IIT-JEE 2012, Maths_Practice Questions

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MATHEMATICS APPLICATION OF DERIVATIVE

(b)

Straight Objective Section-I

(c)

ONLY ONE is correct. Mark your response in OMR sheet against the question number of that question. +3 marks will be given for

intersects the y-axis is :

    1 + 

(a) t (c)

2.



2

k

(

(b) t

k )

2)and tangent to the parabola y = secant to the curve y=

(2−5)

(a)



(b)



(c)



(d)

(d) None

6. The least value of ‘a’ for which the

 +  



equation,

(a) 3

(b) 5

least values of the function f(x) = sin2x –x

−  ,      +

= a has atleast one one

solution on the interval (0,

The difference between the greatest and the

on

is:

(b) 2x 2x + 2y – 3 =0

(c) y -2 =0

dt at

the points points where the graph cuts the x-axis

3.

(a)2x + 2y 2y -5 =0

The angle between the tangent lines to the

is

on [-2, 3]

5. Equation of the line through the point (1/2.

2

(d) none

graph of the function f(x)

7.

For all a, b



(c) 7

/2)

is:

(d) 9

(a)

(b) 0

(a) no no extremum (b) exactly exactly one

(c)

(d)

extremum

4. In which of the following functions Rolle’s theorem is applicable?

4

the function f(x) =3x –

4x3 + 6x2 + ax + b has

−   + 

,

(d)

correct answer and -1 mark for wrong

The angle at which the curve y=

on [

() =   () =  − 2 −−5+6 1 −6, , =1  ≠ 1,,[−2,3]   + 2   4 + 

choices (A), (B), (C) and (D), out of which

1.

−,≤<0 0, =0 −

0]

choice questions, each question has four

answer.

    () =  ,

(c) exactly two extremum (d) three extremum





then the number of critical points on the

13.

graph of the function is (a) 1

(b) 2

Number of critical points of the function, f unction, f(x) =

(c) 3

Which lie in the interval [-2

(d) 4

(a) 2 9.

Number of roots of the function

 ()

f(x)= (a) 0

10.

(c) 2

(c) 4

Given f’(1) = 1 and



(a)

(d) 5

∀

(f(2x)) = f’(x) x>

exists a number c (2, 4) such that f” (c) equals

−

on [1, 4]. The mean

on [1, 4] the number ‘c’ must be

2, then

0. If f’ (x) is differentiable then there



(a) – ¼

(b)

(c) 1/ 4

(d) 1/ 8

15.



(b)



differentiable on

3

the maximum area then the length of the median from the vertex containing the sides ‘a’ and ‘b’ is

(b)

(d)

FUNCTIONS ,then

1, 2] but is not

Multiple correct Section-II

1, 2)

(b)Mean value theorem is not applicable for the function on

(d)

   +      

Consider the function

(a) f is continuous on



‘a’ cm & ‘b’ cm. If the triangle is to have

(a)

1/8

 2 +  () =  3, ,  ≤1> 1 [ − (− [− −

(c)

Two sides of a triangle are to have lengths

(c) 12.

(d) 8

f’(c) is equal to the average slope of f(x)

global maximum value of f(x) is

11.

is:

some number ‘c’ between 1 and 4 so that

(d) more than 2

( − 1), 1≤≤ (b) 2

 

(c) 6

,2]

dt

value theorem says says that there must be

If f(x) =

(a) 1

(b) 4

14. Let f(x) = x+

-3x + sinx is

(b) 1

   −  +   +  2−  

1, 2]

Multiple choice questions, each question has four choices (A), (B), (C) and (D), out of which MULTIPLE (ONE OR MORE) is correct. Mark

(c)Mean value theorem is applicable on

your response in OMR sheet against the

[ 1, 2] and the value of c =1

question number of that question. +4 marks

(d) Mean value theorem is applicable on

 

will be given for correct answer and -2 marks for each wrong answer.





() =  1      + 

integer functions respectively, then

(b)

(x) is

(c)f(x)=

 +   − 

(d) f(x) =

+

+2[-x] where

[.] denotes greatest integer function)

2 []]

17. f(x) =sin

21.



