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1 ABOUT DISHA PUBLICATION
7
CHAPTER
One of the leading publishers in India, Disha Publication provides books and study materials for schools and various competitive exams being continuously held across the country. Disha's sole purpose is to encourage a student to get the best out of preparation. Disha Publication offers an online bookstore to help students buy exam books online with ease. We, at Disha provide a wide array of Bank / Engg./ Medical & Other Competitive Exam books to help all those aspirants who wish to crack their respective different levels of Bank / Engg./ Medical & Other Competitive exams. At Disha Publication, we strive to bring out the best guidebooks that students would find to be the most useful for all kind of competitive exam.
5
7.
MATHEMATICS 1.
Let be a positive angle. If the number of degrees in is divided by the number of radians in , then an irrational number
180
results. If the number of degrees in
multiplied by the number of radians in number (a) (c) 2.
125 9
results. The angle
30° 50°
is
must be equal to
(b) 45° (d) 60°
9.
In a triangle ABC, a (1 3) cm, b = 2 cm and angle C = 60°. Then the other two angles are (a) 45° and 75° (b) 30° and 90° (c) 105° and 15° (d) 100° and 20°
DIRECTIONS: For the next two (2) items that follow. Let be the root of the equation 25cos2 + 5cos – 12 = 0, where 3.
2 What is tan
(a)
4.
.
24 25
cos 1 tan
(b)
3 4
(d)
4 5
(b)
sin equal to? 1 cot
(a) sin – cos (b) sin + cos (c) 2 sin (d) 2 cos DIRECTIONS : For the next three (3) items that follow. 1 1 Consider x = 4 tan , y 5 10. What is x equal to?
tan
1
equal to?
4 3 What is sin 2 equal to?
(a)
What is
(a)
3 4
(c)
8.
, then an irrational
The angles of elevation of the top of a tower standing on a horizontal plane from two points on a line passing through the foot of the tower at distances 49 m and 36 m are 43° and 47° respectively. What is the height of the tower? (a) 40 m (b) 42 m (c) 45 m (d) 47 m (1 – sinA + cos A)2 is equal to (a) 2 (1 – cosA) (1 + sin A) (b) 2 (1 – sin A) (1 + cos A) (c) 2 (1 – cos A) (1 – sin A) (d) None of the above
tan
60 119
90 169 11. What is x – y equal to?
24 25
(c)
tan
1
(a)
tan
1
8281 8450 12. What is x – y + z equal to? (c)
21 5 (d) 25 12 The angle of elevation of the top of a tower from a point 20 m away from its base is 45°. What is the height of the tower? (a) 10 m (b) 20 m (c) 30 m (d) 40 m
828 845
tan
1
1
1 and z 70
tan
1
1 . 99
1 120 (b) tan 119 1 170 (d) tan 169
1 8287 (b) tan 8450 1 8287 (d) tan 8471
(c)
5.
6.
The equation tan–1(1 + x) + tan–1 (1– x)
2
(a)
x=1
(b) x = –1
(c)
x= 0
(d) x
1 2
is satisfied by
(a)
2
(b)
3
(c)
(d) 6 4 DIRECTIONS: For the next three (3) items that follow. Consider the triangle ABC with vertices A (–2, 3), B (2, 1) and C (1, 2). 13. What is the circumcentre of the triangle ABC ? (a) (–2, –2) (b) (2, 2) (c) (–2, 2) (d) (2, –2)
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2 14. What is the centroid of the tirnalge ABC ? (a)
1 ,1 3
(b)
1 ,2 3
2 1 ,3 (d) 3 2 15. What is the foot of the altitude from the vertex A of the triangle ABC? (a) (1, 4) (b) (–1, 3) (c) (–2, 4) (d) (–1, 4) 16. Let X be the set of all persons living in a city. Persons x, y in X are said to be related as x < y if y is at least 5 years older than x. Which one of the following is correct? (a) The relation is an equivalence relation on X (b) The relation is transitive but neither reflexive nor symmetric (c) The relation is reflexive but neither transitive nor symmetric (d) The relation is symmetric but neither transitive nor reflexive 17. Which one of the following matrices is an elementary matrix? (c)
1,
1 0 0
(a)
(c)
0 0 0
1 5 0
(b)
0 0 1
0 2 0
1 0 0
0 0 1
(d)
0 1 0 0 5 2
18. Consider the following statements in respect of the given equation : (x2 + 2)2 + 8x2 = 6x (x2 + 2) 1. All the roots of the equation are complex. 2. The sum of all the roots of the equation is 6. Which of the above statements is/are correct? (a) 1 only (b) 2 only (c) Both 1 and 2 (d) Neither 1 nor 2 19. In solving a problem that reduces to a quadratic equation, one student makes a mistake in the constant term and obtains 8 and 2 for roots. Another student makes a mistake only in the coefficient of first-degree term and finds –9 and –1 for roots. The correct equation is (a) x2 – 10x + 9 = 0 (b) x2 – 10x + 9 = 0 2 (c) x – 10x + 16 = 0 (d) x2 – 8x – 9 = 0 20. If A
2 7 1 5
23. The matrix
then that is A + 3A–1 equal to?
