01. Find the distances between between the following following pairs of of points? (a) t 21 , 2t 1 and t 2 2 , 2t 2 if t 1 and and t 2 are the roots roots of x 2 2 3x 2 0 (b) (a co c os , a sin ) and (a cco os , a sin ) 02. The length of a line segment AB is 10 units. If the coordinates coordinates of one extremity extremity are (2,-3) and the abscissa of the other extremity is 10 then the sum of all possible values of the ordinate of the other extremity is (A) 3 (B) -4 (C) 12 (D) -6 03. A particle begins begins at the origin and moves moves successively in the the manner as shown, shown, 1 unit to the right, 1/2 unit up, 1/4 unit to the right, 1/8 unit down, 1/16 unit to the right etc. The length of each move is half the length of the previous move and movement continues in the ‘zigzag’ ‘zigzag’ manner indefinitely. The coordinates of the point to which the ‘zigzag’ converges is y 1/4 2 / 1

8 / 1

1

1/16 x

0

(A) (4/3, 2/3) (B) (4/3, 2/5)

(C) (3/2, 2/3)

(D) (2, 2/5)

Let ABCD is a square with sides of unit length. Points E and F are taken on sides AB and AD respectively so that AE=AF. Let P be a point inside the square ABCD. 04. The maximum possible area of of quadrilateral CDFE CDFE is (A)

1 8

(B)

1 4

(C)

5 8

(D)

3 8

05. The value of ( PA) 2 ( PB )2 (PC)2 ( PD)2 is equal to (A) 3

(B) 2

(C) 1

(D) 0

06. Let a line passing passing through point A divides divides the square ABCD into into two parts so that area of one portion is double the other, then the the length of portion of line inside the square is (A)

10 3

(B)

13 3

(C)

11 3

(D)

2 3

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07. A stick of length 10 units rests rests against the floor floor and wall of a room. room. If the stick begins to slide on the floor then the locus of its middle point is (A) x 2 y 2 2.5 (B) x 2 y 2 25 (C) x2 y 2 100 (D) none 08. AB is the diameter of a semicircle k, C is an arbitrary arbitrary point on the semicircle (other than A or B) and S is the centre of the circle inscribed into triangle ABC, then measure of C

k S B

A

(A) angle ASB changes as C moves on k. (B) angle ASB is the same for all positions of C but it cannot be determined without knowing the radius. (C) angle ASB= 135 for all C. (D) angle ASB= 150 for all C.

09. Each member of the the family family of parabolas y ax 2 2 x 3 has a maximum or minimum point depending depending upon the value of a. The equation to the locus of the maxima or minima for all possible values of ‘a’ is (A) a straight line with slope 1 and y intercept 3. (B) a straight line with slope 2 and y intercept 2. (C) a straight line with slope 1 and x intercept 3. (D) a straight line with slope 2 and y intercept 3. 10. The area of triangle triangle ABC is 20 cm 2 . The co-ordinates of vertex A are (-5,0) and and B are (3,0). The vertex C lies on the line, x – y=2. The Th e co-ordinates of C are (A) (5,3) (B) (-3, -5) (C) (-5, -7) (D) (7, 5) 11. The diagonals of a parallelogram parallelogram PQRS PQRS are along the lines x+3y=4 and 6x-2y=7. 6x-2y=7. Then PQRS must be a (A) rectangle (B) square (C) cyclic quadrilateral quadrilateral (D) rhombus 12. The greatest slope slope along the graph graph represented represented by the equation 4 x 2 y 2 2 y 1 0 , is (A) -3

(B) -2

(C) 2

(D) 3

13. A ray of light passing passing through the point point A(1,2) is reflected reflected at a point B on the xaxis and then passes through (5,3). Then the equation of AB is (A) 5x+4y=13 (B) 5x-4y=-3 (C) 4x+5y=14 (D) 4x-5y=-6

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07. A stick of length 10 units rests rests against the floor floor and wall of a room. room. If the stick begins to slide on the floor then the locus of its middle point is (A) x 2 y 2 2.5 (B) x 2 y 2 25 (C) x2 y 2 100 (D) none 08. AB is the diameter of a semicircle k, C is an arbitrary arbitrary point on the semicircle (other than A or B) and S is the centre of the circle inscribed into triangle ABC, then measure of C

k S B

A

(A) angle ASB changes as C moves on k. (B) angle ASB is the same for all positions of C but it cannot be determined without knowing the radius. (C) angle ASB= 135 for all C. (D) angle ASB= 150 for all C.

09. Each member of the the family family of parabolas y ax 2 2 x 3 has a maximum or minimum point depending depending upon the value of a. The equation to the locus of the maxima or minima for all possible values of ‘a’ is (A) a straight line with slope 1 and y intercept 3. (B) a straight line with slope 2 and y intercept 2. (C) a straight line with slope 1 and x intercept 3. (D) a straight line with slope 2 and y intercept 3. 10. The area of triangle triangle ABC is 20 cm 2 . The co-ordinates of vertex A are (-5,0) and and B are (3,0). The vertex C lies on the line, x – y=2. The Th e co-ordinates of C are (A) (5,3) (B) (-3, -5) (C) (-5, -7) (D) (7, 5) 11. The diagonals of a parallelogram parallelogram PQRS PQRS are along the lines x+3y=4 and 6x-2y=7. 6x-2y=7. Then PQRS must be a (A) rectangle (B) square (C) cyclic quadrilateral quadrilateral (D) rhombus 12. The greatest slope slope along the graph graph represented represented by the equation 4 x 2 y 2 2 y 1 0 , is (A) -3

(B) -2

(C) 2

(D) 3

13. A ray of light passing passing through the point point A(1,2) is reflected reflected at a point B on the xaxis and then passes through (5,3). Then the equation of AB is (A) 5x+4y=13 (B) 5x-4y=-3 (C) 4x+5y=14 (D) 4x-5y=-6

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In the diagram, a line is drawn through the points A(0,16) and B (8,0). Point P is chosen in the first quadrant on the line through A and B. Points C and D are chosen on the x and y-axis respectively, so that PDOC is a rectangle. y A(0,16 A(0 ,16)) D

P B(8,0) C

O

x

14. Perpendicular Perpendicular distance of the line AB AB from the point (2,2) is (A) 4 (B) 10 (C) 20 (D) 50 15. Sum of the coordinates coordinates of the point P if PDOC is a square is (A)

32

3

(B)

16 3

(C) 16

(D) 11

16. Number of possible possible ordered pair(s) of all all positions of the point P on AB so that the area of the rectangle PDOC is 30 sq. units, is (A) three (B) two (C) one (D) zero 17. Line x a'

x a y

b'

y b

1 cuts the co-ordinate co-ordinate axes at A(a,0) and and B (0,b) and and the line

1 at A`(-a`,0) and B` B` (0,-b`). If the points points A,B, A`, B` are concyclic concyclic then

the orthocentre of the triangle ABA’ is (A) (0,0)

(B) (0.b`)

aa ` b

bb ` a

(C) 0,

(D) 0,

18. Let ( xr , yr ) r=1,2,3 are the coordinates coordinates of the vertices of a triangle ABC. If D is the point on BC dividing it in the ratio of 1:2 reckoning from the vertex B, prove x

y

1

x

y

1

that the equation of the line AD is 2 x1

y1

1 x1

y1

1 0 Also find the

x2

y2

1

y3 1

x3

equation of the line AE in the similar from where E is the harmonic conjugate of D w.r.t. the points B and C. 19. Let ( x1 , y1 ); ( x2 , y2 ) and ( x3 , y3 ) are the vertices of a triangle triangle ABC respectively. respectively. D is a point on BC such that BC=3BD. The equation of the line through A and D, is

:4:

x

y

1

x

y

1

(A) x1

y1

1 2 x1

y1

1 0

x2

y2

1

x3

y3 1

x

y

1

x

y

1

(C) x1

y1

1 3 x1

y1

1 0

x2

y2

1

y3

1

x3

x

y

1

x

y

1

(B) 3 x1

y1

1 x1

y1

1 0

x2

y2

1

x3

y3

1

x

y

1

x

y

1

(D) 2 x1

y1

1 x1

y1

1 0

x2

y2

1

y3

1

x3

20. The graph of (y-x) against against (y+x) is as shown. Which one of the following shows shows the graph of y against x? (y-x)

(y+x)

y y

(A)

O

x

(B)

O y

y

(C)

O

x

x

(D)

O

x

21. Family of lines represented represented by the equation (cos sin ) x (cos sin ) y 3(3 cos sin ) 0 passes through a fixed point M for for

all real values of t he line x-y=0, is . The reflection of M in the (A) (6,3) (B) (3,6) (C) (-6,3) (D) (3,-6) 22. The line (k 1) 2 x ky 2k 2 2 0 passes through a point regardless of the value k. k. Which of the following is the line with slope 2 passing through the point ? (A) y=2x-8 (B) y=2x-5 (C) y=2x-4 (D) y=2x+8 23. m, n are integer integer with 0

:5:

the pentagon ABCDE is (A) 2m(m+n) (B) m(m+3n)

(C) 3(2m+3n)

(D) 2m(m+3n)

24. The parallelogram is bounded by the lines y=ax+c; y=ax+d; y=bx+c and y=bx+d and has the area equal to 18. The parallelogram bounded by the lines y =ax+c; y = ax-d; y=bx+c and y=bx-d has area 72. Given that a, b, c and d are positive integers, find the smallest possible value of (a+b+c+d). 25. A variable line L=0 is drawn through O(0,0) to meet the lines L1 : x 2 y 3 0 and L2 : x 2 y 4 0 at points M and N respectively. A point P is taken on L=0 such

that

1 OP

2

1 OM

2

1 ON 2

. Locus of P is

144

144

144

25

25

25

(A) x 2 4 y 2 (B) ( x 2 y) 2 (C) 4 x 2 y 2

(D) ( x 2 y) 2

144 25

The base of an isosceles triangle is equal to 4, the base angle is equal to 45 . A straight line cuts the extension of the base at a point M at an acute angle and bisects the lateral side of the triangle which is nearest to M.

