doubtion.com · 1000 MOST IMPORTANT QUESTIONS FOR JEE – PART 1 RANK BUILDER SERIES For More Parts and Answer Keys Download the MathonGo App 1. If and are the roots of ax2 bx c 0,

1000 MOST IMPORTANT QUESTIONS FOR JEE – PART 1

RANK BUILDER SERIES For More Parts and Answer Keys Download the MathonGo App

1. If and are the roots of ,02 cbxax then the value of

baba

11is

(a) bc

a (b)

ca

b

(c) ab

c (d) none of these

2. If one root of the equation 0)2()1(22 ixiix is ,2 i then the other root is

(a) i (b) i2 (c) i (d) i2

3. If the equation 023 baxx )0( b has a double root then

(a) 0274 3 ba (b) 0274 3 ba

(c) 0427 3 ba (d) none of these

4. If Za and the equation 01)10)(( xax has integral roots, then the values of ''a are

(a) 8, 10 (b) 10, 12 (c) 12, 8 (d) none of these

5. The number of solutions of the equation xxxe 55)(sin is

(a) 0 (b) 1 (c) 2 (d) infinite 6. If dcba , then the equation 0))((5))((3 dxbxcxax has

(a) real and distinct roots (b) real and equal roots (c) imaginary roots (d) none of these

7. The value of k for which the equation 0)23()1(23 22 kkkxx has roots of opposite

signs, lies in the interval

(a) (–, 0) (b) (–, –1)

(c) (1, 2) (d)

2,

2

3

8. If the roots of the equation 02 qpxx = 0 differ by unity, then

(a) qp 42 (b) 142 qp

(c) 142 qp (d) none of these

9. If the equation 59)1(22 kxkx = 0 has only negative roots, then

(a) k 0 (b) k 0

(c) k 6 (d) k 6



10. The largest interval for which 014912 xxxx is

(a) – < x < (b) – 1 < x < 1

(c) 0 < x < 1 (d) – 4 x 0

11. If 02762 xx and 0432 xx , then

(a) x > 3 (b) x < 4 (c) 3 < x < 4 (d) none of these

12. The equation 06||2 xx has

(a) one root (b) two distinct roots (c) three distinct roots (d) four distinct roots

13. If x is real and k = 1

12

2

xx

xx, then

(a)

3,

3

1k (b) ),3[ k

(c)

3

1,k (d) none of these

14. If , are the roots of the equation 02738 2 xx , then the value of

3/12

3/12

is

(a) 3

1 (b)

4

1

(c) 2

7 (d) 4

15. If 012 xx and 02 2 xx have a common root, then

(a) 0172 (b) 0172

(c) 0172 (d) 0172

16. The solution of the equation 2|6| 2 xxx is

(a) (2, 3) (b) (2, 4) (c) (3, 4) (d) none of these

17. 17 )54(log 27 xxx

, x may have values (a) 2, 3 (b) –2, –3 (c) –2, 3 (d) 2, –3

18. If one root of kxx 2 is square of the other, then k is equal to

(a) 32 (b) 23

(c) 52 (d) 25

19. If 02 cbxax has no real roots, a 0, a, b, c R, then the value of ac is

(a) positive (b) negative (c) zero (d) non-negative



20. If the roots of the equation 012 22 mmxx lie in the interval (–2, 4), then

(a) –1 < m < 3 (b) 1 < m < 5 (c) 1 < m < 3 (d) –1 < m < 5

21. If , are the roots of the equation 032 axx , a R and < 1 < , then

(a) a (–, 2) (b)

4

9,a

(c)

4

9,2a (d) none of these

22. The least integral value of k for which 0)4(8)2( 2 kxxk for all x R, is

(a) 5 (b) 4 (c) 3 (d) none of these 23. If a, b, c denote the sides of a triangle, then both the roots of the equation

0)(22 cxbacx are

(a) real (b) positive (c) negative (d) complex

24. If the equation 010)10(2)5( 2 axaxa has real roots of same sign, then

(a) a > 10 (b) –5 < a < 5

(c) a < –10 or 5 < a 6 (d) none of these

25. If 0log4 22/1

2 axx does not have two distinct real roots, then the maximum value of a

is (a) 1/4 (b) 1/16 (c) –1/16 (d) –1/4

26. If 22 xx is a factor of qpxx 24 , then )( qp equals

(a) 0 (b) 1 (c) –1 (d) 9

27. Number of solutions of the equation upto......666x is

(a) 2 (b) 3 (c) 1 (d) 0

28. If 0)1()23()1( 222 axaaxa have more than two real roots, then ''a is equal to

(a) 2 (b) 1 (c) 0 (d) –1

29. If the roots of the equation 02 cbxx be two consecutive integers, then cb 42

equals (a) 1 (b) 2 (c) 3 (d) –2



30. If the sum of the roots of the equation 02 cbxax is equal to the sum of the squares of

their reciprocals, then a

b

c

a, and

b

c are in

(a) A.P. (b) G.P. (c) H.P. (d) none of these

31. If naaaa ....,, 321 be an A.P. of non-zero terms then nn aaaaaa 13221

1.........

11

(a) naa1

1 (b)

naa

n

1

(c) naa

n

1

1 (d) none of these

32. If the roots of the equation 0283912 23 xxx are in A.P., then their common

difference will be (a) ±1 (b) ±2 (c) ±3 (d) ±4 33. The digits of a positive integer having three digits are in A.P. and their sum is 15. If the

number obtained by reversing the digits is 594 less than the original number then the number is

(a) 352 (b) 652 (c) 852 (d) none of these 34. There are n A.M.’s between 3 and 29 such that 6th mean: )1( n th mean = 3 : 5, then the

value of n, is (a) 10 (b) 11 (c) 12 (d) none of these

35. If the roots of cubic 023 dcxbxax be in G.P., then

(a) dcba 33 (b) 33 cdab

(c) dbac 33 (d) 33 bdca

36. Let ...., 21 SS be squares such that for each ,1n the length of a side of nS equals the length

of a diagonal of 1nS . If the length of a side of 1S is 10 cm, then for which of the following

values of n is the area of nS less than 1 sq. cm.

(a) 7 (b) 8 (c) 19 (d) none of these

37. If ,.....1 32 yyyx then y is

(a) )1( x

x (b)

)1( x

x

(c) x

x 1 (d)

x

x1



38. If cba ,, are in A.P. as well as in G.P., then

(a) cba (b) cba

(c) cba (d) cba

39. If three positive real numbers a, b, c are in A.P. such that abc = 4, then the minimum possible value of b is

(a) 23/2 (b) 22/3 (c) 21/3 (d) 25/2

40. If naaa ....,,, 21 are in A.P. with common difference d 0, then sum of the series

]secsec....secsecsec[secsin 13221 nn aaaaaad is

(a) 1tantan aan (b) 1cotcot aan

(c) 1secsec aan (d) 1coscos ecaecan

41. Let upto....19

444

19

44

19

432

S , then S is equal to

(a) 81

38 (b)

19

4

(c) 171

36 (d) none of these

42. A G.P. consists of an even number of terms. If the sum of all the terms is five times the sum of the terms of occupying odd places, the common ratio will be equal to

(a) 2 (b) 3 (c) 4 (d) 5

43. The value of

324.0 is

(a) 999

419 (b)

990

419

(c) 1000


44. If cba ,, are in A.P. and 222 ,, cba are in H.P., then

(a) cba (b) cab 32

(c) 8

2 acb (d) none of these

45. If the sum of n terms of an A.P. is nn 53 2 , then which of its terms is 164?

(a) 26th (b) 27th (c) 28th (d) none of these

46. If dcba ,,, are in H.P., then

(a) dcba (b) dbca

(c) cbda (d) none of these

47. If cbba zzxx 2/2/ , , then a, b, c are in




48. The sum to n terms of ....321

7

21

5

1

3222222

is

(a) 1

3

n (b)

1

6

n

n

(c) 1

2

n

n (d)

1

12

n

n

49. Value of

upto....

3

1

3

1

3

1log

3225.0

)36.0(y is

(a) 0.9 (b) 0.8 (c) 0.6 (d) 0.25

50. Sum of the series .....531

321

31

21

1

1 333333

16 terms is


51. dcba ,,, are in A.P., then abc, abd, acd, bcd are in


52. If A.M. and G.M. of two numbers are 9 and 4 respectively. Then these numbers are the roots of the equation

(a) x2 + 18x – 16 = 0 (b) x2 – 18x + 16 = 0 (c) x2 + 18x + 16 = 0 (d x2 – 18x – 16 = 0

53. If 01111

bccbaa

and a + c – b 0 then a, b, c are in


54. The minimum value of 4x + 41–x, x R, is (a) 2 (b) 4 (c) 1 (d) none of these

55. The coefficient of x49 in (x – 1) (x – 2) ….. (x – 50) is (a) 1275 (b) 2550 (c) – 2550 (d) – 1275

56. If a, b, c are in H.P., then the straight line 01

cb

y

a

x always passes through a fixed

point and that point is (a) (–1, –2) (b) (–1, 2) (c) (1, –2) (d) (1, –1/2)

57. In a ABC, if a2, b2, c2 are in A.P., then tan A, tan B, tan C are in (a) A.P. (b) G.P. (c) H.P. (d) none of these



58. If one A.M. ‘A’ and two G.M.’s G1 and G2 be inserted between any two numbers, then the

value of 32

31 GG is

(a) A

GG 212 (b) 212 GAG

(c) 22

212 GAG (d) none of these

59. If three distinct numbers x, y, z are in G.P. and axzyx , then

(a)

,

4

3a (b)

,

4

3a

(c)

