit differentiation. a. -16y3/9 b. -16/9y3 c. -9y3/16 d. -9/16y3

866.

Approximate the root of 3x + x – 2 = 0 by Newton’s Method of Approximation. a. 0.420 b. 0.419 c. 0.421 d. 0.418

867.

The volume of a sphere is increasing at the rate of 6 cm3/hr. at what rate is its surface area increasing when the radius is 40 cm? a. 0.30 cm2/hr b. 0.40 cm2/hr c. 0.50 cm2/hr d. 0.60 cm2/hr

868.

If f(x) = ex – e-x – 2x and g(x) = x – sinx, evaluate the limit of f(x)/g(x) as x approaches zero. a. ∞ b. 0 c. 1 d. 2

869.

Find the point of inflection of y = 4 + 3x – x3. a. (1,6) b. (0,4) c. (-2,4) d. (2,2)

870.

Find the volume of the largest right circular cone that can be cut from a sphere of radius R. a. 1.421 R3 b. 1.124 R3 c. 1.241 R3 d. 1.412 R3

871.

If s = x2 + 2y2 + 3z2 and x +y +z = 5, find the minimum value of s. a. 148/11 b. 149/11 c. 150/11 d. 151/11

872.

The cost of fuel per hour in operating a luxury liner is proportional to the square of its speed and is Php. 12,000.00 per hour for a speed of 10-kph. Other costs amount to Php. 48,000.00 per hour independent of the speed. Calculate the speed at which the cost per kilometer is a minimum. a. 35 kph b. 30 kph c. 25 kph d. 20 kph

873.

Find the slope of the tangent to the curve a. b. c. d.

at the point (1,1).

-1/5 -2/5 -3/5 -4/5

874.

If y = 1/2x(sin(lnx) – cos(lnx)), find dy/dx. a. sin(lnx) b. cos(lnx) c. –sin(lnx) d. –cos(lnx)

875.

If x = et and y = 2e-t, find d2 y/dx2. a. 4e-t b. 4e-2t c. 4e-3t d. 4e-4t

876.

Two corridors 6 m and 4 m wide respectively, intersect at right angles. Find the length of the longest ladder that will go horizontally around the corner. a. 13 m b. 14 m c. 15 m d. 16 m

877.

An angle φ of a right triangle is given by the equation φ = arcsin(y/x). If x is increasing at the rate of 1 in/sec and y is decreasing at 0.10 in/sec, how fast is φ changing? a. -0.06892 rad/sec b. -0.08926 rad/sec c. -0.09268 rad/sec d. -0.06928 rad/sec

878.

Find the maximum capacity of a conical vessel whose slant height is 9 cm. a. 293.84 cm3 b. 283.94 cm3 c. 284.93 cm3 d. 294.83 cm3

879.

If the semi-axes of the ellipse 4x2 + 9y2 = 36 are each increased by 0.15 cm, find the approximate increase in its area. a. 2.36 cm2 b. 2.46 cm2 c. 2.56 cm2 d. 2.66 cm2

880.

If y = 4/(2x – 1)3, find y” at x = 1. a. 190 b. 191 c. 192 d. 193

881.

The side of an equilateral triangle increases at the rate of 2 cm/hr. At what rate is the area of the triangle changing at the instant when the side is 4 cm? a. b. 4 c. 5 d. 6

882.

Find the value of x and y which satisfy 2x + 3y = 8 and whose product is a minimum. a. 1 and 2 b. 3 and 2/3 c. 3/2 and 5/3 d. 2 and 4/3

883.

If ln(ln y) + ln y = ln x, find dy/dx. a. b. c. d.

884.

If x = 2sinθ, y = 1 – 4cosθ, then dy/dx is equal to a. 2cotθ b. 2tanθ c. 2cscθ d. 2secθ

885.

The upper and lower edges of a picture frame hanging on a wall are 8 feet and 2 feet above an observer’s eye level respectively. How far from the wall must the observer stand in order that the angle subtended by the picture is a maximum? a. 3.5 ft b. 4 ft c. 4.5 ft d. 5 ft

886.

If x increases at the rate of 30 cm/s, at what rate is the expression (x + 1) 2 increasing when x becomes 6 cm? a. 400 cm2/s b. 410 cm2/s c. 420 cm2/s d. 430 cm2/s

887.

Find the radius of a right circular cylinder of maximum volume that can be inscribed in a right circular cone of radius R. a. R/3 b. R/2 c. 3R/4 d. 2R/3

888.

Find the area of the triangle bounded by the coordinate axes and the tangent to the parabola y = x2 at the point (2,4). a. 2 b. 3 c. 4 d. 5

889.

What is the maximum value of y = 3sinx + 4cosx ? a. 8 b. 7 c. 6 d. 5

890.

Find the maximum point of the curve y = 4 + 3x – x3. a. (-2,6) b. (0,4) c. (1,6) d. (-3,22)

891.

Water flows into a cylindrical tank at the rate of 20 m3/s. How fast is the water surface rising in the tank if the radius of the tank if the radius of the tank is 2 m? a. 5/π b. 6/π c. 3/π d. 4/π

892.

If (0,4) and (1,6) are critical points of y = a + bx + cx3, find the value of c. a. 1 b. 2 c. -1 d. -2

893.

Intensity of light is proportional to the cosine of the angle of incidence and inversely proportional to the square of the distance from the source of light. A lamp is directly over the center of a circular table of radius 3 feet. How high above should the lamp be placed so that there will be maximum illumination around the edge of the table? a. 2.18 ft b. 2.16 ft c. 2.14 ft d. 2.12 ft

894.

Find the value of x so that the determinant given below will have a minimum value.

a. b. c. d.

5 6 7 8

895.

Find the area of the largest triangle that can be formed by the tangent to the curve y = e -x and the coordinate axes. a. 1/e b. 2/e c. 3/e d. 4/e

896.

A bus company planning a tour knows from experience that at Php. 20.00 per person, all 30 seats in the bus will be taken but for each increase of Php. 1.00, two seats will become vacant. The expenses of the tour are Php. 100.00 plus Php. 11.00 per person. What price should the company charge to maximize the profit? a. Php. 23.00 b. Php. 24.00 c. Php. 25.00 d. Php. 26.00

897.

An isosceles triangle has legs 26 cm long. The base decreases at the rate of 12 cm/s. Find the rate of change of the angle at the apex when the base is 48 cm. a. -1.4 cm/s b. -1.3 cm/s c. -1.2 cm/s d. -1.1 cm/s

898.

Find the weight of the heaviest cylinder that can be cut out from a sphere which weighs 12 kg. a. 4.93 kg b. 5.93 kg c. 6.93 kg d. 7.93 kg

899.

If

find dy/dx. a. b. c. d.

eaxcosbx eaxsinbx –eaxcosbx -eaxsinbx

900.

A weight is attached to one end of a 29-m rope passing over a small pulley 17 m above the ground. A man keeping his hand 5 m above the ground holds the other end of the rope and walks away at a rate of 3 m/s. How fast is the weight rising at the instant when the man is 9 m from the point directly below the pulley? a. 1.2 m b. 1.4 m c. 1.6 m d. 1.8 m

901.

A right triangle as a hypotenuse of length 13 and one leg of length 5. Find the area of the largest rectangle that can be inscribed in the triangle if it has one side along the hypotenuse of the triangle. a. 15 b. 16 c. 17 d. 18

902.

Evaluate a. b. c. d.

903.

.

∞ 1 e-2 e2

The sum of two numbers is K. Find the minimum value of the sum of their cubes. a. K3 b. K3/2 c. K3/3 d. K3/4

904.

A chord of a circle 4 m in diameter is increasing at the rate of 0.60 m/min. Find the rate of change of the smaller arc subtended by the chord when the chord is 3 m long. a. 0.81 m/min b. 0.71 m/min c. 0.91 m/min d. 0.61 m/min

905.

A manufacturer estimates that he can sell 1,000 units of a certain product per week if he sets the price per unit at Php. 3.00 and that his sale will rise by 100 units with each Php. 0.10 decrease in price. Find his maximum revenue. a. Php. 3,000 b. Php. 4,000 c. Php. 5,000 d. Php. 6,000

906.

The volume of a pyramid is increasing at the rate of 30 cm3/s and the area of the base is increasing at the rate of 5 cm2/s. How fast is the altitude increasing at the instant when the area of the base is 100 cm2 and the altitude is 8 cm? a. 0.50 cm/s b. 0.40 cm/s c. 0.60 cm/s d. 0.70 cm/s

907.

A closed right circular cylinder has a surface area of 100 cm2. What sould be its radius in order to provide the largest possible volume? a. 3.320 cm b. 2.330 cm c. 3.203 cm d. 2.303 cm

908.

A ship 5 km from a straight shore and travelling at the rate of 36 kph is moving parallel to the shore. How fast is the ship coming closer to a fort on the shore when it is 13 km from the fort? a. 34.24 km b. 33.23 km c. 32.21 km d. 31.20 km

909.

The sum of the base and the altitude of a trapezoid is 36 cm. Find the altitude if its area is to be maximum. a. 18 cm b. 20 cm c. 19 cm d. 17 cm

910.

Find the equation of the line parallel to the line x + 2y = 6 and tangent to the ellipse x 2 + 4y2 = 8 in the first quadrant? a. x + 2y + 4 = 0 b. x – 2y + 4 = 0 c. x + 2y – 4 = 0 d. x – 2y – 4 = 0

911.

A sector with perimeter of 24 cm is to be cut from a circle. What should be the radius of the circle if the area of the sector is to be a maximum? a. 6 cm b. 7 cm c. 5 cm d. 4 cm

912.

Find the equation of the line tangent to the curve y = x3 – 6x2 at its point of inflection. a. 3x + y + 2 = 0 b. 3x – y + 2 = 0 c. 3x + y – 2 = 0 d. 3x – y – 2 = 0

913.

Find the radius of a right circular cylinder of greatest lateral surface area that can be inscribed in a sphere of radius 4. a. 2.53 b. 2.63 c. 2.73 d. 2.83

914.

Evaluate a. b. c. d.

zero one infinity none

.

915.

Two posts 30 m apart are 10 m and 15 m high respectively. A transmission wire passing through the tops of the post is used to brace the posts at a point on level ground between them. How far from the 10-m post must that point be located in order to use the least amount of wire? a. 10 m b. 11 m c. 12 m d. 13 m

916.

Three sides of a trapezoid are each 8 cm long. How long is the fourth side when the area of the trapezoid has the largest value? a. 14 cm b. 15 cm c. 16 cm d. 17 cm

917.

A spherical iron ball 8 inches in diameter is coated with a layer of ice of uniform thickness. If the ice melts at a rate of 10 cu in per min, how fast is the outer surface of the ice decreasing when the ice is 2 inches thick? a. -3.39 in2/min b. -3.33 in2/min c. -3.36 in2/min d. -3.31 in2/min

918.

A circular filter paper of radius 15 cm is folded into a conical filter, the radius of whose base is x. Find the value of x for which the conical filter will have the greatest volume. a. 11.25 cm b. 12.25 cm c. 13.25 cm d. 14.25 cm

919.

Water flows out of a hemispherical tank at the constant rate of 18 cu cm per min. If the radius of the tank is 8 cm, how fast is the water level falling when the water is 4 cm deep? a. -0.1491 cm/min b. -0.1941 cm/min c. -0.1194 cm/min d. -0.1149 cm/min

920.

Find the area of the largest isosceles triangle that can be inscribed in a circle of radius 2. a. 5.2 b. 6.3 c. 4.1 d. 3.8

921.

Sand is poured at the rate of 10 ft 3/min so as to form a conical pile whose altitude is always equal to the radius of its base. At what rate is the area of the base increasing when its radius is 5 ft? a. 3 ft3/min b. 4 ft3/min c. 5 ft3/min d. 6 ft3/min

922.

Find the altitude of the largest right circular cone that can be cut from a sphere of radius R. a. 7R/3 b. 5R/3 c. 4R/3 d. 8R/3

923.

A light is placed 3 ft above the ground and 32 ft from a building. A man 6 ft tall walks from the light toward the building at the rate of 6 ft/sec. Find the rate at which the length of his shadow is decreasing when he is 8 ft. a. -1 fps b. -1.5 fps c. -2 fps d. -2.5 fps

924.

An open box is made by cutting squares of side x inches from four corners of a sheet of cardboard that is 24 inches by 32 inches and then folding up the sides. What should x be to maximize the volume of the box? a. 16.3 in b. 15.2 in c. 13.8 in d. 14.1 in

925.

Let f be a function defined by f(x) = Ax2 + Bx + C with the following properties: f(0) = 2, f’(2) = 10 and f”(10) = 4. Find the value of B. a. 1 b. 2 c. 3 d. 4

926.

A rectangle has its base on the x-axis and its two upper corners on the curve y = 2(1 – x2). What is the maximum perimeter of the rectangle? a. 4 b. 5 c. 6 d. 7

927.

Find the maximum vertical distance between y = cosx and y =

sinx over the interval

[0,2]. a. 1.5 b. 2 c. 2.5 d. 3 928.

A baseball diamond is a square 90 ft on the side. A runner travels from home plate to first base at the rate of 20 ft/sec. how fast is the runner’s distance from the second base changing when the runner is halfway to the first base? a. b. c. d.

929.

If the line to the curve y = x – lnx at x = a, passes through the origin, find a. a. 2.72 b. 2.83 c. 2.91 d. 2.69

930.

Find the radius of curvature of the ellipse 4x2 + 5y2 = 20 at (0,2). a. -1.5 b. -2.5 c. -3.5 d. -4.5

931.

If sin(x/y) = y/x, find dy/dx. a. x/y b. –x/y c. y/x d. –y/x

932.

Water is running into a right circular cone with vertical angle equal to 60 degrees (at the bottom) at the rate of 2 cubic feet per second, and at the same time water is leaking out at a rate which is 4.8 times the square root of its depth. How high will the water rise? a. 0.1637 ft b. 0.1367 ft c. 0.1673 ft d. 0.1736 ft

933.

