248461097 MATHS IB QUESTION BANK Chapter Wise Important Questions for IPE PDF

JUNIOR MATHS IB - Chapter wise important questions for IPE by MN Rao LOCUS SHORT ANSWER TYPE QUESTIONS 1.

2. 3. 4. 5. 6. 7. 8. 9 10.

Find the equation of locus of a point P such that PA2 + PB 2 = 2c 2 , where A = (a, 0), B = (-a, 0)and 0< a < c . Find the equation of locus of P, if the line segment joining (2, 3) and (-1, 5) subtends a right angle at P. The ends of the hypotenuse of a right angled triangle are (0, 6) and (6, 0). Find the equation of locus of its third vertex. Find the equation of locus of P, if A = (4, 0), B = (-4, 0) and PA − PB = 4.

Find the equation of locus of P, if A = (2, 3), B = (2, -3) and PA + PB = 8, A (5, 3) and B (3, -2) are two fixed points. Find the equation of locus of P, so that the area of triangle PAB is 9. If the distance from P to the points (2, 3) and (2, -3) are in the ratio 2 : 3, then find the equation of locus of P. A (1, 2), B (2, -3) and C (-2, 3) are three points. A point P moves such that PA2 + PB 2 = 2PC 2 . Show that the equation to the locus of P is 7x – 7y + 4 = 0. Find the equation of locus of a point P, if the distance of P from A (3, 0) is twice the distance of P from B (-3, 0) A (2, 3) and B (-3, 4) be two given points. Find the equation of locus of P so that the area of the triangle PAB is 8.5

2.

TRANSFORMATION OF AXES

SHORT ANSWER TYPE QUESTIONS :

1.

When the origin is shifted to (-1, 2) by the translation of axes, find the transformed equations of the following 2 x2 + y 2 − 4x + 4 y = 0 2. The point to which the origin is shifted and the transformed equation are given below. Find the original equation. ( −1, 2) ; x 2 + 2 y 2 + 16 = 0 3. Find the point to which the origin is to be shifted so as to remove the first degree terms from the equation 4 x 2 + 9 y 2 − 8 x + 36 y + 4 = 0 4. Find the angle through which the axes are to be rotated so as to remove the xy term in the equation x 2 + 4 xy + y 2 − 2 x + 2 y − 6 = 0 5. When the origin is shifted to the point (2,3), the transformed equation of a curve is x 2 + 3xy − 2 y 2 + 17 x − 7 y − 11 = 0 . Find the original equation of the curve. 6. When the axes are rotated through an angle 45°, the transformed equation of a curve is 17 x 2 − 16 xy + 17 y 2 = 225. Find the original equation of the curve. π 7. When the axes are rotated through and angle , find the transformed equation of x 2 + 2 3xy − y 2 = 2a 2 6 π 8. When the axes are rotated through an angle , find the transformed equation of 3x 2 + 10 xy + 3 y 2 = 9 4 1  2h  9. Show that the axes are to be rotated through an angle of Tan−1   so as to remove the xy term 2  a −b  π from the equation ax 2 + 2hxy + by 2 = 0 , if a ≠ b and through the angle , if a = b 4 Vanithatv – GUIDE education and Career Live show daily at 8.30pm 1

3.

STRAIGHT LINE

VERY SHORT ANSWER TYPE QUESTION : 1. Find the equation of the line containing the points (2, -3) and (0, -3). 2. Find the value of x, if the slope of the line passing through (2, 5) and (x, 3) is2. 3. Find the equations of the straight lines which make the following angles with the positive X-axis in the positive direction and which pass through the points given below. 135° and (3, -2) 4. Find the equations of the straight lines which make the following angles with the positive X-axis in the positive direction and which pass through the points given below. 150° and (-2, -1) 5. Find the equations of the straight lines passing through the origin and making equal angles with the coordinate axes. 6. The angle made by a straight line with the positive X-axis in the positive direction and the Y-intercept cut off by it are given below. Find the equation of the straight line. 2 Tan −1   , 3 3 7. Find the equation of the straight line passing through (-4, 5) and cutting off equal nonzero intercepts on the coordinate axes. 8. Find the equation of the straight line passing through (-2, 4) and making non-zero intercepts whose sum is zero 9. Find the equation of the straight line passing through the point (3, -4) and making X and Y-intercepts which are in the ratio 2 : 3. 10 In what follows, p denotes the distance of the straight line from the origin and α denotes the angle suur made by the normal ray drawn from the origin to the straight line with OX measured in the anticlockwise sense. Find the equations of the straight line with the following values of p and α 7π ii) p = 6, α = 150° iii) p = 1, α = i) p = 5, α = 60° 4 suur 11. A straight line passing through A(-2, 1) makes an angle of 30° with OX in the positive direction. Find the points on the straight line whose distance from A is 4 units. 12. Find the ratios in which the following straight lines divide the line segments joining the given points. State whether the points lie on the same side or on either side of the straight line. 3x – 4y = 7 13. Transform the following equations into the form L1 + λ L2 = 0 and find the point of concurrency of the family of straight lines represented by the equation. (k + 1)x + (k + 2)y + 5 = 0 14. Find the area of the triangle formed by the following straight lines and the coordinate axes. x – 4y + 2 = 0 1 2 15. y = − 3x + 5, y = x− 3 3 16. ax + by = a + b, a(x – y) + b(x + y) = 2b 17. Find the length of the perpendicular drawn from the point given against the following straight line 5x – 2y + 4 = 0, (-2, -3) 18. Find the distance between the following parallel lines 5x – 3y – 4 = 0, 10x – 6y – 9 = 0 19. Find the equation of the straight line parallel to the line 2x + 3y + 7 = 0 and passing through the point (4, -3) 20. Find the equation of the straight line perpendicular to the line 5x – 3y + 1 = 0 and passing through the point (4, -3) 21. Find the value of k, if the straight lines 6x – 10y + 3 = 0 and kx – 5y + 8 = 0 are parallel. 22. Find the value of p, if the straight lines 3x + 7y – 1 = 0 and 7x – py + 3 = 0are mutually perpendicular