(a) 3I (b) 5I (c) 7I (d) None of these where I is the identity matrix order 2. 21. In a class of 60 students, 45 students like music, 50 students like dancing, 5 students like neither. Then the number of students in the class who like both music and dancing is (a) 35 (b) 40 (c) 50 (d) 55
0
4 i
4 i
0
is
(a) symmetric (b) skew-symmetric (c) Hermitian (d) skew-Hermitian 24. Let Z be the set of integers and aRb, where a, b Z if and only if (a – b) is divisible by 5. Consider the following statements: 1. The relation R partitions Z into five equivalent classes. 2. Any two equivalent classes are either equal or disjoint. Which of the above statements is/are correct? (a) 1 only (b) 2 only (c) Both 1 and 2 (d) Neither 1 nor 2 25. If
2 1 2i
z
3 i
where
1, then argument
i
of z is (a)
0 1 0
0 0 1
1 0 0
22. If log10 2, log10 (2x – 1) and log10(2x + 3) are three consecutive terms of an A.P, then the value of x is (a) 1 (b) log5 2 (c) log2 5 (d) log10 5
3 4
(b)
4
5 3 (d) 6 4 26. If m and n are the roots of the equation (x + p) (x + q) – k = 0, then the roots of the equation (x – m) (x – n) + k = 0 are
(c)
(a)
P and q
(b)
1 1 and p q
(c) –p and –q (d) p + q and p – q 27. What is the sum of the series 0.5 + 0.55 + 0.555 + ... to n terms? (a)
5 n 9
2 1 1 9 10n
1 2 1 (b) 9 5 9 1 10n
(c)
1 n 9
5 1 1 9 10n
5 (d) 9 n
28. If 1, of (1
,
2
1 1 1 9 10n
are the cube roots of unity, then the value 2
4
8
) is (a) –1 (b) 0 (c) 1 (d) 2 29. Let A = {1, 2 , 3, 4, 5, 6, 7, 8, 9, 10}. Then the number of subsets of A containing exactly two elements is (a) 20 (b) 40 (c) 45 (d) 90 )(1
)(1
)(1
30. What is the square root of i, where i
1?
(a)
1 i 2
(b)
1 i 2
(c)
1 i 2
(d) None of these
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3 31.
The point on the parabola y2 =
4ax nearest to the focus has
its abscissa (a) x = 0
(b) x = a
a (d) x = 2a 2 32. A line passes through (2, 2) and is perpendicular to the line 3x + y = 3. Its y-intercept is
(c)
(a)
(c)
3 4
(b)
1 3
4 3
12 4 3 , , 13 13 13
(b)
(c)
12 4 3 , , 13 13 13
(d)
(d) 3 x2
y2
2
2
a
b
1 passes through the point
3 units 2
(b)
3 unit 10
(c)
3 unit 4
(d)
2 unit 7
4 units. The 3
35. The area of a triangle, whose vertices are (3, 4), (5, 2) and the point of intersection of the lines x = a and y = 5, is 3 square units. What is the value of a? (a) 2 (b) 3 (c) 4 (d) 5 36. The length of perpendicular from the origin to a line is 5 units and the line makes an angle 120° with the positive direction of x-axis.The equation of the line is
3y
5
(b)
y
10
(d) None of these
3x
y
10
37. The equation of the line joining the origin to the point of interesection of the lines
(a) (c)
(a)
3x
x a
y b
12 4 3 , , 13 13 13
x 2
y
2
z
3
3 4
. Let Q be the foot of
the perpendicular. 40. What are the direction ratios of the line segment PQ?
length of the conjugate axis is (a) 2 units (b) 3 units (c) 4 units (d) 5 units 34. The Perpendicular distance between the straight lines 6x + 8y + 15 = 0 and 3x + 4y + 9 = 0 is
x
12 4 3 , , 13 13 13
DIRECTIONS: For the next two (2) items that follow. From the point P(3, – 1, 11), a perpendicular is drawn on the line L given by the equation
(3 5,1) and the length of its latus rectum is
(c)
(a)
x
33. The hyperbola
(a)