,

4

3a (d)

,

4

3a – {3}

60. If a, b, c are in A.P.; a, x, b are in G.P. and b, y, c are in G.P. then x2, b2, y2 are in (a) H.P. (b) G.P. (c) A.P. (d) none of these

61. A ray of light coming from the point (1, 2) is reflected at a point B on the y-axis and then passes through the point (5, 3). The co-ordinates of the point B is

(a)

8

13,0 (b)

8

17,0

(c)

8

13,0 (d) none of these

62. The equation of the line through (5, 4) such that its segment intercepted by the lines

2

1

2 y

x and

2

1

2 y

xis of length

2

5 is

(a) 2x y + 14 = 0 (b) 2x y 14 = 0

(c) 2x y + 1 = 0 (d) 2x y + 13 = 0

63. A straight line which makes an acute angle with the positive direction of x-axis is drawn through P (4, 5) to meet x = 7 at R and y = 9 at S. Then

(a) sec4PR (b) ecPS cos5

(c)

2sin

)cos4sin3(2PSPR (d) 2

16922

PSPR

64. A line intersects the x-axis at A(9, 0) and y-axis at B(0, –7). A variable line perpendicular to AB cuts x-axis at P and y-axis at Q. If AQ and BP intersects at R, then the locus of R is

(a) a straight line (b) a straight line parallel to x-axis

(c) a straight line parallel to y-axis (d) a circle

65. If 0 , the line 023 yx passes through the fixed point

(a)

3

2,2 (b)

2,

3

2

(c)

3

2,2 (d) none of these



66. The lines 1sincos pyx and 2sincos pyx will be perpendicular if

(a) 2

(b)

2

(c) 2

||

(d) =

67. The sum of the abscissas of all the points on the line x + y = 4 that lie at a unit distance from the line 01034 yx , is

(a) 3 (b) –3 (c) 4 (d) –4

68. Through the point P(, ), where > 0, the straight line 1b

y

a

x is drawn so as to form

with coordinate axes a triangle of area S. If ab > 0, then the least value of S is

(a) (b) 2

(c) 4 (d) none of these

69. The range of values of in the interval (0, ) such that the points (3, 5) and )cos,(sin lie

on the same side of the line x + y – 1 = 0, is

(a)

2,0 (b)

4,0

(c)

2,

4 (d) none of these

70. If P and Q are two points on the line 03034 yx such that OP = OQ = 10, where O is

the origin, then the area of the OPQ is

(a) 48 (b) 16


71. The medians AD and BE of a triangle with vertices A(0, b), B(0, 0) and C (a, 0) are perpendicular to each other if

(a) 2

ba (b)

2

ab

(c) ab = 1 (d) ba 2

72. A point equidistant from the lines 01034 yx , 026125 yx and 050247 yx

is

(a) (1, –1) (b) (1, 1)

(c) (0, 0) (d) (0, 1)

73. If the straight line drawn through the point 2,3P and making an angle 6

with the x-axis

meets the line 0843 yx at Q, then the length of PQ is

(a) 4 (b) 5


74. The number of integer values of m, for which the x-coordinate of the point of intersection of the lines 943 yx and 1mxy is also an integer, is



(a) –2 (b) 0

(c) 4 (d) 1

75. The separate equations of the straight lines whose joint equation is 065 22 yxyx , are

(a) ,02 yx x – 3y = 0 (b) ,02 yx x – 3y = 0

(c) ,02 yx x + 3y = 0 (d) ,02 yx x – 3y = 0

76. If the lines joining the origin to the points of intersection of 1mxy with 122 yx are

perpendicular, then m is equal to

(a) 2 (b) 1

(c) 5 (d) –2

77. The coordinates of a point on the line 4yx that lies at a unit distance from the line

01034 yx are

(a) (3, 1) (b) (–7, 3)

(c) (3, –1) (d) (7, –11)

78. If the gradient of one of the lines 02 22 yhxyx is twice that of the other, then h =

(a) ± 2 (b) ± 3

(c) 1 (d) 2

3

79. The number of lines that are parallel to 0762 yx and have an intercept 10 units

between the coordinate axes is (a) 1 (b) 2

(c) 4 (d) infinitely many 80. The medians AD and BE of a triangle with vertices at A(0, b), B (0, 0) and C (a, 0) are

perpendicular to each other if

(a) ab 2 (b) ab 2

(c) ba 2 (d) ba 2

81. A(a, b), B(x1, y1) and C(x2, y2) are the vertices of a triangle. If a, x1, x2 are in G.P. with

common ratio r and b, y1, y2 are in G.P. with common ratio s, then area of ABC is

(a) ))(1)(1( rssrab (b) ))(1)(1(2

1rssrab

(c) ))(1)(1(2

1rssrab (d) ))(1)(1( srsrab

82. Two opposite vertices of a rectangle are (1, 3) and (5, 1). If the equation of a diagonal of this rectangle is y = 2x + c, then the value of c is

(a) –4 (b) 1 (c) –9 (d) none of these

83. In a ABC, if A is the point (1, 2) and equations of the median through B and C are respectively x + y = 5 and x = 4, then B is

(a) (1, 4) (b) (7, –2) (c) (4, 1) (d) (–2, 7)



84. The straight lines 045 yx , 0102 yx and 052 yx are

(a) concurrent (b) the sides of an equilateral triangle (c) the sides of a right angled triangle (d) none of these

85. Let )0,0(),0,1( QP and )33,3(R be three points. Then the equation of the

bisector of angle PQR is

(a) 02

3 yx (b) 03 yx

(c) 03 yx (d) 02

3 yx

86. A triangle is formed by the points O(0, 0), A (0, 21) and B (21, 0). The number of points

having integral coordinates (both x and y) and lying on or inside the triangle is (a) 285 (b) 105

(c) 305 (d) none of these 87. If the equation of the locus of a point equidistant from the points (a1, b1) and (a2, b2) is

0)()( 2121 cybbxaa , then the value of ‘c’ is

(a) 22

22

21

21 baba (b) )(

2

1 21

21

22

22 baba

(c) 22

21

22

21 bbaa (d) 2

221

22

21

2

1bbaa

88. For the triangle whose sides are along the lines x = 0, y = 0 and 186

yx, the

circumcentre is (a) (3, 4) (b) (2, 2)

(c) (2, 3) (d) (3, 2) 89. Area bounded by the lines 14|27||52| yx

(a) 91 (b) 57 (c) 79 (d) none of these 90. The middle points of the sides of a triangle are (–4, 2), (6, –2) and (7, 9). The area of the

triangle is (a) 21 (b) 45


91. The straight line x cos + y sin = 2 will touch the circle 0222 xyx if

(a) Inn , (b) Inn

,2

12

(c) Inn ,2 (d) none of these

92. A triangle is formed by the lines whose combined equation is given by (x + y – 4)(xy – 2x – y + 2) = 0. The equation of its circumcircle is

(a) 083522 yxyx (b) 085322 yxyx

(c) 085322 yxyx (d) none of these



93. If the chord of contact of the tangents from a point on the circle 222 ayx to the circle

222 byx touch the circle ,222 cyx then the roots of the equation ax2 + 2bx + c = 0

are necessarily

(a) imaginary (b) real and equal

(c) real and unequal (d) rational

94. The equation of the circle passing through (1, 0) and (0, 1) and having smallest possible radius is

(a) 022 22 yxyx (b) 022 22 yxyx

(c) 022 yxyx (d) 022 yxxx

95. If chord x cos + y sin = p of 222 ayx subtends a right angle at the origin, then

(a) 22 pa (b) 22 2pa

(c) 22 3pa (d) none of these

96. Two circles, each of radius 5, have a common tangent at (1, 1) whose equation is 3x + 4y – 7 = 0, then their centres are (a) (4, –5), (–2, 3) (b) (4, –3), (–2, 5) (c) (4, 5), (–2, –3) (d) none of these

97. The equation of the circumcircle of the regular hexagon whose two consecutive vertices have the coordinates (–1, 0) and (1, 0) and which lies wholly above the x-axis, is

(a) 013222 yyx (b) 01322 yyx

(c) 013222 yx (d) none of these

98. If the angle of intersection of the circles 022 yxyx and 022 yxyx is ,

then equation of the line passing through (1, 2) and making an angle with the y-axis is (a) x = 1 (b) y = 2 (c) x + y = 3 (d) x – y = 3

99. If p and q be the longest distance and the shortest distance respectively of the point (–7, 2)

from any point (a, b) on the curve whose equation is 051141022 yyx then GM of

p and q is equal to

(a) 112 (b) 55


100. Locus of the middle-points of the line segment joining )1,0( 2 ttP and )1,2( 2 tttQ

Q cuts an intercept of length a on the line x + y = 1, then a is equal to

(a) 2

1 (b) 2


101. If (2, 4) is a point interior to the circle 010622 yxyx and circle does not cut the

axes at any point then belongs to the interval

(a) (25, 32) (b) (9, 32)

(c) (32, ) (d) (9, 25)



102. If a line segment AM = a, moves in the plane XOY remaining parallel to OX so that the left

end point A slides along the circle 222 ayx , then locus of M is

(a) 222 4ayx (b) axyx 222

(c) ayyx 222 (d) 02222 ayaxyx

103. The shortest distance of the chord of contact of tangents from the point (10, 3) to the circle

014222 yxyx is

(a) 109

99 (b)

106

99

(c) 109

97 (d)