26. The area ‘A’ of the quadrilateral which the straight line cuts off from given triangle is (A)

3 tan

1 tan

(B)

3 2 tan

1 tan

(C)

3 tan

1 tan

(D)

3 5 tan 1 tan

27. The range of values of ‘A’ for different values of , lie in the interval, 5 7

(A) , 2 2

(B) (4,5)

9

(C) 4, 2

(D) (3,4)

28. The length of portion of straight line inside the triangle may lie in the range: (A) (2,4)

3

(B) , 3 2

(C)

2, 2

(D)

2, 3

Consider two points A (1, 2) and B (3, 1) . Let M be a point on the straight line L x y 0 .

29. If M be a point on the line L=0 such that AM+BM is minimum, then the reflection of M in the x=y is (A) (1,-1) (B) (-1,1) (C) (2,-2) (D) (-2,2) 30. If M be a point on the line L=0 such that |AM-BM| is maximum, then the distance of M from N (1,1) is (A) 5 2

(B) 7

(C) 3 5

(D) 10

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31. If M be a point on the line L=0 such that |AM-BM| is minimum, then the area of ∆AMB equals (A)

13

(B)

4

13 2

(C)

13 6

(D)

13 8

32. Suppose that a ray of light leaves the point (3,4), reflects off the y-axis towards the x-axis, reflects off the x-axis, and finally arrives at the point (8,2). The value of x, is y (3,4) (8,2)

(0,y)

x

(x,0)

(A) x 4

1

1

2

(B) x 4

2

(C) x 4

3

(D) x 5

3

1 3

33. Let B(1, -3) and D(0,4) represent two vertices of rhombus ABCD in (x,y) plane, then coordinates of vertex A if BAD 60 can be equal to

1 7 3 1 3 , 2 2

(B)

1 7 3 1 3 , 2 2

1 14 3 1 2 3 , 2 2

(D)

(A)

1 14 3 1 2 3 , 2 2

(C)

34. (a) If the tangent at the point P on the circle x 2 y 2 6 x 6 y 2 meets the straight line 5 x 2 y 6 0 at a point Q on the y-axis, then the length of the PQ us (a)

(b)

(c)

(d)

(b)If a>2b>0 then the positive value of m for which y mx b 1 m2 is a common tangent to x 2 y 2 b 2 and ( x a) 2 y 2 b2 is (A)

2b a 2 4b2

(B)

a 2 4b2 2b

(C)

2b

a 2b

(D)

b a 2b

35. A circle is given by x 2 ( y 1)2 1, another circle C touches it extremely and also the x-axis, then the locus of its centre is (A) {( x, y) : x2 4 y} {( x, y ) : y 0} (B) {( x, y ) : x2 ( y 1) 2 4} {( x, y ) : y 0} (C) {( x, y ) : x 2 y} {(0, y) : y 0}

(D) {( x, y ) : x 2 4 y} {(0, y) : y 0}

36. Comprehension (3 questions together) A circle C of radius 1 is inscribed in an equilateral triangle PQR. The

:7:

points of contact of C with the sides PQ, QR, RP are D, E, F respectively. The 3 3 3 line PQ is given by the equation 3 x y 6 0 and the point D is , 2 2 Further, it is given that the origin and the centre of C are on the same side of the line PQ.

(i)The equation of circle C is 2

(A) x 2 3 y 1 1 2

2

(C) x 3 y 1 1 2

2

(D) x 3 y 1 1

(ii) Points E and F are given by 3 (A) , , 3, 0 (B) 2 2 3 3 3 1 , , , 2 2 2 2

2

3 1 , , 3, 0 2 2 3

(C)

2

1 (B) x 2 3 y 1 2 2

(D) , 2

3 3 1 , , 2 2 2

(iii) Equations of the sides RP, RQ are (A) y

2

(C) y

3

3 2

x 1, y

2

x 1, y

3

3 2

1

x 1

(B) y

x 1

(D) y 3 x, y 0

3

x, y 0

1 1 1 1 37. If a, , b, , c, and d , are four distinct points on a circle of radius 4 a b c d units then, abcd is equal to (A) 4 (B) 1/4 (C) 1 (D) 16

38. Four unit circles pass through the origin and have their centres on the coordinate axes. The area of the quadrilateral whose vertices are the points of intersection (in pairs) of the circles, is (A) 1 sq. unit (B) 2 2 sq. units (C) 4 sq. units (D) can not be uniquely determined, insufficient data 39. Consider 3 non collinear points A, B, C with co-ordinates (0,6), (5,5) and (-1,1) respectively. Equation of a line tangent to the circle circumscribing the triangle ABC and passing through the origin is (A) 2x-3y=0 (B) 3x+2y=0 (C) 3x-2y=0 (D) 2x+3y=0

:8:

x y 2

40. If

x y

2

4 , then all possible values of (x-y) is given by

(A) 2 2 , 2 2

(B) {-4, 4}

(C) [-4, 4]

(D) [-2, 2]

41. The value of ‘c’ for which the set, {( x, y) | x 2 y 2 2 x 1} {( x, y) | x y c 0} contains only one point in common is (A) ( , 1] [3, ) (B) {-1, 3}

(C) {-3}

(D) {-1}

42. In a right triangle ABC, right angled at A, on the leg AC as diameter, a semicircle is described. The chord joining A with the point of intersection of D of the hypotenuse and the semicircle, then the length AC equals to (A)

AB. AD AB 2 AD 2

(B)

AB. AD

(C) AB. AD

AB AD

(D)

AB. AD AB 2 AD 2

43. Locus of all point P (x,y) satisfying x3 y 3 3 xy 1 consists of union of (A) a line and a an isolated point (B) a line pair and an isolated point (C) a line and a circle (D) a circle and an isolated point 44. Prove that the length of the common chord of the two circles x 2 y 2 a 2 and ( x c) 2 y 2 b2 is

1 c

{(a b c)(a b c)( a b c)( a b c)}

In the diagram as shown, a circle is drawn with centre y R Q C (1,1)

O (0,0)

L P

x

C(1,1) and radius 1 and a line L. The line L is tangential to the circle at Q. Further L meet the y-axis at R and the x-axis at P in such a way that the angle OPQ equals where 0

2

.

45. The coordinates of Q are (a) (1 cos ,1 sin ) (b) (sin ,cos ) (c) (1 sin , cos ) (d) (1 sin ,1 cos )

:9:

46. Equation of the line PR is (a) x cos y sin sin cos 1 (b) x sin y cos cos sin 1

(c) x sin y cos cos sin 1 (d) x tan y 1 cot 2 47. If the area bounded by the circle, the x-axis and PQ is A ( ), then A equals 4 3

(a) 2 1

8

(b)

3 2 1 8

(c) 2 1

2 1

(d)

8

8

48. If H represent the harmonic mean between the abscissa, and K that between the ordinates of the points, in which a circle x 2 y 2 c2 is cut by a chord lx my , where l and m are the direction cosines of the unit vector in the xy plane then lH+mK has the value equal to (Take: l 2 m2 1 ) c2

(A) 2

c2

(B)

2

2c 2

(C)

(D) 2

c2 2

49. As shown in figure, three circles which have the same radius r, have centres at (0,0); (1,1) and (2,1). If they have a common tangent line, as shown then, Their radius ‘r’ is y C1

1

r

r

r o

1

C2

x

2

C

(A)

5 1 2

(B)

5 10

(C)

1 2

(D)

3 1 2

50. Consider the circles, x 2 y 2 25 and x 2 y 2 9 . From the point A (0,5) two segments are drawn touching the inner circle at the points B and C while intersecting the outer circle at the points D and E. If ‘O’ is the centre of both the circles then the length of the segment OF that is perpendicular to DE, is (A) 7/5 (B) 7/2 (C) 5/2 (D) 3 51. Two circles with centres at A and B, touch at T.BD is the tangent at D and TC is a common tangent. AT has length 3 and BT has length 2. The length CD is