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23.

Find the equation of the straight line passing through the point ( at12 , 2at1 ) and ( at22 , 2at2 )

24.

If the area of the triangle formed by the straight lines x = 0, y = 0 and 3x + 4y = a (a > 0) is 6. Find the value of a. State whether (3, 2) and (-4, -3) are on the same side or on opposite sides of the straight line 2x – 3y + 4 =0 Find the ratios in which (i) the X – axis and (ii) the Y-axis divide the line segment AB joining A (2,-3) and B (3, -6) Find the distance between the parallel straight line 3x + 4y – 3 = 0 and 6x + 8y – 1 = 0

25. 26. 27.

SHORT ANSWER QUESTION 1. Show that following sets of points are collinear and find the equation of the line L containing them. i) (-5, 1), (5, 5), (10, 7) ii) (1, 3), (-2, -6), (2, 6) iii) (a, b+c), (b, c + a ), (c, a + b) 2. A (10, 4), B (-4, 9) and C (-2, -1) are the vertices of a triangle. Find the equations of suur i) ABsuurii) the median through A iii) the altitude through B iv) the perpendicular bisector of the side AB 3. If the portion of a straight line intercepted between the axes of coordinates is bisected at (2p, 2q), write the equation of the straight line. 4. Transform the following equations into (a) slope-intercept form (b) intercept form and (c) normal form iii) x + y + 2 = 0 iv) 3 x + y + 10 = 0 1) ex + 4y = 5 ii) 3 x + y = 4 5. A straight line L is drawn through the point A(2, 1) such that its point of intersection with the straight line x + y = 9 is at a distance of 3 2 from A. Find the angle which the line L makes with the positive direction of the X-axis Show that the straight line (a – b)x + (b – c)y = c – a, (b – c)x + (c – a)y = a – b and (c – a)x + (a – b)y = b – c are concurrent. 6. A triangle of area 24 sq. units is formed by a straight line and the coordinate axes in the first quadrant. Find the equation of the straight line, if it passes through (3, 4) 7. Show that the lines 2x + y – 3 = 0, 3x + 2y – 2 = 0 and 2x – 3y – 23 = 0 are concurrent and find the point of concurrency. 8. Find the value of p, if the following lines are concurrent. 3x + 4y = 5, 2x + 3y = 4, px + 4y = 6 9. If 3a + 2b + 4c = 0, then show that the equation ax + by + c = 0 represents a family of concurrent straight lines and find the point of concurrency. 10. Find the point on the straight line 3x + y + 4 = 0 which is equidistant from the points (-5, 6) and (3, 2) 11. A straight line through P(3, 4) makes an angle of 60° with the positive direction of the X-axis. Find the coordinates of the points on the line which are 5 units away from P

12.

13. 14. 15. 16. 17. 18. 19.

(

)

3, 2 makes an angle

π

with the positive direction of the X-axis. If the 6 straight line intersects the line 3 x − 4 y + 8 = 0 at P, find the distance PQ. Find the equations of the straight lines passing through (1,3) and (i) parallel to (ii) perpendicular to the line passing through the points (3, -5) and (-6, 1) Find the foot of the perpendicular drawn from (4, 1) upon the straight line 3x – 4y + 12 = 0 x – 3y – 5 = 0 is the perpendicular bisector of the line segment joining the point A, B If A = (-1, -3), find the coordinates of B. Find the image of the point (1, 2) in the straight line 3x + 4y – 1 = 0 Find the locus of the foot of the perpendicular from the origin to a variable straight line which always passes through a fixed point (a, b) Show that the lines x – 7y – 22 = 0, 3x + 4y + 9 = 0 and 7x + y – 54 = 0 form a right angled isosceles triangle. Find the equations of the straight lines passing through the point (-3, 2) and making an angle of 45° with the straight line 3x – y + 4 = 0 A straight line through Q

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20. 21. 22. 23. 24. 25. 26. 27.

28.