106

100

104. The area of a quadrilateral formed by a pair of tangents from the point (4, 5) to the circle

16)1()2( 22 yx with a pair of radii where tangents touch the circle is

(a) 2 (b) 4

(c) 8 (d) 16

105. The radical centre of the three circle described on the three sides of a triangle as diameter is

(a) orthocentre (b) circumcentre

(c) incentre (d) centroid

106. Let 0 < < 2

be a fixed angle. If )sin,(cos P and )sin(),cos( Q , then Q

is obtained from P by

(a) clockwise rotation around origin through an angle

(b) anticlockwise rotation around origin through an angle

(c) reflection in the line through origin with slope tan

(d) reflection in the line through origin with slope 2

tan

107. The locus of mid-points of the chords of the circle 0122 22 yyxx which are of unit

length is

(a) 4

3)1()1( 22 yx (b) 2)1()1( 22 yx

(c) 4)1()1( 22 yx (d) none of these

108. If radii of the smallest and the largest circles passing through the point )2,3( and

touching the circle 022222 yyx and r1 and r2 respectively, then the mean of r1, r2

is

(a) 1 (b) 2

(c) 3 (d) 2



109. All the circle which cut the circle 422 yx orthogonally and pass through )2,1( also

pass through another fixed point, having coordinates

(a) )1,2( (b)

3

22,

3

4

(c)

3

22,

3

2 (d)

3

2,

3

1

110. The locus of the centre of the circle which bisects the circumferences of the circles

422 yx and 016222 yxyx

(a) a straight line (b) a circle

(c) a parabola (d) none of these

111. The locus of the mid points of the chords of the circle 022 byaxyx which subtend

a right angle at

2,

2

ba is

(a) 0byax (b) 22 babyax

(c) 08

2222

babyaxyx (d) 0

8

2222

babyaxyx

112. A rhombus is inscribed in the region common to the two circles 012422 xyx and

012422 xyx with two of its vertices on the line joining the centres of the circles.

The area of the rhombus is

(a) 38 sq. units (b) 34 sq. units

(c) 316 sq. units (d) none

113. The points A(a, 0), B(0, b), C(c, 0) and D(0, d) are such that ac = bd and a, b, c, d are all non-zero. Then the points

(a) form a parallelogram (b) do not lie on a circle

(c) form a trapezium (d) are concyclic

114. The locus of the centers of the circles which cut the circles 096422 yxyx and

024522 yxyx orthogonally is

(a) 07109 yx (b) 02 yx

(c) 011109 yx (d) 07109 yx

115. If the two circles 022 1122 yfxgyx and 022 22

22 yfxgyx touch each then

(a) 2211 gfgf (b) 2

2

1

1

g

f

g

f

(c) 2121 ggff (d) none of these



116. Two circles whose radii are equal to 4 and 8 intersect at right angles. The length of their common chord is

(a) 5

16 (b) 8

(c) 64 (d) 5

58

117. A circle of constant radius a passes through O and cuts the axes of co-ordinates in points P and Q, then the equation of the locus of the foot of perpendicular from O to PQ is

(a) 2

22

22 411

)( ayx

yx

(b) 2

22

222 11)( a

yxyx

(c) 2

22

222 411

)( ayx

yx

(d) 2

22

22 11)( a

yxyx

118. The equation of the image of the circle 0183241622 yxyx by the line mirror

01374 yx is

(a) 023543222 yxyx (b) 023543222 yxyx

(c) 023543222 yxyx (d) 023543222 yxyx

119. Let x and y be the real numbers satisfying the equation 034 22 yxx . If the

maximum and minimum values of 22 yx and M and m respectively, then the numerical

value of M – m is

(a) 2 (b) 8


120. The circle having 052 yx and 0152 yx as tangents and (–5, –5) is one of the

points of contact of one of them, then the equation of circle is

(a) 0154622 yxyx (b) 0952422 yxyx

(c) 0106222 yxyx (d) 01054622 yxyx

121. Circle drawn having it’s diameter equal to focal distance of any point lying on the parabola

x2 – 4x + 6y + 10 = 0, will touch a fixed line whose equation is (a) y = 2 (b) y = –1

(c) x + y = 2 (d) x – y = 2

122. ‘t1’ and ‘t2’ are two points on the parabola y2 = 4x. If the chord joining them is a normal to

the parabola at ‘t1’, then

(a) t1 + t2 = 0 (b) t1(t1 + t2) = 1

(c) t1(t1 + t2) + 2 = 0 (d) t1t2 + 1 = 0

123. Two parabolas y2 = 16 (x – k) and x2 = 16 (y – l) always touch each other (where k, l are variable parameters). Their point of contact lies on



(a) a straight line (b) a parabola (c) a circle (d) none of these

124. If the line joining the points )2,( 121 atatA and )2,( 2

22 atatB passes through C (0, b), then

(a) b(t1 + t2) = 2at1t2 (b) 2b(t1 +t2) = at1t2

(c) b(t1 + t2) = at1t2 (d) none of these

125. The set of points on the axis of the parabola y2 = 4x + 8 from which the 3 normals to the parabola are all real and different is

(a) {(k, 0) | k –2} (b) {(k, 0) |k > –2} (c) {(0, k) | k > –2} (d) none of these

126. The normal chord at a point ‘t’ on the parabola 16y2 = x subtends a right angle at the vertex. Then t is equal to

(a) 2 (b) 2

(c) 64

1 (d) none of these

127. The total number of chords that can be drawn from the point (a, a) to the circle

x2 + y2 = 2a2 such that they are bisected by the parabola y2 = 4ax is (a) 1 (b) 4 (c) 2 (d) 0

128. A (x1, y1) and B (x2, y2) are any two points on the parabola y = cx2 + bx + a. If P (x3, y3) be

the point on the arc AB where the tangent is parallel to the chord AB, then (a) x2 is the A.M. between x1 and x3 (b) x2 is the G.M. between x1 and x3

(c) x2 is the H.M. between x1 and x3 (d) none of these

129. Tangents drawn to parabolas y2 = 4ax at the points A and B intersect at C. Ordinate of A, C and B forms

(a) a A.P. (b) a G.P. (c) a H.P. (d) none of these

130. Consider the parabola 3y2 + 4y – 6x + 8 = 0. The points on the axis of this parabola from where 3 distinct normals can be drawn are given by

(a)

h,

3

2, where

18

29h (b) ,

3

1,

h where

18

19h

(c) ,3

2,

h where

18

29h (d) none of these

131. If (2, –8) is one end of a focal chord of the parabola y2 = 32x, then the other end of the chord is

(a) (32, 32) (b) (32, –32)

(c) (–2, 8) (d) none of these

132. The HM of the segments of a focal chord of the parabola axy 42 is

(a) 4a (b) 2a

(c) a (d) a2



133. AB is a chord of the parabola axy 42 . If its equation is cmxy and it subtends a right

angle at the vertex of the parabola then

(a) amc 4 (b) mca 4

(c) amc 4 (d) 04 mca

134. The point )2,( aa is an interior point of the region bounded by the parabola xy 162 and

the double ordinate through the focus. Then a belongs to the open interval

(a) a < 4 (b) 0 < a < 4

(c) 0 < a < 2 (d) a > 4

135. The range of values of for which the point (, –1) is exterior to both the parabolas

||2 xy is

(a) (0, 1) (b) (–1, 1)

(c) (–1, 0) (d) none of these

136. If )(1 axmby and )(2 axmby are two tangents to the parabola axy 42 , then

(a) 021 mm (b) 121 mm

(c) 121 mm (d) none of these

137. The equation of the common tangent to the equal parabolas axy 42 and ayx 42 is

(a) 0 ayx (b) ayx

(c) ayx (d) none of these

138. If the line kxy is a normal to the parabola xy 42 , then k can have the value

(a) 22 (b) 4

(c) –3 (d) 3

139. If two of the three feet of normals drawn from a point to the parabola xy 42 be (1, 2) and

(1, –2), then the third foot is

(a) 22,2 (b) 22,2

(c) (0, 0) (d) none of these

140. The locus of the middle points of parallel chords of a parabola ayx 42 is a

(a) straight line parallel to the x-axis

(b) straight line parallel to the y-axis

(c) circle

(d) straight line parallel to a bisector of the angles between the axes

141. If the vertex and focus of a parabola are (3, 3) and (–3, 3) respectively, then its equation is



(a) 0632462 yxx (b) 0812462 yxx

(c) 0632462 xyy (d) 0812462 xyy

142. Equation of the parabola whose axis is parallel to y-axis and which passes through the points (1, 0), (0, 0) and (–2, 4) is

(a) yxx 322 2 (b) yxx 322 2

(c) yxx 22 2 (d) yxx 22 2

143. The triangle formed by the tangent to the curve bbxxxf 2)( at the point (1, 1) and the

co-ordinate axes lies in the first quadrant. If its area is 2, then the value of b is

(a) –1 (b) 3

(c) –3 (d) 1

144. If the normals at two points P and Q of a parabola axy 42 intersect at a third point R on

the curve, then the product of ordinates of P and Q is

(a) 24a (b) 22a

(c) 24a (d) 28a

145. The equation of the parabola whose vertex and focus lie on the axis of x at distances a and

1a from the origin respectively is

(a) xaay )(4 12 (b) ))((4 1

2 axaay

(c) ))((4 112 axaay (d) none of these

146. The point on the curve 2y x, the tangent at which makes angle 45° with x-axis will be

given by

(a) (2, 4) (b) (1/2, 1/2)

(c) (1/2, 1/4) (d) (1/4, 1/2)

