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GRAND SUCCESS PACKAGE : MATHEMATICS

LEVEL – I

1. The domain of f xx x

( ){ [ ]}

1, where [.] is G.I.F

is(1) R – I (2) R(3) I (4) None

2. The domain of the function

f x x x( ) cos | | (log( ))F

HGIKJ 1 12

43 , is

(1) [– 2, 6] (2) [– 6, 2) (2, 3)(3) [– 6, 2] (4) [– 2, 2] [2, 3]

3. The domain of f x xx

( ) | || |

12

, is

(1) (- , ) – [– 2, 2](2) (– , ) – [– 1, 1]

(3) [– 1, 1] (- , - 2) (2, ) ( 4 )None

4. The domain of the function f xx

x x( )[ ]

( ) 1 2 ,

is where [.] is greatest integer(1) [0, 2] (2) [0, 1)(3) (1, 2] (4) [1, 2]

5. Let f : R R, g : R R be two functions givenby f (x) = 2x – 3, g (x) = x3 + 5. Then ( fog) – 1 (x)is equal to

(1) x F

HGIKJ

7

2

13

(2) x FHG

IKJ

7

2

13

(3) x F

HGIKJ

2

7

13

(4) x F

HGIKJ

7

2

13

6. The function f : R R defined byf (x) = (x – 1) (x – 2) (x + 3) is

(1) one-one but not onto

ASSIGNMENTOBJECTIVE TYPE QUESTION

1. Sets, Relations & Functions

(2) onto but not one-one

(3) both one-one and ont(4) neither one-one nor onto

7. The inverse of the function f xe e

e e

x x

x x( )

1 is

(1) log10 2

x

xFHG

IKJ (2) n

x

x2 FHG

IKJ

(3) nx

x2

1 2

FHG

IKJ

/

(4) 12 2

logx

x

FHG

IKJ

8. f x xx

xx

( ) |sin |cos

sin|cos |

FHG

IKJ

12

, is periodic with period

(1) (2) 2(3) / 2 (4) none

9. If f (x) is an even function, then the curve y = f (x)is symmetric about(1) x-axis (2) y - axis(3) both the axes (4) None

10. Which of the following is an even function

(1) x aa

x

x

11

(2) tan x

(3) a ax x

2(4)

aa

x

x

11

11. Which of the following function is not odd

(1) log x x FH IK1 2 (2) log11

FHG

IKJ

xx

(3) x aa

x

x

11

(4) None

12. The range of the function f xx x

( ){ [ ]}

1 , is

(where [·] is G.I.F)(1) (1, ) (2) [1, ](3) [1, [ (4) None

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13. The range of the function f xx

( )cos

14 3

, is

(1) [ , ]1 7 1 (2) ] , ]1 7 1

(3) ( , ]1 7 1 (4) None

14. The range of the function f x Pxx( )

73 , is

(1) {1, 2, 3} (2) {1, 2, 3, 4, 5, 6}(3) {1, 2, 3, 4} (4) {1, 2, 3, 4, 5}

15. Range of the function f (x) = sin2 (x4) + cos2 (x4) is(1) (– , 1) (2) {1}(3) (– 1, 1) (4) (0, 1)

16. The domain of f xx

xx( ) sin

F

HGIKJ

12

21

21 is

(1) {1} (2) (– 1, 1)(3) {1, – 1} (4) None

17. The domain of definition of the function y x givenby the equation 2 2 2x y is(1) 0 1x (2) 0 1x ( 3 )

0x (4) 1x 18. Let R = {(3,3), (6,6), (9,9), (12,12), (6,12), (3,9),

(3,12), (3,6)} be a relation on the A = {3, 6, 9, 12}.The relation is(1) reflexive and symmetric only (2) anequivalence relation(3) reflexive only(4) reflexive and transitive only

19. If the function f(x) and g(x) are defined from R R such that

0, x rational 0, x irrationalf(x) , g(x)

x, x irrational x, x rational

,

then (f – g)(x) is(1) one one and onto(2) neither one-one nor onto(3) one-one but not onto(4) onto but not one-one

20. Let f : R R be a function defined by

f x e ee e

x x

x x( )| |

then

(1) f is a bijection(2) f is an injection only(3) f is a surjection(4) f is neither injection nor a surjection

LEVEL – II

1. If f(x) cos (log x), then f(x) f(y) – 12

[f(x/y) +

f(xy)]=(1) [ 1, ) (2) ( , 1)

(3) ( , 1] (4) (1, )

2. The domain of the function f x x x( ) 1 6 ,is(1) (1, 6) (2) [1, 6](3) [1, – ) (4) (– , 6]

3. The domain of the function f xx x

( )| |

1 , is

(1) R + (2) R –

(3) R0 (4) R

4. The domain of 1(| | )x x

, is

(1) ] - , 0 [ (2) [– , 0](3) ] – , 0] (4) [- , 0 [

5. The domain of the function cos

[ ]

1 xx , is where [.] is

G.I.(1) [–1, 0) {1} (2) (– , 0](3) [0, ] (4) None

6. The domain of the function f x xx

( )sin ( )

42

2

1, is

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

7. The domain of sin FHG

IKJ

1 2 13

x is

(1) (– 2, 1) (2) [– 2, 1](3) R (4) [– 1, 1]

8. The interval for which sin cos 1 1

2x x

holds(1) [0, ] (2) [0, 3](3) [0, 1] (4) [0, 2]





9. Which of the following function is periodic -

(1) f (x) = x + sin x (2) f x x( ) cos

(3) f (x) = cos x2 (4) f (x) = cos2 x10. The range of the function f (x) = 3 cos 4x, is

(1) R (2) { : }x x 1 1

(3) { : }x x 4 4 (4) { : }x x 3 3

Assertion – Reason TypeDirection : The questions given below consist of an‘Statement 1’ and the ‘Statement 2’. Use the followingkey to choose the appropriate answer.

(1) Statement-1 is True, Statement-2 is True;Statement-2 is a correct explanation forStatement-1.

(2) Statement-1 is True, Statement-2 is True;Statement-2 is NOT a correct explanation forStatement-1.

(3) Statement-1 is True, Statement-2 is False.(4) Statement-1 is False, Statement-2 is True.

1. STATEMENT-1 : The period of the function f(x) =

sin [x] cos x cot [x]4 2 3

is 24

andSTATEMENT-2 : The period of sin x, cos x is 2and period (f(x) + g(x) = 1.c.m (period of f(x), periodof g(x))

2. STATEMENT-1 : The range of x x

x xe ef (x)e e

defined on [0, ) is [0, )andSTATEMENT-2 : Range of ex defind R is [0, ) .

3. STATEMENT-1 : The set of all values of a for

wich the function f(x) = cot(sinx) +2x

a

whose

domin is [–4, 4] is an odd function is R [ 16,16]andSTATEMENT-2 : The set of all zeros of the greatestinteger function is (–1, 1)

4. STATEMENT-1 : The period of the function f(x) =2 2ocs[2 ]x cos[ 2 ]x {x} is , [x] being

greatest integer function and {x} is fractional partof x is .andSTATEMENT-2 : Cosine function is periodic withperiod 2 .

Linked Comprehension TypePassage - IA right cone is inscribed in a sphere of radius R. Let S =f(x) be the functional relationship between the lateralsurface area S of the cone and its generatrix x.1. The domain by f(x) is

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

2. The value of f(R) is given by

(1) 23 R (2) 23 R2

(3) 2R (4) 2 R3. S2 is a polynomal of degree

(1) 4 (2) 5(3) 3 (4) 6

Passage - IILet f(x) be a function defined on–2, 2] such that f(x) = –1, –2 x 0 ; f(x) = x – 1, 0 < x 2 and g(x) = |x|. Formthe composite function h(x) = f o g(x) + g o f(x).4. The range of h(x) is

(1) [0, 1] (2) [–2, 2](3) [0, 2] (4) [1, 2]

5. The function h(x)(1) decreases on [–2, 2](2) decreases strictly on [–2, 1](3) increases on [–2, 2](4) increases on [1, 2]

6. The function h(x) is(1) one-one(2) one-one on [–1, 1](3) a linear function on [–2, 1](4) a linear function on [1, 2]





2. Mathematical Induction

LEVEL – I

1. n n 210 3.4 5, n N is divisible by

(1) 5 (2) 7

(3) 9 (4) 11

2. For every natural number n, n (n 1)(2n 1) isdivisible by

(1) 6 (2) 12

(3) 24 (4) 5

3. For every natural number n, 2n 23 8n 9 isdivisible by

(1) 16 (2) 128

(3) 256 (4) None of these

4. n49 16n 1 is divisible by

(1) 3 (2) 19

(3) 64 (4) 29

5. For all positive integral values of n, 4n2 1 isdivisible by

(1) 8 (2) 16


6. For all positive integral values of n, 2n3 2n 1 isdivisible by

(1) 2 (2) 4

(3) 8 (4) 12

7. If n N , then 2n 1 2n 1x y is divisible by

(1) x y (2) x y

(3) 2 2x y (4) 2x xy

8. For each n N , 3n2 7n 1 is divisible by

(1) 23 (2) 41

(3) 49 (4) 98

9. For each n N , n nx y is divisible by

(1) x y (2) x y

(3) 2 2x y (4) 2 2x y

10. If n N , then the greatest integer which dividesn(n – 1)(n – 2) is

(1) 2 (2) 3

(3) 6 (4) 8

11. If n N , then 2n 3n 37 2 . n 13 is always divisibleby

(1) 25 (2) 35


12. If n N , then n 2 2n 111 12 is divisible by

(1) 113 (2) 123


13. For every natural number n, 2n (n 1) is divisibleby

(1) 4 (2) 6

(3) 10 (4) None of these14. The difference between an integer and its cube is

divisible by(1) 4 (2) 6(3) 9 (4) None of these

15. For every natural number n

(1) nn 2 (2) nn 2

(3) nn 2 (4) nn 2

16. For each n N , the correct statement is

(1) n2 n (2) 2n 2n

(3) 4 nn 10 (4) 3n2 7n 1

17. For natural number n, n n2 (n 1)! n , if

(1) n < 2 (2) n > 2(3) n 2 (4) Never

18. If n is a natural number then nn 1 n!

2

is true

when(1) n > 1 (2) n 1(3) n > 2 (4) n 2





19. For positive integer n, n 210 81n , if

(1) n > 5 (2) n 5(3) n < 5 (4) n > 6

20. For every positive integer n, n2 n! when

(1) n < 4 (2) n 4(3) n < 3 (4) None of these

LEVEL – II1. For every positive integral value of n, n 33 n when

(1) n > 2 (2) n 3(3) n 4 (4) n < 4

2. For natural number n, 2 n(n!) n , if

(1) n > 3 (2) n > 4(3) n 4 (4) n 3

3. The value of the n natural numbers n such that theinequality n2 2n 1 is valid is

(1) For n 3 (2) For n < 3(3) For mn (4) For any n

4. Let P(n) denote the statement that 2n n is odd.It is seen that P(n) P(n 1) , nP is true for all

(1) n > 1 (2) n(3) n > 2 (4) None of these

5. If {x} denotes the fractional part of x then2n3 ,n N8

, is

(1) 3/8 (2) 7/8(3) 1/8 (4) None of these

6. If p is a prime number, then pn n is divisible by pwhen n is a(1) Natural number greater than 1(2) Irrational number(3) Complex number(4) Odd number

7. n 1 n 1 nx(x na ) a (n 1) is divisible by 2(x a)for(1) n > 1 (2) n > 2(3) All n N (4) None of these

8. Let P(n) be a statement and let P(n) p(n + 1)for all natural numbers n, then P(n) is true(1) For all n(2) For all n > 1(3) For all n > m , m being a fixed positive integer(4) Nothing can be said

9. If P(n) = 2 + 4 + 6 +….+ 2n, n N, then P(k) = k(k+ 1) + 2 P(k + 1) = (k + 1)(k + 2) + 2 for all k N. So we can conclude that P(n) = n(n + 1) + 2 for(1) All n N (2) n > 1

(3) n > 2 (4) Nothing can be said

10. For every natural number n, n(n + 1) is always

(1) Even (2) Odd

(3) Multiple of 3 (4) Multiple of 4





3. Complex Number

LEVEL - I

1. A complex number z is such that z 2argz 2 3

,

then the points representing this complex numberwill lie on(1) an ellipse (2) a parabola(3) a circle (4) a straight line

2. The equation z i z i represent

(1) 2 2 (2) 2 (3) 0 2 (4) none of these

3. If a ib r cos i sin , then a ibtan i loga ib

is equal to

(1) ab (2) 2 2

2aba b

(3)2 2a b2ab

(4) 2 2

2aba b

4. The area of the triangle on the complex plane formedby the complex numbers, z, iz and z+iz is

(1) |z|2 (2)2

z

(3)2z

2(4)

21 z4

5. If z, be the complex numbers, such that z 1

and Arg z Arg2

, then z is equal to

(1) –1 (2) 1(3) –i (4) i

6. Let z = x –i y and 1/ 3z p iq ,then 2 2

x yp q

p q

is equal

to(1) 1 (2) –1(3) 2 (4) –2

7. If 1 1 2 2 n nx iy x iy ... x iy A iB , then

1 1 2 2 n n

2 2 2 2 2 2x y x y ... x y is equal to

(1) A2+B2 (2) A2 – B2

(3) –A2 + B2 (4) –A2 – B2

8. If 1 1 2 2 n nx iy x iy ..... x iy A iB , then

1 1 11 2 n1

2 n

y y ytan tan ... tanx x x

is

(1) 1 BtanA

(2) 1 AtanB

(3) 1 AtanB

(4) 1 BtanA

9. If 21, , are the three cube roots of unity then

2 2 4 4 81 1 1 .... to 2n

factors is(1) 2 n (2) 22n

(3) 24n (4) none of these

10. The value of 81 cos /8 i sin /8

1 cos /8 isin /8

equals

(1) 0 (2) –1(3) 1 (4) 2

11. If and are the roots of the equation2x 2x cos2 1 0 , then the equation whose

roots are n / 2 and n / 2 is

(1) 2x 2nx cos 1 0

(2) 2x 2nx cos 1 0

(3) 2x 2 cosn .x 1 0

(4) 2x 2nx cos 1 0

12. The continued product of the roots of

3/81 3i.2 2

will be(1) –1 (2) 1(3) 0 (4) 2





13. The value of the expression 2 2 21. 2 . 2 2. 3 3 ..... n 1 n n

where ' ' is an imaginary cube root of unity, is

(1) 2

n n 12

(2) 2

n n 1n

2

(3) 2

n n 1n

2

(4) none of these

14. If 1i tan z , z x iy and is constant,then the locus of z is

(1) 2 2x y 2xcot 2 1

(2) 2 2x y cot 2 1 x

(3) 2 2x y 2y tan 2 1

(4) 2 2x y 2xsin 2 1

15. If 2

3

z z 1log 2

2 z

, then the locus of z is

(1) z 5 (2) z 5

(3) z 5 (4) none of these

16. If z1, z2, z3 be the vertices of an equilateral triangle,then

(1) 2 2 21 2 3 1 2 2 3 3 1z z z z z z z z z

(2) 2 2 21 2 3 1 2 2 3 3 1z z z z z z z z z

(3) 2 2 21 2 3 1 2 2 3 3 1z z z z z z z z z

(4) 2 2 21 2 3 1 2 2 3 3 1z z z z z z z z z

17. If is a complex cube root of unity, then the value

of

2

2

2

x 1x 1

1 x

is

(1) x3 (2) 2x3

(3) 3x3 (4) x4

18. If z1, z2 are the two complex numbers such that

1 2

1 2

z z 1z z

and i z1 = kz2, where k R , then the

angle between z1 – z2 and z1 + z2 is

(1) 12

2ktan1 k

(2) 1

2

2ktan1 k

(3) 13tan k (4) 13tan k

19. If z is a complex number such that z 2i 1z 2i

, then

z lies on(1) the real axis (2) the line, Im (z) =3(3) a circle (4) parabola

20. If k >0, | z | | w | k and 2

z wk z w

, then

Re equals

(1) 0 (2) k / 2(3) k (4) k / 4

LEVEL - II1 If n is not multiple of 3, then n 2n is

(1) 0 (2) –1(3) 1 (4) 2

2 If z1, z2 are two non-zero complex numbers such

that 1 2 1 2z z z z , then arg z1 –arg z2 is equal

to

(1) / 2 (2) 0

(3) (4) / 2

3 If r r rx cos isin2 2

, then the value of

x1.x2.x3...... to is

(1) –1 (2) 1(3) 2 (4) –2

4 The value of the sum 13

n n 1

n 1i i

, where i 1 ,

is

(1) i (2) i–1(3) –i (4) 0





5 The value of (1+i)n + (1–i)n is

(1)n 12 n2 cos

4

(2)n 12 n2 sin

4

(3)n2 n2 cos

4

(4)n2 n2 sin

4

6 The complex numbers sin x – i cos 2x and

cos x–i sin 2x are conjugate to each other for

(1) x n (2) x = 0

(3) x 2n 1 / 2 (4) no value of x

7 |z–4| < |z–2| represents the region given by

(1) Re(z) < 0 (2) Re(z) > 0(3) Re (z) > 2 (4) Re(z) > 3

8.z 1 1z 1

represents

(1) a circle (2) an ellipse(3) a straight line (4) parabola

9. If | z i | 2 and z0 = 5 + 3i, then the maximum

value of |iz+z0| is

(1) 2 31

(2) 7

(3) 31 2

(4) none of these

10. If arg z / 2 , then arg z arg z is

(1) (2) (3) / 2 (4) / 2





1. STATEMENT-1 : If 2 2cos isin7 7

,

2 4p , 3 5 6q then theequation where roots are p and q is x2 + x+ 2 = 0andSTATEMENT-2 : If is a root of Z7 = 1, then

2 61 ... 0

2. STATEMENT-1 : 23 ix y and 2x y 4i are

conjugate numbers, then 2 2x y 3 andSTATEMENT-2 : If sum and product of twocomplex numbers is real then they are conjugatecomplex numbers.