If

, a. b. c. d.

and

evaluate

25 26 27 28

934.

Find the area bounded by x = y + 2, x = 1 – y2, y = 1 and y = -1 with or without integration. a. 11/3 b. 8/3 c. 7/3 d. 5/3

935.

Find the upper area bounded by the curves r = cscθ and r = 4sinθ. a. 9.10 b. 10.11 c. 11.12 d. 12.13

936.

If f(x) = x1/2 and g(x) = (2x + 1)5/2, evaluate a. b. c. d.

from x = ½ to x = 4.

37/324 36/324 35/324 43/324

937.

Find the perimeter of the cardioid r = 1 – cosθ. a. 7 b. 9 c. 6 d. 8

938.

Find the centroid of the volume of a cone formed by revolving about the y-axis the part of the line intercepted between the coordinate axes. a. b. c. d.

(0,1) (0,2) (0,3) (0,4)

939.

A barrel has the shape of an ellipsoid of revolution with equal pieces but off ends. If the barrel is 10 units long with circular ends of radius 2 units and the midsection of radius 4 units, find the volume of the barrel with or without integration. a. 100π b. 110π c. 120π d. 130π

940.

Each of the vertical ends of a trough is a parabolic segment with base 4 m and altitude 1 m. Find the force against one of the trough if it is full of water. a. 11.43 kN b. 12.44 kN c. 11.45 kN d. 10.46 kN

941.

If the trough in problem 444 is 5 m long, how long will it take a 0.50-hp pump to empty the trough by pumping all of the water to the top of the trough? a. 2.1 min b. 1.2 min c. 1.4 min d. 2.4 min

942.

Find axis. a. b. c. d.

943.

944.

(r/π,0) (3r/π,0) (2r/π,0) (4r/π,0)

Find the moment of inertia of the semicircular arc in problem 446 with respect to its diameter. a. r5 b.

r4

c.

r3

d.

r2

If

, find the value of m. a. b. c. d.

945.

the centroid of a semicircular arc of radius r by placing its diameter along the y-

2 3 4 5

A dog is tied to a 4-m circular tank with a cord 3 m long. The point where the cord is attached to the tank is at the same level as the dog’s collar. Compute the total area in which the dog can move. a. 18.64 m2 b. 16.84 m2 c. 14.85 m2 d. 16.48 m2

946.

If

find f(x). a. b. c. d.

x3/3 x4/4 x3 x4

947.

An equilateral triangle of side 8 ft is immersed in water with its plane vertical. If one side is horizontal, and the vertex opposite that side is in the surface of the water, find the force of pressure on the face of the triangle. a. 8,500 lb b. 8,000 lb c. 7,500 lb d. 7,000 lb

948.

The area bounded by y = x2 and y = 2 – x2 is revolved about the x-axis. Find the volume of the solid generated with or without integration. a. 14π/3 b. 16π/3 c. 17π/3 d. 19π/3

949.

Find the perimeter of the curve x2/3 + y2/3 = 4. a. 46 b. 47 c. 48 d. 49

950.

Evaluate the integral of cos4xdx from x = -π/2 to x = π/2. a. 3π/8 b. 4π/5 c. 5π/6 d. 9π/4

951.

Find the surface area generated by revolving the length of the arc of r = 1 + cosθ from 0 to π about the polar axis. a. 23.13 b. 22.15 c. 21.12 d. 20.11

952.

Evaluate a. b. c. d.

.

5/3 7/3 2/3 4/3

953.

Find the area of the region that is inside the curve r = 8cosθ but is outside the curve r = 4cosθ with or without integration. a. 10π b. 11π c. 12π d. 13π

954.

A conoid is a solid having a circular base such that every plane section perpendicular to the diameter of the base is an isosceles triangle. Find the volume of the conoid having a radius of 2 m and the altitude of the triangle is 4 m. a. 6 m3 b. 7 m3 c. 8 m3 d. 9 m3

955.

A rectangular plate 5 ft long and 4 ft wide is submerged in a liquid at an angle of 60 degrees with the vertical. If the liquid weighs w lb per cu ft, find the force of pressure on the plate if the longer edge is parallel to the surface of the liquid and is 2 ft below the surface. a. 45w lb b. 50w lb c. 55w lb d. 60w lb

956.

Find the moment of inertia of the volume of a right circular cylinder with base radius r and altitude h relative to its base. a. b. c. d.

957.

Evaluate a. b. c. d.

.

1/23 1/24 1/25 1/26

958.

Find the area bounded by yx2 = 1, x = 1 and the x-axis. a. ½ b. 1 c. 3/2 d. 2

959.

If a 10-lb weight could be lifted from the surface of the earth to a height of 4000 miles above the surface of the earth, how much work would have to be done? Assume the force of gravitation to vary inversely as the square of the distance from the center of the earth and take the radius of the earth to be 4000 miles. a. 20,000 mi-lb b. 21,000 mi-lb c. 22,000 mi-lb d. 23,000 mi-lb

960.

Find the value of a. b. c. d.

961.

.

0.4049 0.4409 0.4094 0.4904

The cross section of a certain solid made by any plane perpendicular to the x-axis is an equilateral triangle with the ends of one of its sides on the parabolas y = x2 + 5 and y = 2x2 + 1. Find the volume of this solid between the points of intersection of the parabolas. a. 12.76 b. 13.77 c. 14.78 d. 15.79

962.

A hole of radius 3 units is bored through the center of a sphere of radius 5 units. Find the volume of the part of the sphere with or without integration. a. 278.2 b. 268.1 c. 258.4 d. 248.3

963.

Find the x-coordinate (or ) of the centroid of the area in the first quadrant bounded by the curves y = 2 – x2 and y = x2. a. 3/8 b. 1/4 c. 2/3 d. 4/9

964.

Find the length of the arc of the curve r = 2(1 + cosθ) from θ = 0 to θ = π. a. 6 b. 7 c. 8 d. 9

965.

A solid has a circular base of radius 3 units. Find the volume of the solid if every plane section perpendicular to a fixed diameter of the base is an isosceles triangle with its altitude equal to its base. Solve with or without integration. a. 42 b. 52 c. 62 d. 72

966.

A pit is to be dug in the form of an inverted right circular cone, 4 m deep, and 6 m in diameter at the surface of the ground. Find the number of kilojoules of work to be done if the material weighs w kN/cm3. a. 15πw kJ b. 14πw kJ c. 12πw kJ d. 13πw kJ

967.

A trough 6 m long as its vertical cross section in the form of an isosceles trapezoid. The upper and lower bases are 6 m and 4 m respectively and its altitude is 2 m. if the trough is full of liquid with specific weight 9.81 kN per cu m, find the forces against the slant side of the trough. a. 111.41 kN b. 121.51 kN c. 131.61 kN d. 141.71 kN

968.

Evaluate a. b. c. d.

.

4.9348 4.3894 4.4938 4.8439

969.

The stretch of a spring is proportional to the force applied. If a force of 5 pounds produces a stretch of one-tenth the original length, how much work will be done in stretching the spring to double its original length? (Let L = original length) a. 20L b. 22L c. 24L d. 25L

970.

Find the volume of the ring-shaped solid generated by revolving about the x-axis the portion of the plane bounded by the line y = 5 and the parabola y = 9 – x2. a. 342.24 b. 442.34 c. 542.44 d. 642.54

971.

A uniform chain that weighs 4 N/m has a leaky 15-L bucket attached to it. The bucket contains a liquid that weighs 9 N/L. If the bucket is full when 8 m of the chain is out and half full when no chain is out, how much work was done in winding the chain on a windlass. Assume that the liquid leaks out at a uniform rate. a. 893 J b. 938 J c. 398 J d. 839 J

972.

Find the volume of the torus generated by revolving a circle of radius r about a line on the same plane of the circle and whose distance is 2r from the center of the circle. Solve with or without integration. a. 4πr2 b. 4πr3 c. 4π2r2 d. 4π2r3

973.

A plate in the shape of a right triangle is submerged vertically in the water and the base 3 m long is in the surface of the water. Find the altitude of the triangle if the force due to the water pressure against one face of the plate is 50w kN where w is the specific weight of the water. a. 8 m b. 9 m c. 10 m d. 11 m

974.

Find the volume of the torus generated by revolving about the x-axis the area bounded by x2 + (y – 4)2 = 4. a. 315.83 b. 314.73 c. 313.63 d. 312.53

975.

The cross section of a deep well containing mineral water is a circle of radius 1.2 m. the cost of pumping the water to an outlet at the top of the well is 2 pesos per joule of work. The mineral water weighs 9810 newton per cubic meter. If the surface of the water is one meter below the top of the well and the water is sold 50,000 pesos per cubic meter, find the depth to which the water is to be pumped out to realize maximum profit. a. 2.35 m b. 2.45 m c. 2.55 m d. 2.65 m

976.

A wedge is cut from a circular tree whose diameter is 2 m by a horizontal cutting plane up to the vertical axis and another cutting plane which is inclined by 45 degrees from the previous plane. Find the volume of the wedge with or without integration. a. 3/5 m3 b. 2/3 m3 c. 3/4 m3 d. 2/5 m3

977.

Find the moment of inertia of a circle 5 cm in diameter about an axis through its centroid. a. 30.68 cm4 b. 31.58 cm4 c. 32.48 cm4 d. 33.38 cm4

978.

Find the moment of inertia of the circle in problem 481 relative to the line tangent to the circle. a. 76.47 cm4 b. 77.57 cm4 c. 78.67 cm4 d. 79.77 cm4

979.

Find the perimeter of the astroid whose parametric equations are x = acos 3t, y = asin3t. a. 5a b. 6a c. 7a d. 8a

980.

The axes of two right circular cylinders of equal radii 9 cm each intersect at right angles. Find the volume of the common part of the cylinders. a. 3666 cm3 b. 3777 cm3 c. 3888 cm3 d. 3999 cm3

981.

A hemispherical tank is full of oil weighing 7.85 kN/m3. The oil is to be pumped to the top of the tank. Find the work done if the radius of the tank if 0.60 m. a. 0.799 kJ b. 0.688 kJ c. 0.577 kJ d. 0.466 kJ

982.

Find the area of one loop of the curve r2 = 8cos2θ. a. 3 b. 4 c. 5 d. 6

983.

Find

of the centroid of the solid generated by revolving about the y-axis, the first

quadrant area bounded by y2 = 12x, x = 3 and y = 0. a. 2.3 b. 2.5 c. 2.7 d. 2.9

984. The angle between 90 degrees and 180 degrees has A. negative cotangent and cosecant B. negative sine and tangent C. negative secant and tangent D. negative sine and cosine 985. It is defined as the angle subtended by a circular arc whose length is equal to the radius of the circle. A. mil B. radian C. degree D. grade 986. In what quadrant does an angle terminate if its cosine and tangent are both negative? A. first B. second C. third D. fourth

987. Which of the following angles in standard position is a quadrantal angle ? A. 540 degrees B. 480 degrees C. -135 degrees D. -390 degrees 988. It is an angular unit that is equal to 1/6400 of four right angles. A. mil B. grade C. radian D. rpm 989. Relative to a right triangle ABC where C = 90 degrees, which of the following is not true ? A. sin A = cos B B. tan A = cot B

C. cos A = sec B D. csc A = sec B

990. If the value of sin A is a negative fraction, then angle A terminates in A. quadrants II and III B. quadrants I and III

C. quadrants III and IV D. quadrants II and IV

991. The secant is the cofunction of A. sine

B. cosine

C. cotangent

D. cosecant

992. Which of the following is an undirected distance ? A. The distance of a point from the x-axis. B. The distance of a point from the y-axis. C. The distance of a point from the origin. D. The distance of a point from a line. 993. Which of the following systems of angle measurements uses the degree as the unit of measure? A. circular system B. mil system

C. sexagesimal system D. grade system

994. In what quadrant will angle A terminate if sec A is positive and csc A is negative. A. I

B. II

C. III

D. IV

995. Which of the following relations is not true ? A. sinx = (tanx/secx) B. (cotx/cscx) = (sinx/tanx)

C. cotx = cscx cosx D. (secx/tanx) = (cosx/cotx)

996. Within what limits between between 0 degrees and 360 degrees must the angle θ lie if cos θ = -2/5 ? A. between 0 degrees and 180 degrees B. between 90 degrees and 180 degrees C. between 90 degrees and 270 degrees D. between 90 degrees and 360 degrees 997. The coreference angle of any angle A is the positive acute angle determined by the terminal side of A and the y-axis. What is the coreference angle of 290 degrees ? A. 70 degrees B. 50 degrees C. 30 degrees D. 20 degrees 998. A measure of 3200 mils is equal to A. 90 deg B. 45 deg C. 180 deg D. 120 deg 999. The value of vers θ is equal to A. 1 - cosθ B. 1 - sinθ C. 1 + cosθ D. 1 +sinθ

1000. To find the interior angles of a triangle whose sides are given, use the law of A. sine

B. cosine

C. tangent

D. secant

1001. The point P(x,y) where x  0 and y > 0 is located in quadrant A. I or IV

B. II or III

C. I or II

D. III or IV

1002. Which of the following relations is true for any angle θ ? A. sin(-θ) = sin θ B. sec(-θ) = sec θ

C. tan(-θ) = tan θ D. csc(-θ) = csc θ

1003. Coversine A is equal to A. 1 - cosA B. 1 - sin A

C. 1 + cosA D. 1 + sin A

1004. The terminal side of -1,500 degrees will lie in quadrant A. one

B. two

C. three

D. four

1005. Which of the following is false as the angle A increases from 0 degrees to 90 degrees ? A. sin A increases from zero to one B. tan A increases from zero to infinity C. cos A decreases from one to zero D. sec A decreases from one to infinity 1006. Which of the following functions is positive if angle A terminates in the second quadrant ? A. csc A

B. tan A

C. sec A

D. cos A

1007. An angle in standard position and whose terminal side falls along one of the coordinate axes is called a A. reference angle B. vertical angle