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T

B

A

C D

(A) 4/3

(B) 3/2

(C) 5/3

(D) 7/4

52. Three concentric circles which the biggest is x 2 y 2 1, have their radii in A.P. If the line y=x+1 cuts all the circles in real and distinct points. The interval in which the common difference of the A.P. will lie is 2 2 1 1 (A) 0, (B) 0, (C) (D) none 0, 4 4 2 2 Consider the circle S: x 2 y 2 4 x 1 0 and the line L: y 3x 1 . If the line L cuts the circle at A and B then 53. Length of chord AB equal (A) 2 5 (B) 5

(C) 5 2

(D) 10

54. The angle subtended by the chord AB in the minor arc of S is (A)

3 4

(B)

5 6

(C)

2 3

(D)

4

55. Acute angle between the line L and the circle S is (A)

2

(B)

3

(C)

4

(D)

6

56. if the equation of the circle on AB as diameter is of the form x 2 y 2 ax by c 0 then the magnitude of the vector has the value equal to Consider a circle x 2 y 2 4 and a point P(4,2). denotes the angle enclosed by the tangents from P on the circle and A, B are the points of contact of the tangents from P on the circle. 57. The value of lies in the interval (A) (0,15 ) (B) (15 ,30 ) (C) (30 ,45 )

(D) (45 ,60 )

58. The intercept made by a tangent on the x-axis is (a) 9/4 (b) 10/4 (c) 11/4

(d) 12/4

59. Locus of the middle points of the portion of the tangent to the circle terminated by the coordinate axes is (a) x 2 y 2 1 2 (b) x 2 y 2 2 2 (c) x 2 y 2 3 2 (d) x 2 y 2 42

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60. A parabola y ax 2 bx c crosses the x-axis at ( , 0), ( , 0) both to the right of the origin. A circle also passes through these two points. The length of a tangent from the origin to the circle is (A)

bc a

(B) ac 2

(C)

b a

(D)

c a

61. The straight line joining any point P on the parabola y 2 4ax to the vertex and perpendicular from the focus to the tangent at P, intersect at R, then the equation of the locus of R is (A) x 2 2 y 2 ax 0 (B) 2 x 2 y 2 2ax 0 (C) 2 x 2 2 y 2 ay 0

(D) 2 x 2 y 2 2ay 0

62. Consider the parabola whose equation is y x2 4 x and the line y 2 x b . Then which of the following is/are correct? (A) for b=9 the line is a tangent to the parabola (B) for b=7 the line cuts the parabola in A and B such that the AOB is a right angle when ‘O’ is the origin (C) for some b R the line cuts the parabola in C and D such that x-axis bisects the COD (D) for b>9 the line passes outside the parabola 63. Suppose that three points on the parabola y x 2 have the property that their normal lines intersect at a common point (a,b). The sum of their x-coordinates is (A) 0

(B)

2b 1 2

(C)

a 2

(D) a+b

64. PQ is a double ordinate of the parabola y 2 4ax . If the normal at P intersect the line passing through Q and parallel to axis of x at G, then locus of G is a parabola with (A) length of latus rectum equal to 4a. (B) vertex at (4a, 0) (C) directrix as the line x-3a=0 (D) focus as (5a,0) 65. Through the vertex O of the parabola, y 2 4ax two chords OP & OQ are drawn and the circles on OP & OQ as diameters intersect in R. If 1 , 2 & are the angles made with the axis by the tangents at P & Q on the parabola & by OR then the value of, cot 1 cot 2 (A) 2tan

(B) 2 tan( ) (C) 0

(D) 2cot

66. Normals are concurrent drawn at points A, B and C on the parabola y 2 4 x at P(h, k). The locus of the point P if the slope of the line joining the feet of two of

: 12 :

them is 2, is (A) x+y=1

(B) x-y=3

(C) y 2 2( x 1)

1

(D) y 2 2( x ) 2

Consider the curve C : y 2 8 x 4 y 28 0. Tangents TP and TQ are drawn on C at P(5,6) and Q(5,-2). Also normals at P and Q meet at R.

67. The coordinates of circumcentre of ∆PQR is (A) (5,3) (B) (5,2) (C) (5,4)

(D) (5,6)

68. The area of quadrilateral TPRQ, is (A) 8 (B) 16 (C) 32

(D) 64

69. Angle between a pair of tangents drawn at the end points of the chord y+5t=tx+2 of curve C t R is (A)

6

(B)

4

(C)

3

(D)

2

From the point A, common tangents are drawn to the curve C1 : x 2 y 2 8 and C2 : y 2 16x

70. The y-intercept of common tangent between C 1 and C 2 having negative gradient, is (A) -1

(B) -2

(C) -3

(D) -4

71. The radius of circle tangent to C 2 at an upper end of latus rectum and passing through its focus is (A) 3 2 (B) 4 2

(C) 6 2

(D) 8 2

72. Area of quadrilateral formed by the common tangents, the chord of contact of C 1 and C 2 with respect to A, is (A) 20

(B) 40

(C) 60

73. Match the following: Column-I (A) The eccentricity of the ellipse which

(D) 80

Column-II

(P)

1 2

meets the straight line 2x-3y=6 on the X-axis and the straight line 4x+5y=20 on the Y-axis and whose principal axes lie along the coordinate axes, is (B) A bar of length 20 units moves with its

(Q)

1 2

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ends on two fixed straight lines at right angles. A point P marked on the bar at a distance of 8 units from one end describes a conic whose eccentricity is (C) If one extremity of the minor axis of the ellipse

x 2 a2

y2 b2

5

(R)

3

1 and the foci from an equilateral

triangle, then its eccentricity, is (D) There are exactly two points on the ellipse x

2

a

2

y

2

b2

7

(s)

4

1 whose distance from the centre of the

ellipse are greatest and equal to

a 2 2b 2

.

2

Eccentricity of this ellipse is equal to 74. Consider the particle travelling clockwise on the elliptical path

x 2 100

y2 25

1 . The

particle leaves the orbit at the point (-8,3) and travels in a straight line tangent to the ellipse. At what point will the particle cross the y-axis?

(A) 0,

25

3

(B) 0,

23

3

75. Statement I: The ellipse Statement 2: The ellipse

x 2 16

(C) (0,9)

x 2 16

y2 9

1 and

y2 9

(D) 0,

x 2

1 and

9

x 2 9

y2 16

26

3

1 are congruent

y2 16

1 have the same eccentricity

(A) Statement-1 is true, statement-2 is true and statement-2 is correct explanation for statement-1 (B) Statement-1 is true, statement-2 is true and statement-2 is NOT the correct explanation for statement-1 (C) Statement-1 is true, statement-2 is false (D) Statement-1 is false, statement-2 is true Let the two foci of an ellipse be (-1,0) and (3,4) and the foot of perpendicular from the focus (3,4) upon a tangent to the ellipse be (4,6). 76. The foot of perpendicular from the focus (-1,0) upon the same tangent to the ellipse is 12 34 7 11 17 (A) , (B) , (C) 2, (D) (1, 2) 5 5 3 3 4

: 14 :

77. The equation of auxiliary circle of the ellipse of (A) x 2 y 2 2 x 4 y 5 0 (B) x 2 y 2 2 x 4 y 20 0 (C) x 2 y 2 2 x 4 y 20 0

(D) x 2 y 2 2 x 4 y 5 0

78. The length of semi-minor axis of the ellipse is (A) 1 (B) 2 2 (C) 2 2

(D) 19

79. The equations of directrices of the ellipse are (A) x y 2 0, x y 5 0 3

5

2

2

(C) x y 0, x y

0

(B) x y

21

(D) x y

31

2 2

0, x y

17

0, x y

19

2 2

0 0

80. Match the properties given in Column1 with the corresponding curves given in the column 2. Column 1 Column 2 (A) The curve such that product of the (P) circle distances of any of its tangent from two given points is constant, can be (B) A curve for which the length of the (Q) Parabola subnormal at any of its point is equal to 2 and the curve passes through (1,2), can be (R) Ellipse (C) A curve passes through (1,4) and is such that the segment joining any point of P on the curve and the point of intersection of the normal at P with the x-axis is bisected by the y-axis. The curve can be (D) A curve passes through (1,2) is such that the (S) Hyperbola length of the normal at any of its point is equal to 2. The curve can be. 81. The chord PQ of the rectangular hyperbola xy a 2 meets the axis of x at A; C is the mid point of PQ & ‘O’ is the origin. Then the ∆ACO is (A) equilateral(B) isosceles (C) right angled (D) right isosceles 82. AB is a double ordinate of the hyperbola

x 2 a2

y2 b2

1 such that ∆AOB (where ‘O’ is

the origin) is an equilateral triangle, then the eccentricity of the hyperbola satisfies (A) e 3

2

(B) 1 e

3

2

(C) e

3

(D) e

2 3

: 15 :

A conic C passes through the point (2,4) and is such that the segment of any of its tangents at any point contained between the co-ordinate axes is bisected at the point of tangency. Let S denotes circle described on the foci F 1 and F 2 of the conic C as diameter.