3. STATEMENT-1 : If z1 and z2 are two distinct pointsin an Argand plane. If 1 2a | z | b | z | , then

1 2

2 1

az bzbz az

is a point on the line segment [–2, 2] of

the real axis.andSTATEMENT-2 : When arg (z1) = and arg (z2) = +

i i1 2

2 1

az bz e e 2cosbz az

4. STATEMENT-1 : If | z | 2 1 , then2| z 2zcos | is les than 1.

and

STATEMENT-2 : 1 2 1 2| z z | | z | | z | also | cos | 1

Linked Comprehension TypePassage - I

Let 2 2cos isin7 7

and 2 4 and

3 5 6





1. equals(1) 0 (2) –1(3) –2 (4) 1

2. equals(1) –1 (2) 0(3) 1 (4) 2

3. and are roots of the equations(1) x2 + x + 1 = 0 (2) x2 + x + 2 = 0(3) x2 + 3x + 5 = 0 (4) None of these

Passage - IIFor z = x + iy, x, y R , define

z xe e (cos y isin y) ,

iz iz1sin z (e e )2i

and iz iz1cosz (e e )2

4. Number of value of z for which ez = 0 is(1) 0 (2) 1(3) two (4) infinite

5. Number of value of z for which sin z = 0 is(1) 0 (2) 1(3) two (4) infinite

6. Number of values of z for which cos z = 0 is(1) 0 (2) 1(3) two (4) infinite





4. Quadratic Equations

LEVEL - I1. If x2 + kx + 1 = 0 and (q – r) x2 + (r – p) x + (p–q) = 0 have

both the roots common, then p, q, r are in(1) A.P. (2) G.P.(3) H.P. (4) None of these

2. Let , , , R, 0 be a root of the equatonx3 + ax + b = 0, where a,b R . Then the cubicequation with real cofficients one of whose roots is

, is

(1) 3x ax b 0 (2) 3x 2ax b 0 (3) 38x 2ax b 0 (4) 38x 2ax b 0

3. If P(x)=px2+qx+r, Q(x)=–px2+qx+r, where pr 0then the equation P(x). Q(x) = 0 has(1) at least three real roots(2) no real roots(3) at least two real roots(4) two real and two imaginary roots

4. If tanA and tanB are the roots of the quadraticequation (a+c)x2–2bx+(a–c) = 0, then value ofcos2(A+B) is

(1)2

2 2

bb c

(2)2

2 2

cb c

(3)2

2 2

bb c

(4)2

2 2

cb c

5. If 5 {x}=x+[x] and [x]–{x}=1/2, where {x} and [x] arefractional and integral part of x, then the number ofsolutions of the equation is(1) one (2) two(3) three (4) no solution

6. The equation whose roots are the nth power of theroots of the equation 2x 2x cos 1 0 is given by

(1) x2–2x cos n +1=0

(2) x2+2x cos n +1=0

(3) x2–2x sin n +1=0

(4) x2+2x sin n +1 = 0

7. If the roots of the equation x2–2ax+a2+a–3=0 arereal and less than 3, then(1) a < 2 (2) 2 a 3 (2) 3 a 4 (4) a > 4

8. In a triangle PQR, R2

. If Ptan2

and Qtan2

are

the roots of the equation ax2+bx+c=0, a 0 , then

(1) a+b=c (2) b+c=0(3) a+c=b (4) b–c=0

9. If a, b, c are positive and are in AP, then roos of thequadratic equation ax2+bx+c=0 are real for

(1)c 7 4 3a (2)

a 7 4 3c

(3)c 7 4 3a (4) no p and r

10. If , are the roots of the equation px2+qx+r=0

and k , k are the roots of the equation

ax2+bx+c=0, then 2

2

q 4prb 4ac

is equal to

(1)

2ap

(2)2p

a

(3) 1 (4) 011. If , are the roots of x2–ax+1=0 and , are the

roots of the equation x2–bx+1=0, then the value of

is equal to

(1) a2–b2 (2) b2–a2

(3) a2 (4) b2

12. If the quadratic equation ax2+2cx+b=0 and

ax2+2bx+c=0 b c have a common root, thena+4b+4c is equal to(1) –2 (2) –1(3) 0 (4) 1

13. If the equation x2 + 2(k+1) x + 9k – 5 = 0 has onlynegative roots, then

(1) k 0 (2) k 0(3) k 6 (4) k 6

14. The integer k for which the inequalityx2–2(4k–1)x+15k2 –2k–7>0 is valid for any x, is(1) 2 (2) 3(3) 4 (4) 5





15. The number of real solutions of the equationx x xsin (e ) 3 3 is

(1) 0 (2) 1(3) 2 (4) infinitely many

16. The difference beween the corresponding roots ofthe equation x2+rx+q=0 and x2+qx+r=0 is the samethen(1) q + r + 4 = 0 (2) q – r + 4 = 0(3) r – q + 4 = 0 (4) q + r – 4 = 0

17. If b > a then the equation (x–a) (x–b) –1=0 has(1) both roots in [a,b]

(2) both roots in , a

(3) both roots in b,

(4) one root in ( ,a) and other in b,18. The product of all solutions of the equation

2x 2 3 x 2 2 0 is

(1) 2 (2) –4(3) 0 (4) none of these

19. If the equation (cos p–1) x2 +x cos p +sin p =0 in thevariable x, has real roots, then p can take any valuein the interval

(1) 0, 2 (2) ,0

(3) / 2, / 2 (4) 0,20. If the equation ax2+bx+c =0 (a>0) has two roots

and such that 2 and 2 , then

(1) b2 – 4ac > 0 (2) a + |b| + c < 0(3) 4a + 2|b| + c < 0 (4) all of these

LEVEL - II1. The number of real solutions of

2 2

1 1x 3x 9 x 9

is

(1) 0 (2) 1(3) 2 (4) infinite

2. The sum of the real roots of the equation2x | x | 12 0 is

(1) 4 (2) 0(3) 6 (4) 7

3. If a + b + c = 0, then the roots of the equation(a + b – c) x2 + (b + c – a) x + (c + a – 2b) = 0 are(1) rational (2) irrational(3) integers (4) complex

4. If a I (the set of integers) and the equation(x–a) (x–5) + 1 = 0 has integral roots, then the valuesof a are(1) 10, 8 (2) 3, 7(3) 7, 8 (4) 3, 8

5. The condition that x3 + ax2 + bx + c = 0 may havetwo of its roots equal to each other but of oppositesigns is

(1) c ab (2) 2c ab(3) c ab 0 (4) b ac

6. If the roots of the equation px2 + qx + r = 0 (p>0) beeach greater than 2, then(1) p + q + r > 0 (2) 4p + 2q + r > 0(3) 4p – 2q + r > 0 (4) 4p + 2q + r < 0

7. The equation x

24 3x 5x 95

has

(1) no solution (2) one solution(3) two solutions (4) three solutions

8. The number of real values of x which satisfy theequation

2x x| x |x 1 | x 1|

is

(1) 1 (2) 2(3) 5 (4) infinite

9. If x is real and 2

2

x x 1px x 1

, then

(1)1 p 33 (2) p 5

(3) p 0 (4) p 3

10. If px2 + qx + r, p, q, r R , has no real zeros and ifr<0, then(1) p < 0 (2) p + q + r > 0(3) p > 0 (4) none of these




(3) Statement-1 is True, Statement-2 is False.





(4) Statement-1 is False, Statement-2 is True.

1. If P(x), Q(x) and R(x) are polynomials in x suchthat 3 3 2P(x ) xQ(x ) (x x 1)R(x)

STATEMENT-1 : 1 is root of P(x) = 0andSTATEMENT-2 : 1 is root of Q(x) = 0

2. Let p, q, r, s be four real numbers such thatpr = 2(q + s)

STATEMENT-1 : At least one of x2 + px + q = 0and x2 + rx + s = 0 has real roots.andSTATEMENT-2 : x2 + px + q = 0 has real roots ifand only if x2 + rx + s = 0 has imaginary roots.

3. STATEMENT-1 : If a, b, c R and 3c < 4a + 4band the equation ax2 – 2bx – 3c = 0 has no realroots, then c < 0.and

STATEMENT-2 : If a, b, c c R , the equation2ax bx c 0 , has no real roots if 2b 4ac 0 .

4. STATEMENT-1 : Let f : R R be defined by f(x)

= 2

2ax 6x 8

6x 8x

. If f is onto, the [2,14] .

andSTATEMENT-2 : If the expression

2

2x 3x 4y

3x 4x

, x R assumes all real values,

then [1,7] .

Linked Comprehension TypePassage - ILet f(x) = ax2 + bx + c. If x1, x2, x3 are distinct realnumbers such that f(x1) = f(x2) = f(x3) = 0, then f (x) 0 ,i.e. f (x) 0 x R 1. If f and x1, x2, x3 are as in teh above paragraph,

then(1) a 0,b 0,c 0 (2) a b 0,c 0

(3) a = b = c = 0 (4) a 0,b c 0 2. If x1, x2, x3 are three distinct real numbers and

2 3 3 1

1 2 1 3 2 3 2 3

(x x )(x x ) (x x )(x x )g(x)(x x )(x x ) (x x )(x x )

1 2

3 1 3 2

(x x )(x x )(x x )(x x )

then g(x) is identically

equal to(1) 0(2) 1(3) (x – x1) (x – x2) (x – x3)(4) None of these

3. If 2 31

1 2 1 3

(x x )(x x )g(x) x(x x )(x x )

3 1 1 22 3

2 1 2 3 3 1 3 2

(x x )(x x ) (x x )(x x )x x(x x )(x x ) (x x )(x x )

then g(x) is identically equal to(1) 0(2) 1(3) x(4) 2

1 2 3(x x x x ) 2 2 21 2 3 1 2 3(x x x )x x x x

Passage - IILet p(x) be a polynomial.

(a) Positive roots The number of positive roots ofthe equation p(x) = 0 cannot exceed the numberof changes of sign from positive to negative andnegative to positive in terms of the coefficientsof p(x).

(b) Negative roots The number of negative rootsp(x) = 0 cannot exceed the number of changesin the signs of p(–x).

4. The teast number of imaginary roots of the equationP(x) = x9 – x5 + x4 + x2 + 1 = 0 is(1) 2 (2) 8(3) 4 (4) 6

5. The equation p(x) = x5 + x3 – 8x – 5 = 0 has(1) exactly 3 real roots(2) exactly one real root(3) at least one root in (0, 1)(4) none of these

6. The equation p(x) = x5 – x + 16 = 0 has(1) no imaginary roots(2) exactly two imaginary roots(3) exactly four imaginary roots(4) at least one root in ( , 2)





5. Sequence & Series

LEVEL - I

1. If n

2 4 128 r

r 01 x 1 x 1 x ... 1 x x

, then

n is

(1) 255 (2) 127(3) 63 (4) 66

2. If cos (x–y), cosx, cos(x+y) are in H.P., thencos x . sec y/2 is equal to

(1) 2 (2) 1/ 2

(3) 2 (4) 3

3. Two A.M’s A1 and A2, two G.M.’s G1 and G2 andtwo H.M.’s H1 and H2 are inserted between anytwo numbers, then 1 1

1 2H H equals

(1) 1 11 2A A (2) 1 1

1 2G G

(3)1 2

1 2

G GA A (4)

1 2

1 2

A AG G

4. Let an be the nth term of the G.P. of positive numbers.

Let 100

2nn 1

a

and 100

2n 1n 1

a

such that ,

then the common ratio is

(1) / (2) /

(3) / (4) /

5. If x, y, z are in G.P. and tan–1x, tan–1y, tan–1z are inA.P. then

(1) x = y = z or y 1

(2)1xz

(3) x = y = z = 0

(4) x = y = z, but their common value is notnecessarily zero.

6. It is given that 4

4 4 4

1 1 1 .... to1 2 3 90

, then

4 4 4

1 1 1 .... to1 3 5

is equal to

(1)4

96

(2)4

45

(3)489

90

(4)4

55

7. If the H.M between P and Q be H, then

1 1HP Q

is equal to

(1) 2 (2)PQ

P Q

(3)P QPQ

(4)12

8. If 1 1 1 1

b a b c a c

and b a c , then a, b, c

are in(1) A.P. (2) G.P.(3) H.P. (4) None of these

9. If 3log 2 , x3log 2 5 and x

3log 2 7 / 2 are inA.P. then x is equal to(1) 2 (2) 3(3) 4 (4) 2, 3

10. If / 4

n 2n

0

I tan x sec xdx

, then I1, I2, I3 , .... are in

(1) A.P. (2) G.P.(3) H.P. (4) None of these

11. If S1, S2, S3 are the sums of first n natural numbers,their square, their cubes respectively, then

3 122

S 1 8SS

is equal to

(1) 1 (2) 3(3) 9 (4) 10





12. The interior angle of a polygon are in A.P. If thesmaller angle is 120° and the common difference is5°, then number of sides in the polygon is(1) 7 (2) 8(3) 9 (4) 16

13. If the ratio of sums to n terms of two A.P.’s is(5n+3) : (3n+4), then the ratio of their 17th term is(1) 172 : 99 (2) 158 : 97(3) 175: 99 (4) 171 : 97

14. The number of terms common between the series1 + 2 + 4 + 8 +..... to 100 terms and1 + 4 + 7 + 10 +.....to 100 terms is(1) 6 (2) 4(3) 5 (4) 6

15. Let S be the sum, P be the product and R be thesum of the reciprocals of n terms of a G.P., thenP2Rn: Sn is equal to(1) 1 : 1(2) (Common ratio)n : 1(3) (first term)2 : (common ratio)n

(4) 2 : 3

16. If 5cloga

, 3blog5c

and alog

3b are in A.P., where

a, b, c are in G.P., then a, b, c are the lengths ofsides of(1) an isoceles triangle(2) an equilateral triangle(3) a scalene triangle(4) none of these

17. Let a1, a2, a3, ..... a10 be in A.P. and h1, h2 h3, .....,h10 be in H.P. If a1 = h1 = 2 and a10 = h10= 3, thena4h7 is(1) 2 (2) 3(3) 5 (4) 6