147. Tangents are drawn from the point (–8, 3) to the parabola 07862 xyy . The angle

between the tangents is

(a) 60° (b) 90°

(c) 120° (d) none of these



148. The orthocentre of the triangle formed by any three tangents to a parabola axy 42 lies

on the line

(a) x = a (b) x = 2a

(c) x = –a (d) x = –2a

149. The normal at the point (at2, 2at) on the parabola axy 42 cuts the curve again at the

point t1 , then

(a) 0221 tttt (b) 022

1 tttt

(c) 0221 ttt (d) 02

1 tttt

150. A line AB meets the parabola axy 42 in P such that AB is bisected at P. If A is (, )

then locus of B is

(a) )(8)( 2 xay (b) )32(3

4

(c) 2

34 (d) )32(

3

2

151. The eccentricity of the ellipse 9

)4()3(2

22 yyx is

(a) 2

3 (b)

3

1

(c) 23

1 (d)

3

1

152. For an ellipse 149

22

yx

with vertices A and ,'A tangent drawn at the point P in the first

quadrant meets the y-axis in Q and the chord PA' meets the y-axis in M. If ''O is the

origin then 22 MQOQ equals to

(a) 9 (b) 13 (c) 4 (d) 5

153. The line, 0 nmylx will cut the ellipse 12

2

2

2

b

y

a

x in points whose eccentric angles

differ by 2

if

(a) 22222 2mnbla (b) 22222 2nlbma

(c) 22222 2nmbla (d) 22222 2lmbna

154. The area of the rectangle formed by the perpendiculars from the centre of the standard

ellipse to the tangent and normal at its point whose eccentric angle is 4

is

(a) 22

22 )(

ba

abba

(b)

abba

ba

)(

)(22

22

(c) )(

)(22

22

baab

ba

(d)

abba

ba

)( 22

22



155. If 2

2

21 tantanb

a , then the chord joining two points 21 & on the ellipse 1

2

2

2

2

b

y

a

x

will subtend a right angle at (a) focus (b) centre (c) end of the major axis (d) end of the minor axis 156. An ellipse having foci at (3, 3) and (–4, 4) and passing through the origin has eccentricity

equal to

(a) 7

3 (b)

7

2

(c) 7

5 (d)

5

3

157. Length of the perpendicular from the centre of the ellipse 243927 22 yx on a tangent

drawn to it which makes equal intercepts on the coordinates axes is

(a) 2

3 (b)

2

3

(c) 23 (d) 6

158. For each point ),( yx on the ellipse with centre at the origin and principal axes along the

coordinate axes, the sum of the distances from the point ),( yx to the points (±2, 0) is 8.

The positive value of x such that )3,(x lies on the ellipse, is

(a) 3

3 (b) 2

(c) 4 (d) 32

159. Let ''E be the ellipse 149

22

yx

and ''C be the circle .922 yx Let P and Q be the

points (1, 2) and (2, 1) respectively. Then (a) Q lies inside C but outside E (b) Q lies outside both C and E (c) P lies inside both C and E (d) P lies inside C but outside E. 160. If the distance between the foci is equal to the minor axis and latus rectum = 4, then

equation of the ellipse whose centre is at origin and mirror axis is along X-axis, is

(a) 162 22 yx (b) 543 22 yx

(c) 162 22 yx (d) none of these

161. The distance of the point of contact from the origin of the 7 xy with the ellipse

,1243 22 yx is

(a) 3 (b) 2

(c) 7/5 (d) none of these



162. The ellipse 12

2

2

2

b

y

a

x passes through the point (1, –2) and has eccentricity

2

1, then its

latus rectum is equal to

(a) 2 (b) 3

(c) 2 (d) 3

163. Let S1, S2 be the foci of an ellipse and PT, PN be the tangent and the normal respectively

to the ellipse at some point P on it. Then

(a) PN externally bisects 21PSS (b) PT internally bisects 21PSS

(c) PT bisects – 21PSS (d) none of these

164. The eccentric angle of a point on the ellipse 134

22

yx

at a distance of 4

5 units from the

focus on the positive X-axis, is

(a) 3

(b)

4

(c) 6

(d) none of these

165. A point on the ellipse 149

22

yx

where the normal is parallel to the line 32 yx , is

(a)

5

9,

5

8 and

5

9,

5

8 (b)

5

8,

5

7 and

5

8,

5

4

(c)

5

9,

5

8 and

5

9,

5

8 (d)

5

8,

5

9 and

5

8,

5

9

166. The line 0 nmylx is a normal to the ellipse 12

2

2

2

b

y

a

x, if

(a) 2

222

2

2

2

2 )(

n

ba

m

b

l

a (b)

2

222

2

2

2

2 )(

n

ba

l

b

m

a

(c) 22222222 )( nbambla (d) none of these

167. If the mid-point of a chord of the ellipse 12516

22

yx

is (0, 3), then length of the chord is

(a) 5

32 (b) 16

(c) 5

4 (d) 12

168. The line kxy 32 touches the ellipse 1049 22 yx , if k is equal to

(a) 4 (b) 3

1

(c) 2 (d) 9

10



169. If the tangent to the ellipse 164 22 yx at the point P() is a normal to the circle

04822 yxyx , then is equal to

(a) 0 (b) 3

(c) 6

(d)

4

170. Which of the following points lies inside the ellipse 2516)1(9 22 yx

(a)

2

3,

4

1 (b)

4

5,

2

1

(c)

1,

2

3 (d) none of these

171. Equation of the common chord of the ellipse

19

1

4

122

yx

and the circle

4)1()1( 22 yx , is

(a) y + 3 = 0 and x – 1 = 0 (b) x – 3 = 0 and y + 1 = 0 (c) x + 3 = 0 and x – 1 = 0 (d) y + 3 = 0 and y – 1 = 0

172. Equation of the tangent to the hyperbola 632 22 yx which is parallel to the line

43 xy , is

(a) y = 3x + 5 (b) y = 3x – 5

(c) y = 3x 5 (d) none of these

173. If the coordinates of a point are )sec3,tan4( , where is a parameter, then the point

lies on a conic section whose eccentricity is

(a) 3

5 (b)

4

5

(c) 4

3 (d)

5

3

174. If )( 1P and )( 2D be the end points of CP and CD of an ellipse 12

2

2

2

b

y

a

x whose

centre is C. If 2

2

21a

bMM (where 21, MM are slopes of CP and CD) then the 21

(a) 45° (b) 90° (c) 135° (d) none of these

175. The locus of the mid-point of the focal chords of the ellipse 12

2

2

2

b

y

a

x is

(a) a

ex

b

y

a

x

2

2

2

2

(b) a

ex

b

y

a

x

2

2

2

2

(c) 2222 bayx (d) none of these



176. The tangent at a point P(a cos , b sin ) of an ellipse 12

2

2

2

b

y

a

x, meets its auxiliary

circle in two points, the chord joining which subtends a right angle at the centre, then the eccentricity of the ellipse is

(a) 12 )sin1( (b) 2/12 )sin1(

(c) 2/32 )sin1( (d) 22 )sin1(

177. The eccentric angles of extremities of a chord of an ellipse 12

2

2

2

b

y

a

x are 1 and 2. If

this chord passes through the focus, then

(a) 01

1

2tan

2tan 21

e

e (b) )cos(

2cos 21

21

e

(c) )sin(

sinsin

21

21

e (d)

1

1

2cot

2cot 21

e

e

178. Tangents are drawn to ellipse 12

2

2

2

b

y

a

x at points P(1) and P(2), then the point of

intersection of these tangents is

(a)

2cos

2sin

,

2cos

2cos

21

21

21

21 ba (b)

2cos

2sin

,

2cos

2cos

21

21

21

21 ba

(c)

2sin

2cos

,

2cos

2sin

21

21

21

21 ba (d) none of these

179. The area of rectangle formed by perpendiculars from the centre of ellipse 12

2

2

2

b

y

a

x to

the tangent and normal at the point whose eccentric angle is /4 is

(a) abba

ba

22

22

(b) abba

ba

22

22

(c) 22 ba (d) 22 ba

180. The equation of the normal to the ellipse 12

2

2

2

b

x

a

x at the positive end of latus rectum is

(a) 03 aeeyx (b) 03 aeeyx

(c) 02 aeeyx (d) none of these

181. Eccentricity of the hyperbola conjugate to the hyperbola 1124

22

yx

is

(a) 3

2 (b) 2

(c) 3 (d) 3

4



182. The asymptote of the hyperbola 12

2

2

2

b

y

a

x form with any tangent to the hyperbola a

triangle whose area is tan2a in magnitude, then its eccentricity is

(a) sec (b) cosec

(c) 2sec (d) 2cosec

183. Locus of the feet of the perpendiculars drawn from either foci on a variable tangent to the

hyperbola 1916 22 xy is

(a) 922 yx (b) 9

122 yx

(c) 144

722 yx (d) 16

122 yx

184. The locus of the point of intersection of the lines 0343 tyx and

0343 tytx (where t is parameter) is a hyperbola whose eccentricity is

(a) 3 (b) 2

(c) 3

2 (d)

3

4

185. If the eccentricity of the hyperbola 5sec22 yx is 3 times the eccentricity of the

ellipse ,25sec 222 yx then a value of is

(a) 6

(b)

4

(c) 3

(d)

2

186. For all real values of m, the straight line 49 2 mmxy is a tangent to the curve

(a) 3649 22 yx (b) 3694 22 yx

(c) 3649 22 yx (d) 3694 22 yx

187. The foci of the ellipse 116 2

22

b

yx and the hyperbola

25

1

81144

22

yx

coincide. Then the

value of 2b is

(a) 5 (b) 7 (c) 9 (d) 4

188. P is a point on the hyperbola ,12

2

2

2

b

y

a

x N is the foot of the perpendicular from P on the

transverse axis. The tangent to the hyperbola at P meets the transverse axis at T. If O is the centre of the hyperbola, then OT. ON is equal to