Find the equations of the straight lines passing through the point of intersection of the lines 3x + 2y + 4 = 0, 2x + 5y = 1 and whose distance from (2, -1) is 2 Find the area of the parallelogram whose sides are 3x + 4y + 5 = 0, 3x + 4y – 2 = 0, 2x + 3y + 1 = 0 and 2x + 3y – 7 = 0 Find the equation of the straight line parallel to the line 3x + 4y = 7 and passing through the point of intersection of the lines x - 2y – 3 = 0 and x + 3y - 6 = 0 Find the equation of the straight line perpendicular to the line 2x + 3y = 0 and passing through the point of intersection of the lines x + 3y – 1 = 0 and x – 2y + 4 = 0 A straight line parallel to the line y = 3 x passes through Q (2, 3) and cuts the line 2x + 4y – 27 = 0 at P. Find the length of PQ. The ratio in which the straight line L ≡ ax + by + c = 0 divides the line segment joining the points A ( x1 , y1 ) and B ( x2 , y2 ) is − L11 : L22 If the straight lines ax + by + c= 0, bx + cy + a = 0 and cx + ay + b = 0 are concurrent, then prove that a3 + b3 + c3 = 3abc x y A variable straight line drawn through the point of intersection of the straight lines + = 1 and a b x y + = 1 meets the coordinate axes at A and B. Show the locus of the mid point of AB is 2 (a + b) xy b a = ab (x + y) The length of the perpendicular from the point P ( x0 , y0 ) to the straight line ax + by + c = 0 is

ax0 + by0 + c a 2 + b2

LONG ANSWER QUESTIONS

1. 2. 3. 4. 5. 6. 7. 8. 9. 10.

Find the orthocenter of the triangle whose sides are given by x + y + 10 = 0, x – y – 2 = 0 and 2x + y – 7 =0 Find the circum center of the triangle whose vertices are (1, 0), (-1, 2) and (3, 2) Find the circumcenter of the triangle formed by the straight lines x + y = 0, 2x + y + 5 = 0 and x – y = 2 Find the equations of the straight lines passing through the point (1, 2) and making an angle of 60° with the line 3 x + y + 2 = 0 Find the orthocenter of the triangle with the following vertices (-2, -1), (6, -1) and (2, 5) Find the circumcenter of the triangle whose vertices are given below (-2, 3), (2, -1) and (4, 0) Find the orthocenter of the triangle formed by the lines x + 2y = 0, 4x + 3y – 5 = 0 and 3x + y = 0 Find the circumcenter of the triangle whose sides are given by x + y + 2 = 0, 5x – y – 2 = 0 and x – 2y + 5 = 0 If p and q are the lengths of the perpendiculars from the origin t the straight lines x sec α + y cosec α = a and x cos α -y sin α = a cos2 α , prove that 4 p 2 + q 2 = a 2 Find the in centre of the triangle formed by the following straight lines X + 1 = 0, 3x – 4y = 5 and 5x + 12y = 27

4.

PAIR OF STRAIGHT LINES

LONG ANSWER QUESTIONS 2 1. Show that the straight lines represented by ( x + 2a ) − 3 y 2 = 0 and x = a form an equilateral triangle. 2. Find the centroid and the area of the triangle formed by the following lines i) 2 y 2 − xy − 6 x 2 = 0, x + y + 4 = 0 ii) 3 x 2 − 4 xy + y 2 = 0, 2 x − y = 6 3. Find the equation of the pair of lines intersecting at (2, -1) and Vanithatv – GUIDE education and Career Live show daily at 8.30pm

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4. 5.

i) perpendicular to the pair 6 x 2 − 13 xy − 5 y 2 = 0 , and ii) parallel to the pair 6 x 2 − 13 xy − 5 y 2 = 0 Find the equation of the bisector of the acute angle between the lines 3x – 4y + 7 = 0 and 12x + 5y – 2 = 0 2 2 Show that the lines represented by ( lx + my ) − 3 ( mx − ly ) = 0 and lx + my + n = 0 form an equilateral

n2 3 (l 2 + m2 )

triangle with area

6.

7.

Show that the straight lines represented by 3 x 2 + 48 xy + 23 y 2 = 0 and 3x – 2y + 13 = 0 form an 13 equilateral triangle of area sq.unit 3 If one line of the pair of lines ax 2 + 2hxy + by 2 = 0 bisects the angle between the coordinate axes. Prove 2

that ( a + b ) = 4 h 2 . 8.

If (α , β ) is the centroid of the triangle formed by the lines ax 2 + 2hxy + by 2 = 0 and lx + my = 1, prove that

9. 10. 11.

α bl − hm

=

β am − hl

=

2 3 ( bl − 2hlm + am2 ) 2

Find the value of k, if the equation 2 x 2 + kxy − 6 y 2 + 3 x + y + 1 = 0 represents a pair of straight line. Find the point of intersection of the lines and the angle between the straight lines for this value of k. Show that the lines x 2 + 2 xy − 35 y 2 − 4 x + 44 y − 12 = 0 and 5x + 2y – 8 = 0 are concurrent. Show that the two points of lines 3 x 2 = 8 xy − 3 y 2 = 0 and 3 x 2 + 8 xy − 3 y 2 = 2 x − 4 y − 1 = 0 form square.