39. What are the direction cosines of the line segment?
1 and
x b
y a
1 is
(a) x – y = 0 (b) x + y = 0 (c) x = 0 (d) y = 0 DIRECTIONS: For the next two (2) items that follow. The projections of a directed line segment on the coordinate axes are 12, 4, 3 respectively. 38. What is the length of the line segment? (a) 19 units (b) 17 units (c) 15 units (d) 13 units
1, 6, 4 1, 6, 4
(b) (d)
1, 6, 4 2, 6, 4
41. What is the length of the line segment PQ? (a)
47 units
(b) 7 units
(c)
53 units
(d) 8units
DIRECTIONS: For the next two (2) items that follow. A triangular plane ABC with centroid (1, 2, 3) cuts the coordinate axes at A, B, C respectively. 42. What are the intercepts made by the plane ABC on the axes? (a) 3, 6, 9 (b) 1, 2, 3 (c) 1, 4, 9 (d) 2, 4, 6 43. What is the equation of plane ABC? (a) x + 2y + 3z = 1 (b) 3x + 2y + z = 3 (c) 2x + 3y + 6z = 18 (d) 6x + 3y + 2z = 18 DIRECTIONS: For the next two (2) items that follow. A point P (1, 2, 3) is one vertex of a cuboid formed by the coordinate planes and the planes passing through P and parallel to the coordinate planes. 44. What is the length of one of the diagonals of the cuboid? (a)
(b) 14 units 10 units (c) 4 units (d) 5 units 45. What is the equation of the plane passing through P(1, 2, 3) and parallel to xy–plane? (a) x + y = 3 (b) x – y = – 1 (c) z = 3 (d) x + 2y + 3z = 14 46. The decimal number (127. 25)10, when converted to binary nunber, takes the form (a) (1111111.11)2 (b) (1111110.01)2 (c) (1110111.11)2 (d) (1111111.01)2 47. Consider the following in respect of two non-singular matrices A and B of same order: 1. det (A + B) = det A + det B 2. (A + B)–1 = A–1 + B –1 Which of the above is/are correct? (a) 1 only (b) 2 only (c) Both 1 and 2 (d) Neither 1 nor 2
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4 X
48. If
3
4
1
1
5
,B
2
2 1
and A
p
q
r
s
satisfy
the equation AX = B, then the matrix A is equal to 7 26
(a)
(c)
1
5
7
4
7 26
(b)
7 26
(d)
26 13 1
4 17
6 23
n r
Cn equal to?
49. What is r 0 n + 2C 1 n + 3C n
n + 2C n n + 2C n+1
(a) (b) (c) (d) 50. How many words can be formed using all the letters of the word ‘NATION’ so that all the three vowels should never come together? (a) 354 (b) 348 (c) 288 (d) None of these 51. (x3 – 1) can be factorised as (x – 1) (x –
) (x +
2
(b) (x – 1) (x –
) (x –
2
)
(c)
) (x +
2
)
(a)
(x – 1) (x +
)
(d) (x – 1) (x + ) (x – 2 ) where is one of the cube roots of unity.. 52. What is 3
sin sin
i 1 cos
6
i 1 cos
6
where i (a) 1 (c) i 53. Let A
6
(b) –1 (d) –i y
y
2x
x
y
,B
2 1
3
and C
2
6
10
4
(c)
(b)
26 5
6
4
20
(d)
10
5
4
24
5
7
5 20
54. The value of 1
1
1 1 x 1
1
(a) (c)
x+y xy
1 1
58. If E (a)
E(
cos
sin
sin
cos
)
(b) E (
)
(c) E ( ) (d) E( ) 59. Let A = {x, y, z}and B = {p, q, r, s}. What is the number of distinct relations from B to A? (a) 4096 (b) 4094 (c) 128 (d) 126 60. If 2p + 3q = 18 and 4p2 + 4pq – 3q2 – 36 = 0, then what is (2p + q) equal to? (a) 6 (b) 7 (c) 10 (d) 20 DIRECTIONS: For the next two (2) items that follow.
2 x2 3. Ax B 1 x x 63. What is the value of A? (a) –1 (b) 1 (c) 2 (d) 3 64. What is the value of B? (a) –1 (b) 3 (c) –4 (d) –3 65. What is the solution of the differential equation ydx xdy y2
is
1 y
(b) x – y (d) 1 + x + y
1?
, then E ( ) E ( ) is equal to
Given that lim
If AB = C, then what is A2 equal to? (a)
57. What is the real part of (sin x + i cos x)3 where i (a) –cos 3x (b) –sin 3x (c) sin 3x (d) cos 3x
d 1 x2 x4 Ax B. Given that dx 1 x x 2 61. What is the value of A? (a) –1 (b) 1 (c) 2 (d) 4 62. What is the value of B? (a) –1 (b) 1 (c) 2 (d) 4 DIRECTIONS: For the next two (2) items that follow.
6
1, equal to?
x
55. If A = {x : x is a multiple of 3} and B = {x : x is a multiple of 4} and C = {x : x is a multiple of 12}, then which one of the following is a null set? (a) ( A / B) C (b) ( A / B) / C (c) ( A B) C (d) ( A B) / C 56. If (11101011)2 is converted decimal system, then the resulting number is (a) 235 (b) 175 (c) 160 (d) 126
0?
(a) xy = c (b) y = cx (c) x + y = c (d) x – y = c where c is an arbitrary constant.
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5 66. What is the solution of the differential equation sin
dy dx
1.
0?
a
(a) y = x sin–1 a + c (b) x = y sin–1 a + c –1 (c) y = x + x sin a + c (d) y = sin–1 a + c where c is an arbitrary constant. 67. What is the solution of the differential equation dx dy
x y
0?
xe x dx
68. What is
(c)
a, b, c are coplanar..