C. quadrantal angle D. central angle

1008. Which of the following pairs of angles in standard positions are coterminal angles ? A. 710 degrees and -10 degrees B. 120 degrees and 60 degrees C. -240 degrees and 30 degrees D. 325 degrees and -40 degrees

1009. The gradient of the line in the figure is A. tan θ B. -1/tan θ C. -tan θ D. cot θ

1010. Which of the following is true in quadrants III and IV ? A. negative cosecant B. positive sine

C. negative cotangent D. positive tangent

1011. Which of the following is not a first quadrant angle ? A. 450 degrees B. 60 degrees

C. -330 degrees D. -120 degrees

1012. If tan θ > 0 and cosθ < 0, then θ is a A. first quadrant angle B. second quadrant angle

C. third quadrant angle D. fourth quadrant angle

1013. If an angle is in the standard position and its measure is 215 degrees, the its reference angle is A. 25 degrees

B. 30 degrees

C. 35 degrees

D. 40 degrees

1014. In the second quadrant, which of the following is true ? A. The tangent and secant are positive B. The sine and cosecant are positive C. The cotangent and cosecant are positive D. The sine and tangent are positive 1015. In what quadrant can we locate the point (x, -4) if x is positive ? A. I

B. II

C. III

D. IV

1016. In what quadrants do the secant and cosecant of an angle have the same algebraic sign? A. II and IV

B. I and II

C. I and III

D. III and IV

C. 60 degrees

D. 90 degrees

C. 45 degrees

D. 60 degrees

C. 10/13

D. 12/13

1017. If cos 3A + sin A = 0, find the value of A. A. 30 degrees

B. 45 degrees

1018. If tan A = 2 and tan B = 1/2, find A + B. A. 90 degrees

B. 30 degrees

1019. If sin x = 5/13 , find sin 2x. A. 120/169

B. 25/169

1020. If cot θ = square root of 3 and cos θ < 0, find csc θ. A. 2

B. -2

C. 1/2

D. -1/2

1021. If sin A = -5/13 and A in quadrant III, find cot A. A. 12/5

B. -12/5

C. 5/12

D. -5/12

C. 17/19

D. 8/17

C. 4 pi

D. 6 pi

1022. Find the value of sin(Arecos 15/17). A. 8/9

B. 8/2

1023. The cosecant of 960 degrees is equal to A. -2( square root of 3 / 3) B. 2( square root of 3 / 3) C. 1/2 D. -1/2 1024. If sin 3A = cos 6B, then A. A - 2B = 90 degrees B. A + 2B = 90 degrees C. A + B = 180 degrees D. A + 2B = 30 degrees 1025. What is the period of y = 3 sin(x/2) ? A. 2 pi

B. 3 pi

1026. If the product of cot 2θand cot 68 degrees is equal to unity, find θ. A. 13 degrees B. 12 degrees C. 11 degrees D. 10 degrees 1027. Sec A - cos A is identically equal to A. sin A cot A B. cos A tan A C. sin A tan A D. cos A cot A 1028. Simplify ( sin θ/ 1 - cos θ) - ( 1 + cos θ/ sin θ) A. sin²θ

B. cos²θ

C. 1

D. 0

C. 2

D. 1/2

C. 24/25

D. -24/25

C. sinθ

D. 2tanθ

1029. If tan x = 1/2 and tan y = 1/3, find tan (x + y). A. 1

B. 2/3

1030. If cos θ= 3/5 and θ in quadrant IV, find cos2θ A. 7/25

B. -7/25

1031. simplify (sinθ + cosθtanθ)/(cosθ) A. tanθ

B. 2cotθ

1032. If Arcsin(2x) = 30 degrees, find x. A. 0.20

B. 0.25

C. 0.3

D. 0.35

1033. If sin 40 degrees + sin 20 degrees = sin θ, find the value of θ. A. 20 degrees

B. 60 degrees

C. 80 degrees

D.120 degrees

1034. The angle that is equal to one half of its supplement is A. 60 degrees

B. 90 degrees

C. 80 degrees

D. 45 degrees

1035. Find the equivalent value of y in the equation y = (1 + cos 2θ) / (cot θ) A. sin2θ

B. cos2θ

C. sinθ

D. cosθ

1036. If tan A = -3 and tan B = 2/3, find tan(A - B). A. -11/9

B. -10/9

C. -13/9

D. -12/9

1037. If cos 65 degrees + cos 55 degrees = cos θ, find the θ in radians. A. 1.832

B. 1.658

C. 0.7853

D. 0.0873

C. 42 degrees

D. 62 degrees

C. sin 2x

D. cos 2x

1038. If tan (A / 4) = cot A, find A. A. 52 degrees

B. 72 degrees

1039. Simplify cos^4 x- sin^4 x A. cos 4x

B. sin 4x

1040. If tan 4x = cot 6y, then A. 2x - 3y = 45 degrees B. 2x + 3y = 45 degrees 1041. Simplify

C. 4x - 6y = 90 degrees D. 6y - 4x = 90 degrees

Arctan(1/3) + Arctan(1/5)

A. Arctan (7/4) B. Arctan (4/7)

C. Arctan (8/15) D. Arctan (1/15)

1042. If sin A =3.5x and cos A = 5.5x, find angle A. A. 32.47 degrees B. 33.47 degrees

C. 34.47 degrees D. 35.47 degrees

1043. If the tangent of an angle x is 3/4, find the value of the cosine of 2x. A. 0.60

B. 0.28

C. 0.8

D. 0.38

1044. Find the angle which a 9-m ladder will make with the ground if it is leaned against a window still 6m high. A. 21.8 degrees B. 31.8 degrees

C. 41.8 degrees D. 51.8 degrees

1045. The expression (1 -sinx) / (cosx) is equal to A.tanx

B.1

C.(1 - cosx)/sinx

D.(cosx) / (1 + sinx)

1046. A tree 30 m long casts a shadow 36 m long. Find the angle of elevation of the sun. A. 39.41 degrees B. 39.51 degrees

C. 39.81 degrees D. 39.61 degrees

1047. Which of the following is true ? A. tan(180 degrees + θ) = - tanθ B. tan(180 degrees - θ) = -tan θ

C. tan(90 degrees + θ) = -tanθ D. tan(270 degrees - θ) = - tanθ

1048. Express 3i + 5 + (square root of -16) in the standard form. A. 5 - 7i

B. 5 + 7i

C. -5 + 7i

D. -5 - 7i

1049. Write (square root of 2) cis 135 degrees in rectangular form. A. 1 -i

B. -1 + i

C. -1 - i

D. 1 + i

1050. Give the conjugate of 2 + (square root of -25) in the standard form. A. 2 - 5i

B. 2 + 5i

C. -2 + 5i

D. -2 -5i

1051. For the trigometric function y = a sin(bx +c), the absolute value of the ratio c/b is called A. amplitude

B. period

C. argument

D. phase shift

1052. If sin2x sin4x = cos2x cos4x, find the value of x. A. 13°

B. 14°

C. 15°

D. 16°

C. 0.1536

D. 0.1538

C. 19°

D. 17°

1053. If sin θ = 3.5x and cos θ = 5.5x, find x. A. 0.1532

B. 0.1534

1054. Find θ if 2tan θ = ( 1 - tan² θ) cot 56° . A. 18°

B. 16°

1055. Solve for x if Arctan ( 1 – x ) + Arctan ( 1 + x ) = Arctan ( 1/8 ). A. 2

B. 4

C. 6

D. 3

1056. If A + B = 180°, then which of the following is true ? sin A = sin B cos A = cos B tan A = tan B A. (1) only 1057. Simplify

B. (2) only

C. (3) only

D. all of them

(sin ½x – cos ½x) ²

A. 1 + sin x

B. 1 – sin x

C. 1 + cos x

D. 1 – cos x

C. 60°

D. 90°

1058. Find the value of Arctan 2cos(Arcsin √3/ 2) . A. 30°

B. 45°

1059. If sin A = -7/25 where 180° < A < 270°, find tan(A/2). A. -1/5

B. -5

C. -1/7

D. -7

1060. If sin²x + y = m and cos²x + y = n, find y. A. (m + n + 1)/2 B. (m + n – 1)/2 C. (m+n)/2 – 1 D. (m+n)/2 +1 1061. Given cos θ = √3/2, find the value of 1 - tan² θ. A. -2 B. -1/3 C. ½

D. 2/3

1062. What is the value of A between 270° and 360° if 2sin² A – sin A = 1 ? A. 290°

B. 275°

C. 300°

D. 330°

1063. Evaluate ( sin 0° + sin 1° + sin 2° + … + sin 90°) / ( cos 0° + cos 1° + cos 2° + … + cos 90°) A. 0

B. 1

C. 2

D. 3

1064. If the supplement of an angle θ is 5/2 of its complement. Find the value of θ. A. 30°

B. 25°

C. 20°

D. 15°

1065. Express -4 - 4√3 i in trigonometric form. A. 8 cis 120°

B. 8 cis 240°

C. 8 cis 150°

D. 8 cis 300°

1066. If cos A = -15/17 and A is in quadrant III, find cos ½ A. A. 0.29054

B. 0.24125

C. 0.24254

D.0.24354

1067. If sin A = 3/5 and cos B = 5/13, find sin (A + B). A. 0.388 1068. Simplify A. cot x

B. 0.865

C. 0.650

D. 0.969

C. tan 2x

D. 1

(sin 2x) / ( 1 + cos 2x) B. tan x

1069. A pole which leans to the sun by 10° 15’ from the vertical casts a shadow of 9.43 m on the level ground when the angle of elevation of the sun is 54°50’. The length of the pole is A. 15.3 m

B. 16.3 m

C. 17.3 m

D. 18.3 m

1070. Triangle ABC has sides a, b and c. If a = 75 m, b = 100 m and the angle opposite side a is 32°, find the angle opposite side c. A. 93°

B. 80°

C. 103°

D. 100°

1071. If the cosine of angle x is 3/5, then the value of the sine of x/2 is A. 0.500

B. 0.361

C. 0.215

D. 0.447

C. 12°

D. 13°

C.-25/7

D.-24/7

C. 36/85

D.37/85

1072. If 82° + 0.35x = Arctan( cot 0.45x ), find x. A. 11°

B. 10°

1073. If sec A = -5/4, A in quadrant II, find tan 2A. A.24/7

B.25/7

1074. Evaluate cos( Arcsin 3/5 + Arctan 8/15 ) A. 34/85

B. 35/85

1075. If sin x = ¼ , find the value of 4sin(x/2)cos(x/2). A. 1/8

B. 1/3

C. ½

D. 1/6

1076. If Arcsin( x – 2 ) = π/6, find x. A. 5/4

B. 5/3

C. 5/2

D. 5/6

1077. The trigonometric expression ( 1 - tan²x ) / ( 1 + tan²x ) is equal to A. sin1/2x

B. sin2x

C. cos1/2x

D. cos2x

1078. If x + y = 90°, then ( sinx tan y ) / ( sin y tan x ) is equal to A. tanx

B. 1/tanx

C. –tanx

D. -1/tanx

1079. Twelve round holes are bored through a square piece of steel plate. Their centers are equally spaced on the circumference of a circle 18 cm in diameter. Find the distance between the centers of two consecutive holes. A. 4.33 cm

B. 4.44 cm

C. 4.55 cm

D. 4.66 cm

1080. Two sides and the included angle of a triangle are measured to be 11 cm, 20 cm and 112° respectively. Find the length of the third side. A. 26.19 cm

B. 24.14 cm

C. 23.16 cm

D. 22.15 cm

1081. The rationalized value of ( 4 - 4√3 i ) / ( -2√3 + 2i ) is A. √3 + i

B. -√3 + i

C. -√3 – i

D. √3 – i

C. 0.281

D. 0.291

1082. If Arctan(2x) + Arctan(x) = π/4, find x. A. 0.261

B. 0.271

1083. A ladder leans against the wall of a building with its lower end 4 m from the building. How long is the ladder if it makes an angle of 70° with the ground? A. 12.3 m

B. 13.5 m

C. 11.7 m

D. 10.8 m

1084. Find the product of (4cis120°)(2cis30°) in rectangular form. A. -4(√3 + i)

B. -4(√3 – i)

C. 4(√3 + i)

D. 4(√3 – i)

C. 2

D. 1

1085. Solve for x if x = (tanθ + cotθ) ² sinθ - tan²θ A. 4

B. 3

1086. If ysinx = a and ycosx = b, find y in terms of a and b. A. a + b

B. a² + b²

C. √a² + b²

D. √a + b

C. 1/5

D. ½

1087. If tan(Arctanx + Arctan ¼) = 7/11, find x. A. 1/3

B. ¼

1088. if tanθ = √3, θ in quadrant III, find the value of (1 + cosθ) / (1 – cosθ). A. ½

B. ¼

C. 1/3

D. 1/5

1089. From the top of a lighthouse 37 m above sea level, the angle of depression of a boat is 15°. How far is the boat from the lighthouse? A. 138.1 m

B. 137.2 m

C. 136.3 m

D. 135.4 m

1090. The angles B and C of a triangle ABC are 50°30’ and 122°09’ respectively and BC = 9, find the length of AB. A. 57.36

B. 58.46

C. 59.56

D. 60.66

1091. If the product of csc(x/2) and cos(x/3 + 60°) is equal to 1, find the value of x. A. 46°

B. 36°

C. 26°

D. 16°

C. 1/5

D. 1/6

1092. If Arctanx + Arctan(1/3) = 45°, find x. A. ½

B. ¼

1093. If cscθ = 2 and cosθ < 0, then ( secθ + tanθ ) / ( secθ – tanθ ) = A. 2

B. 3

C. 4

D. 5

1094. Evaluate [6( cos80° + isin80° ) / 3( cos35° + isin35° )] A. √2 ( 1 + i )

B. √2 ( 1 – i )

C. 2 ( 1 + i )

D. 2 ( 1- i )

C. 21°

D. 20°

1095. If sin(x + 10°) = cos3x, then x = A. 23°

B. 22°

1096. If cos(x + y) = 0.17 and cosx = 0.50, find sin y. A. 0.2355

B. 0.3455

C. 0.4344

D. 0.4233

1097. If sin A + sin B = 1 and sin A – sin B = 1, find A. A. 60°

B. 70°

C. 80°

D. 90°

1098. At a certain instant, a lighthouse is 4 miles north of a ship which is traveling directly east. If after 10 minutes, the bearing of the lighthouse is found to be North 21 degree 15 minutes West, find the speed of the ship in miles per hour. A. 11.3 mph

B. 10.3 mph

C. 9.3 mph

D. 8.3 mph

C. cot A

D. sin A

B. -8

C. 8i

D. -8i

B. 16i

C. -16

D. 16

1099. Simplify ( sec A + csc A ) / ( 1 + tan A ) A. csc A

B. sec A

1100. Evaluate [2(cos60° + isin60°)]³ A. 8 1101. Evaluate (1 + i)^8 A. -16i

1102. Two buildings with flat roofs are 15 m apart. From the edge of the roof of the lower building, the angle of elevation of the edge of the roof of the taller building is 32°. How high is the taller building if the lower building is 18 m high? A. 26.4 m

B. 27.4 m

C. 28.4 m

D. 29.4 m

1103. If two sides of a triangle are each equal to 8 units and the included angle is 70°, find the third side. A. 6.15

B. 7.16

C. 8.17

D. 9.18

1104. Express sin(2Arccosx) in terms of x. A. 2x√1 + x²

B. 3x√1 + x²

C. 2x√1 - x²

D. 3x√1 - x²

1105. Transform Arctanx + Arctany = pi/4 into an algebraic equation A. x + xy + y = 1

B. x + xy –y = 1

C. x – xy + = 1

D. x – xy-y =1

1106. A tower 28.65 m high is situated on the bank of a river. The angle of depression of an object on the opposite bank of the river is 25°20’. Find the width of the river.