13.

Show that the straight lines y 2 − 4 y + 3 = 0 and x 2 + 4 xy + 4 y 2 + 5 x + 10 y + 4 = 0 form a parallelogram and find the lengths of its sides Show that the product of the perpendicular distances from the origin to the pair of straight lines c represented by ax 2 + 2 hxyy + by 2 + 2 gx + 2 fy + c = 0 is 2 ( a − b ) + 4h 2

14.

If the equation ax 2 + 2hxy + by 2 + 2 gx + 2 fy + c = 0 represents a pair of intersecting lines, then show that

12.

the square of the distance of their point of intersection from the origin is

c (a + b) − f 2 − g2 ab − h 2

. Also show

f 2 + g2 if the given lines are perpendicular. h2 + b 2 Show that the lines joining the origin to the points of intersection of the curve x 2 − xy + y 2 + 3 x + 3 y − 2 = 0 and the straight line x − y − 2 = 0 are mutually perpendicular Find the values of k, if the lines joining the origin to the points of intersection of the curve 2 x 2 − 2 xy + 3 y 2 + 2 x − y − 1 = 0 and the line x + 2y = k are mutually perpendicular Find the angle between the lines joining the origin to the points of intersection of the curve x 2 + 2 xy + y 2 + 2 x + 2 y − 5 = 0 and the line 3x – y + 1 = 0 Find the condition for the chord lx + my = 1 of the circle x 2 + y 2 = a 2 (whose centre is the origin) to subtend a right angle at the origin Find the condition for the lines joining the origin to the points of intersection of the circle x 2 + y 2 = a 2 and the line lx + my = 1 to coincide. Write down the equation of the pair of straight lines joining the origin to the points of intersection of the line 6x – y + 8 = 0 with the pair of straight lines 3 x 2 + 4 xy − 4 y 2 − 11x + 2 y + 6 = 0 Show that the lines so obtained make equal angles with the coordinate axes. If Q (h, k) is the foot of the perpendicular from P ( x1 , y1 ) on the straight line ax = by + c = 0, then that the square of this distance is

15. 16. 17. 18. 19. 20.

21.

( h − x1 ) : a = ( k − y1 ) : b = − ( ax1 + by1 + c ) : ( a 2 + b2 ) Vanithatv – GUIDE education and Career Live show daily at 8.30pm

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22. 23. 24. 25. 26. 27.

If Q (h,k) is the image of the point P ( x1 , y1 ) w.r.t the straight line ax + by + c = 0 Find the orthocenter of the triangle whose vertices are (-5, -7), (13, 2) and (-5, 6) If the equations of the sides of a triangle are 7x + y – 10 = 0, x – 2y + 5 = 0 and x + y + 2 = 0, find the orthocenter of the triangle. Find the circumcenter of the triangle whose vertices are (1, 3), (-3, 5) and (5, -1) Find the circumcenter of the triangle whose sides are 3x – y – 5 = 0, x + 2y – 4 = 0 and 5x + 3y + 1 = 0. Find the incenter of the triangle formed by the straight lines y = 3 x , y = − 3x and y = 3.

28.

Let the equation ax 2 = 2hxy + by 2 = 0 represent a pair of straight lines. Then the angle θ between the a+b lines is given by cos θ = 2 ( a − b ) + 4h 2

29.

If the equation ax 2 + 2hxy + by 2 = 0 represents a pair of distinct (i.e., intersecting) lines, then the combined equation of the pair of bisectors of the angle between these lines is h ( x 2 − y 2 ) = ( a − b ) xy

30. 31. 32.

Find the centroid of the triangle formed by the lines 12 x 2 − 20 xy + 7 y 2 = 0 and 2x – 3y + 4 = 0 Prove that the lines represented by the equations x 2 − 4 xy + y 2 = 0 x + y = 3 form an equilateral triangle. Show that the product of the perpendicular distances from a point (α , β ) to the pair of straight lines 2

2

ax + 2hxy + by = 0 is

aα 2 +2hαβ+bβ 2

( a-b )

2

+4h 2

5. THREE DIMENSIONAL COORDINATES VERY SHORT ANSWER QUESTIONS 1. Find the distance between the points (3, 4, -2) and (1, 0, 7) 2. If (3, 2, -1), (4, 1, 1) and (6, 2, 5) are three vertices and (4, 2, 2) is the centroid of a tetrahedron, find the fourth vertex 3. Show that the point whose distance from Y-axis is thrice its distance from (1, 2, -1) satisfies the equation 8 x 2 + 9 y 2 + 8 z 2 − 18 x − 36 y + 18 z + 54 = 0 4. Find the ratio in which YZ – plane divides the line joining A (2, 4, 5) and B(3, 5, -4). Also find the point of intersection. 5. For what value of t, the points (2, -1, 3), (3, -5, t) and (-1, 11, 9) are collinear ?