18. If 0 , / 2 and 2n

n 0x sin

,

2n

n 0y cos

and

n n

n 0z cos .cos

, then

(1) xyz + 1 = yz –zx (2) xyz –1 = yz+zx(3) xyz–xy = yz–zx (4) xyz +1 = yz+zx

19. If three positive real numbers a, b, c are in A.P.such that abc = 4, then the minimum positive valueof b is(1) 23/2 (2) 22/3

(3) 21/3 (4) 25/2

20. For x , the values of x which satisfy therelation

2 3(1 |cos x| cos x |cos x| .....to ) 425 5 are given by

(1) /3, 2 /3 (2) /3,3 / 4

(3) / 4,3 / 4 (4) 5 / 6, /3

LEVEL - II1. Sum of infinite number of terms of a G.P. is 20 and

sum of their square is 100. The common ratio of theG.P. is

(1) 5 (2) 3/5(3) 8/5 (4) 1/5

2. The coef f icient of x49 in the product(x–1) (x–3).....(x–99) is

(1) –992 (2) 1(3) –2500 (4) none of these

3. 13–23+33–43+.....+93=

(1) 425 (2) –425(3) 475 (4) –475

4. If non-zero numbers a, b, c are in H.P., then the

straight line x y 2 0a b c always passes through

a fixed point. The point is

(1) (–1, 1) (2) (–1, –2)(3) (1, –2) (4) (1, –1/2)

5. If ax = by = cz = dt and a, b, c, d are in G.P. then x,y, z, t are in

(1) A.P. (2) G.P.(3) H.P. (4) none of these

6. If n

n 0x a

, n

n 0y b

, n

n 0z ab

,

where |a|, |b| < 1, then

(1) xyz = x + y + z

(2) xz + yz = xy + z

(3) xy + yz = xz + y

(4) xy + xz = yz + x

7. The angles of a triangle are in A.P. and the leastangle is 30°. The greaterst angle in radian is

(1) / 3 (2) 2 /3

(3) 5 / 6 (4) / 2





8. 0.2 0.22 0.222 ....... to n terms is equal to

(1) n2 1n 1 109 9

(2) n1n 1 109

(3) n2 1n 1 109 9

(4)

29

9. If x, y, z are three real numbers of the same sign,

then the value of x y zy z x lies in the interval

(1) 2, (2) 3,

(3) 3, (4) ,3

10. If a bx b cx c dxa bx b cx c dx

x 0 , then a, b, c, d

are in

(1) A.P. (2) G.P.(3) H.P. (4) none of these





1. Let a, b, c be the three positive real numbers whichare in H.P.

STATEMENT-1 : a b c b 42a b 2c b

and

STATEMENT-2 : If x > 0, then 1x 4x

2. STATEMENT-1 : If a, b, c are three positive realnumbers such that a c b and

1 1 1 1 0a a b c c b

then a, b, c are in H.P..

andSTATEMENT-2 : If a, b, c are distinct positive realnumbers such that

2 2a(b c)x b(c a)xy c(a b)y is a perfectsquare, then a, b, c are in H.P.

3. Let a, b, c be three distinct non-zero real numbers.STATEMENT-1 : If a, b, c are in A. P. and b, c, aare in G.P., then c, a, b are in H.P.andSTATEMENT-2 : If a, b, c are in A.P. are b, c, aare in G.P., then a : b : c = 4: 2 : – 1

4. Let a0, a1, a2, a3, .... be an arithmetic progression.

STATEMENT-1 : 2 4 2nsin a sin a ... sin a

= 1 2n 1

2 1

cosa cosa2sin(a a )

and

STATEMENT-2 : 2 4 2ncosa cosa ... cosa

= 2n 1 1

2 1

cosa cosa2cos(a a )

Linked Comprehension TypePassage - IThe sum of three terms of a strictly increasing G.P. is

S and sum of the squares of these terms is 2S .

1. 2 lies(1) (1/3, 2) (2) (1, 2)(3) (1/3, 3) (4) None of these

2. If 2 2 , then value of r equals

(1)1 (5 3)2

(2)1 (3 5)2

(3)1 ( 5 3)2

(4)1 ( 3 5)3

3. If r 2 , then 2 equals





(1)1 (2 3)7

(2)1 (11 7)4

(3)1 (11 6 2)7

(4)1 (11 6 2)5

Passage - IIA man is to receive Rs Pi at the end of ith year for nyear. Assume that the rate of interest is Rs r per rupeeper year and the interest is compounded annually.

4. If iP P i , then present value of income streamis

(1) nP [1 (1 r) ]r

(2) nP [1 r (1 r) ]r

(3) nP[1 (1 r) ] (4) None of these

5. If iP P i and n , then present value of theincome stream is(1) P (2) 2P(3) P/r (4) P + P/r

6. If iP iP i , then present value of the incomestream is

(1) n2

P(1 r) [1 (1 r) ]r

(2)n

2 nP(1 r) nP[1 (1 r) ]

r r(1 r)

(3)n

2 nP(1 r) nP[1 (1 r) ]

r (1 r)

(4) None of these





6. Permutation & Combinations

LEVEL - I1. The sum of all the four different digitsnumbers that

can be formed with the digits 0, 1, 2, and 3 is(1) 38664 (2) 38663(3) 38662 (4) 38665

2. There are 20 books of which 4 are single volume andthe other are books of 8, 5 and 3 volumes respectively.In how many ways can all these books be arrangedon a shelf so that volumes of the same book are notseparated?(1) 7!8!5!3! (2) 7!7!5!3!(3) 7!5!7!6! (4) 7!8!5!4!

3. Six boys and six girls sit along a line alternately in yways and along a circle again alternatively in x ways,then(1) x = y (2) y = 12 x(3) x = 10 y (4) x = 12 y

4. The number of integral solutions of x+y+z+t = 20,where x, y, z, t are greater than or equal to –1 is(1) 20C4 (2) 23C3

(3) 27C4 (4) 27C3

5. The total number of ways of dividing 15 different thingsinto groups of 8, 4 and 3 respectively is

(1) 2

15!8! 4!(3!) (2)

15!8! 4! 3!

(3)15!

8! 4! (4)15!

6! 5!

6. m parallel lines in a plane are intersected by a familyof n parallel lines. The total number of parallelogramsso formed is

(1)(m 1)(n 1)

4

(2)mn4

(3)mn(m 1)(n 1)

2

(4)mn(m 1)(n 1)

4

7. The number of signals that can be generated by using6 different coloured flags, when any number of themmay be hosisted at a time is(1) 1956 (2) 1957(3) 1958 (4) 1959

8. The total number of ways of selecting six coins outof 20 one rupee coins, 10 fifty paise coins and 7twenty five paise coins, is(1) 28 (2) 26(3) 56 (4) 57

9. A set contains (2n+1) elements. The number ofsubsets of the set which contains at most n elementsis(1) 2n (2) 2n+1

(3) 22n (4) 2n+1

10. A five digit number divisible by 9 is to be formed usingthe digits 0,1,2,7,8 and 9 without repetition. The totalnumber of ways of doing so is(1) 660 (2) 240(3) 216 (4) 3125

11. In an examination of 9 papers, a candidate has topass in more papers than the number of papers inwhich he fails, in order to be successful. The numberof ways in which he can be unsuccessful is(1) 256 (2) 255(3) 193 (4) 319

12. A candidate is required to answer 6 out of 10questions which are divided into two groups eachcontaining 5 questions and he is not permitted toattempt more than 4 from any group. The number ofways he can make up this choice is(1) 150 (2) 200(3) 250 (4) 300

13. In how many ways can the letters of the word‘DIRECTOR’ be arranged so that the three vowelsare never together?(1) 18000 (2) 18001(3) 18036 (4) 18003

14. The total number of arrangements which can be madeout of the letters of the word ‘ALGEBRA’ withoutaltering the relative position of vowels and consonantsis

(1) 4!3! (2)4!3!

2(3) 2(4! 3!) (4) 4! 5!





15. m distinct animals of a circus have to be placed in mcages, one in each cage. If n(<m) are too small toaccommodate p(n<p<m) animals, then the numberof ways of putting the animals into cages is

(1) m npC (2) m n m p

p m pP P

(3) m n m pp m pC C

(4) m n m np m pC C

16. The number of ways in which ten candidates A1, A2,...... A10 can be ranked such that A1 is always aboveA10 is(1) 5! (2) 2(5!)

(3) 10! (4)1 (10!)2

17. Find the total number of squares, which can bechoosen from a chess board (8×8)(1) 201 (2) 402(3) 102 (4) 204

18. In a club election the number of contestants is onemore than the number of maximum candidates forwhich a voter can vote. If the total number of ways inwhich a voter can vote be 62, then the number ofcandidates is(1) 7 (2) 5(3) 6 (4) 7

19. The number of ways in which a pack of 52 cards offour different suits can be distributed equally amongfour players so that each player gets the Ace, King,Queen and Knave of the same suit is

(1) 4

36!4!(9!) (2) 4

36!4!(9!) 4!

(3) 4

52!4!(9!) (4) 5

52! 4!(9!)

20. There are four balls of different colours and four boxesof colours, same as those of the balls. The numberof ways in which the balls, one each in a box, couldbe placed such that a ball does not go to box of itsown colour is(1) 9 (2) 24(3) 12 (4) 16

LEVEL - II1. If n 2

6C + n 17C > n

6C , then

(1) n > 4 (2) n > 12

(3) n 13 (4) n 13

2. In how many ways 7 men and 7 women can be seatedaround a round table such that no two women cansit together?

(1) (7!)2 (2) 7! × 6!

(3) (6!)2 (4) 7!

3. The number of diagonals of a polygon of m sides is

(1)m(m 1)

2!

(2)m(m 2)

2!

(3)m(m 3)

2!

(4)m(m 5)

2!

4. Everybody in a room shakes hands with everybodyin a room. The total number of hand shakes is 66.The total number of persons in the room is

(1) 11 (2) 12

(3) 13 (4) 14

5. 12 persons are to be arranged on a round table. Iftwo particular persons among them are not to sitside by side, the number of arrangements is

(1) 9 (10!) (2) 2 (10!)

(3) 45 (10!) (4) 10!

6. A fruit basket contains 4 oranges, 5 apples and 6mangoes. The number of ways a person makeselection of fruits from among the fruits in the basketis

(1) 210 (2) 209

(3) 208 (4) 210

7. Given 5 different green dyes, 4 different blue dyesand 3 different red dyes. The number of combinationsof dyes which can be chosen taking at least onegreen and one blue dye is

(1) 3600 (2) 3720

(3) 3800 (4) 3660





8. In how many ways 3 prizes can be given away to 7boys when each boy is eligible for any of the prizes?

(1) 343 (2) 342

(3) 341 (4) 346

9. The number of five-digit telephone numbers having atleast one of their digits repeated is

(1) 90000 (2) 100000

(3) 30240 (4) 69760

10. A box contains 2 white balls, 3 black balls and 4 redballs. The number of ways in which three balls canbe drawn from the box so that at least one of theballs is black is

(1) 74 (2) 84

(3) 64 (4) 20





1. STATEMENT-1 : Number of rectangle in a chessboard is 8C2 × 8C2.andSTATEMENT-2 : To form a rectangle we have toselect any two of the horizontal line and any twovertical line.

2. STATEMENT-1 : (n 2)!(n 1)! is divisible by 6.

andSTATEMENT-2 : Product of 3 consecutive integersis divisible by 3!

3. STATEMENT-1 : The number of polynomials ofthe form x3 + ax2 + bx + c where a, b, c {1, 2, 3....10} and which are divisible by x2 + 1 is10.

andSTATEMENT-2 : The given polynomial is divisibleby x2 + 1 for all a, b, c

4. STATEMENT-1 : n n

(2n)!2 .3

is an integer

and

STATEMENT-2 : (2n)!

n!(n 1)! is an integer..


One of the most important techniques of countingis the principle of exclusion and inclusion. let A1, A2, ...,Am be m sets and n(Ai) represents the cardinality of theset Ai (the number of elements in the set Ai), thenaccording to the principle of exclusion and inclusion

n(A1 A2 . . . . Am = m

ii 1

n(A )

i ji 1

n(A A )

i j ki j k

n(A A A )

– ... +

(–1)m+1 n(A1 A2 ... Am)In particular, if A, B, C are three sets, thenn(ABC) = n(A) + n(B) + n(C) – n(AB) –

n(B C) – n(C B C).Principle of exclusion and inclusin must be applied

whenever there is a chance of repeated counting of someof the samples1. The number of numbers from 1 to 100, which are

neither divisible by 3 nor by 5 nor by 7 is(1) 67 (2) 55(3) 45 (4) 33

2. A six letters word is formed using the letters of theword ALMIGHTY with or without repetition. Thenumber of words that contain exactly three differentletters is(1) 15600 (2) 30240(3) 8P6 – 8P3 (4) None of these

3. The number of natureal numebrs less than or equalto 2985984, which are neither perfect square norperfect cubes is(1) 2984124 (2) 2984244(3) 2959595 (4) None of these

Passage - IIThe tournament for ABC Cup is arranged as per





the following rules: in the beginning 16 teams are enteredand divided in 2 groups of 8 teams each where the teamin any group play exactly onece with all the teams in thesame group.

At the end of this round top four teams from eachgroup advance to the next round in which two teamsplay each other and the losing team goes out of thetournament. Then four winning teams play for semi finalsand finally is one final. the rules of the tournament aresuch that every match can result only in a win or a lossand not in a tie.4. What is the total number of matches played in the

tournament?(1) 63 (2) 56(3) 64 (4) 55

5. The maxium number of matches that a team goingout of the tournamnet in the first round itself canwin is(1) 1 (2) 2(3) 3 (4) 4

6. The minimum number of matches that a team mustwin in order to qualify for the second round is(1) 4 (2) 5(3) 6 (4) 7





7. Binomial Theorem and Its Applications

LEVEL - I1. If the ratio of the 7th term from the beginning to the

7th term from the end in the expansion ofx

33

123

is

16

, then x is

(1) 9 (2) 6(3) 12 (4) 13

2. The number of rational terms in the expansion of63(2 2 3) is

(1) 5 (2) 7(3) 11 (4) 14

3. If the coefficients of 2nd, 3rd and 4th terms is theexpansion of (1+x)n are in A.P., then value of n is(1) 7 (2) 3(3) 4 (4) 5

4. The remainder when 337 is divided by 80 is(1) 78 (2) 3(3) 4 (4) 5

5. The greatest value of the term independent of x in

the expansion of 1 10(x sin x cos ) , R , is

(1) 25 (2) 2

10!(5!)

(3) 5 2

1 10!2 (5!)

(4) 4 3

1 10!2 (5!)

6. In the expansion of (x+a)n the sum of even terms is Aand that of odd terms is B, then B2 – A2 is(1) (x2 + a2)n (2) (x2–a2)n

(3) (x–a)2n (4) none of these7. The greatest coefficient in the expansion of (1+x)2n

is

(1) n1.3.5......(2n 1) 2n!

(2) 2n

n 1C

(3) 2nn 1C (4) 2n

n 1C

8. If n is an odd natural number, then rn

nr 0 r

( 1)C

equals

(1) 0 (2)1n

(3) n

n2

(4)1

n 1

9. If (1+x)n = 2 n0 1 2 nC C x C x ...... C x , then

31 2 n

0 1 2 n 1

3CC 2C nC...C C C C

is

(1)n2

n!(2)

n(n 1)n!