 

(b) a =

(c) a =

(d) a =

Range of f(x) =

(

(2 sinx +

≤

(a)[0, 1]

(b) [0, 3]

(c)

(d) none of these

22. Let f(x) be a non-constant twice differentiable function defined

(−∞, ∞ ’ ) ”( / = 0   +  //  + on

. Then

) such that f(x) = f(1-x) and

=0. Then

(a) f(x) is a many –one and into function

(a

(b)f(x)=0 for infinite number of values of

x) vanishes at least twice on [0,

1]

x

(b) f’

(c)f(x) =0 for only two real values

(c)

(d) none of these

→  − 

19. If f:R R, f(x) =

(d)

)) 

is a given

function, then which of the following are correct:

23. Let

(a) f is many –one into function

 = / (1−  ()=

’

dt

’ − ∀ , − − −

(x) (x) > 0 and

(x) (x) < 0 ,

then

∞]∞,0]

(b) f(g( 1)) > f(g(x+ 1))

(c)range of f is [0,

(c)g(f(x+1)) < g (f(x 1))

(d) range of f is (-

If f(x) = sin { [x+5] + {x-{x-{x}}}} for

sin x dx =0

(a) (f(x+1)) > g(f(x 1))

(b) f is many one onto function

20.

(d)

MONOTONICITY

18. Let f(x) = [x] + [x + 1) – 3, where [x] = the greatest integer

(c) si

for

(a) a =

2

(b) si

−∞, 

greatest integer function, has

 

(a) si

cosx)+ 5) is

, where [.] denote the

fundamental period

         {}  {}    5

(d) g(g(x+ 1)) < g(g(x 1))

24.

3

2

If f(x) = x –x + 100x + 1001, then





− −

(c) f(x+ 1) > f(x (d) f(3x

 

(a) tan

(b) tan

5) > f(3x)

(c) cot

(d)

Passage II

Comprehension Section-III

x)2, 2x(1-x)}, where 0 28.

The interval in which f(x) is increasing is

(c) (d)

Let f and g are two functions such that f(x) & g(x) are continuous in [a, b] and

29.

differentiable in (a, b) Then at least one

’

(c)

≠ (≠)() () ’ ≠ 0, ()() = ()

(ii) If f(a)

(iii) If (x) (x)

f(b) and a b,

(c)

(RMVT)

(d)

(LMVT)

 ,    ,  0,  ∪ ,  0,  ∪   , 1 ’   

then a+ b +c is (where c is point point such

(Cauchy theorem)

that (a)

25. The set of values of k, for which equation

∞) 

(a) (1, 4)

(b) (0,

(c)(0, 1)

(d)

(b) (c)

− ≤  − ∀,∀, − −  − ≥  −≥∀,∀−,∀,∀,

(d)none of these

(b)



(c)



31.

 

(d)

Consider the function f(x)=

The interval in which f is increasing is

−∞, − 1)∪(−1, 0 ) −∞,∞∪)−{−1, 1 } 1,∞) →  ℎ   (a) (-1, 1)

(b) (

(c) (

Which of the following is true ? (a)



(c) (c) =0)

Passage III

x3 -3x + k = 0 has two distinct roots in (0,

26.

(b)

30. Let RMVT is applicable for f(x) on (a, b)

then then

1) is

(b)

The interval in which f(x) is decreasing is (a)

()()

(i) if f(a) = f(b), then f’(c)=0

 ,   ,  ∪ ,   ,  ∪   , 1

(a)

Passage I

c (a, b) such that

2

Consider the function f(x) = max (x , (1-

This section contains 2 paragraphs; each has 3 multiple choice questions. Each question has four choices (A), (B), (C) and (D), out of which ONLY ONE is correct. Mark your response in OMR sheet against the question number of that question. +4 marks will be given for correct answer and -1 mark for each wrong answer.



– – ≤≤1  , 

1)

(d) (0, 1)

32.