(a) 2e (b) 2a

(c) 2b (d) 22 /ab



189. If PN is the perpendicular from a point on a rectangular hyperbola 222 ayx on any of

its asymptotes, then the locus of the mid point of PN is (a) a circle (b) a parabola (c) an ellipse (d) a hyperbola

190. Area of the quadrilateral formed with the foci of the hyperbola 12

2

2

2

b

y

a

x and

12

2

2

2

b

y

a

x is

(a) )(4 22 ba (b) )(2 22 ba

(c) )( 22 ba (d) )(2

1 22 ba

191. If AB is a double ordinate of the hyperbola 12

2

2

2

b

y

a

x such that OAB (O is the origin) is

an equilateral triangle, then the eccentricity e of the hyperbola satisfies

(a) 3e (b) 3

21 e

(c) 3

2e (d)

3

2e

192. A normal to the hyperbola 2

2

2

2

b

y

a

x = 1 meets the coordinate axes at A and B. If the

rectangle OABP (O is the origin) is completed, then the locus of P is (a) circle (b) parabola (c) hyperbola (d) ellipse

193. If 0),(` yxS , 0),(2 yxS and 0),(3 yxS represent equations of a hyperbola, its

asymptotes and its conjugate respectively, then for any point (h, k), the quantities S1(h, k),

),(2 khS and ),(3 khS are in


194. The directrices of the hyperbola 12255 22 xy , are

(a) 2y (b) 2x

(c) 3y (d) 3x

195. The equation of the pair of asymptotes of the hyperbola abbyax ))(( , is

(a) 0))(( byax (b) 2

))((ab

byax

(c) xy = 0 (d) none of these

196. The conjugate of the hyperbola abbyax ))(( , is

(a) abbyax 2))(( (b) 0))(( byax

(c) abbyax ))(( (d) none of these



197. The product of lengths of the perpendiculars draw from foci on any tangent to the

hyperbola 12

2

2

2

b

y

a

x is

(a) a2 (b) b2 (c) a2b2 (d) none of these

198. The equation of hyperbola, conjugate to the hyperbola 023232 22 yxyxyx , is

(a) 013232 22 yxyxyx (b) 03232 22 yxyxyx

(c) 043232 22 yxyxyx (d) 043232 22 yxyxyx

199. If chords of the hyperbola 222 ayx touch the parabola axy 42 , then the locus of the

mid-point of these chords is

(a) 322 yayxy (b) 32 )( xaxy

(c) 322 yaxyx (d) none of these

200. If the foci of the ellipse 12

2

22

2

a

y

ak

x and the hyperbola 1

2

2

2

2

a

y

a

x coincide, then the

value of k is equal to

(a) 3 (b) 3

(c) 2 (d) 2

201. The points from where perpendicular tangents can be drawn to the hyperbola

4)2()1( 22 yx , are

(a) only one point (1, 2) (b) two points (0, 0) and (1, 2)

(c) infinite points all lying on the circle 4)1()1( 22 yx

(d) none of these

202. If 321 ,, xxx as well as 321 ,, yyy are in G.P. with the same common ratio, then the points

),( 11 yxA , ),( 22 yxB and ).( 33 yxC

(a) lie on a straight lines (b) lie on an ellipse (c) lie on a circle (d) are vertices of a triangle

203. Let )tan,sec( baP and )tan,sec( baQ where 2

be two points on the

hyperbola 12

2

2

2

b

y

a

x. If (h, k) is the intersection point of the normals of P and Q, then k is

equal to

(a) a

ba 22 (b)

a

ba 22

(c) b

ba 22 (d)

b

ba 22



204. The equation kyxyx 2222 )1()1( will represent a hyperbola for

(a) k (0, 2) (b) k (2, )

(c) k (1, ) (d) k R+

205. The line 112 2 pypx , (| p | < 1) for different values of p, touches

(a) an ellipse of eccentricity 2/3 . (b) an ellipse of eccentricity 3/2 .

(c) a hyperbola of eccentricity 2 (d) an ellipse or a hyperbola depending on p

206. Let )(A and B() be the extremities of a chord of an ellipse. If the slope of AB is equal to

the slope of the tangent at a point C() on the ellipse, then the value of , is

(a) 2

(b)

2

(c)

2 (d) none of these

207. The foci of the hyperbola 01513218169 22 yxyx are

(a) (2, 3), (5, 7) (b) (4, 1), (–6, 1)

(c) (0, 0), (5, 3) (d) none of these

208. If e1 and e2 respectively be the eccentricities of the ellipse 1925

22

yx

and hyperbola

144169 22 yx , then e1e2 is equal to

(a) 25

16 (b) 1

(c) > 1 (d) < 1/2

209. The eccentricity of the hyperbola 12

2

2

2

b

y

a

x is given by

(a) 2

22

a

bae

(b)

2

22

a

bae

(c) 2

22

a

abe

(d)

2

22

b

bae

210. The centre of the hyperbola 02529616369 22 yyxx is

(a) (2, 3) (b) (–2, –3) (c) (–2, 3) (d) (2, –3)

211. If cba ,, are three real numbers not all equal and the vector

kbjaiczkajcibykcjbiax ˆˆˆ,ˆˆˆ,ˆˆˆ are coplanar then xzzyyx ... is

necessarily. (a) positive (b) non-negative (c) non positive (d) negative



212. If G is the centroid of a triangle ,ABC then GCGBGA

(a) GA3 (b) GB3

(c) GC3 (d) none of these

213. A tetrahedron has vertices at )3,1,2(),1,2,1(),0,0,0( CBA and )2,1,1(D then the angle

between the faces ABCand BCD will be

(a) 6

(b)

2

(c)

35

19cos 1

(d)

71

31cos 1

214. The vector moment of three forces kjikji ˆ4ˆ3ˆ2,ˆ3ˆ2ˆ and kji ˆˆˆ acting an a

particle at a point )2,1,0(P about the point )0,2,1( A is

(a) kji ˆ6ˆ5ˆ4 (b) kji ˆ10ˆ4ˆ8

(c) ki ˆ2ˆ7 (d) none of these

215. If G and 'G are centroid of ABC and ''' CBA respectively, then ''' CCBBAA

(a) '3

2GG (b) 'GG

(c) '2GG (d) '3GG

216. If 5|| a and points north east and vector b has magnitude 5 and points north-west, then

|| ba

(a) 25 (b) 5

(c) 37 (d) 25

217. If a vector r of magnitude 63 is directed along the bisector of the angle between the

vectors kjia ˆ4ˆ4ˆ7 and kjib ˆ2ˆ2ˆ2 then r

(a) kji ˆ2ˆ7ˆ (b) kji ˆ2ˆ7ˆ

(c) kji ˆ2ˆ7ˆ (d) kji ˆ2ˆ7ˆ

218. If vector a

lies in the plane of vectors b

and c

, which of the following is correct?

(a) 1 cba

(b) 0 cba

(c) 1 cba

(d) 2 cba

219. If x

and y

are two unit vectors and is the angle between them, then ||2

1yx

is equal to

(a) 0 (b) 2

(c) 2

sin

(d) 2

cos



220. If 0 cba

, 3|| a

, 7||,5|| cb

, then angle between a

and b

is

(a) 6

(b)

3

2

(c) 3

5 (d)

3

221. The vector equally inclined to the vectors kji ˆˆˆ and kji ˆˆˆ in the plane containing

them, is

(a) 3

ˆˆˆ kji (b)

2

i

(c) kj ˆˆ (d) none of these

222. One of the diagonals of a parallelepiped is kj ˆ8ˆ4 . If the two diagonals of one of its faces

are ki ˆ6ˆ6 and kj ˆ2ˆ4 , then its volume is

(a) 60 (b) 80 (c) 100 (d) 120

223. If is the angle between unit vectors a

and b

, then

2sin is

(a) ||2

1ba

(b) ||2

1ba

(c) ||2

1ba

(d) ba1

224. Unit vectors ba

, and c

are coplanar. A unit vector d

is perpendicular to them. If

kjidcba ˆ3

1ˆ3

1ˆ6

1)()(

, and the angle between a

and b

is 30°, then c

is

(a) 3

)ˆ2ˆ2ˆ( kji (b)

3

)ˆ2ˆ2ˆ( kji

(c) 3

)ˆˆ2ˆ2( kji (d)

3

ˆˆ2ˆ kji

225. The vectors b

and c

are in the direction of north-east and north-west respectively and

4|||| cb

. The magnitude and direction of the vector bcd

, are

(a) 24 , towards north (b) 24 , towards west

(c) 4, towards east (d) 4, towards south

226. If cba

23 , cba

324 and ca

710 are the position vectors of three points A, B and

C, then A, B and C are (a) collinear (b) non-collinear (c) vertices of triangle (d) non-coplanar