(3)n(n 1)

2

(4)n (n 1)

2

10. If x+y = 1,then n

n r n rr

r 0. C x y

equals

(1) nxy (2) nx(x+yn)(3) 1 (4) Noneof these

11. The middle term in the expansion ofn

n11 .(1 x)x

is

(1) 2nn

n1 C (2) 2nnC

(3) 2nn 1C (4) None of these

12. Let n be an odd natural number greater than 1, thenthe number of zeros at the end of the sum 99n + 1 is(1) 3 (2) 4(3) 2 (4) 1

13. The coef f icient of x20 in the expansion of(1+x2)40 (x2 + 2 + 1/x2)–5 is(1) 30C10 (2) 30C25(3) 1 (4) 30C15

14. The coefficient of 3 4x y z in the expansion of (1+x+y–z)9 is

(1) 2.9 77 4C . C (2) 9 7

2 32 C . C

(3) 9 72 4C . C (4) 2 9 7

2 3C . C

15. If the coefficient of x7 in 11

2 1axbx

and the

coefficient of x–7 in 11

2

1axbx

are equal, then ab

is equal to(1) 1 (2) 2(3) 3 (4) 4

16. If x is positive then first negative term is the expansion

of 275(1 x) is

(1) 6th term (2) 7th term(3) 5th term (4) 8th term





17. The number of real negative terms in the binomialexpansion of (1+ix)4n–2 , n N,x 0 is(1) n (2) n+1(3) n–1 (4) 2n

18. The coefficient of x6 in

6 7 151 x (1 x) ...... (1 x) is

(1) 169C (2) 16 6

5 5C C

(3) 166C 1 (4) 16

5C19. The number of terms in the expansion of

2n2

2

1x 1 , n Ix

is

(1) 2n (2) 3n(3) 2n+1 (4) 2n–1

20. The sum of the series4 5 6 2 7 3

0 1 2 3C C x C x C x ......to is

(1) 4(1 x) (2) 5

1(1 x)

(3) 5(1 x) (4) 4(1 x)

LEVEL - II1. The number of distinct terms in the expansion of

(x + 2y – 3z)15 is(1) 135 (2) 136(3) 137 (4) 134

2. The sum of the coefficients in the polynomial

expansion of 2 2163(1 x 3x ) is

(1) 1 (2) –1(3) 0 (4) 2

3. The number of irrational terms in the expansion of

152 3 is equal to

(1) 16 (2) 7(3) 12 (4) 15

4. If the sum of coefficients in the expansion of (a+b)n

is 4096, then the greatest coefficients in theexpansion is(1) 924 (2) 925(3) 926 (4) 927

5. Maximum value of 25Cr+2 is equal to(1) 25C11 (2) 25C10(3) 25C12 (4) 25C14

6. The value of 47C4 + 5

52 r3

r 1C

is equal to

(1) 524C (2) 51

3C

(3) 514C (4) 53

5C

7. The expression 7 75 5x 1 x x 1 x is

a polynomial of degree(1) 16 (2) 18(3) 20 (4) 22

8. In the expansion of 8 82 2x x 1 x x 1 ,

the number of distinct terms is(1) 7 (2) 14(3) 5 (4) 4

9. The sum of the rational terms in the expansion of

201/ 52 3 is

(1) 71 (2) 85(3) 97 (4) none of these

10. If the coefficients of the (r +1)th term and (r+3)th termin the expansion of (1+x)30 are equal, then the valueof ‘r’ is(1) 10 (2) 14(3) 13 (4) 12





For n 1, let n n

n kn k j

j 1 k ja

and n 1n nb a 2

1. STATEMENT-1 : n 1 n n 1b 5b 6b n 2

and

STATEMENT-2 : n n n 1a 5a 6a n 2





2. STATEMENT-1 : If 100rC , 100

r 1C , 100r 2C ,

100r 3C are in A.P.P., then r = 49

andSTATEMENT-2 : Four consecutive coefficients ofa binomial can never be in A.P.

3. For m, n N , n m , let

m m 1 m n 1

2 n

1 x 1 x ... 1 x(m,n)

(1 x)(1 x )...(1 x )

STATEMENT-1 : (m, n + 1) = (m – 1n + 1)+ xm–n–1 (m – 1, n)

andSTATEMENT-2 : (m, n) is a polynomial of degreem + n – 2

4. STATEMENT-1 : For non-negative intergers m, n,r and k,

k

m 0

(r m)! (r k 1)! n k(n m)m! k! r 1 r 2

and

STATEMENT-2 : k

r m r k 1m k

m 0

Linked Comprehension TypePassage - IIf m, r, r N , then

m n m n m n0 r 1 r 1 r 0C C C C ... C C

= coefficient of r m nx in (1 x) (1 x)

= coefficient of r m nx in (1 x)

= m nrC

We can use similar techniques to evaluate product ofbinomial coefficients.1. The values of r for which

20 10 20 10 20 10r 0 r 1 1 0 rS C C C C ... C C

is maximum is(1) 7 (2) 8(3) 10 (4) 15

2. The value of r(0 r 30) is20 10 20 10 20 10

r 0 r 1 1 0 1C C C C ... C C is least is(1) 0 (2) 1(3) 10 (4) 15

3. Let n n n n n nn 0 1 1 2 n 1 nA C C C C ... C C and

n 1

n

A 15A 4

, then n equals

(1) 8, 4 (2) 4, 6(3) 2, 4 (4) 8, 6

Passage - II

To find coefficient of rx (0 r n 1) in the expansionof

n 1 n 2(x a) (x a) (x b)

n 2 n 1... (x a)(x b) (x b)

We first sum up the series.

4. The coefficient of rx (0 r n 1) in the expansionn 1 n 2E (x 2) (x 2) (x 1) n 3 2 n 1(x 2) (x 1) ... (x 1)

(1) nrC (2) n r

rC (2 1)

(3) n n rrC 2 (4) None of these

5. The coefficient of xn–1 in the expansion ofn 1 n 2E (2x 1) (2x 1) (x 1) n 1... (x 1) is

(1) 2n (2) 2n – 1(3) 2n + 1 (4) 22n

6. The coefficient of rx (0 r n) in the expansionof

n n 1 n 2 2E 2 2 (x 2) 2 (x 2) n... (x 2)

(1) n 1 n rr 1C 2 (2) n n r

rC 2

(3) n rrC 2 (4) None of these





LEVEL – I1. Three natural numbers are taken at random from

the set of first 100 natural numbers. The probabilitythat their A.M. is 25, is

(1)77

2100

3

CC (2)

252

1003

CC

(3)74

72100

72

CC (4)

254

1003

CC

2. The probability that out of 10 persons, all born inJune, at least two have the same birth day is

(1)

301010

C30 (2)

3010C

30!

(3)

10 3010

10

30 C30

(4)30

20C30!

3. 10 mangoes are to be distributed among 5 persons.The probability that at least one of them will receivenone, is

(1)35

143(2)

108143

(3)18143

(4)125143

4. There are four machines and it is known that exactlytwo of them are faulty. They are tested, one byone, in a random order till both the faulty machinesare identified. Then the probability that only twotests are needed is(1) 1/3 (2) 1/6(3) 1/2 (4) 1/4

5. In convex hexagon two diagonals are drawn atrandom. Thw probability that the diagonals intersectat an interior points of the hexagon is(1) 5/12 (2) 7/12(3) 2/5 (4) 7/13

8. Probability6. Fifteen persons, among whom are A and B, sit down

at random at a round table. The probability that thereare 4 persons between A and B is

(1)9!

14!(2)

10!14!

(3)9!

15!(4)

9!10!

7. The probability that the birth days of six differentpersons will fall in exactly two calender months is

(1) 1/6 (2)6

122 6

2C12

(3)6

122 6

2 1C12

(4) 5

34112

8. The probability that when 12 balls are distributedamong three boxes, the first will contain three ballsis,

(1)9

12

23

(2)12 9

312

C .23

(3)12 12

312

C .23

(4)10

1223

9. A team of 8 couples, (husband and wife) attend alucky draw in which 4 persons picked up for a prize.Then, the probability that there is at least one coupleis(1) 11/39 (2) 12/39(3) 14/39 (4) 15/39

10. Three persons A, B and C are to speak at a functionalong with five others. If they all speak in randomorder, the probability that A speaks before B and Bspeaks before C is(1) 3/8 (2) 1/6(3) 3/5 (4) 1/5

11. A 2×2 square matrix is written down at random usingthe numbers 1, –1 as elements. The probability thatthe matrix is non-singular is(1) 1/2 (2) 3/8(3) 5/8 (4) 1/3





12. A fair coin is tossed repeatedly. If tail appears onfirst four tosses, then the probability of headappearing on fifth toss equals(1) 1/2 (2) 1/32(3) 31/32 (4) 1/5

13. If A and B are two events such that 1P A2

and

2P B3

, then which of the following is correct?

(1) 2P A B3

(2) 1P A B3

(3) 1 1P A B6 2 (4) All of these

14. Four numbers are multiplied together. Then theprobability that the product will be divisible by 5 or10 is(1) 369/625 (2) 399/625(3) 123/625 (4) 133/625

15. A bag contains (n+1) conins. It is known that one ofthese coins shows heads on both sides, whereasthe other coins are fair. One coin is selected atrandom and tossed. If the probability that toss resultsin heads is 7/12, then the value of n is(1) 3 (2) 4(3) 5 (4) 6

16. If 1 3p

2

, 1 4p

3

, 1 p

6

are probabilities of three

mutually exclusive and exhaustive events, then theset of all values of p is(1) (0, 1) (2) [–1/4, 1/3]

(3) (0, 1/3) (4) 0,

17. A person draws a card from a pack of 52 playingcards, replaces it and shuffles the pack. Hecontinues doing this until he draws a spade, thechance that he will fail in the first two draws is(1) 1/16 (2) 9/16(3) 9/64 (4) 1/64

18. n biscuits are distributed among N beggars atrandom. The probability that a particular beggar getsr(<n) biscuits is

(1)r n r

nr

1 N 1Cn N

(2)n

rr

CN

(3) nrC (4)

rn

19. Let 0 < P(A) < 1, O < P(B) < 1 and P A B P A P B P A P B . Then,

(1) P A / B P A P B

(2) C C CP A B P A P B

(3) C C C CP A B P A P B

(4) P B/ A P B P A

20. A committee of the five is to be chosen from a groupof a people. The probability that a certain marriedcouple will either serve together or not at all is(1) 2/3 (2) 4/9

(3) 1/2 (4) 5/9

LEVEL – II1. n 3 persons are sitting in a row. Two of them

are selected at random. The probability that theyare not together is

(1)11n

(2)21n

(3)2

n 1(4)

21n

2. If ten objects are distributed at random among tenpersons, the probability that at least one of themwill not get any thing is

(1)10

10

10 1010

(2)10

10

10 10!10

(3)10

10

10 110

(4)

10

1010 9!

10

3. A number x is chosen from the set nA 33 : n N . Then probability that x has 3 in

units place, is(1) 1/2 (2) 1/3(3) 1/4 (4) 1/5

4. If from each of the three boxes containing 3 whiteand 1 black, 2 white and 2 black, 1 white and 3balck balls, one ball is drawn at random, then theprobability that 2 white and 1 black ball will be drawnis(1) 13/32 (2) 1/4(3) 1/32 (4) 3/16





5. If E and F are the complementary events of eventsE and F respectively and if 0 < P (F) < 1, then

(1) P E / F P E / F 1

(2) P E / F P E / F 1

(3) P E / F P E / F 1

(4)E EP P 1F F

6. Cards are drawn from a pack of 52 playing cardsone by one. The probability that exactly 10 cardswill be drawn before the first ace is

(1)451884

(2)241

1456

(3)1644165

(4)243

14557. Two numbers are selected randomly from the set

S 1,2,3,4,5,6 without replacement one by one.The probability that the minimum of the two numbersis less than 4 is

(1)1

15(2)

1415

(3)15

(4)45

8. A and B are two events such that 3P(A B)4

,

1P A B4

, 2P(A)3

, then P A B

(1)5

12(2)

38

(3)58

(4)14

9. If three distinct numbers are chosen randomly fromthe first 100 natural numbers, then the probabilitythat all three of them are divisible by both 2 and 3 is

(1)425

(2)4

35

(3)433

(4)4

115510. The probabiliy that A speaks the truth is 4/5, while

this probability for B is 3/4. The probability that theycontradict each other when asked to speak on a

fact is

(1)45

(2)15

(3)720

(4)320





1. STATEMENT-1 : Out of 5 tickets consecutivelynumbered, three are drawn at random, the chance

that the numbers on them are in A.P. is 2

15andSTATEMENT-2 : Out of (2n + 1) ticketsconsecutively numbered, three are drawn at random,the chance that the numbers on them are in A.P. is

2

3n4n 1

2. STATEMENT-1 : The probability of an event canbe a rational number only lying between 0 and 1 inall possible sample spaces discrete or continuous.andSTATEMENT-2 : In discrete sample space the

probability of an event m ,n

whree m, n are

integers m 0, n > 0

3. STATEMENT-1 : If P(A/B) < were P(B) < P(A)then P(B/A) < and

STATEMENT-2 :P(A B)P(A / B)

P(B)





P(A B)P(B/ A)P(A)

4. Let 1 2ˆ ˆ ˆ ˆ ˆr i aj k, r j ak

and 3ˆ ˆr ai k

arethe adjacent edges of a parallelepiped.

STATEMENT-1 : The volume of parallelepiped is

minimum when 1a3

andSTATEMENT-2 : If V(a) is volume of parallelepiped

then 2

2d V(a) 10 at a

da 3 and V(a) = 0


Ajay plays a game against an electronic machine.At each round he deposits one rupee in a slot and thenflips a coin which has probability p of sowing a head. Ifhead comes, he gets back the rupee he deposited andone more rupee from the machine. if tail shows he loseshis coin. Suppose he started with 10 one rupee coin.Assume q as the probable event of getting a tail. Now,attempt the following:1. The probability that Ajay is left with no money by

the tenth round or earlier is

(1) q10 (2)10

10 r 10 rr

r 1C p q

(3) pq9 (4) (1 – p10)2. C The probability that his money will end up exactly

in the twelfth round is(1) q12 (2) 1 – p12

(3) 10C1pq11 (4) 12C2p2q10

3. The probability that he is left with no money by the14th round or earlier is(1) q10(1 + 10pq + 45p2q2)(2) q14(p2q + 36pq + 7)(3) q12 + 3pq12 + 3p13q + q12

(4) 1 – 10C1pq11 – 10C2p2q12

Passage - IILet A, B, C be the vertices of a triangle ABC in

which B is taken as origin of reference and positionvectors of A and C are a and c respectively. A line ARARparallel to BC is drawn from A. PR (P is mid poitn of

AB) meets AC at A and area of triangle ACR is 2 timesarea of triangle ABC.4. Positioon vector of R in terms of a and c is

(1) a + 2 c (2) a + 3 c

(3) a + c (4) a + 4 c

5. Position vector of Q is (considering result of ??)

(1)2a 3c

5

(2)2a 2c

5

(3)a 2c

5

(4) None of these

6.PQ AQ.QR QC

is equal to

(1) 1/4 (2) 2/5(3) 3/5 (4) 1/6





9. Matrices & Determinants

Level I1. If A and B are square matrices of same order and

A is non-singular, then for a positive integern(A–1BA)n is equal to(1) A–nBnA (2) AnBnA–n

(3) A–1BnA (4) n(A–1BA)

2.

cos x sin x 0F(x) sin x cos x 0

0 0 1

and

cos x 0 sin xG(x) 0 1 0

sin x 0 cos x

, then [F(x) G(y)]–1 is

equal to(1) F(–x) G(–y) (2) F(x – 1) G(y – 1)(3) G (–y) F(–x) (4) G(y–1) F(x–1)

3. If D = diag [d1, d2 ... dn] is a diagonal matrix with di 0, i = 1, 2, 3, ....m n, then D–1 is(1) d1

–1 d2–1 ...dn

–1 In(2) diag [dn

–1, dn–1–1... d1

–1](3) diag [d1

–1, d2–1... dn

–1](4) None of these

4. The index of the matrix

1 1 35 2 62 1 3

is

(1) 2 (2) 3(3) 4 (4) None of these

5. One factor of

2

2

2

a ab acab b cbca cb c

is

(1) 2

(2) (a2 + ) (b2 + ) (c2 + )

(3)1

(4) None of these

6. If

2

2

2

x 1 1 x 12x 1 1 x 23x 2 1 x 3

= a0 + a1x + a2x2 + a3x3 + ...

then a1 is equal to(1) 1 (2) 2(3) 3 (4) 0

7. The system of equations 6x + 5y + z = 0, 3x – y +4z = 0, x + 2y – 3z = 0 has non-trivial solutions for(1) = 0 (2) = 1(3) = –5 (4) None of these

8. I f

x x c x af (x) x b x x a

x b x b x

, g(x) = ( x)