A. 62.50 m

B. 60.52 m

C. 65.20 m

D. 63.25 m

1107. Two cars start at the same time from the same station and move along straight roads that form an angle of 30°, one car at the rate of 30 kph and the other at the rate of 40 kph. How far apart are the cars at the end of half an hour ? A. 10.17 km

B. 10.27 km

C. 10.37 km

D. 10.47 km

C. -0.80

D. -0.90

1108. Given: sec2θ = √10 and 2θ in quadrant IV Find : cos4θ A. -0.60

B. -0.70

1109. The bearing of B from A is N20°E, the bearing of C from B is S30°E and the bearing of A from C is S40°W. If AB = 10, find the area of triangle ABC. A. 14.95

B. 13.94

C. 12.93

D. 11.92

1110. Two ships start from the same point, one going south and the other North 28° East. If the speed of the first ship is 12 kph and the second ship is 16 kph, find the distance between them after 45 minutes. A. 17.3 km

B. 18.5 km

C. 19.2 km

D. 20.4 km

1111. If tanθ = ´ and θ is in the 1st quadrant, find tan 4θ. A. -24/7

B. -20/7

C. -23/7

D.-22/7

1112. Find the height of a tree if the angle of elevation of its top changes from 20° to 40° as the observer advances 23m toward its base. A. 138.5 m

B. 148.5 m

C. 158.5 m

D. 159.5 m

C. 19°

D. 18°

C. -33/54

D. -33/53

1113 If 77° + (2x/5) = Arccos(sin x/4) , find x. A. 21°

B. 20°

1114. Evaluate tan (Arccos(12/13) – Arcsin(4/5)) A. -33/56

B. -33/55

1115. Three times the sine of an angle is equal to twice the square of the cosine of the same angle. Find the angle. A. 20°

B. 25°

C. 30°

D. 35°

1116. Stations A and B are 1000 m apart on a straight road running from eat to west. From A, the bearing of a tower at C is 32° west of north and from B, the bearing of C is 26° north of east. Find the shortest distance of the tower at C from the road. A. 243.92 m

B. 253.92 m

C. 263.92 m

D. 273.92 m

1117.If tan35° = y, then (tan145° - tan125°) / (1 + tan145°tan125°) = A.(1 + y²) / 2y

B.(1 - y²) / 2y

C.(y²-1) / 2y

D. (2y-1)/2y

1118. A tree stands vertically on a hillside which makes an angle of 22° with the horizontal. From a point 60 ft down the hill directly from the base of the tree, the angle of elevation of the top of the tree is 55°. How high is the tree ? A. 56.97 ft

B. 57.96 ft

C. 59.76 ft

D. 57.69 ft

C. 8m² -8m + 1

D. 8m² - 8m -1

1119. If cos 2A = √m , find cos 8A. A. 8m² + 8m + 1

B. 8m² + 8m – 1

1120. The angle of triangle ABC are in the ratio 5:10:21 and the side opposite the smallest angle is 5. Find the side opposite the largest angle. A. 13.41

B. 14.31

C. 13.14

D. 11.43

1121. On the top of a cliff, the farthest distance that can be seen on the surface of the earth is 60 miles. How high is the cliff if the radius of the earth is taken to be 4000 miles ? A. 0.41 mi

B. 0.43 mi

C. 0.45 mi

D. 0.47

1122. Two towers are of equal height. At a point P on level ground between them, the angle of elevation of the top of the nearer tower is 60° and at a point M 24 meters directly away from point P, the angle of elevation of the top of the nearer tower is 45°. How high is each tower ? A. 20.8 m

B. 19.8 m

C. 18.8 m

D. 17.8 m

1123. A quadrilateral ABCD has its side AB perpendicular to side BC at B and its side AD perpendicular to side CD at D. If angle BAD equals 60°, AB = 10 m and AD = 12 m, find the distance (diagonal) from A to C. A. 11.96 m

B. 12.86 m

C. 13.76 m

D. 14.66 m

1124. The sides of triangle ABC are AB = 5, BC = 12 and AC = 10. Find the length of the line segment drawn from vertex A and bisecting BC.

A. 5.15

B. 5.25

C. 5.35

D.5.45

1125. Express 1/2 (1 - √3 i ) in trigonometric form. A. cis 120°

B. cis 240°

C. cis 300°

D. cis 315°

1126. If versinθ = x and 1 – sinθ = ´ , find x if θ < 90°. A. 0.124

B. 0.134

C. 0.154

D. 0.164

1127. Two points A and B, 150 m apart lie on the same side of a tower on a hill and in a horizontal line passing directly under the tower. The angles of elevation of the top and bottom of the tower viewed from B are 42° and 34° respectively and at A, the angle of elevation of the bottom is 10°. Find the height of the tower. A. 7.3 m

B. 8.3 m

C. 9.3 m

D. 10.3 m

1128. A point P is at a distance of 4, 5 and 6 from the vertices of an equilateral triangle of side of x. Find x. A. 8.5

B. 9.5

C. 7.5

D. 10.5

1129. A quadrilateral ABCD has its sides AB and BC perpendicular to each other at B. Side AD makes an angle of 45° with the vertical while side CD makes an angle of 70° with the horizontal. If AB = 15 and BC = 10, find the length of side CD. A. 31.5

B. 51.5

C. 61.5

D. 41.5

1130. A clock has a minute hand 16 cm long and an hour hand 11 cm long. Find the distance between the outer tips of the hands at 2:30 o’clock. A. 19.6 cm

B. 20.6 cm

C. 21.6 cm

D. 22.6 cm

1131. If rcosxsiny = a, rcosxcosy = b and rsinx = c, find r. A. √a² - b² - c²

B. √a² + b² -c²

C. √a² - b² + c²

D. √a²+b²+c²

1132. From the top of a tower 18 m high, the angles of depression of two objects situated in the horizontal line with the base of the tower and on the same side, are 30 and 45 degrees. Find the distance between the two objects. A. 13.18 m

B. 13.28 m

C. 13.38 m

D. 13.48 m

1133. The sum of the sines of two angles A and B is 3/2 while the sum of the cosines of the angles is √3 /2 . Find A. A. 60°

B. 30°

C. 90°

D. 45°

C. ¾

D. 3/6

1134. Evaluate tan( Arcsec √5 – Arccot 2 ) A. 3/7

B. 3/5

1135. What is the greatest distance on the surface of the earth that can be seen from the top of Mayon volcano which is 2.4 kilometers high if the radius of the earth is 6370 km ? A. 159.7 km

B. 174.8 km

C. 179.7 km

D. 189.7 km

1136. A pole stands on a plane which makes an angle of 15° with the horizontal. A wire from the top of the pole is anchored on a point 8 m from the foot of the pole. If the angle between the wire and the plane is 30 degrees, find the length of the wire. A. 10.93 m

B. 11.93 m

C. 12.93 m

D. 13.93 m

1137. If sin x + sin y = ½ and cos x – cos y = 1, find x. A. 15°

B. 20°

C. 25°

D. 30°

1138. A tower standing on level ground is due north of point A and due east of point B. At A and B, the angles of elevation of the top of the tower are 60° and 45° respectively. If AB = 20 , find the height of the tower. A. 18.32 m

B. 17.32 m

C. 16.32 m

D. 15.32 m

C. 90°

D. 45°

1139. If cot(80° - x/2) cot(2x/3) = 1, find x. A. 30°

B. 60°

1140. If Arctan z = x/2, find cos x in terms of z. A. (1 + z²) / (1 - z²) B. (1 - z²) / (1 + z²) C. (z² + 1) / (z² - 1) D. (z² - 1) / (z² + 1) 1141. A flagstaff stands on the top of a house 15 m high. From a point on the plane on which thee house stands., the angles of elevation of the top and bottom of the flagstaff are found to be 60° and 45° respectively. Find the height of the flagstaff.

A. 10.98 m

B. 11.87 m

C. 12.76 m

D. 13.25 m

1142. Two observers 100m apart and facing each other on a horizontal plane, observer at the same time the angles of elevation of a balloon in their vertical to be 58° and 44°. Find the height of the balloon . A. 60.23 m

B. 59.34 m

C. 61.31 m

D. 58.75 m

1143. From a point outside an equilateral triangle, the distances of the vertices are 10 m, 18 m and 10 m respectively. Find the side of the triangle. A. 19.94 m

B. 20.94 m

C. 21.94 m

D. 22.94 m

1144. A spherical triangle which contains at least one side equal to a right angle is called A. a right triangle B. a polar triangle

C. an isosceles triangle D. a quadrantal triangle

1145. If A, B and C are the angles of a spherical triangle, then which of the following is true ? A. 180° < A + B + C < 360° B. 180° < A + B + C < 540°

C. 0° < A + B + C < 360° D. 0° < A + B + C < 180°

1146. The angular distance of the horizon from the zenith is equal to how many degrees ? A. 45°

B. 60°

C. 90°

D. 180°

1147. The point on the celestial sphere directly above the observer is called the A. zenith

B. nadir

C. pole

D. azimuth

1148. The small circle parallel to the equator is called the A. equinox

B.parallel of latitude C.meridian

D.horizon

1149. If a, b and c are the sides of a spherical triangle, then A. a + b + c < 180°

B. a + b + c < 360°

C. a + b + c < 540°

D. a+b+c< 90°

1150. The point on the celestial sphere diametrically opposite the zenith is called the A. south pole

B. nadir

C. azimuth

D. north pole

1151. It is the angular distance of a heavenly body from the celestial equator. A. declination

B. altitude

C. latitude

D. colatitude

1152. At sunset or at sunrise, the astronomical triangle is A. an isosceles triangle B. a quadrantal triangle

C. a right triangle D. an oblique triangle

1153. The azimuth angle is always less than A. 90°

B. 180°

C. 360°

D. 540°

1154. A great circle which passes through the celestial poles and a heavenly body B is called the ________ of B. A. vertical circle

B. hour circle

C. longitude

D. horizon

1155. The angular distance of a point on the celestial sphere from the horizon is called its A. longitude B. altitude

C. latitude D. polar distance

1156. It is the angle at the zenith from the upper branch of the observer's meridian toward the east to the vertical circle of the heavenly body. A. quadrantal angle B. polar angle

C. hour angle D. azimuth

1157. The zenith distance of a star is the complement of its A. declination B. polar distance

C. altitude D. latitude

1158. Which of the following given sets of parts of a spherical triangle is possible in order to define the triangle ? A. A = 55°, B = 65°, C = 60° B. a = 110°, b = 135°, c = 130° C. A = 160°, B = 65°, C = 90° D. a = 120°, b = 150°, c = 60°

1159. The complement of the declination of a star is called the A. polar distance B. zenith distance

C. longitude D. altitude

1160. A 90-degree arc on the terrestrial sphere is equal to how many nautical miles ? A. 3400

B. 4400

C. 5400

D. 6400

1161. How far in statute miles is a place at latitude 40° N from the equator ? A. 2764.8

B. 2846.7

C. 2684.7

D. 2486.7

1162. Find the distance in nautical miles between A ( 40°30'N, 60°E ) and B (80°20'S, 60°E) A. 6250

B. 7250

C. 8250

D. 9250

C. 4964

D. 4496

1163. Express 82°26' in nautical miles. A. 4946

B. 4694

1164. If a place is 12°S of the equator, find its distance in nautical miles from the north pole. A. 5130

B. 6120

C. 7110

D. 8100

1165. Find the difference in longitude between the following places: M(34°54'33" N, 56°12'51" W) P(30°20'46" N, 87°18'20" W) A. 31°05'29"

B. 31°06'28"

C. 31°07'27"

D. 31°08'26"

1166. Find the difference in latitude between the places given in problem 22. A. 4°32'46"

B. 4°33'47"

C. 4°31'48"

D. 4°30'49"

1167. If an observer is 840 nautical miles south of the equator, find his latitude. A. 12° S

B. 13° S

C. 14° S

D. 15° S

1168. How far apart are two points on the equator one in longitude 40° East and the other in longitude 150° West ? A. 190°