2. a b b c c a Select the correct answer using the code given below. (a) 1 only (b) 2 only (c) Both 1 and 2 (d) Neither 1 nor 2 75. If | a b | | a b |, then which one of the following is correct?
y2
(a) xy = x4 + c (b) xy = y4 + c 4 (c) 4xy = y + c (d) 3xy = y3 + c where c is an arbitrary constant.
(a)
x 1
equal to?
2
(a)
a
(b)
a is parallel to b
(c)
a is perpendicular to b
(d)
a b
ex
(d)
x 1
1
(a)
1
(c)
3i 4 j 3k is applied at the point P, whose
position vector is r
2i 2 j 3k . What is the magnitude
of the moment of the force about the origin? (a) 23 units (b) 19 units (c) 18 units (d) 21 units 71. Given that the vectors
and
of x and y for which
(a) (c)
y ,v
2y
3x and w
x = 2, y = 1 x = – 2, y = 1
w holds t rue if
2
5 , are
(b) x = 1, y = 2 (d) x = – 2, y = – 1
72. If | a | = 7, | b | = 11 and | a b | 10 3, then | a b | is equal to (a)
40
(b) 10
(c)
4 10
(d) 2 10
73. Let
(a) (b) (c) (d)
1.
y
ex
e 2
i
j
k, i
j
k and i
are collinear form an equilateral triangle form a scalene triangle form a right-angled triangle
6
x
is an increasing function on 0,
ex e x is an increasing function on , 2 Which of the above statements is/are correct? (a) 1 only (b) 2 only (c) Both 1 and 2 (d) Neither 1 nor 2 2.
y
j
k
.
.
x . The range |x|
of f is (a) a null set (b) a set consisting of only one element (c) a set consisting of two elements (d) a set consisting of infinitely many elements 79. Consider the following statements: 1. f (x) = [x], where [.] is the greatest integer function, is discontinuous at x = n, where n Z . 2. f (x) = cot x is discontinuous at x = n , where n Z . which of the above statements is/are correct? (a) 1 only (b) 2 only (c) Both 1 and 2 (d) Neither 1 or 2
, , be distinct real numbers. The points with position
vectors
1
(d)
78. For each non-zero real number x, let f x
are non-collinear. The values u v
1 5
6 77. Consider the following statements:
represented by the vectors 2i 3 j 2k and 4i 5 j 2k respectively. The area of the triangle ABC is (a) 6 square units (b) 5 square units (c) 4 square units (d) 3 square units
(b)
2 6
c
2
G x G 1 equal to? x 1 1
25 x 2 , then what is lim
(b) (x + 1)ex + c
ex c x 1
2x
0 x
(x + 1)2 ex + c
70. A force F
b for some scalar
76. If G x
where c is the constant integration. 69. The adjacent sides AB and AC of a triangle ABC are
u
0, then which of the following is/are correct?
74. If a b c
80. What is the derivative of tan
1
1 x2 1 with respect x
to tan–1 x? (a)
0
(c)
1
1 2 (d) x
(b)
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6 81. If f x
log e
then what is g f (a)
3x x 3
1 x ,g x 1 x
2
1 3x 2
and g o f (t) =g(f(t)),
e 1 equal to? e 1 (b) 1
f x
If
ax b If 1
x 0
If
3
x 1
(a)
a2 + b2
(b)
a2
(c)
a+b
(d)
a2 b2
b2
equal to? 1
a b
(b) tan
1
b a
a b 1 a b (d) tan a b a b DIRECTIONS: For the next two (2) items that follow.
(c)
tan
1
Consider the function f ( x)
x2 1
, where x x2 1 87. At what value of x does f(x) attain minimum value? (a) –1 (b) 0 (c) 1 (d) 2 88. What is the minimum value of f(x)?
(a)
0
(c)
–1
1 2 (d) 2
(b)
If
x
2 2
2
, where
is a constant.
? (b) 3 (d) 1
90. What is xlim0 f (x ) equal to?
x 1
x 1 dx is of the form In tan r 2 a cos x b sin x 85. What is r equal to?
tan
x
89. What is the value of (a) 6 (c) 2
0
The integral
(a)
If
Which is continuous at x
where a, b are constants. The function is continuous everywhere. 82. What is the value of a? (a) –1 (b) 0 (c) 1 (d) 2 83. What is the value of b? (a) –1 (b) 1 (c) 0 (d) 2 84. Consider the following functions: 1. f(x) = x3, x 2. f(x) = sin x, 0 < x < 2 3. f(x) = ex, x Which of the above functions have inverse defined on their ranges? (a) 1 and 2 only (b) 2 and 3 only (c) 1 and 3 only (d) 1, 2 and 3 DIRECTIONS: For the next two (2) items that follow.
86. What is
cos x 2x
f ( x)
1 (c) 0 (d) 2 DIRECTIONS: For the next two (2) items that follow. Given a function
1
DIRECTIONS: For the next two (2) items that follow. Consider the function
.