( – x) ( – x), then the value of bg(a) ag(b)

b a

is

(1) f(1) (2) (–1)(3) f(0) (4) None of these

9. If [x] stands for the greatest integer less than orequal to x, then the value of

2

2

2

[e] [ ] [ 6][ ] [ 6] [e]

[ 6] [e] [ ]

is

(1) –8 (2) 8(3) 0 (4) 1

10. If 2 2 2

x y zf x y z

yz zx zy and g = (x – y) ( y – z) (z – x),

then fg is :

(1) xy + yz + zx(2) x2 + y2 + z2

(3) x2 + y2 + z2 – xy – yz – zx(4) None of these

11. If

c

e

e

x log ax 2

3 log a3x 4

5 log a5x

a e xg(x) a e x

a e 1

, then

(1) g(x) + g(–x) = 0 (2) g(x) – g(–x) = 0(3) g(x) × g(–x) = 0 (4) None of these

12. A determinant of second order is made with theelement 0 and 1. The number of determinants with





non-negative values is(1) 3 (2) 10(3) 11 (4) 13

13. Let {D1, D2, D3, ...Dn} be the set of third orderdeterminantes that can be made with the distinctnon-zero real number a1, a2 ... a9 then

(1)n

ii 1

D 1

(2)n

ii 1

D 0

(3) Di = Dj, i, j (4) None of these

14. Coefficient of x in

3

2 2

x (1 sin) cos xf (x) 1 log(1 x) 2

x (1 x ) 0

= 0 is

(1) 0(2) 1(3) –2(4) Can’t be determined

15. If f(x) =

2

2

sin x secx x 1cosecx x sin x cos xtan x x tan x x 1

then

/ 3

/ 3f (x)dx

(1) 1 (2) 13

(3) 0 (4) 13

16. If [p] stands for the greatest integer less than orequal to p, then in order that the set of equations,2x – 3y = 4, 7z – 2y =2, [3]x – [2e]y = [4a] maybe consistent is that a should lie in :

(1)1 ,12

(2) [2, 3]

(3)3 7,2 4

(4)

31,2

17. If a, b, c are non-zeros, then the system of equations:( + a)x + y + z = 0; x + ( + b)y + z = 0;x + y + ( + c)z = 0 has a non - trivial solution if:(1) –1 = –(a–1 + b–1 + c–1)(2) –1 = a + b + c(3) + a + b + c = 1(4) None of the above

18. The value of x, so that [1 × 1] 1 3 2 10 5 1 10 3 2 x

= 0

is

(1)7 35

2

(2)9 35

2

(3) ±2 (4) 019. X, Y and Z are positive number greater than 10

such that Y and Z have respectively 1 and 0 at their

unit’s place and D is the determinant

X 4 1Y 0 1Z 1 0

.

If (D + 1) is divisible by 10 then X has at its unit’splace(1) 1 (2) 0(3) 2 (4) None of these

20. Let w ¹ 1 be complex cube root of unity and n be a

natural number and

n 2n

n 2n

2n n

1 w ww w 1w 1 w

Then

equals(1) 0 (2) 1(3) w (4) w2

Level II

1. Let 2 2 21

ax by czx y z1 1 1

and 2

a b cx y zyz zx xy

,

then 1 – 2 then(1) (x – 1) (y – 1) (z – 1)(2) (x – y) (y – z) (z – x)(3) abc (x – y) (y – z) (z – x)(4) None of these

2. If R,, then the determinant

cos sin 1sin cos 1

cos( ) sin( ) 0

lies in the

interval

(1) [ 2, 2] (2) [–1, 1]

(3) [ 2,1] (4) [ 1, 2]





3. If the system of linear equationsx + 2ay + az = 0x + 3by + bz = 0x + 4cy + cz = 0has a non-zero solution, then a, b, c(1) are G.P.(2) are in H.P.(3) satisfy a + 2b + 3c = 0(4) are in A.P.

4. If a2 + b2 + c2 = –2 and

2 2 2

2 2 2

2 2 2

1 a x (1 b )x (1 c 0xf (x) (1 a )x 1 b x (1 c )x

(1 a )x (1 b )x 1 c x

then f(x) is a polynomial of degree(1) 3 (2) 2(3) 1 (4) 0

5. Let A and B be two 2 × 2 matrices. Consider thestatements(i) AB = O A = O or B = O(ii) AB = I2 A = B–1

(iii) (A + B)2 = A2 + 2AB + B2

Then(1) (i) is false, (ii) and (iii) are true(2) (i) and (iii) are false, (ii) is true(3) (i) and (ii) are false, (iii) is true(4) (ii) and (iii) are false, (i) is true

6. If cos sin

Asin cos

, then

(1) A is an orthogonal matrix(2) A is a symmetric matrix(3) A is a skew symmetric matrix(4) None of these

7. If A and B are two skew symmetric matrices oforder n, then(1) AB is a skew symmetric matrix(2) AB is a symmetric matrix(3) AB is a symmetric matrix if A and B commute(4) None of these

8. The inverse of a skew symmetric matrix (if it exists)is(1) a symmetric matrix(2) A skew symmetric matrix(3) a diagonal matrix(4) None of these

9. If 0 1

A1 0

, then AA4 is

(1)0 11 0

(2)1 10 0

(3)0 01 1

(4)0 11 0

10. If A2 = A, the (I + A)4 equals(1) I + 15A (2) I + 7A(3) I + 8A (4) I + 11A





1. Let X be any matrix of order m × n, where n < mand not all element of X = 0, thenSTATEMENT-1 : XXT will be singular matrixandSTATEMENT-2 : XXT will be product of twodeterminants, where each one of them is zero.

2. STATEMENT-1 : The rank of a unit matrix of ordern is nandSTATEMENT-2 : The rank of a non singular matrixof order n is not n

3. STATEMENT-1 : The determinant of a matrix

0 p q p rq p 0 q rr p r q 0

is zero.

andSTATEMENT-2 : The determinant of a skewsymmetric matrix of odd order is zero.

4. STATEMENT-1 : If x, y, z are different from 0 and





a b y c za x b c z 0a x b y c

, then the value of

the expression a b cx y z is 2

andSTATEMENT-2 : If

1 3cos x 1f (x) sin x 1 3cos x

1 sin x 1 , then maximum

value of f(x) is10


1 0 0A 2 1 0

3 2 1

, if U1, U2 and U3 are columns

matrics satisfying.

1 2 3

1 2 2AU 0 , AU 3 , AU 3

0 0 1

and U is 3 × 3

matrix whose columns are U1, U2, U3 then1. The value of |U| is

(1) 3 (2) –3(3) 3/2 (4) 2

2. The sum of elements of U–1 is(1) –1 (2) 0(3) 1 (4) 3

3. The value of 3

[3 2 0] U 20

is

(1) 5 (2) 5/2(3) 4 (4) 3/2

Passage - IILet A be a square matrix. The transpose matrix of

the matrix for the corresponding cofactors of the elemnetsof A is called the adjoint of A, and is written as adj A.

Now answer the following questions

4. of A = (aij)m × n and |A| 0 then (adj A)–1 =

(1) (aij)m × n (2) m n(adj)| A |

(3) adj A (4) None of these5. Adj (Aq1) =

(1) Adj A (2) A(3) |A|n–2 A (4) None of these

6. adj (adj A) =(1) |A|n – 2 A (2) A(3) |A|n – 1 A (4) None of these





10. The Straight Line

LEVEL - I1. The distance between two parallel lines is unity. A

point P lies between the lines at a distance a fromone of the lines. The length of the side of an equilateral PQR, where Q and R lie in the two parallel lines,is

(1)22 a a 1

3 (2)

22 a a 13

(3)21 a a 1

3 (4)

21 a a 13

2. If the centroid and a vertex of an equilateral triangleare (2, 3) and (4, 3) respectively then other twovertices are

(1) (2,3 3) (2) (1, 2 3)

If(3) (1, 3 3) (4) None of these

3. If , , are the real roots of x3 – 3px2 + 3qx – 1 = 0.The centroid of triangle whose vertices are

1,

, 1,

and 1,

are

(1) (p, –q) (2) (–p, q)

(3) (–p, –q) (4) (p, q)

4. If the image of the point (4, –6) by a line mirror is thepoint (2, 2) then equation of the mirror is

(1) x + 4y – 5 = 0 (2) –4x + y + 11 = 0

(3) x – 4y – 11 = 0 (4) None of these

5. If the point (2a–5, a2) is on the same side of the linex + y – 3 = 0 as that of the origin, then the set ofvalues of a is

(1) (–4, 2) (2) (–2, 4)

(3) (1, –2) (4) (2, –3)

6. If two of the lines ax3 + bx2 y + cxy2 + dy3 = 0 makecomplementary angles with x-axis in anticlockwisesense then

(1) a (a – c) – d (b – d) = 0

(2) a (a – c) + d (b – d) = 0

(3) d (a – c) + a (b – d) = 0

(4) None of these

7. The orthocentre of a triangle whose three sides aregiven by (y – 15) (15x2 + 16xy – 48y2) = 0 is

(1) (0, 33) (2) (0, –33)

(3) (33, 0) (4) (–33, 0)

8. Consider the lines y – y1 = m (x – x1) where m andx1 are fixed, then for different values of y1

(1) the lines will pass through a fixed point

(2) all the lines intersect at x = x1

(3) there will be a set of parallel lines

(4) None of these

9. Given the equation 2 24x 2 3xy 2y 1 , theangle through which the axes be rotated so that theterm in xy can be removed is

(1)25

(2)23

(3)3

(4) None of these

10. A pair of straight lines, drawn through the origin formwith the line 2x + 3y = 6 an isosceles triangle, rightangled at the origin. The equation of the pair of linesis

(1) 5x2 + 24xy + 5y2 = 0

(2) 5x2 – 24xy + 5y2 = 0

(3) 5x2 + 24xy – 5y2 = 0

(4) 5x2 – 24xy – 5y2 = 0

11. The range of values of in the interval (0, ) such

that the points (3, 5) and (sin ,cos ) lie on thesame side of the line x + y – 1 = 0

(1) 0,4

(2) 0,2

(3) ,4 2

(4) 0,2

12. Through the point P ( , ) where 0 the

straight line x y 1a b is drawn so as to form with

coordinate axes a triangle of area s. If ab >0, thenthe least value of s is

(1) (2) 2





(3) 4 (4) 6

13. The vertices of a OBC are 0(0, 0), B(–3, –1) andC(–1, –3). The equation of a line parallel to BC andintersecting sides OB and OC whose distance from

the origin is 12

is

(1)1x y 02

(2)1x y 02

(3)1x y 02

(4)1x y 02

14. The image of the point (4, –13) in the line5x + y + 6 = 0 is

(1) (1, 2) (2) (–4, 15)

(3) (–1, –14) (4) (3, 4)

15. If the slope of one of the l ines given byax2 + 2hxy + by2 = 0 be the square of the other then

(1) ab (a+b) + 3abh + 4h3 = 0

(2) ab (a + b) + 6abh + 8h3 = 0

(3) ab (a + b) – 6abh + 8h3 = 0

(4) none of these

16. If pair of straight lines x2 – 2pxy – y2 = 0 andx2 – 2qxy – y2 = 0 such that each pair bisects theangle between the other pair then

(1) pq = –1 (2) pq = 1

(3) pq = –2 (4) pq = 2

17. The point of intersection of the pair of straight linesgiven by 6x2 + 5xy– 4y2 + 7x + 13y – 3 = 0 is

(1) (–1, –1) (2) (–1, 1)

(3) (1, –1) (4) (1, 1)

18. The equation to the straight line passing through thepoint (2,3) and equally inclined to the lines 3x – 4y –7 = 0 and 12x – 5y + 6 = 0 is

(1) 7x + 9y – 73 = 0

(2) 9x + 7y – 73 = 0

(3) 7x – 9y + 73 = 0

(4) 9x – 7y + 73 = 0

19. A vertex of an equilateral triangle is (2,3) and theopposite side is x + y = 2. Then one of the equationsof other sides is

(1) 2 3 x y 1 2 3

(2) 2 3 x y 1 2 3

(3) 2 3 x y 1 2 3

(4) 2 3 x y 1 2 3

20. Let A be the fixed point (0, 4) and B(2t, 0) be a movingpoint. Let M be the mid-point of AB and theperpendicular bisector of AB meet the y-axis at R.The locus of the mid-point P of MR is

(1) 2 2 1x y4

(2) 2 2y 2 x 4

(3) 2y x 2

(4) 2 23x y 8

LEVEL - II1. In ABC let the vertices A and B are (2, –3) and

(–2, 1) respectively. If the centroid of this trianglemoves on the lines 2x + 3y = 1, then the locus of Cis the line

(1) 3x + 2y = 5 (2) 3x – 2y = 3

(3) 2x + 3y = 9 (4) 2x – 3y = 7

2. The points (1, 1), (2, 2), (3, –1) and (5, 1) are thevertices of a

(1) Rhombus (2) Rectangle

(3) Trapezium (4) None of these

3. Let P (2, 0) and Q(0, 2) be two points and O be theorigin. If A(x, y) is a point such that xy > 0 and x + y< 2 then

(1) A cannot be inside the OPQ

(2) A lies either inside of OPQ or in the thirdquadrant





(3) A lies outside the OPQ

(4) None of these

4. A point equidistant from the lines 4x + 3y + 10 = 0,5x – 12y + 26 =0 and 7x + 24 y – 50 = 0 is

(1) (1, –1) (2) (0, 0)

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

5. The base of a triangle lies along the line x = a and itslength is a. If its area is a2 then third vertex lies onthe line

(1) x = –3a (2) x = a

(3) x = 3a (4) none of these

6. a, b, c are in A.P, then the line ax + by + c = 0

(1) has a fixed direction

(2) always pass through a fixed point

(3) form a triangle with the axes whose area isconstant

(4) none of these

7. The equation 2x2 + kxy + 2y2 = 0 represent a pair ofreal and distinct lines, if

(1) k [ 4,4] (2) k R

(3) k ( , 4) (4, ) (4) none of these

8. The equation of the image of the lines y = |x| by theline x = 2 is(1) y = |x + 4| (2) |y + 4| = y(3) |y – 4| = x (4) y = |x – 4|

9. If one of the lines given by 6x2 – xy + 4cy2 = 0 is3x + 4y = 0 then c is(1) –1 (2) 1(3) –3 (4) 3

10. If (1, 1) and 3 3,2 4

are the orthocentre and

circumcentre of a triangle respectively then itscentroid is

(1)4 5,3 6

(2)4 5,3 6

(3)4 5,5 6

(4)4 5,3 6





1. STATEMENT-1 : The straight line (sin + 3 cos)x

+ ( 3 sin – cos) y + (5 sin – 7 cos) = 0 passesthrough the point of intersection of the lines

x 3 5 0 and 3x y 7 0 for all valuesof except = n/2, n is an integer.andSTATEMENT-2 : L1 + L2 = 0 represents theequation of a line through the points of intersectionof the lines L1 = 0 and L2 = 0 for all non-zero finitevalues of .

2. STATEMENT-1 : One side of a rectangle lies alongthe line 4x + 7y + 5 = 0. Two of its vertices are(–3, 1) and (1, 1). Equation of the side farthest fromthe origin is 7x – 4y + 25 = 0andSTATEMENT-2 : If a and b are constants(a, b 0) and c is a variable, then from the linesax + by + c = 0 and bx – ay – c = 0 the line farthestfrom the origin x + y = 2 one for which |c| is least

3. STATEMENT-1 : If the area of the triangle formedby the lines y = x, x + y = 2 and the lines throughP(h, k) parallel to x-axis is 4h2, then P lies on2x ± y 1 = 0andSTATEMENT-2 : Area of the triangle formed bythe lines y = x, x + y = 2 and the x-axis is equal tohalf the area of the triangle formed by the line x +





y = 2 and the coordinate axes

4. STATEMENT-1 : If the origin is shifted to thecentroid of the triangle with vertices (0, 0), (3, 3)and (3, 6) without rotation of axes then the verticesof the triangle in the new system of coordinatesare (–2, 0), (1, 3) and (1, –3).andSTATEMENT-2 : If the origin is shifted to the point(2, 3) without rotation of the axes then thecoordinates of the point P( – 1, + 1) in the newsystem of coordiantes are ( – 3, – 2).