B. 180°

C. 170°

D. 160°

1169. Express 3^h 11^m 55^s in angle units. A. 45°47'58"

B. 58°47'45"

C. 47°45'58"

D. 47°58'45"

1170. Express 260°34' in time units. A. 17^h 22^m 16^s B. 17^h 16^m 22^s

C. 17^h 26^m 21^s D. 17^h 21^m 26^s

1171. The plane of a small circle on a sphere of radius 25 cm is 7 cm from the center of the sphere. Find the radius of the small circle. A. 22 cm

B. 23 cm

C. 24 cm

D. 25 cm

1172. Find the area of a spherical triangle ABC on the surface of a sphere of raidus 10 where A = 119°37', B = 38°43' and C = 34°23'. A. 23.18

B. 22.19

C. 21.16

D. 24.13

C. 15°

D. 16°

1173. An hour-angle of one hour is equal to A. 14°

B. 13°

1174.The plane of a small circle on a sphere of radius 10 cm is 5 cm from the center of the sphere. Find the area of the small circle. A. 55π

B. 65π

C. 75π

D. 85π

1175. If the radius of the earth is 3960 miles, find the radius of a parallel of latitude 50° north. A. 2445.44 mi

B. 2554.44 mi

C. 2455.44 mi

D. 2545.44 mi

1176. Use Napier's rule to find a formula for finding angle B of a right spherical triangle when angle A and side c are given. A. tan B = cos c tan A B. cot B = sin c tan A

C. cot B = cos c tan A D. tan B = sin c cot A

1177. Given a right triangle with angles A = 63°15' and B = 135°34'. Find side b. A. 134.1°

B. 143.1°

C. 131.4°

D. 141.3°

1178. The two sides of a right spherical triangle are 86°40' and 32°41'. Find the angle opposite the first given side.

A. 88°12'

B. 87°11'

C. 86°10'

D. 85°09'

1179. If the angles of a spherical triangle are A = 74°21' , B = 83°41' and C = 58°39', find side c. A. 55°54'

B. 54°55'

C. 45°55'

D. 55°45'

1180. The sides of an oblique spherical triangle ABC are given as follows: a = 51°31' , b = 36°47' and c = 80°12'. Find A. A. 32.35°

B. 33.45°

C. 34.55°

D. 35.56°

1181. Find the distance of Manila(14°36' N, 121°05' E) from Hongkong(22°18' N, 114°10' E) in kilometers. A. 1123.42 km

B.1124.32 km

C.1231.24 km

D.1321.42km

1182. If a boat sails N 30° W until the departure is 20 miles, what distance does it sail? A. 55 mi

B. 50 mi

C. 45 mi

D. 40 mi

1183. A ship in latitude 50° N sails 80 nautical miles due East. Find the resulting change in longitude. A. 2.05° E

B. 2.07° E

C. 2.09° E

D. 2.03° E

1184. Find the longitude of an observer if his local apparent time is 10:36:41 AM and the local Greenwich time is 4:23:12 AM. A. 93°22'15" E B. 92°22'15" E

C. 91°22'15" E D. 90°22'15" E

1185. A ship in latitude 32° N sails due East intil it has made good a difference in longitude of 2°35' . Find the departure. A. 128.42 nm

B. 129.43 nm

C. 130.44 nm

D. 131.45 nm

1186. Given a spherical triangle ABC with a = 68°27' , b = 87°32' and C = 97°53'. Find c. A. 96.41°

B. 95.14°

C. 94.61°

D. 93.65°

1187. Find the area of a spherical triangle on the surface of a sphere of radius 10 where a = 140°30', b = 70°15' and C = 116°45' A. 301.7

B. 370.2

C. 300.7

D. 307.1

1188. If the difference in longitude between two places A and B on the earth is 50° and their latitudes are each 30° N. Find the distance AB in nautical miles. A. 2589

B. 2598

C. 2985

D. 2895

1189. A ship leaves A(45°15' N, 140°38' W) and arrives at a place B(48°45' N, 137°12' W). Find the distance AB in nautical miles using middle latitude sailing. A. 140.49

B. 140.47

C. 140.45

D. 140.43

1190. An arc of one degree on the surface of the earth is approximately equal to how many statute miles ? A. 67.1

B. 68.1

C. 69.1

D. 70.1

1191. How many miles away is Manila(14°36' N, 121°05' E) from San Francisco(37°48' N, 122°24' W) ? A. 7051

B. 8051

C. 9051

D. 10051

1192. A ship sails on a course between south and east making a difference in latitude of 13 nautical miles and a departure of 20 nautical miles. Find the course of the ship. A. 54°56'43" E B. 55°53'84" E

C. 56°58'34" E D. 58°54'36" E

1193. Leaving point A(49°57' N, 15°16' W) , a ship sails between south and west till the departure is 38 nautical miles and the latitude is 49°38' N. Find the distance traveled. A. 42.49 n miles B. 43.48 n miles

C. 44.47 n miles D. 45.46 n miles

1194. Find the initial course of a flight from Manila(14°36' N, 121°05' E) to Tokyo(35°40' N, 139°46' E). A. 35°06'

B. 36°05'

C. 30°56'

D. 30°65'

1195. Given a quadrantal triangle with B = 117°54', a = 95°42' and c = 90°. Find angle A. A. 95.64°

B. 96.46°

C. 97.54°

D. 94.56°

1196. The initial course of a ship sailing from a place at latitude 40°40' N and longitude 73°58' W is due east. After it has sailed 600 nautical miles on a great-circle track, find its latitude. A. 37°54' N

B. 38°54' N

C. 39°54' N

D. 36°54' N

1197. If an airplane is to fly from Manila ( 14°36' N, 121°05' E) to Hongkong(22°18' N, 114°10' E) at an average speed of 200 nautical miles per hour, how long should the trip take ? A. exactly 3 hours B. less than 3 hours

C. almost 3 hours D. about 3 hours

1198. Find the local apparent time of sunrise at Paris ( lat 48°50' N) when the declination of the sun is 14°38'. A. 4:40:31 AM B. 4:45:30 AM

C. 4:50:41 AM D. 4:55:40 AM

1199. Find the local apparent time when an observer at latitude 37°52' N finds that the sun's altitude in the eastern sky is 50°10' and the sun's declination is 12°30' N. A. 9:46:51 AM B. 9:45:56 AM

C. 9:56:45 AM D. 9:41:56 AM

1200. An airplane leaves Guam ( 13°24' N, 144°38' E) with an initial course of 36°40' for a great-circle track. Locate the point on the great-circle track which is nearest to the north pole. A. (54°09' N, 80°12' W) B. (59°04' N, 82°10' W)

C. (45°10' N, 81°02' W) D. (49°05' N, 80°21' W)

1201. The declination of a star is 22°; its hour angle is 15°10' and the place of observation is Berlin ( 52°32' N, 13°25' E). Find the altitude of the star. A. 56.32°

B. 57.32°

C. 58.32°

D. 59.32°

1202. At 8:56 AM, the altitude and declination of the sun are found to be 36°18' and 14°35' respectively. If the observation is done in the northern hemisphere, find the latitude of the place of observation. A. 52°56' N

B. 53°57' N

C. 54°58' N

D. 55°59' N

1203. An airplane flew from Manila (14°36' N, 121°05' E) at an average speed of 300 mph on a course S 32° E. At what point will it cross the equator ? A. 130°02' E

B. 140°03' E

C. 150°04' E

D. 160°05' E

1204. A ship sails from A( 38° N, 120° W) on a course 300° for a distance of 140 nautical miles to point B. Find the position of B by middle latitude sailing method. A. (107° N, 125°44' W) B. (106° N, 126°54' W)

C. (108° N, 126°55' W) D. (109° N, 127°45' W)

1205. Find the azimuth of a star at 5:30 PM at a place whose latitude is 41° if the star's declination is 24°. A. 284.18°

B. 274.18°

C. 264.18°

D. 254.18°

1026.Which of the following statements is false ? A. The diagonals of a rhombus are perpendicular to each other. B. The diagonals of a rectangle are equal. C. The diagonals of a rhombus are equal. D. The diagonals of a parallelogram bisect each other. 1207. The angle inscribed in a semicircle is A. an obtuse angle B. an acute angle

C. a straight angle D. a right angle

1208. Which of the following points is equidistant from the vertices of a triangle ? A. incenter B. centroid

C. orthocenter D.circumcenter

1209. The point of intersection of the altitudes of a triangle is called the A. incenter B. centroid

C. orthocenter D.circumcenter

1210. The point of concurrency of the angle bisectors of a triangle is called the A. incenter B. centroid

C. orthocenter D.circumcenter

1211. The point inside a triangle that is equidistant from its sides is called the A. incenter B. centroid

C. orthocenter D.circumcenter

1212. The point of intersection of the medians of a triangle is called A. incenter B. centroid

C. orthocenter D.circumcenter

1213.The line segment which joins the midpoints of two sides of a triangle is parallel to the third side and is what part of the third side ? A. one half

B. one third

C. one fourth

D. two thirds

1214. The sum of the interior angles of a convex polygon of n sides is equal to how many right angles ? A. 2(n-1)

B. 2(n-2)

C. 2(n-3)

D.2(n-4)

1215. A convex polygon is a polygon each interior angle of which is less than A. 45°

B. 60°

C. 180°

D. 90°

1216. Which of the following points is two thirds of the distance from each vertex of a triangle to the midpoint of the corresponding opposite side ? A. incenter B. centroid

C. orthocenter D.circumcenter

1217. It is a quadrilateral two and only two of whose sides are parallel A. rectangle B. rhombus

C. trapezoid D. parallelogram

1218. It is a quadrilateral whose four sides are equal and with no angle equal to a right angle. A. rectangle B. rhombus

C. trapezoid D. parallelogram

1219. The area of a circle is 6 times its circumference. What is its radius ? A. 12

B. 11

C. 10

D. 13

1220. In a circle of radius 6, a sector has an area of 15 pi. What is the length of the arc of the sector ? A. 3 pi

B. 4 pi

C. 5 pi

D. 6 pi

1221. If the length of a side of a square is increased by 100%, its perimeter is increased by A. 100 %

B. 150 %

C. 200 %

D. 250 %

1222. The side of a regular hexagon measures 10 cm. The radius of the circumscribing circle is A. 8 cm B. 10 cm C. 12 cm D. 14 cm 1223. The median of a trapezoid is 8 and one base is 5. How long is the other base ? A. 13

B. 12

C. 11

D. 10

1224. What is the value of θ in the figure ? A. 20° B. 10°

C. 30° D. 15°

1225. The area of the triangle inscribed in a circle is 40 sq. cm. and the radius of the circumscribed circle is 7 cm. If two sides of the triangle are 8 cm and 10 cm, find the third side. A. 10 cm B. 12 cm C. 13 cm D. 14 cm 1226. The altitude of an equilateral triangle is 4. Find the length of each side. A. 3.62 B. 4.62 C. 5.62 D. 6.62 1227. Find the side of a square whose area is equal to that of a rectangle with sides 32 and 18 cm. A. 21 cm B. 22 cm C. 23 cm D. 24 cm 1228. A railroad curve is to be laid out on a circle. What radius should be used if the tract is to change direction by 25° in a distance of 36 m ? A. 82.51 m B. 81.52 m C. 85.21 m D. 81.25 m 1229. Find the area of a rhombus whose diagonals are 32 cm and 40 cm. A. 540 cm² B. 340 cm² C. 640 cm²

D. 440 cm²

1230. The altitude of a triangle is half the base. Find the base if the area is 64. A. 15 B. 16 C. 17 D. 18

1231. Find the area of a triangle whose sides are 9, 12 and 15. A. 54 B. 53 C. 52

D. 51

1232. An isosceles trapezoid has two base angles of 45° and its bases are 6 and 10. Find its area. A. 12 B. 14 C. 16 D. 18 1233. Find the altitude of a trapezoid of area 180 cm² if the bases are 16 and 14 cm. A. 11 cm B. 12 cm C. 13 cm D. 14 cm 1234. Find the area of a sector of a circle of radius 10 cm and whose central angle is 15°. A. 193.32 cm² B. 194.33 cm² C. 195.34 cm² D. 196.35 cm² 1235. Find the length of an arc of a circle of radius 20 which subtends a central angle of 30°. A. 10 pi/3 B. 11 pi/3 C. 13 pi/3 D. 14 pi/3 1236. Find the length of a chord which is 2 units from the center of a circle of radius 6 units. A. 6√2 B. 7√2 C. 8√2 D. 9√2 1237. How many sides has a convex polygon if the sum of the measure of its angles is 1080°? A. 8 B. 7 C. 6 D. 5 1238. What is the measure of each interior angle of a regular pentagon ? A. 106° B. 109° C. 107°

D. 108°

1239. What is the radius of a circle if its circumference is equal to its area ? A. 4 B. 3 C. 2 D. 1 1240. What is the radius of a circle if the length of a 72° arc is 4π ? A. 11π B. 10π C. 9π

D. 8π

1241. Find the area of a parallelogram of sides 15 and 16 if one of its angles is 60°. A. 206.82 B. 207.85 C. 208.81 D. 205.83 1242. Each side of a rhombus is 7 and one angle is 42° . What is its area ? A. 30.69 B. 31.59 C. 32.79 D. 33.89 1243. In triangle ABC, if a = 10 and b = 12 and angle C = 150° , find the area of the triangle. A. 30 B. 31 C. 32 D. 33 1244. The diagonals of a rhombus are 6 cm and 8 cm long. Find the perimeter of the rhombus. A. 20 cm B. 24 cm C. 22 cm D. 26 cm 1245. The angles between the diagonals of a rectangle is 30° and each diagonal is 12 cm long. Find the area of the rectangle. A. 26 cm² B. 36 cm² C. 46 cm² D. 56 cm²