SHORT ANSWER QUESTION 1. Show that the points (2, 3, 5), (-1, 5, -1) and (4, -3, 2) form a right angled isosceles triangle. 2. P is a variable point which moves such that 3PA = 2PB. If A = (-2, 2, 3) and B = (13, -3, 13) prove that P satisfies the equation x 2 + y 2 + z 2 + 28 x − 12 y + 10 z − 247 = 0 3. Show that the points (1, 2, 3), (7, 0, 1) and (-2, 3, 4) are collinear 4. Show that the points A (3, 2, -4), B(5, 4, -6) and C(9, 8, -10) are collinear and find the ratio in which B divides AC . 5. A(5, 4, 6), B(1, -1, 3), C(4, 3, 2) are three points. Find the coordinates of the point in which the bisector of BAC meets the side BC . 6. Find the point equidistant from the four points (-1, 1, 3), (2, 1, 2), (0, 5, 6) and (3, 2, 2) 7. Find the ratio in which the line joining (2, 4, 16) and (3, 5, -4) is divided by the plane 2x – 3y + z + 6 = 0

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LONG ANSWER QUESTIONS suur suur 1. Find the point of intersection of the lines AB and CD where A = (7, -6, 1), B = (17, -18, -3), C = (1, 4, -5) and D = (3, -4, 11)

6. DIRECTION COSINES AND DIRECTIONS RATIOS VERY SHORT ANSWER QUESTIONS 1. If a line makes angle α , β , γ with the positive directions of X, Y, Z – axes, what is the value of sin 2 α + sin 2 β + sin 2 γ ?

2. 3. 4. 5.

Find the angle between the lines whose direction ratios are (1,1, 2 ) ,

(

3, − 3, 0

)

 12 −3 −4   4 12 3  Show that the lines with direction cosines  , ,  and  , ,  are perpendicular to each  13 13 13   13 13 13  other. A straight line is inclined to the axes of X and Y at angles of 60° and 45° respectively find its inclination to the Z-axis uuur uuur uuur If aline makes angles α , β , γ with the coordinate axes OX , OY , OZ find the value of cos 2α + cos 2β + cos 2γ .

SHORT ANSWER QUESTIONS

1. 2. 3. 4. 5.

suur suur Show that the lines PQ and RS are parallel where P, Q, R, S are the points (2, 3, 4), (4, 7, 8), (-1, -2, 1) and (1, 2, 5) respectively Find the angle between DC and AB where A = (3, 4, 5), B = (4, 6, 3), C = (-1, 2, 4) and D (1, 0, 5) Show that the points (4, 7, 8), (2, 3, 4), (-1, -2, 1), (1, 2, 5) are vertices of a parallelogram A (1, 8, 4), B(0, -11, 4), C(2, -3, 1) are three points and D is the foot of the perpendicular from A to BC. Find the coordinates of D. suur suur Lines OA, OB are drawn from o with direction cosines proportional to (1, -2, -1); (3, -2, 3). Find the direction cosines of the normal to the plane AOB.

LONG ANSWER QUESTIONS

1. 2. 3. 4. 5. 6. 7. 8.

Find the direction cosines of two lines which are connected by the relations l − 5m + 3n = 0 and 7l 2 + 5m 2 − 3n 2 = 0 . Show that the lines whose d.c’s are given by l + m + n = 0, 2mn + 3nl – 5lm = 0 are perpendicular to each other Find the angle between the lines whose direction cosines satisfy the equations l + m + n = 0, l 2 + m2 − n 2 = 0 If a ray makes angle α , β , γ , δ with the four diagonals of a cube find cos 2 α + cos 2 β + cos 2 γ + cos 2 δ Find the angle between the lines whose direction cosines are given by the equations 3l + m + 5n = 0 and 6mn – 2nl + 5lm = 0 Find the angle between two diagonals of a cube. A(6, -16, -3), B(3, -1, 4), C(-2, 5, 0) are vertices of a triangle. Find the coordinates of the foot of the perpendicular drawn from A to BC Show that the four points (5, -1, 1), (-1, -3, 4), (1, -6, 10) and (7, -4, 7) taken in order form a rhombus

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7.

THE PLANE

VERY SHORT ANSWER QUESTIONS 1. Reduce the equation x + 2y – 3 – 6 = 0 of the plane to the normal form 2. Find the intercepts of the plane 4x + 3y – 2z + 2 = 0 on the coordinate axes. 3. Find the equation of the plane passing through the point (-2, 1, 3) and having (3, -5, 4) as d.r’s of its normal 4. Find the angle between the planes x + 2y + 2z – 5 = 0 and 3x + 3y + 2z – 8 = 0 SHORT ANSWER QUESTIONS 1. Find the equations of the plane passing through the point (1,1,1) and parallel to the plane x + 2y + 3z – 7=0 2. Find the equation of the plane through (-1, 6, 2) and perpendicular to the join of (1, 2, 3) and (-2, 3, 4) 3. Find the equation of the plane through (4, 4, 0) and perpendicular to the planes 2x + y + 2z + 3 = 0 and 3x + 3y + 2z – 8 = 0 4. Find the equation of the plane passing through (2, 0, 1) and (3, -3, 4) and perpendicular to x – 2y + z = 6 LONG ANSWER QUESTIONS 1. Find the equation of the plane through the points (2, 2, -1), (3, 4, 2), (7, 0, 6) 2. Show that the points (0, -1, 0), (2, 1, -1), (1, 1, 1), (3, 3, 0) are coplanar 3. A plane meets the coordinate axes in A, B, C, If the centroid of ∆ABC is (a,b,c) show that the equation x y z to the plane is + + = 3 . a b c 4. Show that the plane through (1,1,1)(1, -1, 1) and (-7, -3, -5) is parallel to Y-axis