227. For non-zero vectors, cba

,, , |||||||)(| cbacba

holds if and only if

(a) 0,0 cbba

(b) 0,0 accb

(c) 0,0 baac

(d) 0 accbba

228. Point A is ba

2 , P is a

and P divides AB in the ratio 2 : 3. The position vector of B is

(a) ba

2 (b) ab

2

(c) ba

3 (d) b

229. If the vectors, ,ˆˆˆ kjia kjbi ˆˆˆ and kcji ˆˆˆ )1,( cba are coplanar, then the value

of cba

1

1

1

1

1

1 is

(a) 1 (b) –1 (c) 2 (d) none of these

230. A particle is displaced from the point )7,5,5( A to the point )2,2,6( B under the action

of forces kjiP ˆ11ˆˆ101

, kjiP ˆ6ˆ5ˆ42

, kjiP ˆ9ˆˆ23

, then the work done is


231. a

and b

are two unit vectors and is the angle between them. Then ba

is a unit vector

if

(a) = /3 (b) = /4

(c) = /2 (d) = 2/3

232. A vector a

has components 2p and 1 with respect to a rectangular Cartesian system. This

system is rotated through a certain angle about the origin in the counter-clockwise sense. If

with respect to new system, a

has components p + 1 and 1, then

(a) 0p (b) 1p or 3

1p

(c) 1p or 3

1p (d) 1p or p = –1

233. The vectors kjia ˆ2ˆ2ˆ3

, kib ˆ2ˆ

are adjacent sides of parallelogram. Then angle

between its diagonals is (a) 4/ (b) 3/

(c) (d) 3/2

234. If jia ˆ6ˆ4

and kjb ˆ4ˆ3

, then the vector form of component of a

along b

is

(a) )ˆ4ˆ3(310

18ki (b) )ˆ4ˆ3(

25

18ki

(c) )ˆ4ˆ3(3

18ki (d) kj ˆ3ˆ3

235. If a

and b

represent the sides AB and BC of a regular hexagon ABCDEF, then FA

(a) ab

(b) ba

(c) ba

(d) none of these



236. If 2|| a

and 3|| b

and 0ba

, then )}]({[ baaaa

is equal to

(a) a

16 (b) b

16

(c) a

16 (d) a

16

237. If kjia ˆˆˆ

, 1ba

and kjba ˆˆ

, then b

is

(a) kji ˆˆˆ (b) kj ˆˆ2

(c) i (d) 2 i

238. Let cba

,, be three non-coplanar vectors and r

be any vector in space such that 1ar

,

2br

and 3cr

. If 1][ cba

, then r

is equal to

(a) cbabacacb

)(3)(2)( (b) )(3)(2)( baaccb

(c) cba


239. If cba

,, are non-coplanar vectors and cbad

, then is equal to

(a) ][

][

cab

cbd

(b) ][

][

acb

dcb

(c) ][

][

cba

cdb

(d) ][

][

cba

dbc

240. The axes of co-ordinates are rotated about z-axis through an angle of /4 is anticlockwise

direction and the component of a vector are 4,23,22 . Then the components of the

same vector in the original system are

(a) 5, –1, 4 (b) 5, –1, 24

(c) –1, –5, 24 (d) –1, 5, 4

241. The plane which passes through the point (3, 2, 0) and the line 4

4

5

6

1

3

zyx is

(a) 1 zyx (b) 5 zyx

(c) 12 zyx (d) 52 zyx

242. The position vector of the centre of the circle 33).(,5|| kjirr

is

(a) )(3 kji

(b) kji

(c) )(3 kji

(d) none of these

243. The lines )( cbar

and )( acbr

will intersect if

(a) cbca

(b) cbca

..

(c) acab

(d) none of these

244. The straight lines whose direction cosines satisfy 0,0 hlmgnlfmnncbmal are

perpendicular if

(a) 0 chbgaf (b) 0222

h

c

g

b

f

a

(c) 0c

h

b

g

a

f (d) 0222 hcgbfa



245. A, B, C and D are four points in space such that .DACDBCAB Then ABCD is a

(a) rectangle (b) rhombus (c) skew quadrilateral (d) nothing can be said 246. Equation of a plane parallel to x-axis is (a) 0 dczbyax (b) 0 dbyax

(c) 0 dczby (d) 0 dczax

247. If (2, 3, –1) is the foot of the from (4, 2, 1) to a plane, then the equation of the plane is (a) 0322 zyx (b) 0922 zyx

(c) 0522 zyx (d) 0122 zyx

248. The plane ,1432

zyx cuts the axes in A, B, C. Then the area of the ABC is

(a) 29 (b) 41


249. The lines 4

1

3

1

2

1

zyx and

121

3 zkyx

intersect if k equals

(a) 3/2 (b) 9/2 (c) –2/9 (d) –3/2

250. The equation of plane containing the line 3

1

x =

1

2

2

3

zy and the point (0, 7, –7) is

(a) 1 zyx (b) x + y + z = 2

(c) x + y + z = 0 (d) none of these

251. The image of the point (1, 3, 4) in the plane 032 zyx is

(a) (3, 5, –2) (b) (–3, 5, 2) (c) (3, –5, 2) (d) (3, 5, 2)

252. Equation of a line passing through (–1, 2, –3) and perpendicular to the plane 0532 zyx is

(a) 1

3

1

2

1

1

zyx (b)

1

3

1

2

1

1

zyx

(c) 1

3

3

2

2

1

zyx (d) none of these

253. Equation of normal to the sphere 0544222 222 zyxzyx at the point (1, 1, 1) is

(a) 3

1

2

1

4

1

zyx (b)

0

1

2

1

3

14

zyx

(c) 2

1

0

1

3

1

zyx (d) none of these

254. If a sphere of constant radius k passes through the origin and meets the axes in A,B, C

then the centroid of ABC lies on

(a) 2222 kzyx (b) 2222 4kzyx

(c) 2222 4)(9 kzyx (d) 2222 )(9 kzyx



255. The plane 1c

z

b

y

a

x meets the coordinate axes at A, B, C respectively. The equation of

the sphere OABC is

(a) 0222 czbyaxzyx

(b) 0222222 czbyaxzyx

(c) 0222 czbyaxzyx

(d) 0222222 czbyaxzyx

256. The ratio in which the plane x + y + z = 1 divides the line joining the points P(–3, –2, –1) and Q (2, 3, 4) is

(a) 7 : 8 (b) 1 : 8 (c) 7 : 1 (d) 2 : 3

257. The distance of origin from the point of intersection of the line 4

3

3

2

2

zyx and the

plane 22 zyx is

(a) 120 (b) 83

(c) 192 (d) 78

258. The equation of plane bisecting the acute angle between the planes 01 zyx and

2 zyx is

(a) 2

3 zx (b) 2y = 1

(c) 3 zyx (d) none of these

259. Equation of sphere through the circle 16222 zyx , 01543 zyx and the point

(2, 3, 4) is

(a) 17543222 zyxzyx (b) 049543333 222 zyxzyx

(c) 15543222 zyxzyx (d) none of these

260. The radius of the circle in which the sphere 5|| r

is cut by the plane 33)ˆˆˆ( kjir

is

(a) 3 (b) 2

(c) 33 (d) 4

261. If a line is equally inclined with the coordinate axes, then the angle of inclination is

(a)

2

1cos 1 (b)

2

1cos 1

(c)

3

1cos 1 (d)

2

3cos 1

262. The equation of plane passing through the points (1, 0, 0), (0, 2, 0) and (0, 0, 3) is given by (a) 132 zyx (b) 223 zyx

(c) 6236 zyx (d) 8236 zyx



263. The condition for the plane 0 dczbyax is perpendicular to xy-plane is

(a) a = 0 (b) b = 0 (c) c = 0 (d) 0 cba

264. The two lines x = ay + b, z = cy + d and byax , dycz will be perpendicular, if

and only if (a) 01 ccbbaa (b) 0 ccbbaa

(c) 01 ccaa (d) one of these

265. The lines k

zyx

4

1

3

1

2 and

1

5

2

41

zy

k

x are coplanar if

(a) k = 0 or –1 (b) k = 1 or –1 (c) k = 0 or –3 (d) k = 3 or –3

266. A tetrahedron has vertices at O(0, 0, 0), A(1, 2, 1), B (2, 1, 3) and C (–1, 1, 2). Then the angle between the faces OAB and ABC will be

(a)

35

19cos 1 (b)

31

17cos 1

(c) 30° (d) 90° 267. If the foot of perpendicular from the origin to the plane is (a, b, c) then the equation of the

plane is

(a) 1c

z

b

y

a

x (b) 1 czbyax

(c) 0 czbyax (d) 222 cbaczbyax

268. An equation of plane passing through the line of intersection of the planes 6 zyx

and 05432 zyx and passing through (1, 1, 1) is

(a) 9432 zyx (b) 3 zyx

(c) 632 zyx (d) 69262320 zyx

269. Equation of plane though (3, 4, –1) which is parallel to the plane 07)ˆ5ˆ3ˆ2( kjir

is

(a) 011)ˆ5ˆ3ˆ2( kjir

(b) 011)ˆˆ4ˆ3( kjir

(c) 07)ˆˆ4ˆ3( kjir

(d) 07)ˆ5ˆ3ˆ2( kjir

270. If the sum of the reciprocals of the intercepts made by the plane 1 czbyax on the

three axes is 1, then the plane always passes through the point (a) (2, –1, 0) (b) (1, 1, 1)

(c) (–1, –1, –1) (d)

2

1,1,

2

1

271. Karl-Pearson’s co-efficient of skewness of a distribution is 0.32. Its S.D. is 6.5 and mean 39.6. Then the median of the distribution is given by

(a) 28.61 (b) 28.81 (c) 29.13 (d) 28.31



272. The relation between the median M, the second quartile 2Q , the fifth decile 5D and the fifth

percentile 50P , of a set of observations is

(a) 5052 PDQM (b) 5052 PDQM

(c) 5052 PDQM (d) none of these

273. The mean deviation from the median is (a) greater than that measured from any other value (b) less than that measured from any other value (c) equal to that measured from any other value (d) maximum if all observations are positive