83. Vertex of the conic is (A) (2,2),(-2,-2)

(B) (2 2, 2 2), (2 2, 2 2) (D) ( 2, 2), ( 2, 2)

(C) (4,4), (-4,-4)

84. Director circle of the conic is (A) x 2 y 2 4 (B) x 2 y 2 8 (C) x 2 y 2 2

(D) none

85. Equation of the circle S is (A) x 2 y 2 16 (B) x 2 y 2 8

(D) x 2 y 2 4

(C) x 2 y 2 32

86. Statement 1: Diagonals of any parallelogram inscribed in an ellipse always intersect at the centre of the ellipse. Statement 2: Centre of the ellipse is the only point at which two chords can bisect each other and every chord passing through the centre of the ellipse gets bisected at the centre. (A) Statement-1 is True, Statement-2, is True; statement-2 is a correct explanation for statement-1 (B) Statement-1, is True, Statement-2 is True; Statement-2 is NOT a correct explanation for Statement-1 (C) Statement-1 is True, Statement-2 is false (D) Statement-1 is false, Statement-2 is True 87. Statement-1: If P(2a,0) be any point on the axis of parabola, then the chord QPR, satisfy

1 ( PQ)

2

1 ( PR)

2

1 4a2

Statement-2: There exists a point on the axis of the parabola (other than vertex), such that

1 ( PQ)

2

1 ( PR) 2

= constant for all chord QPR of the parabola

(A) Statement-1 is True, Statement-2, is True; statement-2 is a correct explanation for statement-1 (B) Statement-1, is True, Statement-2 is True; Statement-2 is NOT a correct explanation for Statement-1 (C) Statement-1 is True, Statement-2 is false (D) Statement-1 is false, Statement-2 is True

: 16 :

88. Column-I (A) The number of real common tangents to the circle 5 x 2 5 y 2 16 and the hyperbola

Column-II (P) 1

2 2 is, 3 x y 48

(B) The number of real common normals to (Q) 2 parabola y 2 4 x and the circle ( x 1)2 ( y 1)2 1 is (C) If P is any point on ellipse

x 2 4

4 y 2 1 whose

(R) 3

foci are at A and B, then the maximum value of (PA)(PB) equals (D) The length of latus-rectum of the parabola (S) 4 defined parametrically by equations x=cos t-sin t and y=sin 2t equals (E) The value of a for the ellipse

x 2 a2

y2 b2

1

(T)

(a>b), if the extremities of the latus-rectum of the ellipse having positive ordinate lies on the parabola x 2 2( y 2) is 89. If f ( x) x2 bx c and f (2 t ) f (2 t ) for all real numbers t, which of the following is true? (A) f (1) f (2) f (4)

(B) f (2) f (1) f (4)

(C) f (2) f (4) f (1)

(D) f (4) f (2) f (1)

90. Given the graphs of the two functions, y=f(x) and y=g(x). In the adjacent figure from point A on the graph of the function y=f(x) corresponding to the given value of the independent variable (say x0 ), a straight line is drawn parallel to the X-axis to intersect the bisector of the first and the third quadrants at point B. From the point B a straight line parallel to the Y-axis is drawn to intersect the graph of function y=g(x) at C. Again a straight line is drawn from the point C parallel to the X-axis, to intersect the line NN ` at D. If the straight line NN ` is parallel to y-axis, then the coordinates of the point D are

: 17 :

N` Y C

D y=g(x)

B

A

y=f(x) X

x0

0

N

(A) f ( x0 ), g ( f ( x0 ))

(B) x0 , g ( x0 )

(C) x0 , g( f ( x0 ))

(D) f ( x0 ), g ( f ( x0 ))

91. Number of elements in the domain of the function f ( x ) (A) 2

(B) 3

(C) 4

1 10

C x 1 3 C x 10

is

(D) 10

92. Let f:[-1,1] onto [3,5] be a linear polynomial. Which of the following can be true? 1 7 2 2

15

1

(A) f

(B) f 1 4 4

(C) f(0)≠4

1 (D) f 2

1 2

f

93. Match the Column [Note: [k], {k} and sgn k denote the largest integer less than or equal to k, fractional part of k and signum function of k respectively.] Column-I Column II | x 1| x 1 (A) f ( x) (P) both many one and odd function x 1 | x 1| 1 x 4 (B) f ( x ) sgn 2 x 1

(Q) Even and periodic function

(C) f ( x) log1.3 (cos{ x})

(R) Bounded function (S) Range contains atleast one integer and atmost three integers

94. Column I (A) Lim n

Column II

(n 1)3 ( n 1)3 (n 1) ( n 1) 2

2

(P) 0

: 18 :

(B) Lim n

(C) Lim

n3 100n 2 1 (n 2)! ( n 1)!

n

(E) Lim

(R) 2

(n 3)!

n

(D) Lim

(Q) 1

100n 2 15n

(n 2)! ( n 1)!

(S) 3

(n 2)! ( n 1)! 1.3 2.4 3.5 .... n( n 2) n 2

n

C n1

(T) DNE

equals 1

(F) Lim n

2n 1 1

2n 1 n

n

k 1

k 1

95. For n N , let an 2 k and bn 2k 1 . Then Lim an bn is equal to (A) 1

(B)

1

n

(C) 0

2

(D) 2

x

x 1 96. Let l Lim then {l } where {x} denotes the fractional part function is x x 1

(A) 8 e 2 97. Lim x 1

(A)

(B) 7 e 2

nx n 1 ( n 1) x n 1

e

(B)

98. The value of Limit x 0

function: (A) is 1

100 e

5050

(C)

(B) is tan 1

number is equal to (A) 1 (B) 2 100.Column I

(B) Lim x 1

4950 e

tan({ x} 1) sin{x} where {x} denotes the fractional part { x} { x } 1 (C) is sin 1

x 0

x 0

(D)

e

99. The natural number n, for which Lim

(A) Lim

(D) e 2 7

where n=100 is equal to

(e x e) sin x 5050

(C) e 2 6

(C) 3

(D) is not existent

(27 x 9 x 3x 1)(cos x e x ) x 2 n1

(D) 4 Column II

1 x sin x cos 2 x x tan 2 2 sin 2 ( x 3 x 2 x 3) 1 cos( x 4 x 3) 2

(P) -3

(Q) 6

is a finite non-zero

: 19 :

2

(C) Lim x 0

6 x (cot x)(csc2 x)

(R) 8

sec cos x tan 1 4sec x

has the value equal to 2(a 3)( x 2) 6 sin 1 ( x 2) tan 1 (5 x 10 (D)If Lim 0 x 2 ( x 2)2

(S) 12

then the value of a is equal to (T) 18 n

2

n n 101.Consider a problem of limit as Lim .e Two children A and B solved this n n 1

problem as follows Mr. A solved the problem as follows: n n n 1

Lim n 2 ln

Let L e

n

n el where l= Lim n n ln 1 (using a x e x ln a ) n n 1

1

Put n as x

n then x 0

= 1 / x 1 1 1 1 ln( x 1) ln 1 Lim ln 1 Lim 1 x 0 x x x 0 x x x 0 x 1/ x 1 x 1 x

Lim

1 1

=

x 2 x3 x x ... x 2 3 x ln(l x ) 1 Lim Lim x 0 2 x 2 x2 x 0 Hence, l

2

L e1/2

Mr. B solved the problem as follows: n 1 n n 1

Limn 2

L e

n

f (x ) 1 and Lim ( x) u sin g the fact Lim x a x a Lim ( x )[ f (x )1] ( x ) then Lim f ( x) e x a x

n

Lim n

=

e

n2

n 1

Lim

e n

n2 n n 2 n 1

Lim

a

n

e n1 e n

ANS:

Which of the following statement is correct? (A) A is right and B is wrong (B) A is wrong and B is right

: 20 :

(C) both A and B are wrong as the correct value of limit is 1/e (D) both A and B are wrong as limit does not exist 3

102.If Lim n

1.2.4

4 2.3.5

5 3.4.6

....

can be expressed as rational in the n( n 1)( n 3) n2

m

lowest form n , find the value of (m+n). 103. y=f(x) is a continuous function such that its graph passes through (a,0). Then

Lim

log e (1 3 f ( x))

x a

2 f ( x)

(A) 1

is

(B) 0

3

2

(C) 2

(D) 3

104. Indicate all correct alternatives if, f(x)= (A) tan( f ( x))& (B) tan( f ( x))&

1 f ( x) 1 f ( x)

x 2

1 , then on the interval [0, ]

are both continuous are both discontinuous

(C) tan( f ( x)) & f 1 ( x) are both continuous (D) tan( f ( x)) is continuous but