(a) (c)
0 3
(b) 3 (d)
6
91. The mean and the variance 10 observations are given to be 4 and 2 respectively. If every observation is multiplied by 2, the mean and the variance of the new series will be respectively (a) 8 and 20 (b) 8 and 4 (c) 8 and 8 (d) 80 and 40 92. Which one of the following measures of central tendency is used in construction of index numbers? (a) Harmonic mean (b) Geometric mean (c) Median (d) Mode 93. The correlation coefficient between two variables X and Y is found to be 0 6. All the observations on X and Y are transformed using the transformations U = 2 – 3X and V = 4 Y + 1. The correlation coefficient between the transformed variables U and V will be (a) (b) 0 5 0 5 (c) (d) 0 6 0 6 94. Which of the following statements is/are correct in respect of regression coefficients? 1. It measures the degree of linear relationship between two variables. 2. It gives the value by which one variable changes for a unit change in the other variable. Select the correct answer using the code given below. (a) 1 only (b) 2 only (c) Both 1 and 2 (d) Neither 1 nor 2 95. A set of annual numerical data, comparable over the years, is given for the last 12 years. 1. The data is best represented by a broken line graph, each corner (turning point) representing the data of one year. 2. Such a graph depicts the chronological change and also enables one to make a short-term forecast. Which of the above statements is/are correct? (a) 1 only (b) 2 only (c) Both 1 and 2 (d) Neither 1 nor 2
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7 1 1 and 3 2 respectively. What is the probability that exactly one of them hits the target?
96. Two men hit at a target with probabilities
(a)
1 2
(b)
1 3
1 2 (d) 6 3 97. Two similar boxes Bi (i = 1, 2) contain (i + 1) red and (5 – i – 1) black balls. One box is chosen at random and two balls are drawn randomly. What is the probability that both the balls are of different colours?
(c)
(a)
1 2
(b)
3 10
2 3 (d) 5 5 98. In an examination, the probability of a candidate solving a
(c)
1 . Out of given 5 questions in the examination, 2 what is the probability that the candidate was able to solve at least 2 questions?
102. If A and B are two events such that P (A P (A
(a)
(a)
3 (b) 16
(c)
1 2
(d)
99. If A (a)
13 16
B , then which one of the following is not correct?
P (A
B)
P (A | B )
(b)
(a)
1 26
2 3
(b)
1 221
4 1 (d) 223 13 DIRECTIONS: For the next two (2) items that follows.
(c)
Consider the line x 3 y and the circle x2 + y2 = 4. 106. What is the area of the region in the first quadrant enclosed by the x-axis, the line x
0 P A P B
2 , then what is P(B) equal to? 3
1 2 (d) 8 9 103. The ‘less than’ ogive curve and the ‘more than’ ogive curve intersect at (a) median (b) mode (c) arithmetic mean (d) None of these 104. In throwing of two dice, the number of exhaustive events that ‘5’ will never appear on any one of the dice is (a) 5 (b) 18 (c) 25 (d) 36 105. Two cards are drawn successively without replacement from a wellshuffled pack of 52 cards. The probability of drawing two aces is
(a) (b)
1 3
3 , 4
(c)
question is
1 64
1 and P (A ) 4
B)
B)
3 2
3
3 and the circle? (b)
3 2
2
1 (d) None of these 3 2 107. What is the area of the region in the first quadrant enclosed
(c)
(c)
(d)
P (B | A )
P (A | (A
P B P A
B ))
by the x-axis, the line x
P A (a)
P B
100. The mean and the variance in a binomial distribution are found to be 2 and 1 respectively. The probability P(X = 0) is (a)
1 2
3 y and the circle?
(b)
1 4
1 1 (d) 8 16 101. The mean of five numbers is 30. If one number is excluded, their mean becomes 28. The excluded number is (a) 28 (b) 30 (c) 35 (d) 38
(b)
3
6
3 (d) None of these 3 2 DIRECTIONS: For the next two (2) items that follow. Consider the curves y = sin x and y = cos x. 108. What is the area of the region bounded by the above two
(c)
(c)
curves and the lines x = 0 and x
4
(a)
2 1`
(b)
(c)
2
(d) 2
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?
2 1
8 109. What is the area of the region bounded by the above two curves and the lines x=
4
and x =
(a)
2
1.
? 2 1
(c)
(d) 2 2 2 DIRECTIONS: For the next two (2) items that follow. Consider the function 0 75x 4
x 3 9x 2
7
110. What is the maximum value of the function? (a) 1 (b) 3 (c) 7 (d) 9 111. Consider the following statements: 1. The function attains local minima at x = – 2 and x = 3. 2. The function increases in the interval (–2, 0). Which of the above statements is/are correct? (a) 1 only (b) 2 only (c) Both 1 and 2 (d) Neither 1 nor 2 DIRECTIONS: For the next three (3) items that follow. Consider the parametric equation a 1 t2 x
2
,y
2at
1 t 1 t2 112. What does the equation represent? (a) It represents a circle of diameter a (b) It represents a circle of radius a (c) It represents a parabola (d) None of the above
113. What is
dy equal to? dx
y x
(b)
y x
(c)
x y
(d)
x y
d 2y dx 2
equal to?
a2
(a)
(c)
y2 a2 x2
2.
dy dx
f (x ) x is of the form y
(b)
a2 x2
a2
(d)
y3
The degree of
2
dy dx
f (x ) is 2.