274. Suppose values taken by a variable X are such that bxa i where ix denotes the value

of X in the ith case for ni ,.....2,1 Then

(a) bXVara )( (b) 22 )( bXVara

(c) )(4

2

XVara

(d) )()( 2 XVarab

275. Mean of 100 observation is 45. It was later found that two observations 19 and 31 were

incorrectly recorded as 91 and 13. The correct mean is (a) 44.0 (b) 44.46 (c) 45.00 (d) 45.54

276. If the mean of the set of numbers nxxxx ...,,,, 321 is x , then the mean of the numbers

niixi 1,2 , is

(a) nx 2 (b) 1nx

(c) 2x (d) nx

277. If a variable takes the discrete values + 4, 2

7 ,

2

5 , – 3, – 2,

2

1 ,

2

1 , + 5

( > 0), then the median is

(a) 4

5 (b)

2

1

(c) – 2 (d) 4

5

278. If in a moderately asymmetrical distribution, mode and mean of the data are 6 and 9 respectively, then median is

(a) 8 (b) 7

(c) 6 (d) 5

279. The weighted mean of first n natural numbers, whose weights are equal to the squares of corresponding numbers, is

(a) 2

1n (b)

)12(2

)1(3

n

nn

(c) 6

)12)(1( nn (d)

2

)1( nn



280. Which of the following is not a measure of central tendency? (a) Mean (b) Median (c) Mode (d) Range

281. The Quartile Deviation of the daily wages (in Rs) of 7 persons given below: 12, 7, 15, 10, 17, 19, 25 is (a) 14.5 (b) 5 (c) 9 (d) 4.5

282. If the coefficient of correlation between x and y is 0.28, covariance between x and y is 7.6, and the variance of x is 9, then the standard deviation of the y series is

(a) 9.8 (b) 10.1 (c) 9.05 (d) 10.05

283. A group of 10 items has mean 6. If the mean of 4 of these items is 7.5, then the mean of the remaining items is

(a) 6.5 (b) 5.5 (c) 4.5 (d) 5.0

284. The variance of first n natural numbers is

(a) 12

12 n (b)

12

12 n

(c) 6

12 n (d)

2

12 n

285. The mean weight of a group of 10 items is 28 and that of another group of n items is 35. The mean of combined group of 10 + n items is found to be 30. The value of n is

(a) 2 (b) 4 (c) 10 (d) 12

286. The following data gives the distribution of height of students:

Height (in cm) 160 150 152 161 156 154 155

Number of Students 12 8 4 4 3 3 7

The median of the distribution is

(a) 154 (b) 155 (c) 160 (d) 161

287. S.D. of a data is 6. When each observation is increased by 1, then the S.D. of new data is (a) 5 (b) 7 (c) 6 (d) 8

288. The mode of the following items is 0, 1, 6, 7, 2, 3, 7, 6, 6, 2, 6, 0, 5, 6, 0 (a) 0 (b) 5 (c) 6 (d) 2

289. The coefficient of correlation between X and Y is 0.6. U and V are two variables defined as

3

2,

2

3

YV

XU , then the coefficient of correlation between U and V is

(a) 0.6 (b) 0.3 (c) 0.2 (d) 1



290. The standard deviation of 25 numbers is 40. If each of the numbers is increased by 5, then the new standard deviation will be

(a) 40 (b) 45

(c) 40 + 25


291. For a moderately skewed distribution, quartile deviation and the standard deviation are related by

(a) ..3

2.. DQDS (b) ..

2

3.. DQDS

(c) ..4

3.. DQDS (d) ..

3

4.. DQDS

292. The mode of the following data:

Marks 1–10 11–20 21–30 31–40 41–50

Number of students 8 15 28 16 2

is

(a) 25.7 (b) 25.9 (c) 25.2 (d) 25.0

293. What is the standard deviation of the following series?

Measurements 0 – 10 10 – 20 20 – 30 30 – 40

Frequency 1 3 4 4

is

(a) 81 (b) 7.6 (c) 9 (d) 2.26

294. Coefficient of correlation between the two variates X and Y is

X 1 2 3 4 5

Y 5 4 3 2 1

is

(a) 0 (b) –1 (c) 1 (d) none of these

295. Consider any set of observations 101321 ....,,,, xxxx ; it being given that

;... 101100321 xxxxx then the mean deviation of this set of observations about a

point k is minimum when k equals

(a) x1 (b) x51

(c) 101

... 10121 xxx (d) x50



296. The median of set of 9 distinct observations is 20.5. If each of the largest 4 observation of the set is increased by 2, then the median of the new set

(a) remains the same as that of the original set. (b) is increased by 2. (c) is decreased by 2. (d) is 2 times the original median.

297. The standard deviation of the observations 22, 26, 28, 20, 24, 30 is (a) 2 (b) 2.4 (c) 3 (d) 3.42

298. Given n = 10, x = 4, y = 3, x2 = 8, y2 = 9 and xy = 3, then coefficient of correlation is

(a) 4

1 (b)

12

7

(c) 4

15 (d)

3

14

299. Covariance (x, y) between x and y if x = 15, y = 40, xy = 110, n = 5 is (a) 22 (b) –2 (c) 2 (d) none of these

300. If the mean of numbers xxxxx 156,107,89,31,27 is 82, then the mean of

x130 , xxxx 1,50,68,126 is

(a) 79 (b) 157 (c) 82 (d) 75



ASSERTION REASONGING BASED QUESTIONS

Directions: Read the following questions and choose

(A) If both the statements are true and statement-2 is the correct explanation of

statement-1.

(B) If both the statements are true but statement-2 is not the correct explanation of

statement-1.

(C) If statement-1 is true and statement-2 is False.

(D) If statement-1 is False and statement-2 is true.

301. Statement 1: If ,21 x then 21212 xxxx .

Statement 2: The middle point of the interval in which 03)(2 22 xx is 1.

(a) A (b) B (c) C (d) D

302. Statement 1: The polynomial 86421)( xxxxxxf when divided by )1( x leaves

a remainder 6.

Statement 2: ,)()()( RafRQaxxf then )( ax is a factor of ).(xf Where Q is

Quotient and R is remainder.

(a) A (b) B (c) C (d) D

303. Statement 1: 10110110131012101 )5(......)5()5()5()( xxxxxf , then 0)( xf

has only one real root.

Statement 2: )(xf is an increasing function.

(a) A (b) B (c) C (d) D

304. Statement 1: In an A.P. of odd number of terms nTTTS ............211 and

nTTTTS ..........5312 , then 12

1

n

n

S

S.

Statement 2 : If 1, 2, 3, ….n be n numbers where n is odd, then 1, 3, 5….. n will be 2

1n

odd numbers and 2, 4, 6……, 1n will be 2

1n even numbers.

(a) A (b) B (c) C (d) D

305. Statement 1: The roots of 032 23 dxxx are in A.P., then 27

38d .

Statement 2 : If is a root of ,023 dcxbxax then 023 dcba .

(a) A (b) B (c) C (d) D



306. Statement 1: If nHHH ,....,, 21 be n harmonic means between a and b, then

nbH

bH

aH

aH

n

n 21

1

.

Statement 2 : ,)1(

,)1(

1bna

abnH

anb

abnH n

by interchanging a and b.

(a) A (b) B (c) C (d) D

307. Statement 1: The equation to the pair of straight lines through the origin and perpendicular

to 0742 22 yxyx is 0247 22 yxyx .

Statement 2 : To find pair of perpendicular lines to 02 22 byhxyax ,write as

02 22 ayhxybx

(a) A (b) B (c) C (d) D

308. Statement 1: If the lines 03,02 bbyxaayx and 04 ccyx are concurrent,

then ba, and c are in A.P. (Where )0abc .

Statement 2 : Concurrent lives always passes through a common point.

(a) A (b) B (c) C (d) D

309. Statement 1: A straight line through the origin O meets the parallel lines 924 yx and

062 yx at the points QP & respectively, then the point O divides the

segment PQ in the ratio 3 : 4.

Statement 2 : To find point which internally divides the line joining ),( 11 yxA and ),( 22 yxB in

the ratio nm : apply

nm

nymy

nm

nxmx 1212 , .

(a) A (b) B (c) C (d) D

310. Statement 1: The circles x2 + y2 – 4x – 6x – 12 = 0 and x2 + y2 + 4x + 6y + 4 = 0 cut orthogonally.

Statement 2 : Since these circles has a common chord.

(a) A (b) B (c) C (d) D

311. A circle C3 touches externally two circles C1 & C2 of equal radii. Then

Statement 1: centre of circle C3 lies on radical axis of circles C

2 and C

1.

Statement 2 : radical axis of circles C1 and C

2 is the perpendicular bisector of the line joining

centres of circles C1 and C

2.

(a) A (b) B (c) C (d) D

312. Statement 1: The vertex A of a ABC, incentre I of triangle and centre I1 of the excircle

opposite vertex A are collinear.

Statement 2 : Excentre I1 is the point of concurrency of two external angular bisectors and

one internal angular bisector of angle A.

(a) A (b) B (c) C (d) D

313. Statement 1: The angle subtended by the latus rectum of the parabola y2 = 4ax at the

vertex is – tan–1(4/3).



Statement 2 : The angle made by the double ordinate of length 8a of parabola y2 = 4ax at

the vertex is /2.

(a) A (b) B (c) C (d) D

314. Statement 1: A tangent is drawn from the point T which lies on x-axis and which touches

the parabola y2 = 16x at P(16, 16). If S be the focus of the parabola then

TPS = tan–12.

Statement 2 : The tangent at any point on a parabola trisect the angle between the focal distance of the point and the perpendicular on the directrix from the point.