8. FUNCTIONS, LIMITS AND CONTINULTY VERY SHORT ANSWER TYPE QUESTIONS 2M.

1..

Compute the following limits.

.

1.

2.,

3.

lim x→a  x+2 f (x) =  2 x

a=3

1,

if x ≤ 1  1  f ( x ) =  2x + 1 if 1 < x ≤ 2;  3x if x >2 

2.

Show that

3.

4.

x2 − a2 x−a if -1< x ≤ 3 ; if 3 < x < 5

a=2

lim x − 2 = −1 x → 2 x−2 lim  2 x  show that + x + 1 = 3  x→2 x 

4.

Compute

1.

Compute

lim x→2+ lim

x→2−

([ x ] + x ) and

lim x→2−

2 − x ( x < 2 ) What is

([ x ] + x ) lim x→2

2− x ?

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lim 5.

1.

2.. 3. 6.

1. 2.

7.

8.

1.

π  2 x−  2  lim sin ( a + bx ) − sin ( a − bx ) x→0 x lim tan ( x − a ) ( a ≠ 0) x → a x2 − a2 lim  x sin a − a sin x   x → a  x−a  lim  cos ax − cos bx   x → a  x2  lim  3x − 1    x → 0  1 + x −1  x→

2.

lim 3 1 + x − 3 1 − x x→0 x

3.

lim sin ( x − a ) tan 2 ( x − a ) 2 x→a ( x2 − a2 )

1.

lim 8 x + 3x x → ∞ 3 x − 2x

2. 3. 9.

cos x

π

1. 2.

10.

1.

11.

1.

2.

3.

lim 11x 3 − 3 x + 4 x → ∞ 13 x 3 − 5 x 2 − 7 lim x2 + x − x x→∞ lim  2 x + 3    x → ∞  x2 −1 

(

)

lim 2 + cos 2 x x → ∞ x + 2007

 sin 2 x if x ≠ 0 Is f defined by f ( x ) =  continuous at 0 ? if x = 0  x Check the continuity or f given by 2 2 ( x − 9 ) / ( x − 2 x − 3) if 0 < x < 5 and x ≠ 3 f ( x) =  at the point 3. 1.5 if x = 3   x −1  x − 1 if x >1  If f is a function defined by f ( x ) =  5 − 3 x if −2 ≤ x ≤1, then discuss the continuity of f.  6 if x < −2   x − 10  k 2 x − k if x ≥1 If f, given by f ( x ) =  , if x < 1  2 is a continuous function on R, then find the values of k.

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12.

1.

2.

13.

 4 − x2 if x ≤ 0   x − 5 if 0 < x ≤ 1 Check the continuity of f given oy f ( x ) =  2 at the point 0, 1 and 2. 4 x − 9 if 1 < x < 2  3 x + 4 if x ≥ 2

 cos ax − cos bx if x ≠ 0  x2 Show that f ( x ) =   1 ( b2 − a 2 ) if x = 0  2 where a and b are real constants, is continuous at 0.

Solved problems : 1. 2. 3.

4.

5. 6. 7. 8. 9.

lim x lim x = 1 and = −1( x ≠ 0) . x→0 x x →0− x lim  e x − 1  Complete   x → 0  1 + x −1 

Show that

Complete

lim a x − 1 (a > 0, b > 0, b ≠ 1) x → 0 b x −1

 x + 1 if  Check the continuity of the function f given below at 1 and at 2. f ( x ) =  2 x if 1 + x 2 if  x + 2 if − 1 < x≤3 lim lim  Find f ( x ) and f ( x ) , where f ( x ) =  2 . if 3 < x < 5 x → 3+ x → 3−  x lim sin x Find x → 0 x cos 2 x

x ≤1 1< x < 2 x≥2

lim  x   x → 0  1 + x − 1 − x  lim x − 3 Show that = −1 x → 3 x −3 lim  1 + x − 1 + x 2    x → 0  1 − x 2 − 1 − x 

9. DIFFERENTIATION VERY SHORT ANSWER TYPE QUESTIONS 2M. 1.

Find the derivatives of the following functions f ( x )

2.

1.

3.

1.