1246. The sides of triangle ABC are a = 14, b = 12 and c = 10. Find the length of the median from vertex A to side a. A. 8.34 B. 8.44 C. 8.54 D.8.64 1247. The minute hand of a large clock is 2 m long. Find the distance traveled by the tip of the minute hand in 5 minutes. A. pi/4 B. pi/2 C. pi/6 D. pi/3 1248. Find the area of a parallelogram whose sides are 128 and 217 if an included angle is 136°. A, 16942.38 B. 17492.83 C. 19294.83 D. 18249.38 1249. The area of a sector of a circle, having a central angle of 60° is 24 pi. Find the radius of the circle. A. 11 B. 12 C. 13 D. 14 1250. Two circles, each of radius 6 units, have their centers 8 units apart. Find the length of their common chord. A. 2√5 B. 3√5 C. 4√5 D. 5√5 1251. What is the apothem of a regular polygon having an area 225 sq. cm. and a perimeter 60 cm? A. 7.5 cm B. 6.5 cm C. 8.5 cm D. 4.5 cm 1252. Find the area of a regular hexagon of side 3 cm. A. 22.28 cm² B. 23.38 cm² C. 24.48 cm²

D. 25.58 cm²

1253. A triangle has sides 3, 6 and 9. Find the shortest side of a similar triangle whose longest side is 15. A. 6 B. 10 C. 8 D. 5 1254. The perimeter of an octagon is 32 and its longest side is 6. What is the longest side of a similar octagon whose perimeter is 24 ? A. 3.5 B. 4 C. 4.5 D. 5

1255. In the figure AB = AC. The value of θ is A. 31 C. 33 B. 32 D. 34

1256. A hexagon is circumscribed about a circle of radius 5. If the perimeter of the hexagon is 38, what is the area of the hexagon ? A. 75 B. 65 C. 85 D. 95 1257. The circumference of two circles are 6π and 10π. What is the ratio of their areas ? A. 9/25 B. 8/25 C. 7/25 D.6/25 1258. If a regular polygon has 54 diagonals, then it has how many sides ? A. 10 B. 11 C. 14 D. 12 1259. If AB is parallel to CD where CD is the diameter of the circle as shown in the figure, find angle θ. A. 20° B. 10° C. 25° D. 15°

1260. What is the diameter of a circle that is circumscribed about an equilateral triangle of side 7.4 cm. A. 8.64 cm B. 8.54 cm C. 9.54 cm D. 9.64 cm 1261. If the perimeter of a rhombus is 40 and one of its diagonals is 12, find the other diagonal. A. 16 B. 15 C. 18 D. 17 1262. Given a circle as shown. The length of arc AB is A. 1.527 B. 1.725 C. 1.257 D. 1.275

1263. Find the area of the annulus bounded by the inscribed and circumscribed circles of an equilateral triangle with a side of length 6. A. 11π B. 8π C. 10π D. 9π 1264. Find the area of a regular octagon inscribed in a circle whose radius is 10 cm. A. 822.8 cm² B. 282.8 cm² C. 828.2 cm² D. 228.8 cm²

1265. In the figure shown, find the shaded area. A. 4π B. 5π C. 6π D. 7π

1266. If the perimeter of a regular hexagon is 24, what is the apothem ? A. 3√3 B. 4√3 C. 2√3

D. 5√3

1267. The ratio of the sum of the exterior angles to the sum of the interior angles of a polygon is 1:3. Identify the polygon. A. hexagon B. heptagon C. octagon D. nonagon 1268. A circular sector has a radius of 6 cm and whose central angle is 60°. If it is bent to form a right circular cone, the radius of the cone is A. 1 cm B. 2 cm C. 3 cm D. 4 cm 1269. A square is inscribed in a 90° sector of a circle as shown. Find the area of the shaded region. A. 1.214 B. 1.412 C. 1.124 D. 1.142

1270. A regular octagon is inscribed in a circle of radius 6. Find the perimeter of the octagon. A. 34.54 B. 35.64 C. 36.74 D. 37.84 1271. If four angles of a pentagon have measures 100°, 96°, 87° and 97°, find the measure of the fifth angle. A. 150° B. 160° C. 140° D. 130° 1272. If the sum two exterior angles of a triangle is 230°, find the measure of the third exterior angle. A. 130° B. 120° C. 110° D. 100° 1273. Given are two concentric circles with line segment AB = 10 cm which is always tangent to the small circle. Find the area of the shaded region (see figure). A. 50 pi cm² B. 45 pi cm² C. 25 pi cm² D. 30 pi cm²

1274. A circle whose radius is 10 cm is inscribed in a regular hexagon. The area of the hexagon is A. 346.4 cm² B. 634.4 cm² C. 364.4 cm² D. 436.6 cm² 1275. The area of a parabolic segment having a base width of 10 cm and a height of 27 cm is A. 270 cm² B. 150 cm² C. 210 cm² D. 180 cm² 1276. A side of a regular hexagon is 6. What is the circumference of its circumscribed circle? A. 12 pi B. 11 pi C. 13 pi D. 10 pi

1277. Two chords PQ and RS of a circle meet when extended through Q and S at a point T. If QP = 7, TQ = 9 , TS = 6, find SR. A. 16 B. 17 C. 18 D. 19 1278. What is the angle at the center of a circle if the subtending chord is equal to two thirds of the radius. A. 39.95° B. 38.94° C. 37.93° D. 36.92° 1279. The area of a rhombus is 250 and one of the angles is 37°25'. What is the length of each side? A. 20.18 B. 20.28 C. 20.38 D. 20.48 1280. If in triangle ABC, A = 76°30', B = 81°40' and c = 368, find the diameter of the circumscribed circle. A. 989.5 B. 959.8 C. 395.8 D. 958.9 1281. Given a parallelogram ABCD such that AB = 7, AC = 10 and angle BAC = 36°07'. Find the length of BC. A. 4.992 B. 5.992 C. 6.992 D. 7.992 1282. What is the diameter of the circle that is circumscribed about an isosceles triangle whose vertical angle is 18° and the sum of the two equal sides is 18 units ? A. 7.11 B. 8.11 C.9.11 D.10.11 1283. The diagonals of a quadrilateral are 34 and 56 intersecting at an angle of 67°. Find its area. A. 837.62 B. 863.72 C. 826.37 D. 876.32 1284. Find the radius of a circle in which is inscribed a regular nonagon whose perimeter is 417.6 cm. A. 68.37 cm B. 67.83 cm C. 63.87 cm D. 68.73 cm 1285. If each interior angle of a regular polygon has a measure of 160°, how many sides has the polygon ? A. 16 B. 17 C. 19 D. 18 1286. The sides of a right triangle are a, b and c where c is the hypotenuse. Find the radius of the circle that is inscribed in the triangle. A. 1/2 (a+b+c) C. 1/2(a-b+c) B. 1/2(a+b-c) D. 1/2(a-b-c) 1287. Two sides of a parallelogram are 20 and 30 and the included angle is 36°. Find the length of the longer diagonal. A. 74.65 B. 64.75 C. 57.46 D. 47.65

1288. The sides of a triangle are 17, 21 and 28. Find the length of the line segment bisecting the longest side and drawn from the opposite angle. A. 11 B. 12 C. 13 D. 14 1289. Two tangent circles of radii 6 and 2 have a common external tangent as shown in the figure. Find the length of this external tangent. A. 4√3 B. 5√3 C. 6√3 D. 7√3

1290. PQ and RS are secants of a circle which when extended beyond Q and S at a point T outside the circle. Given that arc PR = 105° and arc QS = 67°, find the angle QTS. A. 18° B. 19° C. 20° D. 21°

1291. A bridge across a river is in the form of an arc of a circle. If the span is 40 ft and the midpoint of the arc is to be 8 ft higher than the ends, what is the radius of the circle? A. 27 ft C. 29 ft B. 28 ft D. 30 ft 1292. Find the angle formed by the secant and tangent to a circle if one intercepted arc is 30° more than the other and the secant passes through the center of the circle. A. 15° B. 16° C. 17° D. 18° 1293. Find the radius of a circle whose area is equal to the area of the annulus formed by two consecutive circles with radii 5 and 13. A. 10 B. 11 C. 12 D. 13 1294. A circle in inscribed in an equilateral triangle. If the circumference of the circle is 3, find the perimeter of the equilateral triangle. A. 9.246

B. 6.294

C. 2.946

D. 4.962

1295. Given a square ABCD as shown where E is the midpoint of side AD and G is the midpoint of side BC. If arc DF has its center at E and arc FB has its center at G, find the shaded area. A. 6

B. 8

C. 10

D. 12

1296. Two concentric circles each contains an inscribed square. The larger square is also circumscribed about the smaller circle. If the circumference of the larger circle is 12 pi, what is the circumference of the smaller circle ? A. 6√2 pi

B. 5√2 pi

C. 4√2 pi

D. 3√2 pi

1297. A regular cross is inscribed in a circle as shown. Find the area ( shaded) between the regular cross and the circle. A. 43.44

B. 44.55

C. 45.66

D. 46.77

1298. Point P is a point on the minor arc AB of a circle with center at 0 as shown. If the angle APB is x degree and angle A0E is y degrees, find an equation connecting x and y. A. 2x - y = 360°

B. 2x + y = 360°

C. x - 2y = 360°

D. x+2y=360°

1299. The quadrilateral ABCD is inscribed in a circle and its diagonal AC is drawn so that angle DAC = 34°, angle CAB = 38° and angle DBA = 65°. Find arc AB. A. 96°

B. 86°

C. 76°

D. 66°

1300. PORS is a quadrilateral that is inscribed in a circle. If angle SQR = 23° and the angle between the side SP and the tangent line through the point P is 64°, find angle PSR. A. 86°

B. 87°

C. 88°

D. 89°

1301. The lines TA and TB are tangent to a circle at points A and B respectively. IF angle T = 42° and P is a point on the major arc AB, find angle APB. A. 69°

B. 68°

C. 67°

D. 66°

1302. A secant and a tangent to a circle intersect an angle of 38° degrees. If the measures of the arc intercepted between the secant and tangent are in the ratio 2:1, find the measure of the third arc. A. 129°

B. 130°

C. 131°

D. 132°

1303. Compute the difference between the perimeters of a regular pentagon and a regular hexagon if the area of each is 12. A. 0.31

B. 0.21

C.0.41

D.0.51

1304. The area of the sector of a circle having a central angle of 60° is 24π. Find the perimeter of the sector. A. 34.4

B. 35.5

C. 36.6

D.37.7

1305. Find the common area of two intersecting circles of radii 12 and 18 respectively if their common chord is 14 long. A. 34.19

B. 35.29

C. 36.39

D. 37.49

1306. In a parallelogram ABCD, the diagonal AC makes with the angle 27°10' and 32°43' respectively. If side AB is 2.8 m long, what is the area of the parallelogram ? A. 4.7 m²

B. 5.7 m²

C. 6.7 m²

D. 8.7m²

1307. The sum of the sides of a triangle is 100. The angle at A double that of B and the angle at B is double that of C. Find side c. A. 41.5

B. 42.5

C. 43.5

D. 44.5

1308. A diagonal of a parallelogram is 56.38 ft long and makes an angle of 26°13' and 16°24' respectively with the sides. Find the area of the parallelogram. A. 595 ft²

B. 585 ft²

C. 575 ft²

D. 565 ft²

1309. Find the area of a regular five-pointed star that is inscribed in a circle of radius 10. A. 121.62

B. 112.26

C. 122.16

D. 126.21

1310. What is the difference in the areas between an inscribed and a circumscribed regular octagon if the radius of the circle is 6? A. 15.27

B. 16.37

C. 17.47

D. 18.57

1311. If BC = 2(AB), what fraction of the circle is shaded? A. 1/4

B. 1/3

C. 1/2

D. 1/5

1312. In the figure, the small circle is tangent to 4 circular arcs. Find the area of the shaded region if the radius of the larger circle is 10. A. 34.94

B. 35.49

C. 31.94

D. 32.49

1313. A regular five-pointed star is inscribed in a circle of radius b cm. Find the area between the circle and the star. A. 4.04 b²

B. 3.03 b²

C. 1.01 b²

D. 2.02 b²

1314. From a point outside of an equilateral triangle, the distances of the vertices are 12, 20 and 12 respectively. Find the length of each side of the triangle. A. 23.95

B. 22.85

C. 21.78

D. 20.68

1315. Using the vertices of a square, four arcs are drawn as shown in the figure. If each edge is 10 units long, find the shaded portion (common area). A. 21.5

B. 31. 5

C. 41.5

D. 51.5

1316. Assuming that the earth is a perfect sphere of radius 6370 kilometers, a person at a point T on top of a tower 60 meters high looks at a point P on the surface of the earth. What is the approximate distance of P from T ? A. 24.3 km

B. 25.4 km

C. 26.5 km

D. 27.6 km

1317. Each of four circles ( see figure ) is tangent to the other three. If the radius of each of the smaller circles is 3, what is the radius of the largest circle ? A. 6.46

B. 6.64

C. 4.64

D.4.46

1318. In the figure, if arc AB = 50°, arc BC = 80° and arc AD = 90°, find θ. A. 85°

B. 65°

C. 95°

D. 75°

1319. In the figure, if PB = 6, PC = 10, PA = 5 and θ = 30°, find the area of the quadrilateral ABCD. A. 21.5

B. 22.5

C. 23.5

D. 24.5

1320. The solid formed by revolving a circle about an external axis in its plane is called. A. annulus

B. conoid

C. torus

D. prismatoid

1321. The intersection of two faces of a pyramid is called the A. lateral edge

B. slant height

C. altitude

D. hypotenuse

1322. It is a polyhedron of which one face is a polygon and the other faces are triangles which have a common vertex. A. prism

B. pyramid

C. cone

D. prismatoid

1323. The altitude of any of the lateral faces of a regular pyramid is called the A lateral edge

B.altitude

C.median

D.slant height

1324. It is a polyhedron whose faces are all squares. A. tetrahedron B. hexahedron

C. octahedron D.dodecahedron

1325. The dihedral angle is the angle between two intersecting A.lines

B. arcs

C.planes

D.chords

1326. Which of the regular polyhedrons has faces that are regular pentagons ? A. tetrahedron B. dodecahedron

C. octahedron D. icosahedron

1327. If the base of a solid is a circle and every section perpendicular to the base is an isosceles triangle, the solid is called A. conicoid