1 f ( x )

is not

105. Which of the following functions are continuous at x=0? [Note: sgn x denotes signum function of x.] (A) cos sgn | x | sgn | x | 2

(C) sin sgn | x | sgn | x | 2 106. If f ( x)

e 2 x (1 4 x )1/2 ln(1 x 2 )

(B) cos sgn | x | sgn | x | 2

(D) sin sgn | x | sgn | x | 2

for x ≠ 0 then f has

(A) an irremovable discontinuity at x=0 (B) a removable discontinuity at x=0 and f(0) =-4 (C) a removable continuity at x=0 and f(0)=-1/4 (D) a removable discontinuity at x=0 and f(0)=4 107. Let function f be defined as f : R R and function g is defined as g : R R .Functions f and g are continuous in their domain. Suppose

: 21 :

function h( x) Lim n

x

n 1

f ( x) x

2

x n g ( x)

, x 0 . If h(x) is continuous in its domain then

f(1).g(1) us equal to (A) 2

(B) 1

(C)

1

(D) 0

2

x[ x]2 log (1 x ) 2 for 1 x 0 108.Consider ln e x 2 {x} for 0 x 1 tan x

2

where [*] & {*} are the greatest integer function & fractional part function respectively, then (A) f (0) ln 2 f is continous at x 0 (B) f (0) 2 f is continous at x 0 (C) f (0) e2 f is continous at x 0 (D) f has an irremovable discontinuity at x=0 2 Consider a function, f ( x) ln 3/ 2 (sin x+cos x) for x 0, 3 2

109. The value of Lim x

(A)

2 3

2

f ( x) 3/ 2

2 x

(B) is 1

is

(C) is

2 3

(2) 3/ 4

(D) non existent

2/ 3 3 f ( x) x 0, 2 2 110. The function g(x) is defined as g(x)= 3 Then g(x) e 2 f x 2 ; x , 2 2 2/3

(A) is continuous at x for g =0 2 2 (B) has a removable discontinuity at x (C) g(x) is discontinuous at x

2

2

and jump of discontinuity is equals to 2

(D) has a non-removable discontinuity at x

2

111. Column I contains 4 functions and Column II contains comments w.r.t their continuity and differentiability at x=0. Note that column-I may have more than one matching options in Column II.

: 22 :

Column-I (A) f ( x) [ x] |1 x | [] denotes the greatest

Column-II (P) continuous

integer function (B) g ( x) | x 2 | | x |

(Q) derivability

(C) h( x) [tan 2 x ] [] denotes the greatest integer

(R) discontinuous

function

x(3e1/ x 4) (2 e1/ x ) x 0 l ( x ) (D) 0 x 0

(S) non derivable

112. Let f(x) be a real valued function such that f(a)=0. If g(x)=(x-a) f(x) is continuous but not differentiable at x=a and h( x) ( x a) 2 f ( x) is continuous and differentiable at x=a. Then f(x) (A) must be continuous and differentiable at x=a. (B) must be continuous but not differentiable at x=a (C) may or may not be continuous at x=a. (D) must be discontinuous at x=a

`

Suppose f, g and h be three real valued function defined on R. Let 1 f ( x ) 2 x | x |, g ( x) (2 x | x |) and h( x) f g ( x) 3

113. The range of the function k ( x) 1 1 7

(A) , 4 4

5 11

(B) , 4 4

1

cos ( h( x)) cot ( h( x)) is equal to 1

1 5

(C) , 4 4

1

7 11

(D) , 4 4

114. The domain of definition of the function l ( x) sin 1 f ( x ) g ( x ) is equal to 3 (A) , 8

(B) ,1

3 (C) , 8

3 (D) , 8

115. The function T(x)=f(g(f(x)))+g(f(g(x))), is (A) continuous and differentiable in (, ) (B) continuous but not derivable x R (C) neither continuous nor derivable x R (D) an odd function 116. (A) The total number of local maxima and local minima of the function (2 x)3 , 3 x 1 f ( x) 2/ 3 is x , 1 x 2

: 23 :

(A) 0 (B) 1 (C) 2 (D) 3 (B) Comprehension: consider the function f : (, ) (, ) defined by f ( x )

x 2 ax 1 x 2 ax 1

,0 a 2

(i) Which of the following is true? (A) (2 a )2 f "(1) (2 a) 2 f "( 1) 0 (B) (2 a ) 2 f "(1) (2 a) 2 f "( 1) 0 (C) f '(1) f '(1) (2 a)2

(D) f '(1) f '(1) (2 a) 2

(ii) Which of the is true? (A) f(x) is decreasing on (-1,1) and has a local minimum at x=1 (B) f(x) is increasing on (-1,1) and has a local maximum at x=1 (C) f(x) is increasing on (-1,1) out has neither a local maximum and nor a local minimum at x=1 (D) f(x) is decreasing on (-1,1) but has neither a local maximum and nor a local minimum at x=1 e x

(iii) Let g ( x )

f '(t )

1 t

2

dt which of the following is true?

0

(A) g’(x) is positive on (, 0) and negative on (0, ) (B) g’(x) is negative on (, 0) and positive on (0, ) (C) g’(x) changes sign on both (, 0) and (0, ) (D) g’(x) does not change sign on (, ) 117. The angle between the tangent lines to the graph of the function x

f ( x) (2t 5) dt at the points where the graph cuts the x-axis is 2

(A)

6

(B)

4

(C)

3

(D)

2

118. If a

(B)c+d-b-a

(C)c+d-b+a

(D) c-d+b+a

Min { f (t ) : 0 t x}; 0 x 1 then 3 x ;1 x 2

119. If f ( x) 4 x3 x 2 2 x 1 and g ( x)

1 3 5 g g g has the value equal to 4 4 4

(A)

7 4

(B)

9 4

(C)

13 4

(D)

5 2

: 24 :

1 1 120. f ( x) 2 dx then f is 2 2 1 x 1 x (A) increasing in (0, ) and decreasing in (, 0)

(B) increasing in (, 0) and decreasing in (0, ) (C) increasing in (, )

(D) decreasing in (, )

121.Let f :[ 1, 2] R be differentiable such that 0 f '(t ) 1 for t [ 1,0] and 1 f '(t ) 0 for t [0,2] . Then

(A) 2 f (2) f (1) 1

(B) 1 f (2) f ( 1) 1

(C) 3 f (2) f ( 1) 0

(D) 2 f (2) f ( 1) 0

122. Let h be a twice continuously differentiable positive function on an open interval J. Let g ( x) ln h( x) for each x J 2

Suppose h '( x) h "( x) h( x) for each k J . Then (A) g is increasing on J (C) g is concave up on J

(B) g is decreasing on J (D) g is concave down on J

123. Read the following mathematical statements carefully: I. A differentiable function ‘f’ with maximum at x c f "(C ) 0 II. Antiderivative of a periodic function is also a periodic function. T

T

0

0

III. If f has a period T then for any a R . f ( x) dx f ( x a) dx IV. If f(x) has a maxima at x=c, then ‘f’ is increasing in (c-h,c) and decreasing in (c,c+h) as h 0 for h>0. Now indicate the correct Alternative (A) exactly one statement is correct (B) exactly two statements are correct (C) exactly three statements are correct (D) All the four statements are correct 124. If the point of minima of the function, f ( x) 1 a 2 x x3 satisfy the inequality x 2 x 2 x 2 5 x 6

0 , then ‘a’ must lie in the interval:

(A) 3 3 , 3 3 (B) 2 3 , 3 3 (C) 2 3 , 3 3 (D) 3 3, 2 3 2 3,3 3

: 25 :

f ( x) |1 x |1 x 2

125. Consider

g ( x) f ( x) b sin

2

x,

and 1 x 2

then, which of the following is correct?

(A) Rolles theorem is applicable to both f, g and b=

3 2

(B) LMVT is not applicable to f and Rolles theorem if applicable to g with b=

1 2

(C) LMVT is applicable to f and Rolles theorem is applicable to g with b=1 (D) Rolles theorem is not applicable to both f, g for any real b. 126.Carefully read the following five statements (a) The function f(x)=sec x attains a maximum on the interval ,

2 2

(b) If a function is differentiable at x=c then it is continuous at x=c. (c) The equation x5 10 x sin 5 x 0 has at least one non zero solution. (d) If f is a polynomial such that f’(3)=0 and f”(3)≠0 then f has a critical point at x=3 which is either a local minimum or a local maximum (e) If f is a polynomial such that f’(2)=0 and f”(2)=0 then f has a critical point at x=2 which is neither a local minimum nor a local maximum out of these 5 statements. (A) Exactly 1 is true and 4 are false (B) Exactly 2 are true and 3 are false (C) Exactly 3 are true and 2 are false (D) Exactly 4 are true and 1 is false. 127. For cubic, f ( x) x3 3 x 2 6 x 2006, the statement which does not hold, is (A) f(x) is monotonic increasing x R (B) f : R R is injective as well as surjective (C) Slope of the tangent at the point of inflection is 3. (D) f(x) is non monotic with exactly one real root. x 3 cos1 a,0 x 1 128. The range of values of a for which the function f ( x) has x , 1 x 3 the smallest values at x=1, is (A)[cos 2,0] (B) [-1, cos 2] (C) [0,1] (D) [-1,1]

129. The values of α for which the points of extremum of the function f ( x) x3 3 x 2 3( 2 1) x 1 lie in the interval (-2,4) will be equal to (A) (-1,3)

(B) (3.4)

(C) (-4,-2)

(D) (-2,-1)

: 26 :

130. The lateral edge of a regular hexagonal pyramid is 1cm. If the volume is maximum, then its height must be equal to (A)

1 3

(B)

2 3

(C)

1

(D) 1

3

Let P(x)be a polynomial of degree 4, vanishes at x=0. Given P(-1)=55 and P(x) has relative maximum/relative minimum at x=1,2,3.