3 ax + b c + d ≠ 0) ii) x + 2 x 4 + 3x 6 ( x > 0 ) ( cx + d 1 − cos 2 x I) ii) Tan -1 ( log x ) iii) log sin -1 ( e x ) 1 + cos 2 x v) cot -1 (cosec 3x) Find the derivatives of the following functions.

1.

5

I)

(

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)

iv)

cos x sin x + cos x

10

 a−x  ii) Tan−1    1 + ax  1   1  iv) Sec −1  2   0 < x <  2  2x −1  

i) Cos −1 ( 4 x 3 − 3 x ) iii) Tan −1

1 − cos x 1 + cos x

(

v) sin Tan−1 ( e − x ) 2.

)

Differentiate f ( x ) with respect to g ( x ) for the following.  2x   2x  ii) f ( x ) = Tan −1  , g ( x ) = Sin −1  2  2   1− x   1+ x  Find the derivatives of the following functions.  1 + x2 − 1  tan x i) Tan −1  ii) ( log x )    x   iii) ( x x ) x iv) 20log (tan x ) 2x + 3 If y = then find y " 4x + 5 Find the second order derivatives of the following functions f ( x ) .

i) f ( x ) = e x , g ( x ) = x

4..

1.

5.

1.

6.

1.

7.

2.

ii) e x sin x cos 2 x i) log (4x 2 − 9) Prove the following. i) If y = ax n +1 + bx n then x 2 y '' = n(n + 1) y

1.

i) If ax 2 + 2hxy + by 2 = 1 then prove that iv) If

8.

y = e−k /2 x

d2 y h 2 − ab = dx 2 (hx + by )2

(a cos nx +b sin nx) then prove that

y "+ k y '+  n 

2

+

k2 

y=0

4 

Solved problems : 1. If f ( x ) = x e x sin x, then find f ' ( x ) dy dx

2.

If y = ( sin ( log x ) ) , find

3.

If y = x x ( x > 0), then find

4.

π  If y = (tan x)sin x  0 < x <  2 

5.

If x = a cos3 t , y = a sin 3 t , we shall find

6.

If x 3 + y 3 − 3axy = 0, find

7.

If x = a (t − sin t ), y = a (1 + cos t ) find

8.

dy . dx

dy . dx

dy . dx

d2y . dx 2  2x  Find the second order derivative of y = Tan −1  . 2   1− x 

9.

If x = 2cos t + cos 2t + 1, y = 2 sin t + sin 2t then find

10.

If e x+ y = xy then find

dy . dx

dy . dx

SHORT ANSWER TYPE QUESTIONS 4 M.

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1.

1.

2.

2.

3.

Find the derivatives of the following functions. 1− x x i) ( x > 0) 1+ x x Find the derivatives of the following functions f ( x ) from the first principles.

i) x 4 + 4 ii) x + 1 v) cot x vi) sec 3 x Find the derivatives of the following functions.  b + a sin x  Sin -1  1.  ( a > 0, b > 0)  a + b sin x   b + a cos x  2. C os -1   (a > 0, b > 0)  a + b cos x   cos x  Tan -1  3.   1 + cos x  2.

iii) sin 2x vii) cos 2 x

iv) tan 2x

dy for the following functions. dx i) x = 3cos t − 2cos 3 t , y = 3sin t − 2sin 3 t ii) x = a (cos t + t sin t ), y = a (sin t − t cos t ) Find

3.

Differentiate f ( x ) with respect to g ( x ) for the following.

4.

 1+ x2 −1  i) f ( x ) = Tan −1   , g ( x ) = Tan −1 x   x   Find the derivative of the function y defined implicitly by each of the following equations. i) y x = x sin y 1 If f ( x ) = log x ( x > 0), then prove that f ' ( x ) = x

1.

LONG ANSWER TYPE QUESTIONS 7 M.

1.

Establish the following. dy 1− y2 = dx 1 − x2 dy y 1 − log x log y  ii) x log y = log x then =  dx x  log 2 x  dy y ( x log y − y ) iii) x y = y x then = dx x ( y log x − x) Find the derivatives of the following functions. 1. i) (sin x)log x + x sin x ii) (sin x) x + x sin x 2. Establish the following  yx y −1 + y 2 log y  dy i) If x y + y x = a b then =− y x −1  dx  x log x + xy  i)

2.

If 1 − x 2 + 1 − y 2 = a ( x − y ) then

ii) If f ( x ) = sin −1

x−β x−β and g ( x ) = Tan−1 then α −β α −x

f ' ( x) = g '( x) ( β < x < α )

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 1+ x2 + 1 − x2  dy  for 0 < x < 1, find . dx  1 + x 2 − 1 − x 2  cos x dy 2. If y = x tan x + ( sin x ) , find . dx dy log x 1. If x y = e x− y , then show that . = dx (1 + log x )2

1.