L1 : 3x + 4y + 8 = 0L2 : 2x + 7y – 1 = 0

1. If L1, L2 represent the sides AB and AC of theisosceles triangle ABC and AB = AC = 2 then thecoordiantes of(1) B are (28/5, – 11/5)(2) B are (28/5, 1/5)

(3)14 4 53 4 4 5C are ,

53 5

(4)14 4 53 4 4 5C are ,

53 5

2. Equation of the line through B parallel to AC is(1) 2x + 7y + 21 = 0 (2) 10x + 35y = 63(3) 10x + 35y + 21 = 0 (4) 2x + 7y = 63

3. If D is the mid-point of BC and E is the mid-pointof CA then DE is equal to(1) 1/4 (2) 1/2(3) 1 (4) 2

Passage - IITwo adjacent sides of a parallelogram are

4x + 5y = 0 and 7x + 2y = 04. If an equation to one of the diagonals is

11x + 7y – 9 = 0, then equation of the other diagonalis(1) x + y = 0 (2) 7x – 11y = 0(3) x – y = 0 (4) 11x + 7y = 0

5. Vertices of the parallelogram are(1) (0, 0) (2) (1, 1)(3) (5/3, –4/3) (4) (–2/3, 7/3)

6. Area of the parallelogram is(1) 3 (2) 3/2(3) 6 (4) 2 85 / 3





11. Circle

LEVEL - I1. The equation ax2 + by2 + 2hxy + 2gx + c = 0

represents a circle, the condition will be(1) a = b and c = 0 (2) f = g and h = 0(3) a = b and h = 0 (4) f = g and c = 0

2. The equation x2 + y2 + 4x + 6y + 13 = 0 represents(1) a circle(2) a pair of two distinct straight lines(3) a pair of coincident straight line(4) a point circle

3. The area of circle centred at (1, 2) and passing through(4, 6) is(1) 5 (2) 10(3) 25 (4) None of these

4. If (–3, 2) lies on the circle x2 + y2 + 2gx + 2fy + c =0 which is concentric with the circlex2 + y2 + 6x + 8y – 5 = 0, then c is(1) 11 (2) –11(3) 24 (4) None of these

5. Equation of a circle through (–1, –2) and concentricwith the circle x2 + y2 – 3x + 4y – c = 0 is(1) x2 + y2 – 3x + 4y – 1 = 0 (2) x2 + y2 – 3x +4y = 0(3) x2 + y2 – 3x + 4y + 2 = 0 (4) None of these

6. Points (2, 0), (0, 1), (4, 5) and (0, a) are concyclic fora =(1) 14/3 or 1 (2) 14 or 1/3(3) –14/3 or –1 (4) None of these

7. The equation of a circle passing through the point(4, 5) having the centre at (2, 2) is(1) x2 + y2 + 4x + 4y – 5 = 0(2) x2 + y2 – 4x – 4y – 5 = 0(3) x2 + y2 – 4x = 13(4) x2 + y2 – 4x – 4y + 5 = 0

8. If the line y = x + 3 meets the circle x2 + y2 = a2 at A and B,then equation of the circle which has AB as diameter is

(1) x2 + y2 + 3x – 3y – a2 + 9 = 0(2) x2 + y2 – 3x + 3y – a2 + 9 = 0(3) x2 + y2 + 3x + 3y – a2 + 9 = 0(4) None of these

9. If the coordinates at one end of a diameter of thecircle x2 + y2 – 8x – 4y + c = 0 are (–3, 2), then thecoordinates at the other end are(1) (5, 3) (2) (6, 2)(3) (1, –8) (4) (11, 2)

10. The locus of the centre of a circle of radius 3 whichrol ls on the outside of the circlex2 + y2 + 3x – 6y – 9 = 0 is(1) x2 + y2 + 3x – 6y + 5 = 0(2) x2 + y2 + 3x – 6y – 31 = 0(3) x2 + y2 – 6y + 29/4 = 0(4) None of these

11. The equation of the circle which has two normals (x– 1) (y – 2) = 0 and a tangent 3x + 4y = 6 is(1) x2 + y2 – 2x – 4y + 4 = 0(2) x2 + y2 + 2x – 4y + 5 = 0(3) x2 + y2 = 5(4) (x – 3)2 + (y – 4)2 = 5

12. Equation of the circle through origin which cutsintercepts of length a and b on axes is(1) x2 + y2 + ax + by = 0(2) x2 + y2 – ax – by = 0(3) x2 + y2 + bx + ay = 0(4) None of these

13. Equation of the diameter of the circle x2 + y2 – 2x +4y = 0 which passes through the origin is(1) x + 2y = 0 (2) x – 2y = 0(3) 2x + y = 0 (4) 2x – y = 0

14. The line x + 3y = 0 is the diameter of the circle(1) x2 + y2 + 6x + 2y = 0(2) x2 + y2 – 6x + 2y = 0(3) x2 + y2 – 6x – 2y = 0(4) x2 + y2 + 8x – 2y = 0

15. The radius of the circle passing through the point(6, 2), two of whose diameters are x + y = 6 andx + 2y = 4 is

(1) 10 (2) 2 5(3) 6 (4) 4

16. The line joining (5, 0) to (10 cos, 10 sin) is dividedinternally in the ratio 2 : 3 at P. If varies, then thelocus of P is(1) a pair of straight lines(2) a circle(3) a straight line (4) None of these

17. If (x, 3) and (3, 5) are the extremities of a diameter ofa circle with centre at (2, y), then the value of x andy are(1) x = 1, y = 4 (2) x = 4, y = 1(3) x = 8, y = 2 (4) None of these





18. Centre of a circle is (2, 3). If the line x + y = 1 touchesthe circle then its equation is(1) x2 + y2 – 4x – 6y + 4 = 0(2) x2 + y2 – 4x – 6y + 5 = 0(3) x2 + y2 – 4x – 6y – 5 = 0(4) None of these

19. The equation of a circle whose centre lies on 3x – y– 4 = 0 and x + 3y + 2 = 0 and has an area 154square units is(1) x2 + y2 – 2x + 2y – 47 = 0(2) x2 + y2 + 2x – 2y – 47 = 0(3) x2 + y2 – 2x + 2y + 47 = 0(4) None of these

20. Length of the chord on the line 4x – 3y – 10 = 0 cutoff by the circle x2 + y2 – 2x + 4y – 20 = 0 is(1) 10 (2) 6(3) 12 (4) None of these

LEVEL - II1. The equation of the line passing through the points

of intersection of the circles x2+ y2 – 2x – 4y – 4 = 0and x2 + y2 – 8x – 12y + 51 = 0 is(1) 6x + 8y – 55 = 0 (2) 6x + 8y + 55 = 0(3) No such line (4) None of these

2. If 3x + y = 0 is a tangent to the circle which has itscentre at the point (2, –1), then the equation of theother tangent to the circle from the origin is(1) x – 3y = 0 (2) x + 3y = 0(3) 3x – y = 0 (4) x + 2y = 0

3. A variable chord is drawn through the origin to thecircle x2 + y2 – 2ax = 0. Locus of the centre of thecircle drawn on this chord as diameter is(1) x2 + y2 + ay = 0 (2) x2 + y2 – ay = 0(3) x2 + y2 – ax = 0 (4) x2 + y2 + ax = 0

4. The length of the chord of the circle x2 + y2 = 25,joining the points, tangents at which intersect at anangle of 120o is(1) 5 (2) 5/2(3) 10 (4) None of these

5. Equation of the circle cutting orthogonally the threecircles x2 + y2 – 2x + 3y – 7 = 0, x2 + y2 + 5x – 5y +9 = 0 and x2 + y2 + 7x – 9y + 29 = 0 is(1) x2 + y2 – 16x – 18y = 4(2) x2 + y2 – 7x + 11y + 6 = 0(3) x2 + y2 + 2x – 8y + 9 = 0(4) None of these

6. Equation of circle drawn on the chord 2x + 3y = 13 ofthe circle x2 + y2 = 13 as diameter is(1) x2 + y2 – 4x – 6y + 13 = 0(2) x2 + y2 + 4x + 6y + 13 = 0(3) x2 + y2 – 4x + 6y + 13 = 0(4) None of these

7. The equation of the circle on the chord x cos + ysin – p = 0, of the circle x2 + y2 – a2 = 0, (0 < p < a)as diameter, is(1) x2 + y2 – a2 + 2p (x cos + y sin – p) = 0(2) x2 + y2 – a2 – 2p (x cos + y sin – p) = 0(3) x2 + y2 + a2 – 4p (x cos + y sin + p) = 0(4) x2 + y2 – a2 + 4p (x cos + y sin – p) = 0

8. The circle x2 + y2 – 6x – 10y + p = 0 does not touchor intersect the axes and the point (1, 4) is insidethe circle, then(1) 0 < p < 29 (2) 25 < p < 35(3) 25 < p < 29 (4) None of these

9. Values of p and q for which the equation (x + y – 41)2

– (x + 7y – 7) (px + py + 1) = 0 represents a circle,are given by(1) p = 7/25, p = 1/25 (2) p = 1/25, p = 7/25(3) p = –7/25, p = –1/25 (4) None of these

10. If OA and OB are the tangents from origin to thecircle x2 + y2 + 2gx + 2fy + c = 0 (c < 0) and C is thecentre of the circle, the area of the quadrilateral OACBis

(1) 2 21 [c(g f c)]2

(2) 2 2[c(g f c)]

(3) 2 2c (g f c) (4) None of these





1. STATEMENT-1 : (n 3) for n circles the value ofn for which the number of radical axes if equal to





the number of radical centre is 5andSTATEMENT-2 : If no two of n circle areconcentric, No three of the centre are collinear thennumber of possible radical centres is nc3

2. STATEMENT-1 : The circle x2 + y2 + 2ax + c = 0,

x2 + y2 + 2by + c = 0 touch if 2 2

1 1 1ca b

andSTATEMENT-2 : Two circles with centre C1, C2and radii r1, r2 touch each other if r1 ± r2 = C1 C2

3. STATEMENT-1 : Number of circles passingthrough (1, 4) (2, 3), (–1, 6) is 1andSTATEMENT-2 : Through 3 non collinear points ina plane only one circle can be drawn

4. STATEMENT-1 : The circle described on thesegment joining the points (–2, –1) 0, –3) asdiameter cuts the circle x2 + y2 + 5x + y + 4 = 0orthogonally.andSTATEMENT-2 : (–2, –1) and (0, –3) are conjugatepoints with respect to the circlex2 + y2 + 5x + y + 4 = 0


C1 : x2 + y2 – 4 = 0C2 : x2 + y2 – 14x + 40 = 0

1. P is point on C1 farthest from the circle C2. Anequation of a pair of tangents from P to C2 is(1) x2 + 8y2 – 4x + 4 = 0(2) x2 – 8y2 + 4x + 4 = 0(3) x2 – 8y2 + 4x – 4 = 0(4) None of these

2. Angle of intersection between the tangents from Pto C2 is given by

(1) tan–1 (4/9) (2) 1tan (2 2 /3)

(3) 1tan (4 2 / 7) (4) tan–1 (2)3. Equation of the pair of the lines through the centre

of C2 perpendicular to the pair of tangents from Pto C2 is(1) 72x2 – 9y2 – 1008x + 3538 = 0(2) 72x2 – 9y2 + 1008x – 3538 = 0

(3) 72x2 – 9y2 – 1008x – 3546 = 0(4) None of these

Passage - IIL : 6x2 + xy – 12y2 = 0C : x2 + y2 = 1, S : x2 – y2 = 1

4. A circle of radius 3 passes through the centre of C,its centre lies on a line of L which makes an acuteangle with the positive direcion of x-axis, thecoordinates of the centre of the circle are(1) (12/5, 9/5) (2) (9/5, 12/5)

(3) (9 / 13, 6 / 13) (4) (6 / 13, 9 / 13)5. Parametric coordinates of a point on S are (sec ,

tan). If Pi (sec i, tan i), i = 1, 2, 3, 4 are the

points of intersection of L and S. Then 4

ii 1

tan

is

equal to(1) 73/35 (2) 73/36(3) 0 (4) None of these

6. Area of the rectangle formed by joining the pointsof intersection of the lines L and the circle C is

(1) 14 / 5 3 sq. units (2) 17 /5 3 sq. units

(3) 34 / 5 3 sq. units (4) 7 / 5 3 sq. units





12. Conic SectionsLEVEL - I

1. If the line x – 1 = 0 is the directrix of the parabolay2 – kx + 8 = 0 then one of the values of k is(1) 4 (2) 1/4(3) 8 (4) –8

2. The normal drawn at a point (t1) of the parabolay2 = 4ax meets it again at the point (t2) then

(1) 21 1 2t t t 2 0 (2) 2

1 1 2t t t 2 0

(3) 21 1 2t t t 2 0 (4) 2

1 1 2t t t 2 0 3. A is a point on parabola y2 = 4ax. The normal at A

cuts the parabola again at B. If AB subtends a rightangle at the vertex then slope of AB is

(1) 5 (2) 2

(3) 3 (4) 2

4. The normals at the three points P, Q, R of the parabolay2 = 4ax meet in (h, k). The centroid of PQR lieson(1) x = a (2) y = a(3) y = 0 (4) x = 0

5. If y x 3 3 cuts the parabola 2y x 2 at P

and Q and A is 3,0 then AP.AQ is

(1)4 (2 3)3

(2)4 (2 3)3

(3)2 (2 3)3

(4)2 (2 3)3

6. The latus rectum of a parabola whose focal chordPSQ is such that SP = 3 and SQ = 2 is given by(1) 24/5 (2) 12/5(3) 6/5 (4) 48/5

7. If tangents at A,B on parabola y2 = 4ax intersect atT, then ordinates of A, T and B are in(1) A.P. (2) G.P.(3) H.P. (4) None of these

8. The locus of the points of trisection of the doubleordinates of the parabola y2 = 4ax is(1) 4x2 = 9ay (2) 9x2 = 4ay(3) 9y2 = 4ax (4) none of these

9. The intersection of tangents at the ends of the latusrectum of the parabola y2 = 4x is(1) (0, 0) (2) (1, 1)(3) (–1,0) (4) (–1,1)

10. If focus is at S(–1, 2) and directrix is x – y–1 = 0,then the equation of the parabola is(1) x2 + y2 + 3x – 7 = 0(2) x2 + y2 + 2xy + 6x –10 y + 9 = 0(3) y2 + 6x + y + 7 = 0(4) none of these

11. If e1,e2 are the eccentricities of 9x2 + 4y2 = 36 and9x2 – 4y2 = 36 then

(1) 2 21 2e e 3 (2) 2 2

2e e 4

(3) 2 21 2e e 2 (4) none of these

12. Let C is the circle x2 + y2 = 9 and E is the ellipse4x2 + 9y2 = 36. P and Q are the points having thecoordinates (1, 2) and (2, 1) respectively. Then(1) P lies inside both C and E

(2) P lies inside C but outise E(3) Q lies inside but outside E(4) none of these

13. The radius of the circle passing through the foci of

the ellipse 2 2x y 1

16 9 , having its centre (0, 3) is

(1) 3 (2) 4(3) 5 (4) 6

14. A man running round a race course notes that thesum of the distances of two flag-posts from him isalways 10 metres and distance between the flagposts is 8 metres. The area of the path he enclosesin square metres is

(1) 8 (2) 10

(3) 13 (4) none of these

15. If P(x, y), A(3, 0), B(–3, 0) and 16x2 + 25y2 = 400then PA + PB =(1) 5 (2) 10(3) 15 (4) 20

16. The angle between the pair of tangents drawn to theellipse 3x2 + 2y2 = 5 from the point (1, 2) istan–1() then is(1) 12/5 (2) 8/5(3) 12/ 5 (4) 8/ 5





17. Which of the following points lies inside the ellipse2 29(x 1) 16y 25

(1)1 3,4 2

(2)3 ,12

(3)1 5,2 4

(4) None f these

18. If the mid point of a chord of the ellipse 2 2x y 1

16 25

is (0,3) then the length of the chord is(1) 4/5 (2) 12(3) 32/5 (4) 16

19. Tangents are drawn from the points on the linex – y + 2 = 0 to the ellipse x2 + 2y2 = 4, then all thechord of contact pass through the point(1) (0, –2) (2) (1, 1)(3) (–2, 1) (4) (–1, 2)