131. Area of triangle formed by extremum points of P(x), is (A)

1 2

(B)

1 4

(C)

1

(D) 1

8

1

132. The value of definite integral

p( x) p( x) dx is 1

(A)

252 15

(B)

452 15

(C)

652 15

(D)

752 15

133. Which one of the following statement is correct? (A) P(x) has two relative maximum points and one relative and one relative minimum point. (B) Range of P(x) contains 9 negatives integers. (C) Sum of real roots of P(x)=0 is 5. (D) P(x) has exactly one inflection point. 134. A rectangle with one side lying along the x-axis is to be inscribed in the closed region of the xy plane bounded by the lines y=0, y=3x, and y=30-2x. The largest area of such a rectangle is (A)

135 8

(B) 45

(C)

135 2

(D) 90

135. Which of the following six statements are true about the cubic polynomial P(x)= 2 x 3 x 2 3 x 2? (i) It has exactly one positive real root (ii) It has either one or three negative roots. (iii) It has a root between 0 and 1. (iv) It must have exactly two real roots. (v) It has a negative root between -2 and -1. (vi) It has no complex roots. (A) only (i), (iii) and (iv) (B) only (ii), (iii) and (iv) (C) only (i) and (iii) (D) only (iii), (iv) and (v)

: 27 :

136. Consider f and g be two real-valued function defined on R. Let f(x)=2x-cos x, g ( x) e x , u c 1 c and v c c 1 where c>1. Statement-1: gof(u)>gof(v) Statement-2: f is increasing function and g is decreasing function (A) Statement-1 is true, statement-2 is true and statement-2 is correct explanation for statement-1 (B) Statement-1 is true, statement-2 is true and statement-2 is NOT the correct explanation for statement -1. (C) Statement-1, is true, statement-2 is false (D) Statement-1 is false, statement-2 is true 137. Match the Column Column-I

Column II

x 3/5 if x 1 (A) Let f ( x) Then the number 3 ( x 2) if x 1

(P) 5

of critical points on the graph of the function is (B) Number of real solution of the equation, log 2 2 x ( x 1) log 2 x 6 2 x is

(Q) 4

(C) The number of values of c such that the straight line 3x+4y=c touches the curve x 2 2

(R) 3

x y is x 2

(D) If f ( x) (t 1) dt,1 x 2 Then global

(S) 2

x

maximum value of f(x) is (T) 1 138. Write the correct order sequence in respect of the statement given below. F stands for false and T stands for true I. Suppose that g is continuous with g(1)=5 and g(5)=10. Then the equation g(C)=7 must have a solution such that c (1,5) 10

II. If f(x)>x for all x then

f ( x)dx 25 0

III. If f’(x)=g’(x) then f(x)=g(x) IV. Suppose that f is differentiable and f(2)=f(6). Then there must be at least one point c (2,6) with f’(C)=0.

b b V. f ( x ) g ( x )dx f ( x )dx g ( x)dx a a a b

(A) T T T F F (B)T F F T F

(C) F T F T T

(D) T T F T F

: 28 :

x (1 x cos x ln x sin x)dx sin x

139.

is equal to

/2

(A)

2 2

(B)

2

(C)

x

140. If F(x)=

2

t

f (t )dt wheref(t) 1

7

(A)

4 17

1

(B)

15

17

4 2 4

1 u

(D)

(D)

k

C0 1

C1 2

C 2 3

68

xf x(1 x) dx; I 2

f x(1 x) dx, where

1 k

I 2

(B) 1/2

15 17

I 1

(A) k 142. If

1

k

1 k

2k-1>0. Then

2

du then the value of F” (2) equals

257

141. Let f be a positive function. Let I1

2

u

(C)

(C) 1

(D) 2

0, where C0 , C1 , C 2 are all real, the equation C2 x 2 C1 x C0 0 has

(A) atleast one root in (0,1) (B) One root in (1,2) and other in (3,4) (C) one root in (-1,1) & the other in (-5,-2) (D) both roots imaginary 37

143. The value of the definite integral

{ x}

2

3(sin 2 x) dx where {x} denotes the

19

fractional part function. (A) 0 (B) 6 144. Let f ( x) (A)

2

/2

sin x

, then

x

f ( x )dx

0

(C) 9

(B) f ( x )dx 0

x dx = 2

f ( x) f

0

(D) 18

(C) f ( x )dx 0

(D)

1

f ( x)dx

0

1

145. Let f(x) be a continuous function on R. If f ( x) f (2 x) dx 5 and 0 2

1

0

0

f ( x) f (4 x) dx =10 then the value of f ( x) f (8 x) dx is equal to (A) 0

(B) 5

(C) 10

(D) 15

: 29 :

1

2 3 n 1 n n n n 146. Lim (1 n) n 1 1 .....2 is equal to n 2 3

(A) e

1

1

(B) e 2

(C) e 4 sin x

Consider a function f ( x)

cos x

1

(D) e 2 3 , , 2 2

e t dt

; x (0, 2 )

1 t 2

5 147. The value of f’ is 4 1

1

1

(B) 2e

(A) 2e

2

1

(D) 2e

(C) 2e

2

2

2

143.f(x) is decreasing in the interval

(A) 0, 2

(C) 0,

(B) , 2

(D) none

144. For x 0, , if g is the inverse of f is then the value of g’(0) is 2 1

1

(A) e

(B) e

2

(C)

2

1 2

1

e

2

1

(D) 2e

2

2

Let g be a continuous function on R and satisfies g ( x ) 2 sin x.cos tg (t ) dt sin x 0

145. The value of g’ is equal to 3 (A)

1 2

(B)

1 2

(C)

3 2

(D)

1 4

146. Which one of the following is correct? (A) Lim x 0

g ( x) x

/2

1 (B)

g ( x) dx 1 0

2 1 (C) g 3 3

(D) g 2 ( x) g 2 x 1 2

147. Equation of the tangent to the curve y=g(x) at the point whose abscissa is (A) a line parallel to x-axis (C) a line having gradient

(B) a line parallel to y-axis 1 2

(D) a line having gradient 2.

148. Given f(x)= sin 3 x and P(x) is a quadric polynomial with leading coefficient unity.

2

, is

: 30 :

2

Statement-1: P ( x). f ' '( x) dx vanishes 0 2

Statement-2:

f ( x ) dx vanishes 0

(A) Statement-1, is true statement-2 is true and statement-2 is correct explanation for statement-1. (B) Statement-1, is true, statement-2 is true and statement-2 is NOT the correct explanation for statement-1. (C) Statement-1, is true, statement-2 is false (D) Statement-1 is false statement-2, is true k

k

x

x 3 149. Let L= Lim tan( y sin x) dx, and l= Lim 1 k .e dx, , then k y 0 y 0 0 1

(A) 3l=4L

(B) 3L=4l

(C) L+2l=5

(D) 2(l+L)=7

150. The function f is continuous and has the property f f ( x) 1 x for all x [0,1] 1

and J f ( x ) dx then 0

1 (A) f 4

3 f 1 4

1

(B) the value of J equal to /2

1 2 (C) f . f 1 3 3

(D)

sin xdx

sin x cos x

3

2

has the same value as J

0

151. Which of the following definite integral vanishes?

(A)

ln x

1 x

2

2

dx

x cos x

8 sin x dx

(B)

2

0

(C)

0

sin 2012 x

152. Match The Column Column I

1 (2008) x e 2008

2008

x

sin x

dx

(D)

x sin x

10 sin 0

Column II

1

(A)

dx equals

(P) e

1

0

(B) The value of the definite integrals 1

e 0

(Q) e

1/4

1/ e

x 2

dx

ln x dx is equal to

1 1

11.22.33...(n 1) n 1.nn n (C) Lim equals n n1 2 3. .. .. n 2

(R) e1/2 (S) e

2

x

dx

: 31 :