Which of the above statements is/are correct? (a) 1 only (b) 2 only (c) Both 1 and 2 (d) Neither 1 nor 2
dx 116. What is
(a)
ln
(c)
ln
x2 a x+ x 2 a x2
equal to?
a2
x2 a
(b) ln
c
a2
c
x
x2 a
a2
c
(d) None of these
where c is the constant of integration. DIRECTIONS: For the next four (4) items that follow. Consider the integral I m
0
sin 2 mx dx , where m is a positive sin x
integer. 117. What is I1 equal to? (a)
(a)
114. What is
The general solution of
= g (x) + c, where c is an arbitrary constant. (b)
2 –1
f x
115. Consider the following statements:
0
(b)
1 2
(c) 1 (d) 2 118. What is I2 + I3 equal to? (a) 4 (b) 2 (c) 1 (d) 0 119. What is Im equal to? (a) 0 (b) 1 (c) m (d) 2m 120. Consider the following: 1. Im – Im–1 is equal to 0. 2. I2m > Im Which of the above is/are correct? (a) 1 only (b) 2 only (c) Both 1 and 2 (d) Neither 1 nor 2
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9
Hints & Solutions Now as a > b A> B Now from Sine Rule,
MATHEMATICS 1.
(c)
From going by the options, option (a), = 30°, as we know that 180° = radian
sin A a
sin B b
30 radian 180 Now according to question,
1
30
From option (a),
30° =
180
sin A
180
3
sin 75
30
1
Now number of degree in is multiplied by number of radians in . 30 900 180 180 From option (b), = 45°
30° ×
10 =5 2
3
2
4
45 180 180 45 Now number of degree in is multiplied by number of radian in .
For (3-4): 3. (a) Here is the root of equation 25 cos2 + 5 cos – 12 = 0 25 cos2 + 5 cos – 12 = 0 25 cos2 + 20 cos – 15 cos – 12 = 0 5 cos (5 cos + 4) – 3(5 cos + 4) = 0 (5 cos – 3) (5 cos + 4) = 0
45 45 125 180 4 9 From option (c), = 50° As we know that 180° = radian
cos
50 radian 180 Now according to question
cos
3 or cos 5
=
Here,
50° =
50 2500 125 180 180 9 Option (c) is correct.
=
3 5
tan
=
sin cos
4.
(b)
sin 2 = 2 sin
b = 2 cm
=2
60°
=
(a)
(1
3 5
–5 4
1–
–3 4
Option (a) is correct.
A
a
= 1 – cos2
sin
50° ×
–4 5
–4 5 ( In 2nd quadrant, cos
=
Now, sin
Now number of degree in ‘ ’ is multiplied by number of radian in .
=
< <
2
180
50
B
2 2
3
4+ 4 3 =4+ 4 3 Option (a) is correct.
45° ×
2.
1
2 12 + 4 = 4 + 4 3
45 radian 180 Now according to question,
180
sin 45 2
6
125 9
45° =
50
sin B 2
3 cm
C
3 5
. cos
–4 5
6 – 4 –24 5 5 25 Option (b) is correct.
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value is negative) 16 25
10 5.
(b)
A
In ABC, AB = h BC = 20 m C = 45°
h 1= [ tan 45° = 1] 20 h = 20 m Height of the tower = 20 m Option (b) is correct.
tan–1 (1 + x) + tan–1 (1 – x) = tan –1
1 x
(b)
C
8.
(b)
9.
(b)
2
1– x
1– 1 x 1– x
1 x 1– x 1– 1 x 1– x
7.
20 m
tan
h h 36 49
tan 47º. cot 47º
45° B
(c)
... (ii)
Multiplying equations (i) and (ii)
h
AB tan 45° = BC
6.
h 49
cot 47° =
2
2
2 1 1– 1 x 1– x 0 1 – (1 + x) (1 – x) = 0 (1 + x) (1 – x) = 1 1 – x2 = 1 x2 = 0 x= 0 Option (c) is correct. AB = h (height of the tower) BD = 36 m BC = 49 m D = 47° C = 43°
h2 36 49
1
h = 6 × 7 = 42 m Option (b) is correct (1 – sin A + cos A)2 = 1 + sin2 A + cos2 A – 2 sin A – 2 sin A . cos A + 2 cos A = 2 – 2 sin A – 2 sin A cos A + 2 cos A = 2(1 – sin A) + 2 cos A(1 – sin A) = 2(1 + cos A) (1 – sin A) Option (b) is correct. cos 1 – tan
=
sin 1 – cot
cos sin 1– cos
sin cos 1– sin
=
cos 2 cos – sin
=
cos 2 cos – sin
=
cos 2 cos
sin 2 sin – cos –
sin 2 cos – sin
– sin 2 – sin
cos
– sin cos
cos – sin
sin
= cos + sin Option (b) is correct.