If

y = Tan −1 

2. If sin y = x sin (a + y), then show that

dy sin 2 (a + y ) = (a is not a multiple of π ) . dx sin a

10. APPLICATIONS OF DIFFERENTIATION VERY SHORT ANSWER QUESTIONS 1. y = x 2 + 3x + 6, when x = 10, ∆x = 0.01 2. y = e x when x = 0, ∆x = 0.1 1 3. y = when x = 2, ∆x = 0.002 x 4. y = x 2 + 2 x , when x = 5, ∆x = −0.1 5. If the increase in the side of a square is 1% find the percentage of change in the area of the square.  π  6. Find the approximate value of sin 60°1'  = 0.0175   180  x −1 7. Find the slope of the tangent to the curve y = , x ≠ 2 at x = 10 x−2 8. Find the slope of the tangent to the curve y = x 3 − 3x + 2 at the point whose x-coordinate is 3

9. 10. 11. 12.

13. 14. 15.

16.

17. 18. 19. 20.

Find the slope of the normal to the curve x = a cos 3 θ , y = a sin 3 θ at θ =

π

4 Find the equations of tangent and normal to the curve xy = 10 at (2, 5) Find the equations of tangent and normal to the curve y = x3 + 4 x 2 at (-1, 3) At time t the distance s of a particle moving in a straight line is given by s = −4t 2 + 2t . Find the velocity 1 and acceleration when t = seconds. 2 The displacement s of a particle traveling in a straight line in t seconds is given by s = 45t + 11t 2 − t 3 Find the time when the particle comes to rest The radius of a circular plate is increasing in length at 0.01 cm/sec. What is the rate at which the area is increasing, when the radius is 12 cm State the points at which the following functions are increasing and the points at which they are decreasing 2 ii) x 3 ( x − 2 ) i) x 3 − 3 x 2 Determine the intervals in which the functions are increasing and the intervals in which they are decreasing ln t i) ii) 25 − 4x 2 t If an error of 3% occurs in measuring the side of a cube, find the percentage error in its volume Show that the relative error in the nth power of a number is n times the relative error in that number. Find an approximate value of 3 123 Show that the length of the subnormal at any point on the curve y 2 = 4ax is a constant.

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SHORT ANSWER QUESTIONS 1. Sand is poured from a pipe at the rate of 12 cc./ sec. The falling sand forms a cone on the ground in such a way that the height of the cone is always one-sixth of the radius of the base. How fast is the height of the sand – cone increasing when the height is 4 cm. 2, Water is dripping out from a conical funnel, at a uniform rate of 2cm3 ./ sec through a tiny hole at the vertex at the bottom. When the slant height of the water is 4 cm. find the rate of decrease of the slant height of the water given that the vertical angle of the funnel is 120° 3. A man 6 ft. high walks at a uniform rate of 4 miles per hour away from a lamp 20 ft. high. Find the rate at which the length of his shadow increases. (1 mile = 5280 ft.) 4. A man 180 cm. high walks at a uniform rate of 12 km. per hour away from a lamp post of 450 cm. high. Find the rate at which the length of his shadow increases. 5. A light is at the top of a pole 64 m. high. A ball is dropped from the same height (64 m.) from a point 20 m. from the light. Assuming that the ball falls according to the law s = 5t 2 , how fast is the shadow of the ball moving along the ground 2 seconds later ? LONG ANSWER QEUSTIONS

1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14.

−1

Show that the tangent at P ( x1 , y1 ) on the curve 2 3

−1

1

x + y = a is yy1 2 + xx1 2 = a 2 2 3

2 3

If the tangent at any point on the curve x + y = a intersects the coordinate axes in A and B, then show that the length AB is a constant If the tangent at any point P on the curve x m y n = a m+ n ( mn ≠ 0 ) meets the coordinate axes in A,B, then show that AP : BP is a constant. At any point t on the curve x = a (t + sin t), y = a (1 – cos t), find the lengths of tangent, normal, subtangent and subnormal. Find the lengths of subtangent, subnormal at a point t on the curve x = a(cos t + t sin t), y = a(sin t – t cos t) Find the angle between the curves y 2 = 4 x; x 2 + y 2 = 5 Find the dimensions of the right circular cylinder with the greatest volume that can be inscribed in a sphere of radius a. Show that when the curved surface of right circular cylinder inscribed in a sphere of radius R is maximum, then the height of the cylinder is 2R Show that the semi-vertical angle of the right circular cone of maximum voume and of given slant height is Tan −1 2 . Show that the area of a rectangle inscribed in a circle is maximum when it is a square. Find the rectangle of maximum perimeter that can be inscribed in a circle π Show that f (x) = sin x (1 + cosx) has a maximum value at x = 3 m+n m− n 2 n Show that at any point on the curve x = a y ( a > 0, m + n ≠ 0 ) , m th power of the length of the subtangent varies as the nth power of length of the subnormal. Show that the condition for the orthogonality of the curves ax 2 + by 2 = 1 and a1 x 2 + b1 y 2 = 1 is 1 1 1 1 − = − . a b a1 b1

15.

Show that the curves y 2 = 4 ( x + 1) and y 2 = 36 ( 9 − x ) intersect orthogonally

16.

Find the maximum and minimum value of 2 sin x + sin 2x over [ 0, 2π ]

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