20. The ellipse 2 2

2 2

x y 1a b

passes through the point

(1, –2) and has eccentricity. 1/ 2 then its latusrectum is(1) 2 (2) 3(3) 2 (4) 3

21. If the foci of the ellipse 2 2

2

x y 116 b

and the hyperbola2 2x y 1

144 81 25 coincides, then the value of b2 =

(1) 7 (2) 5(3) 3 (4) 1

22. If 2 2

2 2

x y 1a b

(a > b) and x2 – y2 = c2 cut at right

angles then

(1) a2 – b2 = 2c2 (2) 2 2 2a b 2c

(3) 2 2 2b a 2c (4) none of these23. If x = 9 is the chord of contact of the hyperbola

x2 – y2 = 9, then the equation of the coorespondingpair of tangents is(1) 2 29x 8y 18x 9 0 (2) 2 29x 8y 18x 9 0 (3) 2 29x 8y 18x 9 0 (4) None of these

24. If e is the eccentricity of the hyperbola 2 2

2 2

x y 1a b

and is the angle between the asymptotes then

cos 2

is

(1) e (2) e

(3)1e

(4)1e

25. The equation of the tangents at the point whoseparametric angle is / 6 on the hyperbola4x2 – 9y2 = 3 is(1) 4x + 3y = 3 (2) 4x – 3y = 3(3) 3x – 4y = 3 (4) none of these

26. If the coordinates of a point are (4 tan ,3sec )

where is parameter, then point lies on a conicsection whose eccentricity is

(1)53

(2)34

(3)35

(4) None of these

27. Equation of tangent to the hyperbola 2x2 – 3y2 = 6which is parallel to the line y = 3x + 4(1) y = 3x – 5 (2) y = 3x + 5(3) y = 3x ± 5 (4) none of these

28. If eccentricities of xy = 1, xy = –4, 2 2x y 1 and2 2y x 1 are respectively e1, e2, e3 & e4. Then

1 4 2 3

1 3 2 4

e e e ee e e e

(1) 1 (2) 0

(3) 1/2 (4) 229. The equation of common tangent to the parabolas y2

= 2x and x2 = 16y is(1) 2x + y + 1 = 0 (2) x + 2y + 2 = 0(3) x + y + 3 = 0 (4) x – y + 4 = 0

30. If the normal at (ct1, c/t1) on the rectangular hyperbolaxy = c2 meets the curve again at a point (ct2, c/t2)then

(1) 2 31

1tt

(2) 2 31

1tt

(3) 2 21

1tt

(4) 2 21

1tt

LEVEL - II1. If (0, 4) and (0, 2) are respectively the vertex and

focus of a parabola, then its equation is(1) x2 – 8y = 32 (2) x2 + 8y = 32(3) y2 – 8x = 32 (4) y2 – 8x = 32

2. The curved described parametrically by x = t2+t +1,y = t2 – t + 1 represents(1) pair of straight lines (2) parabola(3) ellipse (4) hyperbola





3. Two tangents are drawn from the point (–2, –1) to aparabola y2 = 4x. If is the angle between them,then tan (1) 1/3 (2) 3(3) 1/2 (4) 2

4. The tangents at the points 21 1(at ,2at ) and

22 2(at ,2at ) on the parabola y2 = 4ax are at right

angles if t1 t2 =(1) 2 (2) –2(3) 1 (4) –1

5. If (2, –8) is one end of a focal chord of the parabolay2 = 32x, then other end of the chord is(1) (32, 32) (2) (–32, –32)(3) (–32, 32) (4) none of these

6. The foci of the ellipse 25(x + 1)2 + 9 (y+2)2 = 225 areat(1) (–1, –2) and (–2, –1)(2) (–1,2) and (–1, –6)(3) (–2, 1) and (–2, 6)(4) none of these

7. In an ellipse the distance between the foci is 8 andthe distance between the directrices is 25. The lengthof the major axis is

(1) 5 2 (2) 10 2

(3) 15 2 (4) 20 2

8. If 2 2x y 1

10 a 4 a

represents an ellipse then

(1) a > 4 (2) 4 < a < 10(3) a < 4 (4) none of these

9. On the ellipse 4x2 + 9y2 = 1, the points at which thetangents are parallel to the line 8x = 9y are

(1)2 1,

5 5

(2)

2 1,5 5

(3)2 1,5 5

(4) none of these

10. S and S' are the foci of an ellipse and B is an end ofthe minor axis. If SS' B is an equilateral triangle thenits eccentricity is(1) 1/4 (2) 1/3(3) 1/2 (4) none of these

11. Which of the following is independent of in the

hyperbola 2 2

2 2

x y 1cos sin

(0 / 2)

(1) abscissa of foci (2) directrix(3) vertex (4) none of these

12. The centre of the hyperbola9x2 – 36x – 16y2 + 96y – 252 = 0(1) (–2, 3) (2) (2, 3)(3) (2, –3) (4) (–2, 3)

13. If e and e ' are the eccentricities of two conics S andS' such that 2 2e e ' 3 then S and S' are the(1) parabolas (2) hyperbolas(3) ellipse (4) none of these

14. The equation of the tangent to the hyperbola4y2 = x2 – 1 at the point (1,0) is(1) x = –2 (2) y = –2(3) x = –1 (4) x = 1

15. Equation of the chord of the hyperbola25x2 – 16y2 = 400 which is bisected at the point(6, 2) is(1) 16x – 75y = 418 (2) 16x + 75y = 418(3) 75x – 16y = 418 (4) 75x + 16y = 418



(2) Statement-1 is True, Statement-2 is True;Statement-2 is not a correct explanation forStatement-1.


1. STATEMENT-1 : The curve 2xy x 1

2 is

symertical with respect to the line x = 1andSTATEMENT-2 : A parabola is symmetric aboutits axis.

2. y2 = 4x is the equation of a parabola

STATEMENT-1 : Through (, + 1), 3 normalscan be drawn to the parabola, if < 2and





STATEMENT-2 : The point (, + 1) lies outsidethe parabola for all –1

3. The tangent at a point P on the ellipse 2 2

2 2x y 1a b

,

which is not an extremity of major axis meets adirectrix at T.STATEMENT-1 : The circle on PT as diameterpasses through the focus of the ellipse correspondingto the diretrix on which T lies.andSTATEMENT-2 : PT subtends a right angle of thefocus of the ellipse corresponding to the directrixon which T lies.

4. STATEMENT-1 : If the normal at an end L of alatus ractum of the ellipse x2/a2 + y2/b2 = 1 meetsthe major axis at G, O is the centre of the ellipse,then OG = ae3, e being the eccentricity of the ellipse.andSTATEMENT-2 : The normal at a point on theellipse x2/a2 + y2/b2 = 1 never passes through thefoci.

5. STATEMENT-1 : If the foci of an hyperbola are atpoints (4, 1) and (–6, 1), eccentricity is 5/4, then thelength of the transverse axis is 4.andSTATEMENT-2 : Distance between the foci of ahyperbola is equal to the product of its eccentricityand the length of the transverse axis.


C : y = x2 – 3, D : y = kx2,L1 : x = a, L2 : x = 1. 9a 0)

1. If the parabolas C and D intersect at a point Aonthe line L1, then equation of the tangent line L at Ato the parabola D is(1) 2(a3 – 3) x – ay + a3 – 3a = 0(2) 2(a3 – 3) x – ay – a3 + 3a = 0(3) (a3 – 3) x – 2ay – 2a3 – 6a = 0(4) None of these

2. If the line L meets the parabola C at a point B onthe Line L2, other than A then a is equal to(1) –3 (2) –2(3) 2 (4) 3

3. If a > 0, the angle subtended by the chord AB atthe vertex of the parabola C is

(1) tan–1 (5/7) (2) tan–1 (1/2)(3) tan–1 (2) (4) tan–1(1/8)

Passage - II

P : 2 2

2 x yy 8x, E : 14 15

4. Equation of a tangent common to both the parabolaP and the ellipse E is(1) x – 2y + 8 = 0 (2) x + 2y + 8 = 0(3) x + 2y – 8 = 0 (4) x – 2y – 8 = 0

5. Equation of the normal at the point of contact ofthe common tangent, which makes an acute anglewith the positive direction of x-axis, to the parabolaP is(1) 2x + y = 24 (2) 2x + y + 24 = 0(3) 2x + y = 48 (4) 2x + y + 48 = 0

6. Point of contact of a common tangent to P and Eon the ellipse is(1) (1/2, 15/4) (2) (–1/2, 15/4)(3) (1/2, –15/4) (4) (–1/2, –15/4)

Passage - III

H : x2 – y2 = 9, P : y2 = 4 (x – 5), L : x = 9

7. If L is the chord of contact of the hyperbola H,then the equation of the corresponding pair oftangents is(1) 9x2 – 8y2 + 18x – 9 = 0(2) 9x2 – 8y2 – 18x + 9 = 0(3) 9x2 – 8y2 – 18x – 9 = 0(4) 9x2 – 8y2 + 18x + 9 = 0

8. If R is the point of intersection of the tangents to Hat the extremities of the chord :, then equation ofthe chord of contact of R with respect to theparabola P is(1) x = 7 (2) x = 9(3) y = 7 (4) y = 9

9. If the chord of contact of R with respect to theparabola P meets the parab ola at T and T, S is thefocus of the parabola, then Area of the triangle STTis equal to(1) 8 sq. units (2) 9 sq. units(3) 12 sq. units (4) 16 sq. units





13. Vector Algebra

LEVEL – I1. For three vectors u, v,w which of the folowing

expressions is not equal to any of the remainingthree?

(1) u.(v w) (2) (v w).u

(3) v.(u w) (4) (u v).w

2. The value of x for which the angle between2ˆ ˆ â 2x i 4x j k

and ˆ ˆ ˆb 7i 2 j xk

is obtuse

and the angle between b

and the z-axis is acuteand less than / 6 , are(1) – < x < 1/2 (2) 1/2 < x < 15(3) x > 1/2 or x<0 (4) none of these

3. Let O be the circumcenter, G be the centroid andO' be the orthocentre of a ABC . Three vectorsare taken through O and are represented by

a OA,b OB and c OC

then a b c is

(1) OG

(2) 2OG

(3) OO'

(4) none of these

4. Let a,b and c be three non-zero vectors, no two

of which are collinear. If the vector a 2b is

collinear with c and 2b 3c is collinear with a ,

then a 2b 6c is equal to

(1) a (2) b

(3) c (4) 0

5. The vectors

ˆ ˆ â xi (x 1) j (x 2)k

ˆ ˆ ˆb (x 3)i (x 4) j (x 5)k

and

ˆ ˆ ˆc (x 6)i (x 7) j (x 8)k are coplanar for

(1) all values of x (2) x<0 only(3) x >0 only (4) none of these

6. If the vector ˆ ˆ î j k bisects the angle between

the vector c and the vector ˆ ˆ3i 4 j , then the unitvector in the direction of c is

(1)1 ˆ ˆ ˆ(11i 10j 2k)

15

(2)1 ˆ ˆ ˆ(11i 10j 2k)

15

(3)1 ˆ ˆ ˆ(11i 10j 2k)

15

(4)1 ˆ ˆ ˆ(11i 10j 2k)

15

7. If a vector a is expressed as the sum of two vectors

and

along and perpendicular to a given vector

b then

is equal to

(1) 2

(a b) b| b | (2) 2

b (a b)| b |

(3) 2

a (a b)| b |

(4) 2

a.b .b| b |

8. The vectors a = ˆ ˆ ˆ3i 2 j 2k and ˆ ˆb i 2k are

the adjacent sides of a parallelogram. Then the acuteangle between its diagonals is(1) / 4 (2) / 3(3) 3 / 4 (4) 2 / 3

9. The length of the longer diagonal of the parallelogramconstructed on 5a 2b

and a 3b if it is given

that | a | 2 2,| b | 3 and angle between

a and b is4

is

(1) 15 (2) 113(3) 593 (4) 369

10. If three vectors a,b,c are such that a 0

and

a b 2(a c),| a | | c | 1,| b | 4 and the angle

between b and c is cos–1 1

4

, then b 2c a

where is equal to(1) ± 4 (2) – 2(3) ± 3 (4) –2





11. Let ˆ ˆ â 2i j 2k and ˆ ˆb i j

. If c is a vector

such that a.c | c |,| c a | 2 2 and the angle

between a b and c is 30°, then | (a b) c |

=

(1) 2/3 (2) 3/2(3) 2 (4) 3

12. Unit vectors coplanar with ˆ ˆ î j 2k and ˆ ˆ î 2 j k

and perpendicular to ˆ ˆ î j k are

(1)1 ˆ ˆ( j k)2

(2)1 ˆ ˆ(i k)2

(3)1 ˆ ˆ(i j)2

(4)1 ˆ ˆ( j k)2

13. If a, b, c are the pth, qth and rth terms of a G.P. then

the angle between the vector ˆ û (loga)i (log c)k

and ˆ ˆ ˆv (q r)i (r p) j (p q)k is

(1)3

(2)6

(3) (4)2

14. If ˆ â i j , ˆ ˆb 2 j k

and r a b a

,

r b a b then a unit vector in the direction of r is

(1)1 ˆ ˆ ˆ(i 3j k)11

(2)1 ˆ ˆ ˆ(i 3j k)11

(3)1 ˆ ˆ ˆ(i j k)11

(4) None of these

15. Let ˆ ˆ â 2i 3j k and ˆ ˆ ˆb i 2 j 3k

. Then the

value of for which the vectorˆc i j (2 1)k

is parallel to the plane

containing a and b , is

(1) 1 (2) 0(3) –1 (4) 2

16. The component vector of the vector a along thevector b

is

(1)(a.b) b| b |

(2) (a.b)b

(3) 2

(a.b) b| b |

(4) None of these

17. If a,b,c are three non-coplanar vectors represented

by non-current edges of a parallelopiped of volume4 units, then the value of

(a b).(b c) (b c).(c a) (c a).(a b) is

(1) 12 (2) 4(3) ±12 (4) 0

18. Let ˆ ˆ ˆb 4i 3j and c be a vector perpendicular to

b and lying in the xy-plane. A vector in the xy-

plane having projections 1 and 2 along b and c is

(1) ˆ ˆ2i j (2) ˆ î 2 j

(3) ˆ ˆ2i 11j (4) None of these

19. If a (a b) b (b c) and a.b 0

, then

[a b c]

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

20. Consider points A, B, C and D with position vectorsˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ7i 4 j 7k, i 6 j 10k, i 3j 4k and

ˆ ˆ ˆ5i j k respectively. Then ABCD is a

(1) Parallelogram but not a rhombus(2) square(3) rhombus(4) rectangle

LEVEL – II1. The number of vectors of unit length perpendicular

to the vectors ˆ â i j and ˆ ˆb j k

is

(1) 1 (2) 2(3) 4 (4) Infinite

2. The values of x for which the angle between the

vectors ˆ ˆ â xi 3j k and ˆ ˆ ˆb 2xi xj k

is

acute, and the angle between the vector b and the

axis of ordinates is obtuse, are





(1)1 ,22

(2) 2,3

(3) all x < 0 (4) all x > 0

3. If x(a b) y(b c) z(c a) and

1a b c8

then x + y + z =

(1) 8 .(a b c) (2) .(a b c)

(3) 8(a b c) (4) none of these

4. Let ˆ ˆ ÔA i 3j 2k

and ˆ ˆ ÔB 3i j 2k

. The

vector OC

bisecting the angle AOB and C being apoint on the line AB is

(1) ˆ ˆ ˆ4(i j k) (2) ˆ ˆ ˆ2(i j k)

(3) ˆ ˆ î j k (4) none of these

5. If 1e (1,1,1) and 2e (1,1, 1)

and a and b are

two unit vectors such that 2e a 2b , then angle

between a and b is

(1)1 7cos

9

(2)1 7cos

11

(3)23

(4)3

6. Let ˆ ˆ ˆ ˆ ˆ â 2i j k,b i 2 j k and a unit vector c

be coplanar If c is perpendicular to a then c =

(1)1 ˆ ˆ( j k)2

(2)1 ˆ ˆ ˆ( i j k)3

(3)1 ˆ ˆ(i 2 j)5

(4)1 ˆ ˆ ˆ(i j k)3

7. For any vector 2 2 2ˆ ˆ ˆr,(r.i) (r.j) (r.k)

(1) 2| r | (2) 22 | r |

(3) 2ˆ ˆ ˆ| r.(i j k) | (4) 3

8. Let a,b,c be three vectors such that a b c

and

c a b , then

(1) 2a.b | c | (2) 2c.a . | b |

(3) b.c |a | (4) a || (b c)

9. Let ˆ ˆ ˆ â i j, b 2 j k , ˆ ˆc j k

. If b is a vector

satisfying a b c and a.b 3

then b

is

(1)1 ˆ ˆ ˆ(5i 2j 2k)3

(2)1 ˆ ˆ ˆ(5i 2j 2k)3

(3) ˆ ˆ ˆ3i j k (4) None of these

10. If a,b,c are any three vectors such that

(a b).c (a b).c 0 , then (a b) c

is

(1) 0 (2) a

(3) b (4) none of these





1. Let a, band c be the position vectors of the vertices

A, B, C of a triangle ABC respectivelySTATEMENT-1 : if for some non-zero vector

1r, r.a r.b r.c2

then the area of the triangle

ABC is [a b c] | r |

andSTATEMENT-2 : The area of the triangle ABC is

given by 1 | a b b c c a |2

2. STATEMENT-1 : If I is the incentre of ABC then

| BC | IA | CA | IB | AB | IC 0 andSTATEMENT-2 : The position vector of centroid





of ABC is OA OB OC

3

3. STATEMENT-1 : If a i j and b j k then

(a b, a b) 90 and

STATEMENT-2 : Projection of a b on a b iszero.