A 10. (b)
x = 4 tan–1
1 5
h
43°
47° B
36 m
D 49 m
C
Now, in ABD, h tan 47° = 36 m
2 5 2 tan –1 1 1– 5
1 = 2 2 tan –1 5
... (i)
2 5
= 2 tan–1
25 24
= 2 tan–1
10 5 = 2 tan–1 24 12
and in ABC, tan 43° =
h 49 m
tan(90° – 47°) =
5 12 = tan–1 120 . 25 119 1– 144 2
= tan–1 h 49
2
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11 11. (c)
= tan
= tan
–1
–1
–1 = tan
Slope of DE = 1
120 1 – tan –1 119 70
x – y = tan –1
Now, equation of ED is y –
120 1 – 119 70 120 1 1 119 70
2y – 3 = 2x – 3 x= y
x – y = tan –1
= tan –1
–1 = tan
–2 1 3 2 , 2 2 –
1 3
Now, equation of EF is y – tan –1
8281 8450
281 8450
tan –1
5 =3 x 2
8281 8450
tan –1
1 99
14. (b)
Centroid of the triangle =
x1
x2 3
x3
,
y1
y2 3
y3
A (x2y1) (–2, 3)
828269 836550 =1 828269 836550
(x2y2)
4 Option (d) is correct.
2 1 3 1 2 1 , ,2 = 3 3 3 Option (b) is correct.
A circumcentre is a point at which perpendicular bisectors meet each other. Here, ‘E’ represents circumcentre
=
1 5 (x, y) F – 2 , 2 E
D 3 3 , 2 2 2 1 1 2 , 2 2
2 –1 Slope of BC = 1 – 2 = –1
–2
(–2, 3) 15. (d)
Mid-point of BC =
C (1, 2) (x3y3)
B (2, 1)
A
1 2
2y – 5 = 6x + 3 ... (ii) From equations (i) and (ii), x = –2 and y = –2 Hence, circumcentre of ABC is (x, y) = (–2, –2) Option (a) is correct.
8281 1 8450 99 8281 1 1– 8450 99
B (2, 1)
1 5 – , 2 2
Slope of EF = 3
tan–1 = (1) =
13. (a)
3–2 –2 – 1
Slope of AC =
option (c) is correct. 12. (d)
... (i)
Now, mid-point of AC =
8400 – 119 8330 120 1 8330 8281 8330 8450 8330
3 3 =1 x– 2 2
Slope of BC =
2 –1 = –1 1– 2
A
(–2, 3)
C (1, 2) (2, 1) B
3 3 , 2 2
D (x, y)
Slope of AD = 1 Now, equation of BC is y – 2 = –1(x – 1)
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C (1, 2)
12 y – 2 = –x + 1 x + y– 3 = 0 ... (i) and equation of AD is (y – 3) = 1(x + 2) x – y+ 5 = 0 ... (ii) From equations (i) and (ii), x = –1 and y = 4 Foot of altitude from the vertex A of the triangle ABC is (–1, 4) Option (d) is correct. 16. (b)
17. (b)
Given that x < y if y x + 5 For Reflexive: x /x Hence, relation is not reflexive. For Symmetry: if x < y, then y / x Hence, relation is not symmetry. For Transitive: if x < y and y < z, then x < z Hence, relation is transitive.
19. (a)
–b b = –10 a a Quadratic equation according to second student ax2 + b1 x + c = 0 and roots are –9 and –1
8+2=
c c =9 a a Putting value in original equation
(–9) × (– 1) =
b c x + =0 a a x2 – 10x + 9 = 0.
x2 +
20. (c)
1 5 0 0 1 0 0 0 1
y=
6x
6x
36x 2 – 32x 2 2
2 7 1 5
1 adj (A) A 5
–7
–1
2
=
1 10 – 7
=
1 5 –7 3 –1 2
3A–1 =
(x2 + 2)2 + 8x2 = 6x(x2 + 2) Let x2 + 2 = y y2 + 8x2 = 6xy y2 – 6xy + 8x2 = 0 y=
A=
Now, A–1 =
An elementary matrix has each diagonal element 1. So, option (b) is correct answer. 18. (b)
Let correct equation is ax2 + bx + c = 0 According to first student, equation is: ax2 + bx + c1 = 0 and roots are 8 and 2
5
–7
–1
2
Now, A + 3A–1 = =
21. (b)
7 0 0 7
7
2 7
5
–7
1 5
–1
2
1 0 0 1
= 7I where I is Identity Matrix.
Dancing
Music
2x
= 3x ± x 2 y = 4x, 2x At y = 4x, x2 + 2 = 4x x2 – 4x + 2 = 0 Discriminant, D = 16 – 8 = 8 > 0 Roots are real. Sum of roots = –(–4) = 4 At x = 2x, x2 + 2 = 2x x2 – 2x + 2 = 0 D=4–8=–4 0 For x = 3,
2
f
y dx
Area of PAB 3
2
x 4 – x2 2
4 – x 2 dx 3
2
3 2 – 2 3
–
2
3 = 9(3)2 – 6(3) – 18 = 81 – 36 = 45 > 0 Statement 1 is correct From Statement 2: + f
3
3 3 [ 4 – 3] – 2sin –1 2 2
–
2
4 –1 x sin 2 2
3
–2 0 3 The function increases in the interval (–2, 0) Option (c) is correct.