B. prismoid

C. conoid

D. astroid

1328. The radius of a sphere that is inscribed in a regular hexahedron of edge e is equal to A. e/2

B. e/3

C. e/4

D. e/5

1329. It is a polyhedron of which two faces are equal polygons in parallel plane and the other faces are parallelogram. A.tetrahedron

B. prism

C.pyramid

D.prismoid

1330. A spherical wooden ball 15 cm in diameter sinks to a depth of 12 cm in a certain liquid. The area exposed above the liquid is A. 54 pi cm²

B. 15 pi cm²

C. 45 pi cm²

D. 35 pi cm²

1331. What is the total area of a cube whose edge is 5 cm? A. 150 cm²

B. 145 cm²

C. 140 cm²

D. 135 cm²

1332. Find the volume of the frustum of a right circular cone whose altitude is 6 and whose base radii are 2 and 3. A. 35π

B. 36π

C. 37π

D. 38π

1333. The angle of a lune is 60° and the radius of the sphere is 15 cm. Find the volume of the spherical wedge whose base is the given lune. A. 750π cm³

B. 700π cm³

C. 650π cm³

D. 600π cm³

1334. A sphere of radius R is inscribed in a cube of edge e. What is the ratio of the volume of the sphere to the volume of the cube? A. 0.6523

B. 0.5236

C. 0.3652

D. 0.2635

1335. The slant height of a right circular cone is 13 and the altitude is 12. Find the radius of the base. A. 8

B. 7

C. 6

D. 5

1336. A hemispherical bowl of radius 10 cm is filled with water to a depth of 5 cm. Find the volume of the water. A. 615π/3 cm³

B. 620π/3 cm³

C. 625π/3 cm³

D. 630π/3 cm³

1337. The area of a lune is 4π m² and the radius of the sphere is 3 m. Find the angle of the lune. A. 40°

B. 45°

C. 50°

D. 55°

1338. The volume of a sphere whose diameter is 20 cm is A. 4198.87 cm³

B. 4179.88 cm³

C. 4188.79 cm³

D.4187.89 cm³

1339 Find the length of the diagonal of a rectangular box whose edges are 6, 8 and 10. A. 7√2

B. 8√2

C. 9√2

D. 10√2

1340. Find the area of the base of a prism whose volume is 516.6 cu. ft and whose height is 16.4 in. A. 372 ft²

B. 374 ft²

C. 376 ft²

D. 378 ft²

1341. Find the slant height of a regular pyramid each of whose faces is enclosed by an equilateral triangle with side 8. A. 6.73

B. 6.93

C. 6.83

D. 6.63

1342. What is the volume of a pyramid whose altitude is 27 and whose base is a square 8 on a side ? A. 756

B. 657

C. 576

D. 675

1343. A concrete pedestal is in the form of a frustum of a regular square pyramid whose altitude is 1.2 cm and base edges 0.40 m and 0.70 m respectively. Find the volume of the pedestal. A. 0.372 m³

B. 0.327 m³

C. 0.273 m³

D. 0.723 m³

1344. The base radii of the frustum of a cone are 6 cm and 10 cm respectively. Find the altitude of the frustum if its volume is 1176π cu. cm ? A. 16 cm

B. 17 cm

C. 18 cm

D. 19 cm

1345. What is the diameter of a sphere for which its volume is equal to its surface area? A. 7

B. 6

C. 5

D. 4

1346. Find the volume of a spherical wedge whose angle is 36° on a sphere of radius 6 cm. A. 28.8π cm³

B. 27.7π cm³

C. 26.6π cm³

D. 25.5π cm³

1347. Find the volume of a right circular cone whose base radius is 8 cm and whose altitude is 15 cm. A. 320 pi cm³

B. 330 pi cm³

C. 340 pi cm³

D. 350 pi cm³

C. 3.872 m³

D. 7.238 m³

1348. The volume of a sphere of radius 1.2 m is A. 8.372 m³

B. 2.783 m³

1349. The volume of a square pyramid is 384 cm³ and its altitude is 8 cm. How long is an edge of the base? A. 11 cm

B. 12 cm

C. 13 cm

D. 14 cm

1350. Find the altitude of a right prism flow which the area of the lateral surface is 338 and the perimeter of the base is 13. A. 25

B. 26

C. 27

D. 28

1351. A conical tank is 10.5 m deep and its circular top has a radius of 5 cm. How many liters of water will it hold? A. 260500π

B. 261500π

C. 262500π

D. 263500π

1352. Find the diameter of a sphere whose surface area is 324π. A. 16

B. 17

C. 18

D. 19

1353. Find the area of a zone of a sphere whose radius is 6 if the altitude of the zone is 2. A. 21 pi

B. 22 pi

C. 23 pi

D. 24 pi

1354. The volume of a 10-cm high conical paper weight is 180 cm³. The radius of the base is A. 4.15 cm

B. 4.17 cm

C. 4.19 cm

D. 4.21 cm

1355. The volume of the frustum of a cone which is 25 cm high and whose base radii are 7.5 cm and 5 cm long respectively is A. 3108.87 cm³

B. 3107.88 cm³

C. 3170.88 cm³

D.3180.78 cm³

1356. Find the volume of a cube whose total area is 384 cm². A. 212 cm³

B. 312 cm³

C. 412 cm³

D. 512 cm³

1357. Find the total area of a tetrahedron 3 units on an edge. A. 8√3

B. 9√3

C. 10√3

D. 11√3

1358. The volume of a pyramid is 256 cm³ and its altitude is 24 cm. Find its base area. A. 52 cm²

B. 42 cm²

C. 32 cm²

D. 22 cm²

1359. The radii of the bases of the frustum of a right circular cone are 6 and 9 respectively and the altitude is 4. Find the lateral area. A. 75π

B. 65π

C. 95π

D. 85π

1360. The altitude of a parallelepiped is 20 and the base is a rhombus with diagonals 10 and 16. Find the volume of the parallelepiped. A. 1500

B. 1600

C. 1700

D. 1800

1361. A sphere of radius r just fits into a cylindrical box. Find the empty space inside the box. A. 2π r³/3

B. 8π r³/9

C. 4π r³/9

D. 20π r³/27

1362. Find the volume of a pyramid having a pentagonal base with sides each equal to 12 cm and an apothem of 3 cm. The altitude of the pyramid is 36 cm. A. 2660 cm³

B. 2770 cm³

C. 2880 cm³

D. 2990 cm³

1363. Find the volume of the frustum of a regular triangular pyramid whose altitude is 3 and whose base edges are 4 and 8 respectively. A. 25√3

B. 26√3

C. 27√3

D. 28√3

1364. The lateral area of a right circular cone with a radius of 20 cm and a height of 30 cm is A. 2265.43 cm²

B. 2236.45 cm²

C. 2245.63 cm²

D.2253.46 cm²

1365. The base of a prism is a rectangle with sides 3 and 5. If its lateral area is 64, find its altitude. A. 3

B. 4

C. 5

D. 6

1366. Find the number of degrees on a dihedral angle of a regular tetrahedron. A. 68.33°

B. 69.43°

C. 70.53°

D. 71.63°

1367. Find the volume of a spherical cone in a sphere of radius 17 cm if the radius of the zone is 8 cm. A.1126π/3 cm³

B.1136π/3 cm³C.1146π/3cm³

D.1156π/3 cm³

1368. Find the volume of a regular square pyramid whose slant height is 10 and whose base edge is 12. A. 384

B. 374

C. 364

D. 354

1369. The base of a prism is a rhombus whose sides are each 10 cm and whose shorter diagonal is 12 cm. If the altitude is 12 cm, find its volume. A. 1132 cm³

B. 1142 cm³

C. 1152 cm³

D. 1162 cm³

1370. Find the volume of a triangular prism whose altitude is 20 cm and whose sides are 6 cm, 8 cm and 12 cm. A. 426.61 cm³

B. 421.66 cm³

C. 461.26 cm³

D. 416.62 cm³

C. 2660

D. 2770

1371. Find the volume of the solid as shown. A. 2330

B. 2440

1372. Find the volume of a spherical segment if the radii of the bases are 3 and 4 respectively and its altitude is 2. A. 83.27

B. 87.32

C. 83.72

D. 82.73

1373. A stone is dropped into a circular tub 40 inches in diameter, causing the water therein to rise 20 inches. What is the volume of the stone ? A. 6000π in³

B. 7000π in³

C. 8000π in³

D. 9000π in³

1374. The base of a right parallelepiped is a rhombus whose sides are each 10 cm long and one of whose angles is 60 degrees. If the altitude of the parallelepiped is 4 cm, find its volume . A. 100√3 cm³

B. 200√3 cm³

C. 300√3 cm³

D. 400√3 cm³

1375. Find the volume of the largest cube that can be out from a circular log whose radius is 30. A. 76367.53

B. 75567.33

C. 73675.36

D. 77653.36

1376. Find the lateral area of a pyramid whose altitude is 27 cm and whose base is a square 8 cm on a side. A. 437.62 cm²

B. 436.72 cm²

C. 432.76 cm²

D. 427.63 cm²

1377. The diagonal of a cube is 2√3. Find its volume. A. 9

B. 7

C. 8

D. 6

1378. Find the lateral area of the frustum of a regular square pyramid whose base edges are 6 and 12 and whose altitude is 4. A. 150

B. 160

C. 170

D. 180

1379. If the radius of a sphere is 8 and if a plane passes through the sphere at a distance of 5 from its center. what is the area of the circle of intersection ? A. 38 pi

B. 39 pi

C. 40 pi

D. 41 pi

1380. Find the lateral area of a right circular cone that can be inscribed in a cube whose volume is 64. A. 28.1

B. 26.1

C. 24.1

D.22.1

1381. Find the lateral edge of a regular square pyramid whose slant height is 8 and whose base edge is 6. A. 6.54

B. 7.54

C. 8.54

D.9.54

1382. The base edges of a triangular pyramid are 12, 14 and 16. If its altitude is 22, what is the volume of the pyramid ? A. 594.64

B. 564.94

C. 544.69

D. 596.44

1383. The volume of the frustum of a right circular cone is 78 pi. The upper base radius is 2 and the lower base radius is 5. What is the altitude of the frustum ? A. 5

B. 6

C. 7

D. 8

1384. The volume of a right circular cone having a slant height of 13 and altitude 12 is A. 100π

B. 150π

C. 200π

D. 250π

1385. Find the lateral area of a regular triangular pyramid whose base edge is 4 and its lateral edge is 6. A. 21√2

B. 22√2

C. 23√2

D. 24√2

1386. Find the height of a pyramid whose volume is 35 and whose base is a triangle with sides 4, 7 and 5. A. 11.72

B. 10.72

C. 8.72

D. 9.72

1387. The radii of the bases of the frustum of a right circular cone are 6 and 9 respectively and its altitude is 4. Find its lateral area. A. 75π

B. 85π

C. 95π

D. 65π

1388. Find the volume of a sphere whose surface area is 64π. A. 256π/3

B. 254π/3

C. 252π/3

D. 250π/3

1389. What is the area of a sphere if a zone on it having an area of 18 and has an altitude of 2? A. 79 pi

B. 80 pi

C. 81 pi

D. 82 pi

1390. A spherical bowl of radius 8 inches contains water to a depth of 3 inches. Find the volume of the water in the bowl. A. 199.72 in³

B. 197.92 in³

C. 179.29 in³

D. 192.27 in³

1391. The volume of a pyramid is 256 cm³ and its altitude is 24 cm. Find the area of its base. A. 32 cm²

B. 34 cm²

C. 31 cm²

D. 33 cm²

1392. Find the lateral area of a right circular cone if its slant height is 22 and the circumference of its base is 8. A. 55

B. 66

C. 77

D. 88

1393. What is the diameter of a sphere for which its volume is equal to its surface area ? A. 5

B. 6

C. 7

D. 8

1394. The lateral area of a regular pyramid is 2048 and the perimeter of the base is 128. Find the slant height. A. 42

B. 22

C. 32

D. 52

1395. The area of the base of a right circular cone is 144π . If its altitude is 14, find its slant height. A. 18.44

B. 17.33

C. 16.22

D. 15.11

1396. Find the approximate change in the volume of a cube if each edge x of the cube is increased by one percent. A. 0.02 x³

B. 0.03 x³

C. 0.04 x³

D. 0.05 x³

1397. The area of a diagonal section of a cube is 4√2 cm². Find the edge of the cube. A. 3 cm

B. 2 cm

C. 4 cm

D. 1 cm

1398. Find the volume of the largest circular cylinder that can be inscribed in a cube whose volume is 64 cu. cm. A. 13π cm³

B. 14π cm³

C. 15π cm³

D. 16π cm³

1399. The altitude of a square pyramid is 10 and a side of the base is 15. Find the area of a cross section at a distance of 6 from the vertex. A. 81

B. 82

C. 83

D. 84

1400. The diameter of one solid ball is 3 times the diameter of another ball of the same material. If the weight of the smaller ball is 250 pounds, what is the weight of the larger ball ? A. 6957 lb

B. 6750 lb

C. 6507 lb

D. 6570 lb

1401. Find the volume of a regular tetrahedron whose edges are each equal to 6. A. 16√2

B. 17√2

C. 18√2

D. 19√2

1402. The lateral area of a regular pyramid is 514.5 and the slant height is 42. Find the perimeter of the base. A. 24.5

B. 26.5

C. 22.5

D.28.5

1403. A wedge is cut from a circular tree whose diameter is 2 m by a horizontal plane up to the vertical axis and another cutting plane which is inclined at 45 degrees from the previous plane. The volume of the wedge is A. 1/4