153. Column I

Column II

3

(A)If I ( x 1)3 (4 x) 3 x cos xdx, , then

(P) 0

2 2

| 50 I | is equal to 10

(B) If J sgn(sin x) dx , then 10J is equal to,

(Q) 100

0

where sgn x denotes signum function of x 102

(C) If K

cot

1

x dx then [K] is equal to,

(R)50

0

where [y] denotes largest integer less than or equal to y. 51

[ x 25]dx

(D) If L 510

then

{ x 25}dx

L 2

is equal to, where

(S) 70

0

[y] and {y} denote greatest integer function and fractional part function respectively. 154. The solution of the differential equation, e x ( x 1)dx ( ye y xe x ) dy 0 with initial condition f(0)=0, is (A) xe x 2 y 2e y 0 (B) 2 xe x y 2e y 0 (C) xe x 2 y 2e y 0 (D) 2 xe x y 2 e y 0 155. A function y=f(x) satisfies the condition f’(x) sinx+f(x) cosx=1, f(x) being /2

bounded where x p If I

f ( x ) dx then

0

2 I (A) 2 4

2 I (B) 4 2

(C) 1 I

2

(D) 0

156. Area enclosed by the curve y ( x 2 2 x)e x and the positive x-axis is (A) 1

(B) 2

(C) 4

(D) 6

157. The slope of the tangent to a curve y=f(x) at (x,f(x)) is 2x+1. If the curve passes through the point (1,2) then the area of the region bounded by the curve, the x-axis and the line x=1 is (A)

5 6

(B)

6 5

(C)

1 6

(D) 1

158. In a chemical reaction a substance changes into another such that the rate of decomposition of a chemical substance x present at instant t is proportional to x itself i.e.., amount of unchanged substance still present. If half of the substance

: 32 :

present initially has been converted at the end of 1 minute then the time t in minutes at the end of which 99% of the substance will have changed lies in the interval (A) 5 and 6 (B) 6 and 7 (C) 7 and 8 (D) more than 10. 159. In the shown figure, half a period of sin x from 0 to is split into two regions (light and dark shaded) of equal area by a line through the origin. If the line and the sine function intersect at a point whose x co-ordinate is k, then k satisfies the equation y k, sin k

O

(A) k cos k + 2 sin k=0 (C) k sin k +2 cos k-2=0

k

x

(B) k sin k +2 cos k=0 (D) 2 cos k+k sin k+2=0

160. Let k be a real number such that k ≠ 0. If α and β are non zero complex numbers satisfying 2k and 2 2 4k 2 2k then a quadratic equation having and as its roots is equal to

(A) 4 x 2 4kx k 0 (C) 4kx 2 4 x k 0 161. Let p( x ) x 2 B

4 x 3

(B) x 2 4kx 4k 0 (D) 4kx 2 4kx 1 0 12

log10 (4.9), A P( ai ) where a1 , a2 ,....., a12 are positive reals and i 1

13

P( b ) where b , b ,...., b j

1

2

13

are non-positive reals, then which one of the

j 1

following is always correct? (A) A>0,B>0 (B) A>0,B<0

(C) A<0,B>0

(D) A<0,B<0

162. Let a1 and a2 be two values of a for which the expression f(x,y)= 2 x 2 3 xy y 2 ay 3 x 1 can be factorised into two linear factors then the product

(a1a2 ) is equal to

(A) 1

(B) 3

(C) 5

(D) 7

: 33 :

163. If c 2 4d and the two equations x 2 ax b 0 and x 2 cx d 0 have one common root, then the value of 2(b+d) is equal to (A)

a c

(B) ac

(C) 2ac

(D) a+c

164. The expression y ax 2 bx c (a, b ,c R and a ≠ 0) represents a parabola which cuts the x-axis at the points which are roots of the equation ax 2 bx c 0 . Column II contains values which correspond to the nature of roots mentioned in column-I Column I Column II (A) For a=1, c=4, if both roots are greater (P) 4 than 2 then b can be equal to (B) For a=-1, b=5, if roots lie on either side (Q) 8 of -1 then c can be equal to (C) For b=6, c=1, if one root is less than -1 and (R) 10 the other root greater than

1 2

then a can be

equal to (S) no real value 165. In an A.P with first term ‘a’ and the common difference d(a, d ≠0), the ratio ' ' of the sum of the first n terms to sum of n terms succeeding them does not depend on n. Then the ratio (A)

1 1 , 2 4

1

(B) 2, 3

a d

and the ratio ' ' respectively are (C)

1 1 , 2 3

(D)

1 2

,2

166. Infinite number of triangles are formed as shown in figure. If total area of these triangles A then 8A is equal to y

1

1 O

1

9

2

3

(A)3

(B) 4

(C) 1

3 x 1 27

(D) 2

............

: 34 :

167. if Sn 2na (A)

ab 2

n 2b

is the sum of first n terms of an A.P.., then common difference is

4

(B)

a 2b 2

(C)

2a b 2

(D)

b 2

Let the sum of first 10 terms of an arithmetic progression is equal to 155 and the sum of first two terms of a geometric progression is 9. Also the first term of the arithmetic progression is equal to the common ratio of the geometric progression and the first term of the geometric progression is equal to the common difference of the arithmetic progression.

168. The common difference of arithmetic progression is (A)

1 2

,3

(B)

1 2 , 3 3

(C)

1 3

,2

(D)

2 3

,3

169. The common ratio of geometric progression is (A) 1,

25 2

(B) 2,

25 2

(C) 1,2

(D)

1 2 , 2 25

Let g n be a decreasing geometric progression of positive numbers. The difference between the first and fifth term of a G.P. is 15 and sum of the first and third term of a G.P. is 20. Also the fifth term of G.P. lies between the roots of the equation K 2 4 K 5 X 2 K 2 X K 2 16 K 2 0 5

170. The value of

g is equal to i 1

(A) 23

i

(B) 31

(C) 35

171. The largest integral value of k is (A) 2 (B) 3 (C) 4

(D) 28

(D)5

172. The value of (R-r) in a triangle whose side lengths are g3 1 , g3 , g 3 1 , is (A)

1

(B)

2

3 2

(C)

1 4

(D)

5 2

[Note: R and r denotes circum radius of the triangle] 173.If X.Y R satisfy the equation x 2 y 2 4 x 2 y 5 0 , then compute the value of 99

the sum

x ry r 0

174.Consider the graph of a cubic polynomial y x3 ax2 bx c as shown in the figure. If roots of the cubic equation x3 ax 2 bx c 0 are , 1, , 1 such that , 1, (in that order) from the first three terms of an arithmetic

: 35 :

progression, then find its 5001th term. y (0,3)

175. Match the Column Column I (A) In an A.P the series containing 99 terms, the sum of all the odd numbered terms is 2550. The sum of all the 99 terms of the A.P. is (B) f is a function for which f(1)=1 and f(n)= n+f(n-1) for each natural number n ≥ 2. The value of f(100) is (C) Suppose, f (n) log2 (3).log3 (4).log4 (5)....logn 1 ( n)

O

x

Column II (P) 5010

(Q) 5049

(R) 5050

100

then the sum

f (2 ) equals k

k 2

(D) Concentric circles radii 1,2,3…100 cms are drawn. The interior of the smallest circle is coloured red and the annular regions are coloured alternately green and red, so that no two adjacent regions are of the same colour. The total area of the green regions in sq.cm is k then ‘k’ equals to

(S) 5100

176. Statement 1: The coefficient of t 49 in the expression t 1 t 2 t 3 ....... t 50 is equal to 1075. 2n

Statement 2: The value of

K n 2n 1 . k 1

(A) Statement-I is true, statement -2 is true and statement-2 is correct explanation for statement-1. (B) Statement-1 is true, statement -2 is true and statement -2 is NOT the correct explanation for statement-1. (C)Stement-1 is true, statement -2 is false. (D) Statement -1 is false, statement -2 is true.

: 36 :

100

177. The coefficient of in the expansion of

100

Ck ( x 2)100 k 3k is also equal to

k 0

(A) number of ways in which 50 identical books can be distributed in 100 students, if each student can get atmost one book. (B) number of ways in which 100 different white balls and 50 identical red balls can be arranged in a circle, if no two red balls are together. (C) number of dissimilar terms in ( x1 x2 x3 ...... x50 )51 (D)

2 6 10 14 .....198 50!