3 2
–
+
106. (a) 107. (a)
Area enclosed by x-axis, the line x2 + y2 = 4 in the first quadrant =
3 2
3
–
3 2
3y , and the circle 112. (b)
2
a 1 – t2
2
x +y =
3 =
1
t2
1
y = sin x
a2 1
0
4a 2 t 2 1 t2
t 4 – 2t 2
=
Area of shaded region =
4 0
a2 1 t4
= 1 t
= sin x = = 109. (a) 110. (c)
111. (c)
1 2
cos x 1 2
4 0
113. (d)
2 – 1 sq. units.
From Statement 1: Function attain local minima at x = –2 and x = 3 As we have,
2
4a 2 t 2 2
2t 2
a2 1 t2
2 2
1 t
2 2
2
= a2
Hence, the given equation represent a circle of radius (a) Option (b) is correct.
– 0 1
f(x) = 0.75x4 – x3 – 9x2 + 7 f(0) = 7 f(–2) = 0.75(–2)4 – (–2)3 – 9(–2)2 + 7 = 2 + 8 – 36 + 7 = –9 f(3) = 0.75(3)4 – (3)3 – 9(3)2 + 7 = 60.75 – 27 – 81 + 7 = – 40.25 maximum value of f(x) is 7 Option (c) is correct.
t2
1
cos x – sin x dx
1 t2
2
2
y = cos x
2
2at
1 t2
a 2 1 – t2
108. (a)
2
Here, x2 + y2 = a2 2x + 2y
dy =0 dx
dy 2x x – dx –2y y Option (d) is correct.
114. (d)
d2 y
y
dx 2
–y
=
x y2
d dy –x – –x dx dx 2 y
dy dx
–y
x
–x y
y2
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26
=
– x2
–y2 – x 2 y3
y2 –
y3
a2 y3
Option (d) is correct. 115. (c)
x2
x
= ln
Option (a) is correct. 117. (a)
dy The general solution of = f(x) + x dx Now integrating on both side dy f x x dx dx y = f(x) + C Statement 1 is correct. Statement 2:
f x
c
Im =
0
sin 2 mx dx sin x
I1 =
0
sin 2x dx sin x
=2
0
cos x dx
118. (d)
2
I2 =
2
x
a
=4
0
=4
0
cos x 1 – 2 sin 2 x dx cos x dx – 8
tan 2 u
a2
=
sec 2 u du a2 1
a a
tan 2 u
sec2 u du sec u
= =
=
0
sec2 u du sec u
0
= ln
1
a2
x a
0
a2
x
sec x dx
2
tan x
0
= –2 c
dx
2 3 – 4 sin 2 x cos 3x dx 6 cos 3x dx –
8 2
–8 1 2 2
c
a2
x2 a
cos x sin 2 x dx
sin x
6 sin 3x 3
=6– =
x
0
sec u du
= ln[tan (u) + sec (u)] + c
= ln
0
sin 6x dx sin x
a sec u du
=a
2 sin 2x cos 2x dx sin x
0
2 3 sin x – 4 sin 3 x cos 3x
=
2
a2
– sin 0
Let sin x = t cos x dx = dt = 4(sin – sin 0) – 0 = 0 I3 =
2
Let x = a tan u dx = a sec2 u du =
2 sin
2 sin x cos x cos 2x dx sin x
0
dx 2
2 sin x 0
sin 4x dx sin x
0
=2
dy 2 f x 2c f x c2 dx Hence, the differential equation is of order –1 and degree –2. both Statements 1 and 2 are correct. Option (c) is correct. 116. (a)
2 sin x cos x dx sin x
0
= 2(0) = 0 Option (a) is correct.
dy = f(x) + c dx Squaring both sides, 2
c
a
Statement 1:
dy dx
a2
0
0
0
–8
1 2
0
8 sin 2 x cos 3x dx
0
sin x sin 4x – sin 2x dx
sin x sin 4x – sin x sin 2x dx
cos 3x – cos 5x – cos x
cos 3x dx
2 sin 3x sin 5x – – sin x = 0 3 5 0
Hence, I2 + I3 = 0 + 0 = 0 Option (d) is correct.
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27 119. (a)
Im = =
=
=
0
sin 2m 0
sin
–x –x
dx
0
sin 2m – 2mx dx sin x
0
– sin 2mx dx sin x
Im = – 2Im = 0 120. (a)
sin 2mx dx sin x
0
sin 2mx dx sin x
Im = 0.
I2m > Im is wrong statement Because, Im = Im – 1 = ..... = In Thus, Im – Im – 1 = 0 is the only correct statement. Option (a) is correct.
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