4. STATEMENT-1 :andSTATEMENT-2 :


Suppose that a rigid body is free to rotate about afixed point O. if a force F acts on the body at a point P,then the body will tend to rotate about an axis through O.This effect is measured by the torque vector which isgiven by

= r × FWhere r is the position vector OP. The straight line

through O determined by is the axis of rotation. Thevector r, F and form a right handed system.1. The magnitude of the torque exerted at origin by

the force F = i + 2j + k at the point (1, 1, 1) is(1) 1 (2) 5

(3) 3 (4) 22. A bolt is being tightened by a 20 N force applied to

a 25 cm long wrench at an angle of 60° with thewrench. The magnitude of the torque is(1) 100 3 (2) 500 3

(3) 250 3 (4) 2503. is the torque, F the force and r the position vector

in above examples, then [, r, F] is equal to(1) 6250 (2) 187500(3) 18750 (4) 62500

Passage - IIEquations of Bisectors of the Angle between Two

Vectors : let two unit vectors along two lines OA and

OB be â and b respectively. Take their point of

intersection as the origin and let P be any point on thebisector of angle between the lines OA and OB. DrawPM parallel to AO cutting OB at M.

M P

B

OA

AOP POM OPM

and hence OM = PM

But ÔM tb

and ˆMP ta

(since ˆ ÔM || b and MP || a and their magnitudes aresame)

Then ˆ ÔP r OM MP t(b a) ...(i)

For external bisector OP

, the angle between OBand OA is the same as the internal bisector of the anglebetween the unit vectors along them being b and aand hence the equation of OP

be

ˆÔP r t(a b) ...(ii)

For any two vectors â and b the equations (i) and(ii) reduce to

a br t| a | | b |

4. A vector c , directed along the internal bisector of

the angle between the vectors ˆ ˆ â 7i 4 j 4k and

ˆ ˆ ˆb 2i j 2k

with | c | 5 / 6 is

(1)5 ˆ ˆ ˆ(i 7 j 2k)3

(2)5 ˆ ˆ ˆ(5i 5j 2k)3

(3)5 ˆ ˆ ˆ(i 7 j 2k)3

(4)5 ˆ ˆ ˆ( 5i 5j 2k)3

5. Let ABC be a triangle and a,b,c be the position

vectors of the point A, B, C respectively. Externalbisectors of B and C meet at P with the sidesof the triangle as a, b, c the position vectors of Pbecomes

(1)( b)b ( c)c

(b c)

(2)aa ( b)b ( c)c

(a b c)

(3)a b c (abc)

3

(4)aa bb cc(a b c)





6. If the interior and exterior bisectors of the angle Aof a triangle ABC meet the base BC at D and E,then(1) 2BC = BD + BE (2) BC2 = BD × BE

(3)2 1 1

BC BD BE (4) None of these





LEVEL – I1. If a plane meets the coordinate axes at A, B and C,

in such a way that the centroid of ABC is at thepoint (1, 2, 3), then equation of the plane is

(1)x y z 11 2 3 (2)

x y z 13 6 9

(3)x y z 11 2 3 3 (4) none of these

2. The image of the point P(1, 3, 4) in the plane2x –y +z +3=0 is(1) (3, 5, –2) (2) (–3, 5, 2)(3) (3, –5, 2) (4) (3, 5, 2)

3. A non-zero vector a

is parallel to the line ofintersection of the plane determined by the vectors

i , ˆ î j and the plane determined by the vectors

ˆ î j , ˆ î k . The angle between a

and ˆ ˆ î 2 j 2k is(1) / 3 (2) / 4(3) / 6 (4) none of these

4. The centre of the circle given by ˆ ˆ ˆr. i 2 j 2k 15

and ˆ ˆr j 2k 4

is

(1) (0, 1, 2) (2) (1, 3, 4)(3) (–1, 3, 4) (4) none of these

5. The distance between the line

ˆ ˆ ˆ ˆ ˆ ˆr 2i 2j 3k i j 4k

and the plane

ˆ ˆ ˆr. i 5j k 5

is

(1)109

(2)10

3 3

(3)103

(4) none of these

6. If the plane x y z 12 3 4 , cuts the coordinate axes

in A, B, C then the area of ABC is

(1) 29sq.units (2) 41sq.units

(3) 61sq.units (4) none of these

14. Three Dimensional Geometry7. The equation of the sphere passing through the point

(1, 3, –2) and the circle x2+y2+z2 = 25, x = 0 is

(1) 2 2 2x y z 11x 25 0

(2) 2 2 2x y z 11x 25 0

(3) 2 2 2x y z 11x 25 0

(4) 2 2 2x y z 11x 25 0

8. Let the pair of vector a , b and c , d

eachdetermine a plane. Then the planes are parallel, if

(1) a c b d 0

(2) a c b d 0

(3) a b c d 0

(4) a b . c d 0

9. The coplanar points A, B, C,D are (2–x, 2, 2),(2, 2–y, 2), (2, 2, 2–z) and (1, 1, 1) respectively, then

(1)1 1 1 1x y z (2) x + y + z =1

(3)1 1 1 1

1 x 1 y 1 z

(4) none of these

10. Equation of the plane that contains the lines

ˆ ˆ ˆ ˆ ˆr i j i 2 j k

and

ˆ ˆ ˆ ˆ ˆr i j i j 2k

is

(1) ˆ ˆ ˆr. 2i j 3k 4

(2) ˆ ˆ ˆr i j k 0

(3) ˆ ˆ ˆr. i j k 0

(4) none of these

11. The distance of the point (1, –2, 3) from the plne

x–y +z=5 measured parallel to the line x y z 12 3 6

is

(1) 1 (2) 2(3) 4 (4) none of these

12. The equation of the sphere inscribed in a tetrahedronwhose faces are x = 0, y = 0, z = 0 and x +2y+2z=4is

(1) 2 2 2x y z 4x 4y 4z 8 0

(2) 2 2 2x y z 4x 4y 4z 8 0

(3) 2 2 2x y z 4x 4y 4z 8 0

(4) none of these





13. The point of contact of the plane ˆ ˆ ˆr. 2i 2j k 6

and the sphere ˆ ˆ ˆr i 2 j k 6

is

(1) ˆ3i (2) ˆ ˆ ˆ3i 2 j k

(3) ˆ ˆ î 4 j 2k (4) none of these

14. Distance of the point P c from the line r a b

is

(1) c a b

b

(2) c a .b

b

(3)

2

c a b

b

(4) none of these

15. If a , b and c are three non-coplanar vectors, then

the length of projection of vector a in the plane ofthe vectors b

and c may be given by

(1) a. b c

b c

(2) a b c

b c

(3)a bc

b.c

(4) none of these

16. If P(0,1,0) snf Q (0, 0, 1) are two points, then the

projection of PQ

on the plane x + y + z = 3 is

(1) 2 (2) 3

(3) 2 (4) 3

17. A plane which passes through the point (3, 2, 0) and

contains the line x 3 y 6 z 4

1 5 4

is

(1) x y z 1 (2) x y z 5

(3) x 2y z 1 (4) 2x y z 5

18. Two system of reactangular axes have the sameorigin. If a plane cuts them at distances a, b, c anda ' , b , c from the origin then

(1) 2 2 2 2 2 2

1 1 1 1 1 1 0a b c a b c

(2) 2 2 2 2 2 2

1 1 1 1 1 1 0a b c a b c

(3) 2 2 2 2 2 2

1 1 1 1 1 1 0a b c a b c

(4) 2 2 2 2 2 2

1 1 1 1 1 1 0a b c a b c

19. If a line makes angle , , , with four diagonals

of a cube, then 2 2 2 2cos cos cos cos isequal to(1) 1/3 (2) 2/3(3) 4/3 (4) 8/3

20. The shortest distance between the lines

ˆ ˆ ˆ ˆ ˆ ˆr 5i 7 j 3k 5i 16j 7k

and

ˆ ˆ ˆ ˆ ˆ ˆr 9i 13j 15k 3i 8j 5k

(1) 0 (2) 10 units(3) 14 units (4) 12 units

LEVEL – II1. The points A (5, –1, 1), B (7, –4, 7), C (1, –6, 10)

and D(–1, –3, 4) are the vertices of a(1) trapezium (2) rectangle(3) rhombus (4) square

2. The equation ax + by + c = 0 represents a planeperpendicular to the(1) xy -plane (2) yz-plane(3) zx-plane (4) none of these

3. The plane 2x 1 y 3 z 0 passes throughthe intersection of the planes(1) 2x –y = 0 and y + 3z = 0(2) 2x – y= 0 and y –3z = 0(3) 2x+3z=0 and y = 0(4) none of these

4. If P(3, 2, –4), Q(5, 4, –6) and R(8, 9, –10) arecollinear, then R divides PQ in the ratio(1) 3 : 2 internally (2) 3 : 2 externally(3) 2 : 1 internally (4) 2 : 1 externally

5. A (3, 2, 0), B (5, 3, 2) and C(–9, 6, –3) are thevertices of a triangle ABC. If the bisector of ABCmeets BC at D, then coordinates of D are





(1) (19/8, 57/16, 17/16)(2) (–19/8, 57/16, 17/16)(3) (19/8, –57/16, 17/16)(4) none of these

6. Equation of a line passing through (1, –2, 3) andparallel to the plane 2x +3y+z+5=0 is

(1)x 1 y 2 z 3

1 1 1

(2)x 1 y 2 z 3

2 3 1

(3)x 1 y 2 z 3

1 1 1

(4) none of these7. Equation of the line passing through (1, 1, 1) and

parallel to the plane 2x + 3y + z + 5 = 0 is

(1)x 1 y 1 z 1

1 2 1

(2)x 1 y 1 z 1

1 1 1

(3)x 1 y 1 z 1

3 2 1

(4)x 1 y 1 z 1

2 3 1

8. If the lines x 1 y 2 z 1

2 3 4

and

x 3 y k z1 2 1

intersect, then the value of k is

(1)32

(2)92

(3)29

(4)32

9. The lines x 2 y 3 z 4

1 1 k

and

x 1 y 4 z 5k 2 1

are coplanar, if

(1) k =3 or –3 (2) k = 0 or –1(3) k = 1 or –1 (4) k = 0 or –3

10. The radius of the circle in which the sphere2 2 2x y z 2x 2y 4z 19 0 is cut by the

plane x 2y 2z 7 0 is(1) 4 (2) 1(3) 2 (4) 3





1. STATEMENT-1 : If point () lies above theplane (a2 + 1)x + (b + 1)y + (c2 + c + 1)z + d = 0,then (a2 + 1) + (b + 1) + (c2 + c + 1) + d > 0andSTATEMENT-2 : If the point () lies abovethe plane ax + by + cz + d = 0 then

a b c d 0c

2. STATEMENT-1 : The shortest distance betweenthe skew lines r a b and r c d is

| [a c b d] || b d |

andSTATEMENT-2 : Two lines are skew lines if thereexists no plane passing through them.

3. STATEMENT-1 : Consider the planes3x – 6y – 2z = 15 and 2x + y – 2z = 5andSTATEMENT-2 : The parametric equation of theline of intersection of the given planes arex = 3 + 14t, y = 1 + 2t, z = 15t being the parameter

4. STATEMENT-1 : The integer value of a for whichthe shortest distance between the lines

x 3 y 5 z 7a 2 1

and x 1 y 1 z 7

7 6 a

is equal to 116 is a root of the equation3a2 – 34a + 31 = 0andSTATEMENT-2 : The distance between the parallel





lines x 2 y 3 z 4

1 2 2

and x 2 y 3 z 4

1 2 2

is equal to the distance between the points (2, –3, 4)and (–2, 3, –4)


A circle is the locus of a point in a plane such thatits distance from a fixed point in the plane is constant.Anologously, a sphere is the locus of a point in spacesuch that its distance from a fixed point in space isconstant. The fixed point is called the centre and theconstant distance is called the radius of the circle/sphere.In anology with the equation of the circle |z – c| = a, theequation of a sphere of radius a is | r c |

= a, where c

is the position vector of the centre and r is the positionvector of any point on the surface of the sphere. InCartesian system, the equation of the sphere, with centreat (–g, –f, –h) is

x2 + y2 + y2 + 2gx + 2fy + 2hz + c = 0 amd its radius

is 2 2 2f g h c

1. Radius of the sphere, with (2, –3, 4) and (–5, 6, –7)as extremities of a diameter, is

(1)2512

(2)2513

(3)2514

(4)2515

2. The centre of the sphere (x – 4) (x + 4) + (y – 3)(y + 3) + z2 = 0 is(1) (4, 3, 0) (2) (–4, –3, 0 )(3) (0, 0, 0) (4) None of these

3. Equation of the sphere having centre at (3, 6, –4)

and touching the plane ˆ ˆ ˆr.(2i 2 j k) 10 , is

(x – 3)2 + (y – 6)2 + (z + 4)2 = k2, where k is equalto(1) 3 (2) 4(3) 6 (4) 17

Passage - IIA tetrahedron is a three dimensional figure bounded

by four non coplanar triangular planes. So, a tetrahedronhas four non-coplanar points as its vertices.

Suppose a tetrahedron has points A, B, C, D as its

vertices, which have coordinates (x1, y1, z1) (x2, y2, z2), (x3,y3, z3) and (x4, y4, z4) respectively in a rectangular three-dimensional space. Then the coordinates of its centroid are

1 2 3 4 1 2 3 4 1 2 3 4x x x x y y y y z z z z, ,4 4 4

. The circumcentre of the tetrahedron is the centre of asphere passing through its vertices. So, this is a pointequidistant from each of the vertices of tetrahedron.

Let a tetrahedron has three of its verticesrepresented by the point (0, 0, 0), (6, –5, –1) and(–4, 1, 3) and its centroid lies at the point (1, –2, 5). Nowanswer the following questions.4. The coordiante of the fourth vertex of the

tetrahedron is(1) (2, –4, 18) (2) (1, –12, 13)(3) (–2, 4, –2) (4) (1, –1, 1)

5. The equation of the triangular plane of tetrahedronthat contains the given vertices is(1) x – 2y + z = 0 (2) 5x – 3y – 2z = 0(3) x + y + z = 0 (4) x + 2y + 3z = 0

6. The coordinates of the centre of the spherecircumscring the tetrahedron is

(1)18 47, ,87 7

(2)18 47, , 87 7

(3)18 45, ,87 7

(4)18 45, ,87 7



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