178. Which of the following statement(s) is (are) correct? 100

(A) The coefficient of in the expansion of

r 0

100

r

Cr ( x 4)100 r 5 is equal to 4950 8

8 (B) If the sixth term in the expansion of x 3 x2 log10 x is 5600, then x is equal

to 100. (C) Let An n C0 nC1 n C1 n C2 .......n C n 1n C n and

An 1 An

15 4

then the sum of possible

values of n is equal to 6. n

(D) If Ak

n

C k

n 1

Ck n C k 1

and 3

A

k

4 , then n is equal to 128.

k 0

179. (i) Find the number of four letter word that can be formed from the letters of the word HISTORY. (each letter to be used at most once) (ii) How many of them contain only consonants? (iii) How many of them begin & end in a consonant? (iv) How many of them begin with a vowel? (v) How many contains the letters Y? (vi) How many begin with T & end in a vowel? (vii) How many begin with T & also contain S? (viii) How many contain both vowels? 180. The number of six digit numbers that can be formed from the digits 1,2,3,4,5,6 & 7 so that digits do not repeat and the terminal are even is (A) 144 (B) 72 (C) 288 (D) 720 181. The interior angles of a regular polygon measure 150 each. The number of diagonals of the polygon is (A) 35 (B) 44 (C) 54 (D) 78

: 37 :

182. Number of 4 digit numbers of the form N=abcd which satisfy following three conditions (i) 4000 N 6000 (ii) N is a multiple of 5 (iii) 3 b c 6 is equal to (A) 12 (B) 18 (C) 24 (D) 48 183. Seven different coins are to be divided amongst three persons. If no of the persons receive the same number of coins but each receives atleast one coin & none is left over, then the number of ways in which the division may be made is (A) 420 (B) 630 (C) 710 (D) none 184. Let there be 9 fixed points on the circumference of a circle. Each of these points is joined to every one of the remaining 8 points by a straight line and the points are so positioned on the circumference that atmost 2 straight lines meet in any interior point of the circle. The number of such interior intersection points is (A) 126 (B) 351 (C) 756 (D) none of these 185. A women has 11 close friends. Find the number of ways in which she can invite 5 of them to dinner, if two particular of them are not on speaking terms & will not attend together. 186. Number of three digit number with atleast one 3 and at least one 2 is (A) 58 (B) 56 (C) 54 (D) 52 187. Let P n denotes the number of ways of selecting 3 people out of ‘n’ sitting in a row, if no two of them are consecutive and Qn is the corresponding figure when they are in a circle. If Pn Qn 6 , then ‘n’ is equal to (A) 8

(B) 9

(C) 10

(D) 12

188. The number of ways in which five different books to be distributed among 3 persons so that each persons gets at least one book, is also equal to the number of ways in which (A) 5 persons are allotted 3 different residential flats so that and each person is allotted at most one flat and no two persons are allotted the same flat (B) number of parallelograms (some of which may be overlapping)formed by one set of 6 parallel lines and the other set of 5 parallel lines that goes in other direction. (C) 5 different toys are to be distributed among 3 children, so that each child gets at least one toy. (D) 3 mathematics professors are assigned five different lectures to be delivered, so that each professor gets at least one lecturer.

: 38 :

189. Column I (A) Four different movies are running in a town. Ten students go to watch these four movies. The number of ways in which every movie is watched by atleast one student, is (Assume each way differs only by number of students watching a movie) (B) Consider 8 vertices of a regular octagon and its centre. If T denotes the number of triangles and S denotes the number of straight lines that can be formed with these 9 points then the value of (T-S) equals (C) In an examination, 5 children were found to have their mobiles in their pocket. The invigilator fired them and took their mobiles in his possession. Towards the end of the test, Invigilator randomly returned their mobiles. The number of ways in which at most two children did not get their own mobiles is (D) The product of the digits of 3214 is 24. The number of 4 digit natural numbers such that the products of their digits is 12, is (E) The number of ways in which a mixed double tennis game can be arranged from amongst 5 married couple if no husband & wife plays in the same game, is

Column II (P) 11

(Q) 36

(R) 52

(S) 60

(T) 84

Let U 1 and U 2 be two urns such that U 1 contains 3 white and 2 red balls, and U 2 contains only 1 white ball. A fair coin is tossed. If head appears then 1 ball is drawn at random from U 1 and put into U 2 However, if tail appears then 2 balls are drawn at random from . Now 1 ball is drawn at random from U 1 and U 2

190. The probability of the drawn ball from U 2 being white is (A)

13 30

(B)

23 30

(C)

19 30

(D)

11 30

: 39 :

191. Given that the drawn ball from is U 2 is white, the probability that head appeared on the coin is (A)

17 23

(B)

11 23

(C)

15 23

(D)

12 23

192. Mr. A forgot to write down a very important phone number. All he remembers is that it started with 713 and that the next set of 4 digit involved are 1, 7 and 9 with one of these numbers appearing twice. He guesses a phone number and dails randomly. The odds in favour of dialing the correct telephone number is (A) 1:35 (B) 1:71 (C) 1:23 (D) 1:36 193. There are 8 students from 4 schools A, B, C, D 2 students from each school. Let these 8 students enter in 4 rooms R1 , R2 , R3 , R4 , so that each room will have 2 students. The probability that each room have students from the same school, is (A)

1 105

(B)

2 105

(C)

3 105

(D)

4 105

194. let S denote the set of nine digit numbers whose digits are 1,2,3,4,5,6,7,8,9 such that each of these digit occurs exactly once. A nine digit number is chosen randomly. Five events defined on S are E-1, E-2, E-3, E-4 and E-5 which are described in Column I. Match the probabilities of these events which are given in Column II. Column I Column II (A) E-1:The number chosen is even

(P)

(B) E-2: The first, fifth and ninth digits

(Q)

4 63 7 9

of the chosen number will be odd (C) E-3: The number will be greater than 3.108

(R)

(D) E-4: The sum of digits used in the

(S)

4 9 5 42

number on first two places from the left, equals the digit used in the left place (E) E-5: The number chosen is divisible by 6 195. A bag contains 3R & 3 G balls and a person draws out 3 at random. He then drops 3 blue balls into the bag & again draws out 3 at random. The chance that the 3 later balls being all of different colours is (A) 15% (B) 20% (C) 27% (D) 40%

: 40 :

196. An urn contains 10 balls coloured either black or red. When selecting two balls from the urn at random, the probability that a ball of each colour is selected is 8/15. Assuming that the urn contains more black balls than red balls, the probability that at least one black ball is selected, when selecting two balls, is (A)

18 45

(B)

30 45

(C)

39 45

(D)

41 45

197. A fair coin is tossed a large number if times. Assuming the tosses are independent which one of the following statement, is True? (A) Once the number of flips is large enough, the number of heads will always be exactly half of the total number of tosses. For example, after 10,000 tosses one should have exactly 5,000 heads. (B) The proportion of heads will be about ½ and this proportion will tend to get closer to ½ as the number of tosses increases. (C) As the number of tosses increases, any long run of heads will be balanced by a corresponding run of tails so that the overall proportion of heads is exactly 1/2. (D) all of the above. 198. An ant is situated at the vertex A of the triangle ABC. Every movement of the ant consists of moving to one of other two adjacent vertices from the vertex where it is situated. The probability of going to any of the other two adjacent vertices. of the triangle is equal. The probability that at the end of the fourth movement the ant will be back to the vertex A, is (A)

4 16

(B)

6 16

(C)

7 16

(D)

8 16

199. Four children A, B, C and D have 1,3,5 and 7 identical unbiased dice respectively and roll them with the condition that one who obtains an even score, wins. They keep playing till some one or the other wins. Statement-1: All the four children are equally likely to win provided they roll their dice simultaneously. Statement-2:The child A is most probable to win the game if they roll their dice in order of A, B, C and D respectively. (A) Statement-1 is true, statement-2 is true and statement-2 is correct explanation for statement-1. (B) Statement-1, is true, statement-2 is true and statement-2 is NOT the correct explanation for statement-1

: 41 :

(C) Statement-1 is true, statement-2 is false. (D) Statement-1 is false, statement-2 is true. 200. Indicate the correct order sequence in respect of the following: I. If the probability that a computer will fail during the first hour of operation is 0.01, then if we turn on 100 computers, exactly one will fail in the first hour of operation. II.A,B,C simultaneously satisfy P(ABC)=P(A).P(B).P(C) and __ __ __ ___ P AB C P A .P( B).P C and P A B C P ( A).P(B).P C and ___

__

P A BC P(A).P( B).P C then A, B, C are independent.

III. Given the events A and B in a sample space. If P(A)=1, then A and B are independent. IV. When a fair six sided die is tossed on a table top, the bottom face can not be seen. The probability that the product of the numbers on the five faces that can be seen is divisible by 6 is one. (A) FTFT

(B) FTTT

(C) TFTF

(D) TFFF

201. Let A, B,C be three events in a probability space. Suppose that P(A)=0.5, P(B)=0.3, P(C)=0.2, P A B =0.15, P A C =0.1 and P B C =0.06. The smallest possible value of P Ac B c C c is (A)0.31 (B)0.25 (C)0 [Note: Ac denotes compliment of event A]

(D) 0.26

202.A coin has probability ‘p’ of showing head when tossed. It is tossed n times. Let P n denotes the probability that no two (or more) consecutive heads occur. The value of P 4 is given by (A) 1 4 P 2 4 P 3 (B) 1 3 P 2 2 P 3 (C) 1 2 P 2 8 P 3

(D) 1 P 2 6 P 3

Suppose you have 10 keys and you wish to open a door and try the keys one at a time, randomly until you open the door. Only one of the keys will open the door.

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