(a) A (b) B (c) C (d) D

315. Statement 1: A ray of light is coming along the line y = b from the positive directrix of x-axis

and strikes a concave mirror whose intersection with the xy-plane is the parabola y2 = 4ax. The slope of reflected ray is

Statement 2 : All ray of light coming from positive direction of x-axis and parallel to axis of parabola after reflection pass through the focus of the parabola

(a) A (b) B (c) C (d) D

316. Statement 1: An ellipse has major and minor axis along x and y-axis respectively. If the

product of semi major and semi minor axis is 20 then the maximum value of the product of abscissa and ordinate of any point on the ellipse is 10.

Statement 2 : Arithmetic mean of two positive numbers is always greater than their geometric mean.

(a) A (b) B (c) C (d) D

317. Statement 1: For all values of , the two tangents drawn from the point

)sin13,cos13( to the ellipse, 3649 22 yx are mutually

perpendicular.

Statement 2 : Tangents drawn from any point on auxiliary circle to an ellipse are mutually perpendicular.

(a) A (b) B (c) C (d) D

318. Statement 1: 1C is a circle contained in the circle 2C . If a circle C moves such that C

touches C1 externally and 2C internally then locus of centre of C is

hyperbola.

Statement 2 : Locus of a point moving such that sum of its distances from two fixed points is always equal to a given constant is an ellipse.

(a) A (b) B (c) C (d) D

319. Statement 1: From a point P two tangents are drawn to the hyperbola 141

22

yx

then the

least value of the angle between these tangents which contains the

hyperbola is 3

4tan 1 .



Statement 2 : The least angle between the two tangents from a point to hyperbola such that they contain hyperbola is the angle between two asymptotes of the hyperbola.

(a) A (b) B (c) C (d) D

320. Statement 1: There is no point in the plane of hyperbola 1169

22

yx

from where two

mutually perpendicular tangents can be drawn to hyperbola.

Statement 2 : Locus of point of intersection of two perpendicular tangents to the hyperbola is the directrix of hyperbola.

(a) A (b) B (c) C (d) D

321. Statement 1: The equation 1)( xyx represents a hyperbola having xy and y-axis as

asymptotes.

Statement 2 : The equation of form 2222111 ))(( cybxacybxa represents a

hyperbola with asymptotes 0111 cybxa and .0222 cybxa

(a) A (b) B (c) C (d) D

322. Statement 1: If kbia ˆ2,3 and kic ˆ2ˆ3 , then a and b are linearly independent but

cba ,, are linearly dependent.

Statement 2 : If a and b are linearly dependent and c is any vector, then cba ,, are

linearly dependent.

(a) A (b) B (c) C (d) D

323. Statement 1: 65

ˆ7ˆ4 ji is a unit vector bisecting angle between ji ˆ4ˆ3 and ji ˆ12ˆ5 .

Statement 2 : Let a and b be two non-collinear vectors then vector || ba

ba

is unit vector

along the bisector of angle between .&ba

(a) A (b) B (c) C (d) D

324. Statement 1: If b and c are two non-collinear vectors such that 4).( cba and

cybxxcba )(sin)62()( 2 where x and y are real, then point

),( yx lies on .1x

Statement 2 : The vector a lies in the plane of .&cb

(a) A (b) B (c) C (d) D

325. Statement 1: A plane is drawn having intercepts CBA sin,sin,sin on the co-ordinate axes

where CBA ,, are angle of ,ABC then maximum volume of tetrahedron

formed by plane and co-ordinate axes is 16

3(unit)3.

Statement 2 : The maximum value of CBA sinsinsin is 2

33.



(a) A (b) B (c) C (d) D

326. Statement 1: If point ),,( lies above the plane 0)1()1()1( 22 dzccybxa ,

then 0)1()1()1( 22 dccba

Statement 2 : If the point ),,( lies above the plane 0 dczbyax , then

0

c

dcba

(a) A (b) B (c) C (d) D

327. Statement 1: Two perpendicular non-intersecting lines are not coplanar.

Statement 2 : Two skew lines are not coplanar.

(a) A (b) B (c) C (d) D

328. Statement 1: For the frequency distribution of the given data

Value )( ix : 1 2 3 4

Frequency )( if : 5 4 6 f

If the mean is known to be 3, then the value of f is 16.

Statement 2 : To calculate mean use formula mean i

ii

f

fx

.

(a) A (b) B (c) C (d) D

329. Statement 1: If is the mean of distribution ],,{ ii fy then )( ii yf is equal to mean

deviation.

Statement 2 : Mean deviation can be written as i

ii

f

Mxf

||, where M represents mean of

distribution.

(a) A (b) B (c) C (d) D

330. Statement 1: The mode of the distribution

Marks ix : 4 5 6 7 8

No. of students if : 6 7 10 8 3

is 6.

Statement 2 : The middle term of the data distribution is mode.

(a) A (b) B (c) C (d) D



PASSAGE BASED PROBLEMS Directions:

This section contains paragraphs. Based upon these paragraphs, multiple choice

questions have to be answered. Each question has 4 choices (a), (b), (c) and (d), out of

which ONLY ONE choice is correct.

Passage-I

If two concentric ellipse be such that the foci of the one be on the other and their major axis

are equal (where foci of first are S and ,'S and foci of other are H and 'H ) then,

331. The value of 'HSHS is

(a) a (b) a2

(c) a4 (d) a6

332. If the angle between the axis be , then the maximum value of 'HS is

(a) )( 21 eea (b) )( 21 eea

(c) || 12 eea (d) none of these

333. If the angle between the axis be , then the minimum value of 'HS is

(a) )( 21 eea (b) |)(| 12 eea

(c) || 12 eea (d) none of these

334. If the angle between the axes be , then the maximum value of HS is

(a) )( 22

21

2 eea (b) 22

21 eea

(c) )( 21 eea (d) none of these

335. If the angle between the axes be , then the value of cos is

(a) 22

21

22

21

111

eeee (b)

22

21

22

21

211

eeee

(c) 22

21

22

21

211

eeee (d)

22

21

22

21

111

eeee

Passage–II

If the vectors cba ,, and x are being defined to satisfy the following conditions

(i) bxbx 325

(ii) cb 32 is perpendicular to x

(iii) the ratio of c b to is 4:33

(iv) a is a vector perpendicular to the plane containing cb & and 102 ca

336. The angle between x and b is

(a) 0° (b) 30° (c) 60° (d) 75°



337. The angle between b and c is

(a) 0° (b) 30° (c) 60° (d) 15°

338. The volume of the parallelopiped formed by c a x and, is

(a) 4

33 (b)

4

315 (c)

4

327 (d) 32

Passage–III

For the location of roots of quadratic equation 02 cbxax , Rcbaa ,,,0 we use

the graph of ,)( 2 cbxaxxf which is a parabola opening upwards if 0a and opening

downwards if .0a )(xf is always positive if it is positive for a value and its roots are non

real and )(xf is always negative if it is negative for a value and its roots are non real. Also

we use the results obtained from graph of )(xf for position of roots.

Now answer the following questions:

339. If the equation )0(,0352 2 abxax does not have any real root, then the value of

)52( ba is always

(a) less than 3 (b) greater than 3 (c) equal to 3 (d) none of these

340. If the equation )0(,0523 2 abxax does not have any distinct real root then

minimum value of ba 412 is

(a) 5 (b) – 5 (c) 10 (d) – 10

341. If ‘2’ lies between the roots of the equation ,0232 2 xpxp then ‘p’ lies in,

(a) (2, 5) (b) (3, 6) (c) (2, 6) (d) (4, 6)

342. If exactly one root of the equation 0622 xmmx lies in 3,1 , then ‘m’ lies in

(a) (1, 4) (b) (2, 4) (c) (1, 3) (d) (2, 3)

Passage–IV

In case of parabola we can reduce the calculations for the tedious question by using its properties, like for any parabola the foot of perpendicular from focus upon any tangent always lie at the tangent at vertex.

Now, considering the parabola 2412

xy , answer the following questions:

343. Equation of the tangent to above parabola at the point )5,6( is

(a) 062 yx (b) 042 yx

(c) 022 yx (d) 042 yx

344. The reflection of focus of above parabola w.r.t. above tangent is (a) (2, 5) (b) (3, 5) (c) (1, 5) (d) (2, 3) 345. Locus of point of intersection of any two perpendicular tangents of above parabola is (a) 01x (b) 0x (c) 02x (d) none of these

346. If a focal chord of the above parabola makes an angle of 45° with positive direction of

x-axis, then its extremity which is farthest from origin is



(a) 223,225 (b) 223,225

(c) 224,225 (d) 224,225

Passage–V

The planes ‘’ and '' k are defined as follows,

222 zyx and ,28 czbyaxk where '' k is the family of parallel planes

such that ,221

kcba

and 0,914

k

kk

347. If ‘d’ is the length of the intercept on the line 3

1

2

2

1

1

zyx between the planes ‘’

and variable plane '' k . The minimum value of ‘d’ is

(a) 3

14 (b)

3

142 (c)

3

147 (d)

3

148

348. If ‘r’ is the radius of sphere which touches both ‘’ and '' k , then the greatest value of r will

be (a) 8 (b) 6 (c) 4 (d) 2

349. The ratio, in which the plane parallel to ‘’ and containing the centre of the smallest sphere

which touches both ‘’ and ,'' k divides the line joining the points )7,3,2( and

)4,2,1( is

(a) 8 : 9 (b) 9 : 11 (c) 13 : 14 (d) 15 : 16 Answer keys are included inside the MathonGo App

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doubtion.com · 1000 MOST IMPORTANT QUESTIONS FOR JEE – PART 1 RANK BUILDER SERIES For More Parts and Answer Keys Download the MathonGo App 1. If and are the roots of ax2 bx c 0,