PRACTICE PAPER FOR IITJEE (LAGRARITHAM, QUADRATIC EQUATION AND BASIC CALCULUS )

Single Answer Questions

1. If a,b,c,d are real numbers such that 2 2 24 0b d a c , then the equations 2 0x ax b and 2 0x cx d have

(a) both have complex roots (b) both must have real roots

(c) at least one of the equations has real roots (d) none of these

2. If 10log 1x then

(a) 0.1 10x (b) 1 10x (c) 0 10x (d) 10x

3. If 32x dy

y thendx

is

(a) 32 ln 2.ln3x

(b) 8 ln8x (c) 33 .2 ln 2x x (d) 33 .2 .ln 2.ln3xx

4. The general solution of the differential equation y x y xdye e

dx

is

(a) y x xe e e c (b) y x xe e e c (c) y x xe e e c (d) y x xe e e c

5. 4

0

tan xdx

equals to

(a) ln 2 (b) 0 (c) 2ln 2 (d) none of these

One or more than one correct option questions

6. Consider the equation 2x x a 0,a N . If the equation has integral roots then ‘a’ can be

(a) 2 (b) 6 (c) 12 (d) 20

7. The inequation 2 2log 2 log 2 log 4 0x x x

(a) has a meaning for all x (b) has a meaning if x 1,

(c) is satisfied for x in 2 12 ,

2

(d) is satisfied for x in 1 1

, U 1,4 2

8. If 1, 3 9

8log 3 2 ,2log 3

3

x x

are in AP then the value of x can be

(a) 1 (b) 3

4log

3

(c) ln 4

ln 3 (d) 3log 4

9. If 1

11 2

x

x

then x satisfies

(a) 1,1 (b) 1 1

,2 2

(c) 2,3 (d) 1,5

Comprehension – 1

Let a, b, x are positive integers and a 1, b 1.

(i) b blog x log aa x

(ii) alog xa x

(iii) a

b

a

log xlog x

log b

(iv) k aa

1log b log b

k

10. If 5 5log 2 log x3x 2 64, then value of x is

(a) 25 (b) 625 (c) 64 (d) 6

11. If 10

2

10

log x 3 1,

2log x 21

then x =

(a) 10 (b) 5 (c) 7 (d) – 6

12. The number of solutions of 4 2log x 1 log x 3 is

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

Comprehension -1I

Let 2 3 0,x m x m m R be a quadratic equation

13. If roots are greater than 2 then values of m

(a) 8,10 (b) 9,10 (c) 0,1 (d) ,1

14. One root is greater than 2 and other is less than 1 then values of m

(a) m R (b) 9,10m (c) m (d) none of these

15. Both roots lie in the interval (1,2) then

(a) m R (b) 9,10m (c) m (d) none of these

Numerical

N1. 0

sin x dx

equals to _______

N2. Number of integers present in the range of the function f x 3 sinx 2 cosx is _________.

N3. The number of values of ‘a’ for which 2 2 2 23 2 5 6 4 0 a a x a a x a is an identity ____

N4. Let 2 , , , , 0.f x ax bx c a b c R a If 1 2 0f f then the number of distinct real roots of 0f x is

_______

N5. If ln 3 ln 3 x dx A x B x Cx D then A B C equals to _____

N6. If the least value of 2x 6x 5 in [2, 4] is ‘a’, then a is __________.

N7. If ‘x’ is an integer satisfying 2x 6x 5 0 and 2x 2x 0. Then the number of

possible values of ‘x’ is __________.

N8. The number of integral values of ‘x’ satisfying x 1 1 1 is __________.

Match the following

M1. Match the range of functions of Column 1 with Column 2.

COLUMN – I COLUMN – II

A 2

f x x 3 1 P All real positive number

B 2x 1

f xx 1

Q

C f x x x R 1,

D 2

1f x

x S 2

M2. Match the derivative values of Column 1 at given points with Column 2.

COLUMN – I COLUMN – II

A If xy e 1, then the value of

dy

dx at

x = 0

P 1

2

B If y sin x cos x, then dy

dx at x = 0 Q 1

C If 1y tan x, then dy

dx at x = 1 R 1

D If ey log x, then

2

2

d y

dx at x = 1 S 0

ANSWER KEY:

1.C 2.A 3.D 4.B 5.A

6.A,B,C,D 7.B,C,D 8.A,C,D 9.A,B 10.B

11. B 12.C 13.B 14.C 15.C

N1.0002 N2.0006 N3.0001 N4.0002 N5.0005

N6: 0004 N7: 0003 N8: 0005

M1: A – R B – S C – Q D – P

M2: A – R B – R C – P D – Q