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Questions and Solutions

PAPER - 1 : MATHEMATICS, PHYSICS & CHEMISTRY

PART- A : MATHEMATICS

1. The equation sinx sinxe e 4 0 has :

(1) infinite number of real roots (2) no real roots

(3) exactly one real root (4) exactly four real roots

1. (2)

esin x

e-sin x

4 = 0

Let esin x = t

t1

t4 = 0

t2

4t 1 = 0

t =4 16 4

2

t = 2 5

esin x

= 2 + 5 Not possible [2.7 < e < 2.8]

esin x

= 2 5 Not possible [Never Negative]

2. Let a and b be two unit vectors. If the vectors c a 2b and d 5a 4b are perpendicular

to each other, then the angle between a and b is :

(1)6

(2)2

(3)3

(4)4

2. (3)

c d = 0

(a 2b) (5a 4b) = 0

5 6a b 8 = 0

a b =1

2

a b cos =1

2

cos =3

3. A spherical balloon is filled with 4500 cubic meters of helium gas. If a leak in the balloon

causes the gas to escape at the rate of 72 cubic meters per minute, then the rate (in meters per

minute) at which the radius of the balloon decreases 49 minutes after the leakage began is :

(1) 9/7 (2) 7/9 (3) 2/9 (4) 9/2

Test Booklet

Code - C

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3. (3)

34R

3= 4500

R3

= 3 1125 R = 15m

dv

dt= 2

4 dr3 r

3 dt= 72

drdt

= 218r (1)

Also, 4500 72 49 = 34

r3

r = 9m

dr

dt=

18

81=

2

9m/minute

4. Statement 1: The sum of the series 1 + (1 + 2 + 4 ) + (4 + 6 + 9) + (9 + 12 + 16) + + (361 +

380 + 400) is 8000.

Statement 2:n

3 3 3

k 1

k (k 1) n , for any natural number n.

(1) Statement 1 is false, Statement 2 is true.

(2) Statement 1 is true, Statement 2 is true; Statement 2 is a correct explanation for Statement 1.

(3) Statement 1 is true, Statement 2 is true; Statement 2 is not a correct explanation for

statement 1.

(4) Statement 1 is true, Statement 2 is false.

4. (2)

Statement 2:n

3 3

K 1

K (K 1)

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

2 2= n

3

Statement 2 is correct

Statement 1:

(13

03) + (2

31

3) + (3

32

3) + (20

319)

3= 8000

Statement 1 is correct

and statement 2 explain statement 1

5. The negation of the statement

"If I become a teacher, then I will open a school", is :

(1) I will become a teacher and I will not open a school.

(2) Either I will not become a teacher or I will not open a school.

(3) Neither I will become a teacher nor I will open a school.

(4) I will not become a teacher or I will open a school.

5. (1)

6. If the integral

5tan x

dx x a n sin x 2cos x k tan x 2 then a is equal to :

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

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6. (4)

5tanxdx

tanx 2= x + a ln | sin x 2 cos x | + K

Differentiating on both side

5tanx

tanx 2= 1 +

a[cosx 2sinx]

sinx 2cos

5sinx

sinx 2cosx =

sinx 2cosx a(cosx 2sinx)

sinx 2cosx

Equating co-efficient of both side

sinx

5 1 2a ,

cosx

0 2 a

a = 2

7. Statement 1 : An equation of a common tangent to the parabola 2y 16 3 x and the ellipse

2x2

+ y2

= 4 is y = 2x + 2 3 .

Statement 2 : If the line4 3

y mx ,m

(m 0) is a common tangent to the parabola

2y 16 3 x and the ellipse 2x

2+ y

2= 4, then m satisfies m

4+ 2m

2= 24.

(1) Statement 1 is false, Statement 2 is true.

(2) Statement 1 is true, Statement 2 is true, Statement 2 is a correct explanation for Statement 1.

(3) Statement 1 is true, Statement 2 is true, Statement 2 is not a correct explanation for

Statement 1.

(4) Statement 1 is true, Statement 2 is false.

7. (2)

Put y = 2 24 3

mx in 2x y 4m

2

2 4 32x mx

m= 4 2 2

2

482 m x 8 3 x 4

m= 0

y =4 3

mxm

is a tangent, discriminant of the above quadratic equation must be zero.

28 3 =

2

2

484 2 m 4

m

m4 + 2m2 24 = 0 (m2 + 6) (m2 4) = 0m = 2

Statement (2) is a correct explanation of statement (1).

8. Let

1 0 0

A 2 1 0 .

3 2 1

If u1 and u2 are column matrices such that Au1 =

1

0

0

and Au2 =

0

1 ,

0

then

u1 + u2 is equal to :

(1)

1

10

(2)

1

11

(3)

1

10

(4)

1

11

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8. (4)

AU1 =

1 0 0 a

2 1 0 b

3 2 1 c

=

1

0

0

a

2a b

3a 2b c

=

1

0

0

a = 1; b = 2; c = 1

U1 =

1

2

1

AU2 =

1 0 0 x

2 1 0 y

3 2 1 z

=

0

1

0

x

2x y

3x 2y z

=

0

1

0

x = 0; y = 1; z = 2

U2 =

0

1

2

U1 + U2 =

1

2

1

+

0

1

2

=

1

1

1

9. If n is a positive integer, then2n 2n

3 1 3 1 is :

(1) an irrational number

(2) an odd positive integer

(3) an even positive integer

(4) a rational number other than positive integers

9. (1)2n 2n

3 1 3 1

1 3 2n 1

2n 1 2n 3

C C C2 2n 3 2n 3 ......2n 3

which is Irrational Number

10. If 100 times the 100th

term of an AP with non zero common difference equals the 50 times its

50

th

term, then the 150

th

term of this AP is :(1) 150 (2) 150 times its 50th

term

(3) 150 (4) zero

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10. (4)

Let first term a and common difference d

100 [a + 99 d] = 50 [a + 49 d]

a = 149 d

Now,

T150 = a + 149 d

T150 = 0

11. In a PQR, if 3 sin P + 4 cos Q = 6 and 4 sin Q + 3 cos P = 1, then the angle R is equal to :

(1)5

6(2)

6(3)

4(4)

3

4

11. (2)

P + Q + R =

3 sin P + 4 cos Q = 6 . (i)

4 sin Q + 3 cos P = 1 . (ii)

squaring and adding (i) and (ii)

9 + 16 + 24 sin (P + Q) = 37

sin (P + Q) =1

2

P + Q =6

;5

6

if P + Q =6

0 < sin P