(

If f is defined from R – {-1, { -1, 1}

(a) injective but not surjective





33. F has

(c) both local maxima and local minima

(a) local maxima but no local minima

(d) neither local maxi a nor local

(b) local minima but no loc al maxima

minima

MAXIMA & MINIMA

Matrix Section-IV Each question contains statement given in two columns which have to be matched. Statements (A, B, C, D) in Column-I have to be matc hed with statements (P, Q, R, S) in Column-II. The answers to these questions have to be appropriat ly bubbled as illustrated in the following exam le. If the correct matches are A-P, A-S, B-Q, B-R, C- , C-Q and D-S, then the correctly bubbled 4 x 4 m atrix should be as follows:

Mark your response in OMR sheet against the question number of that question in s ction-IV. +6 marks will be given for correct answer and No negative marks for wrong answer. Ho ever, +1 mark will be given for a correctly marke answer in any row.

34. Column I

Colu n II

(a) Number of points which are local extrema of

( )1

                    (b) If a+b=1; a>0, b > 0, th n the minimum value of

      

 

is

(c)The maximum value attained by y = 10

2



( )2

   - 1    is 2

(d) If P(t , 2t), t [0, 2] is a arbitrary point on parabola y =4x. Q is foot

() 3

(s) 4

of perpendicular from focu s S on the tangent at P, then maximum area of triangle PQS is (t) 5







(a) The minimum value of

×

(b) Let A and B be 3

3

 is 

(p) 0

matrices of real numbers, where A is

(q) 1

symmetric, B is skew –symmetric and (A+B) A-B) = (A-B) t

k

t

(A +B). If (AB) = (-1) AB, where (AB) is the transpose of the matrix AB, then the possible values of k are

2() < 2,  ±− )

(c) Let a = log3 log32. An integer k satisfying 1 < must be less than

=

(d) If sin

, then the possible values of (

are

36. Column I

,,  then 3+

(t)4

greatest integer and least integer functions respectively) (c) If x2 + y2 =1 and maximum value of x + y is

+  + +, − 

(p) 4

is equal to

(b) Number of solution of 2[x] = x + 2{x} is (where [.], {.} are

(d)f

(s) 3

Column II

(a)If smallest positive integral value of x for which x 2 –x- sin-1 (sin 2) < 0 is

(r)2

,

= f (x) for all x

 

, then



is equal to

then period of f(x) is

37. Column I (a) If function f(x) is defined in [-2, 2], then domain of

(q) 1

(r) 2 (s) 0 (t) 3

Column II (p) (-

∞,−4)





Integer Type Section-V Integer Answer type This section contains 3 questions. he answer to each of the questions is a single digit integer, ranging from 0 to 9. The appropriate bub les below the respective question numbers in th OMR have to be darkened. For example, if the c rrect answers to question number X, Y and Z (say) are 6, 0, 9 respectively, then the correct darkening of bubbles will look like the following

2

2

38. The smallest value of k, for which both the equation equation x -8kx + 16 (k –k + 1) = are real , distinct and have values at least 4, i s

 dt. Then the value of f(In 5) is  Let p(x) be a polynomial d gree 4 having extremum at x = 1, 2 and      =2. Then  

39. Let f: R R be a continuous function which satisfies f(x)= 40.

the value of p(2) is





ANSWER KEY

Q.NO

1

2

3

4

5

6

7

8

9

10

ANS

B

D

A

D

A

D

B

C

C

C

Q.NO

11

12

13

14

15

16

17

18

19

20

ANS

B

D

B

B

A

A,B,C,D

A

A,B

A,D

A,B,C

Q.NO

21

22

23

24

25

26

27

28

29

30

ANS

D

A,B,C

B,C

B,C

D

A

D

D

C

D

Q.NO

31

32

33

ANS

B

D

A

A-q, B-r

37

Q.NO ANS

34

A-q

B-t

C-r

D-s

35

36

C-r,D-t

A-r, B-q,s

C-r,s, D-p,r

38

39

40

2

0

0

A-p, B-t

C-t,

D-t

IIT-JEE 2012, Maths_Practice Questions

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