B. 1/2

C. 2/3

D. 3/4

1404. The zone of a spherical cone has a altitude of 2 cm and a radius of 4 cm. Find the volume of the spherical cone. A. 115π/3 cm³

B. 110π/3 cm³

C. 105π/3 cm³

D. 100π/3 cm³

1405. The base of a prism is the triangle ABC with A = 35 degrees, B = 68 degrees and c = 25. If the altitude of the prism is 10, find the volume of the prism. A. 1607.5

B. 1705.6

C. 1507.6

D. 1076.5

1406. The capacities of two hemispherical tanks are in the ratio 64:125. If 4.8 kg of paint is required to paint the outer surface of the smaller tank, then how many kg of paint would be needed to paint the outer surface of the larger tank ? A. 6.5 kg

B. 7.5 kg

C. 8.5 kg

D. 9.5 kg

1407. Find the volume of a sphere that is circumscribed about a cube of edge 4. A. 30√3 π

B. 32√3 π

C. 34√3 π

D. 36√3 π

1408. A sphere is inscribed in a right circular cone. The slant height of the cone is equal to the diameter of its base. If the altitude of the cone is 15, find the surface area of the sphere. A. 125π

B. 120π

C. 110π

D. 100π

1409. The base of a tetrahedron is a triangle whose sides are 10, 24 and 26. If the altitude of the tetrahedron is 20, find the area of a cross-section whose distance from the base is 15. A. 9.5

B. 8.5

C. 7.5

D. 6.5

1410. If the length of each edge of a cube is increased by 3 cm, its volume is increased by 387 cu cm. Find the length of each edge of the original cube. A. 5 cm

B. 6 cm

C. 7 cm

D. 8 cm

1411.The lateral area of a regular pyramid is 2048 and the perimeter of the base is 128. If its base is a regular octagon, find the altitude of the pyramid. A. 24.5

B. 25.5

C. 26.5

D. 27.5

1412. Find the area illuminated by a candle h meters from the surface of a ball r meters in radius. A. (2πrh²) / (r+h) B. (2πrh²) / (r-h)

C. (2πr²h) / (r+h) D. (2πr²h) / (r-h)

1413. Find the volume of the frustum of a pyramid whose bases are regular hexagons with base edges 5 cm and 10 cm respectively and the altitude is 15 cm. A. 2273.31 cm³ B. 2171.33 cm³

C. 2327.13 cm³ D. 2713.32 cm³

1414. What is the volume of the cube if the number of cubic units in its volume is twice the number of square units in its total surface area ? A. 1827

B. 1287

C. 1872

D. 1728

1415. Find the lateral area of a regular hexagonal pyramid whose lateral edges are each 13 cm and whose base has sides 10 cm each. A. 350

B. 360

C. 370

D. 380

1416. The ratio of the volume of two spheres is 8:27. What is the ratio of their surface areas? A. 2/9

B. 4/9

C. 5/9

D. 7/9

1417. Each edge of the upper base of the frustum of a regular quadrangular pyramid is 2 less than an edge of the lower base. Find the edge of the lower base if the slant height is 10 and the total area is 160. A. 3

B. 4

C. 5

D. 6

1418. Find the volume of a solid formed by revolving an equilateral triangle with side e about an altitude. A. (√3π e³) / 24 B. (√3π e³) / 12

C. (√2π e³) / 24 D. (√2π e³) / 12

1419. If the diameter of a sphere is increased by 40 percent, by what percent is the volume increased ? A. 144.7%

B. 147.4%

C. 177.4%

D. 174.4%

1420. The radii of two spheres are in the ratio 3:4 and the sum of their surfaces is 2500. Find the radius of the smaller sphere. A. 14

B. 15

C. 16

D. 17

1421. If the ratio of the lateral area of the frustum of a cone to its volume is 15:28, find the altitude of the frustum if its base radii are 3 and 6 respectively. A. 6

B. 5

C. 4

D. 3

1422. The lateral area of a right circular cone is 3 times the area of its base. Find the angle at which the slant height of the cone is inclined with the base. A. 71.35°

B. 72.15°

C. 70.53°

D. 73.25°

1423. The volume of a rectangular parallelepiped is 162. The three dimensions are in the ratio 1:2:3. Find the total area. A. 198

B. 197

C. 196

D. 195

1424. The base edge of a square pyramid is 3 m and its altitude is 10 m. Find the area of a section parallel to the base and 6 m from it. A. 1.22 m²

B. 1.33 m²

C. 1.44 m²

D. 1.55 m²

1425. The area of the base of a pyramid is 25 and its altitude is 10. What is the distance from the base of a section parallel to the base whose area is 9 ? A. 4

B. 3

C. 5

D. 2

1426. The edge of a regular tetrahedron is 5. Find the edge of a cube which has the same volume as the tetrahedron.

A. 2.35

B. 2.45

C. 2.55

D. 2.65

1427. The segment of a paraboloid of revolution( see figure ) is a solid in which every section parallel to the base is a circle the radius R of which is the mean proportional between the distance H from the vertex and the radius r of the base. Find the volume of the segment of altitude h. A. 1/2 πr²h

B. 1/3 πr²h

C. 1/2 πrh²

D. 1/3 πrh²

1428. A right circular cone whose slant height is 18 cm and the circumference of whose base is 6 cm is cut by a plane parallel to the base such that the cone is cut off, has a slant height of 4 cm. Find the lateral area of the frustum formed. A. 48.3

B. 49.1

C. 50.2

D. 51.3

1429. A tank has the form of a cylinder of revolution whose diameter is 60 cm and whose height is 244 cm. The tank is in horizontal position and is filled with water to a depth of 46 cm. Find the approximate number of liters of water in the tank. A. 566

B. 567

C. 568

D. 569

1430. A solid gas a circular base of radius 20. Find the volume of the solid if every section perpendicular to a certain diameter is an equilateral triangle. A. 18475.21

B. 14871.52

C. 17845.12

D. 15781.25

1431. In a cone of altitude h and elliptic base A, every section parallel to the base has an area Ay = Ay² / h² where y is the distance from the vertex to the section ( see figure ). Find the volume of the elliptic cone. A. πabh / 3

B. πabh / 2

C. πabh / 4

D. πabh / 5

1432. Find the total area of a regular hexagonal pyramid whose slant height is 5 ft and whose base is 4 ft. A. 105.71 ft²

B. 107.15 ft²

C. 101.57 ft²

D. 110.75 ft²

1433. For the solid shown, every section perpendicular to the edge AB is a circle. If arc ACB is a semicircle of diameter 18, find the volume of the solid ( see figure ). A. 342 pi

B. 423 pi

C. 432 pi

D. 243pi

1434. A solid consists of a hemisphere surmounted by a right circular cone. Find the vertical angle of the cone if the volume of the conical and spherical portions are equal. A. 51.13°

B. 52.13°

C. 53.13°

D. 54.13°

1435. The slant height of the frustum of a right circular cone makes an angle of 60° with the larger base. If the slant height is 30 cm and the radius of the smaller base is 5 cm, find the volume of the frustum. A. 15283.7 cm³

B. 14283.7 cm³

C. 13283.7 cm³

D.12283.7 cm³

1436. The lateral area of the frustum of a regular pyramid is 336 sq cm. If the lower base is a square having a side of 8 cm; the upper base is a square of side x cm and its slant height is 12 cm, find the value of x. A. 6

B. 4

C. 7

D. 5

1437. If the area of the base of a regular hexagonal prism is 3√3 / 2 sq cm and the total area is a 45√3 sq cm, find the volume of the prism. A. 20.5 cm³

B. 31.5 cm³

C. 21.5 cm³

D. 30.5 cm³

1438. If a cylinder has a lateral area of 88 pi and a volume of 176 pi, what is its total area ? A. 120 pi

B. 125 pi

C. 130 pi

D. 135 pi

1439. A rectangular prism has a width of 2 cm, a height of 4 cm and a length of 3√3 cm. If its volume is equal to the volume of a cube with diagonal d, find the value of d. A. 8 cm

B. 7 cm

C. 6 cm

D. 5 cm

1440. The axes of two right circular cylinders of equal radii 3 m long, intersect at right angles. Find the volume of their common part. A. 122 m³

B. 133 m³

C. 144 m³

D. 155 m³

1441. Which of the following statements is false ? A. Any two integrals of a given function differ by a constant. B. The integral of secnxdx where n is an odd integer requires integration by parts. C. If f(x) is an even function, then the integral of f(x)dx from x = -a to x = a is equal to zero. D. The key connection between the derivative and integral is known as the fundamental theorem of calculus.

1442. Which of the following differentials must be integrated by parts ? A. (lnx/x)dx

B. sin²(3x)dx

C. x²cos(x³)dx

D. (lnx)²dx

1443. The process of finding the function f(x) whose differential f'(x)dx is given, is called integration or A. involution

B. evolution

C.antidifferentiation D. exponentiation

B. xex – 1 + c

C. ex – x + c

1444. Evaluate ∫xexdx A. ex(x-1) + c

D. xex – x + c

1445. For some constant k, the antiderivative of xk is equal to A. (xk+1)/(k+1)

B. [(xk+1)/(k+1)]+c C. [(x2k) / 2k] +c

D. A or B

1446. The mathematician who first give a modern definition of the definite integral is A. Riemann

B. Leibniz

C. Newton

D. Gauss

1447. To integrate ∫(xdx) / (1+x4) by the u-substitution method, let u = A. 1 + x²

B. x²

C. 1 + x4

D. x4

1448. Which of the following is correct ? A. ∫cos2xdx = -sin2x + c B. ∫sin2xdx = sin2x + c

C. ∫sin3xdx = [(sin4x) / 4] + c D. ∫excosxdx = exsinx + c

1449. Who proved that the area under a parabolic arch is 2bh/3 where b is the width of the base of the arch and h is the height ? A. Wallis

B. Newton

C. Riemann

D. Archimedes

C. -20/3

D. -28/3

C. lncoshu + c

D. lncothu + c

1450. Evaluate ∫1-1(x2 – 4) dx A. -25/3

B. -22/3

1451. The antiderivative of tanhudu is A. lnsinhu + c

B. lnsechu + c

1452. using the theorem of Pappus, find the volume of the torus generated by revolving the area of the circle x2 + y2 = a2 about the line x = b where b > a. A. 2π a2b

B. 2π ab2

C. 2π2ab2

D. 2π2a2b

1453. If f(x) = x3 – 1and g(x) = x – 1, evaluate ∫10 [f(x) / g(x) ] dx. A. 11/6

B. 13/6

C. 10/6

D. 14/6

1454. Find the area bounded by the curve y = e x , the lines x = -1, x = 1 and the x-axis. A. 2.15

B. 2.25

C. 2.35

D. 2.45

1455. If the area bounded by y = x2 and y = 2 – x2 is revolved about the x-axis and a vertical rectangular element is taken, the element of volume generated is a A. disk

B. washer

C. shell

D. torus

1456. If ∫5-2 f(x)dx = 18, ∫5-2 g(x)dx = 5, and ∫5-2 h(x)dx = -11, evaluate∫ 5-2 [f(x)+g(x)h(x)]dx. A. 32

B. 33

C. 34

D. 35

1457. Find the length of the curve y = coshx from x = -1 to x = 1. A. 2.15

B. 2.25

C. 2.35

D. 2.45

1458. Evaluate ∫2-1 (2x-(2/x)+(x/2) dx A. 2.3637

B. 2.3763

C. 2.3367

D. 2.6733

1459. If the area bounded by the parabola y = x2 and the line y = x is revolved about the x-axis, the volume of the solid formed may be found by using which of the following methods ? A. washer method only B. washer or disk method

C. shell or washer method D. shell or disk method

1460. The differential xnex^2dx is integrable if n is A. an even integer B. an odd integer

C. any positive integer D. any whole number

1461. If a vertical element of area is used in finding the area bounded by the parabolas y = x2 – 7 and y = 1 – x2, then the elemental area dA = A. (2x2 – 8)dx

B. (8 – 2x2)dx

C. (2x2 – 6)dx

D. (6 – 2x2)dx

C. pi/3

D. pi/4

1462. If ∫x0 sin2ycos2ydy = ¼, then x is equal to A. pi/2

B. pi/6

1463. If the area bounded by the ellipse 9x2 + 4y2 = 36 is revolved about the line 2x + y = 8 and a horizontal rectangular element is taken, the element of volume generated is a A. washer or circular ring B. cylindrical shell

C. circular disk D. none of A, B or C

1464. Which of the following cannot be evaluated by the power rule formula ? A. ∫ (dx) / x2 (1 + (2/x))3 B. ∫ (√1 + sinx)dx / (secx)

C. ∫ (ln(x+1)dx) / (x+1) D. ∫ (x2 √x2 + 4 ) dx

1465. Evaluate ∫2π0∫ 10 rdrdθ A. 3π/4

B. π/4

C. π/6

D. 2π/3

1466. Find the area bounded by y = x2, the x-axis and the lines x = 1, x = 3. A. 26/3

B. 25/3

C. 23/3

D. 20/3

1467.Evaluate the integral of xsin(x2)dx from x = 0 to x = √π A. -1

B. 0

C. -1/2

D. 1/3

1468. Find the volume of the solid generated by revolving about the x-axis, the area bounded by y = x3, the x-axis and the line x = 1. A.π/3

B. π/5

C. π/7

D. π/9

C. ¼

D. 1/5

1469. The integral of e4lnx dx from x = 0 to x = 1 is A. ½

B. 1/3

1470. Evaluate ∫ sec2xtanx dx A. ½ tan2 x + c B. 1/3 sec3 x + c

C. ½ sec2 x + c D. A or C

1471. To integrate ∫ x2 ex dx by parts, it is wise to choose u = A. x

B. x2

C. ex

D. xex

1472. Evaluate ∫sin2 xdx C. 1/3sin3 x + c D. A or B

A. 1/2(x-sinxcosx) + c B. ½ x - ¼ sin2x + c 1473. Evaluate ∫10 2x 3x dx A. 2.971

B. 2.719

C. 2.197

D. 2.791