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IITJEE MATHEMATICS SAMPLE PAPER – IV
SECTION – I
Straight Objective Type
This section contains 8 multiple choice questions numbered 1 to 8. Each question has
4 choices (a), (b), (c) and (d), out of which ONLY ONE choice is correct.
1.
Solution set of the equation 32 x
2
x 6
32( x
6)
0 is
(d) {1, –6}
(c) {–2, 3}
[cot 1 x] [cos 1 x] 0 , where x is non-negative real and [ ] is greatest integer function, then
complete set of solution to the equation is
(a) (cos 1, 1]
3.
2 3x
(b) {6, –1}
(a) {–3, 2}
2.
2
If f ( x)
(b) (cos 1, cot1)
(c) (cot 1, 1]
(d) none of these
( x 1)( x 2)
, then number of local extremas of g x
( x 3)( x 4)
(a) 3
(b) 4
f | x | is
(c) 5
(d) none of these
2
2
4.
1
Area bounded by the curves y sin (sin x) and y
2
(a)
4
4
2
2
1 (unit) 2
(b)
1 (unit) 2
(d) none of these
2
2
1 (unit) 2
2
(c)
2
x
4
Space for rough work
2
is
x
5.
Given f x is a periodic function with period ‘T’ and if
g ( x)
f t dt is a periodic
a
function for any real value of a, then
T
(a) no such g (x) exists
f t dt
(b)
0
0
T
f t dt
(c)
0
(d)
g x will always be periodic
a
6.
If a point P moves such that sum of square of distance from the lines
3x
(a)
1
(b)
2
(c)
3
2
3
(d)
1
2
An even function f x is differentiable and intersecting x-axis at x = 1, x = 2, then the
equation f ( x)
(a) 5 roots
8.
y 3 0 and
y 2 0 is constant, then eccentricity of locus of P is
3
7.
3x
2
f ( x) f
x
0 has at least
(b) 4 roots
(c) 6 roots
(d)
none of these
Top three students of each of three streams from Art, Commerce and Science are to be
given prizes. The number of ways it can be done such that no student of lower rank can
receive the prize before a student of better rank in his own stream, is
(a) 280
(b) 1680
(c) 91
(d) none of these
Space for rough work
SECTION II
Reasoning Type
This section contains 4 questions numbered 9 to 12. Each question contains Assertion and
Reason. Each question has 4 choices (a), (b), (c) and (d), out of which ONLY ONE choice is
correct.
Directions: Read the following questions and choose
(A) If both the statements are true and statement-2 is the correct explanation of
statement-1.
(B) If both the statements are true but statement-2 is not the correct explanation of
statement-1.
(C) If statement-1 is True and statement-2 is False.
(D) If statement-1 is False and statement-2 is True.
9.
Statement-1: lim {n ! e} is 0 ({ } denotes fraction part of x).
n
n
Statement-2: e
(a)
10.
y
A
lim
n
r
1
.
0 r!
(b) B
(c)
C
(d)
D
f (x) is monotonic discontinuous function such that | f ( x) | is continuous function.
Statement-1: f (x ) must have a single point of discontinuity.
Statement-2:
(a)
f (x) can be continuous only if at point of discontinuity of f (x ) sum of
left hand and right limit must be zero.
A
Space for rough work
(b) B
(c)
C
(d)
D
11.
Statement-2: The solution Tr
(a)
12.
x) n there cannot be more than two numerically
Statement-1: In binomial expansion of (a
greatest term.
A
Tr is a linear inequality in r.
1
(b) B
(c)
C
(d)
D
Statement-1: If the maximum value of | z1 z 0 | is k, where z0 is a fixed point and z1 lies an
|z| = r, then maximum value of | z1 z 0 | is also k.
Statement-2: If z1 lies on | z | = r so does –z1.
(a)
A
(b) B
(c)
C
(d)
D
SECTION III
Linked Comprehension Type
This section contains 2 paragraphs M13 M18. Based upon each paragraph, 6 multiple choice
questions have to be answered. Each question has 4 choices (a), (b), (c) and (d), out of which
ONLY ONE choice is correct.
Passage-I
1
, where x R and let f ) , f ( ) , f ( ) , f ( ) be four points on the
x i
argand plane. Now answer the following questions:
Let f ( x)
13.
The maximum value of f
(a)
Space for rough work
(b)
f
1
1
is
(c)
1
(d) 2
14.
If a triangle is formed by joining the points f ( ) , f ( ) , f ( ) , then maximum value of the
area of triangle is
(a) 3 3
15.
Points f
is
1
(a)
2
(b)
, f
, f
3 3
4
3 3
16
(d) none of these
are chosen such that they form a square, the length of square
, f
(b)
(c)
1
2
(c)
1
(d) none of these
Passage-II
Let f x be a function satisfying following condition,
(i)
f x 2
f x
2 ; (ii) f x 5
f x
5 ; (iii) f 0
0
Then
10
16.
f x is
The value of
x 1
(a) 10
(c) cannot be determined
17.
If g ( x)
(b) 55
(d) 1155
f ( x) x , then g (x) must be
(a) even
(c) periodic
18.
(b) odd
(d) none of these
If f x is differentiable then which of statement is true?
(a)
f (c) 0 ,
(c)
f (c) 1 for some c
Space for rough work
c
(1, 2)
(1, 2)
(b)
f (c) 1 ,
(d) none of these
c
(1, 2)
SECTION IV
Matrix Match Type
This section contains 2 questions. Each question contains statements given in two columns
which have to be matched. Statements (A), (B), (C), (D) in Column I have to be matched
with statements (1, 2, 3, 4) in Column II. One statement in first column has one or more than
one match with the statements in second column. The answers to these questions have to be
appropriately bubbled as illustrated in the following example.
If the correct matches are A 1,3, B 3, C 2,3 and D 2,4, then the correctly bubbled
4 × 4 matrix should be as follows:
1 2 3 4
A
B
C
D
19.
1
2
3
4
1
2
3
4
1
2
3
4
1
2
3
4
Let P be a moving point such that OP makes a constant angle of 30° with the line x = y = z.
Let locus of P is S (where O is origin)
Column I
(A) The locus of point of intersection of S and
x y z 0 is
(B) The locus of point of intersection of S and
x y z 3 is
(C) The locus of point of intersection of S and
3x 2 y z 3 is
(D) The locus of point of intersection of S and
x y z 6 3 0 is
Space for rough work
Column II
1.
Hyperbola
2.
Parabola
3.
Point
4.
Circle
20.
Column I
Column II
sin x a sin 3x sin 5 x
is periodic function with fundamental
1.
cos x a cos 3x cos 5 x
period being , then a can be
(B) The number of integral solution of equation ab b a 1 0 are
2.
(A)
f ( x)
(C) Number of values of x, where cos 2 x cos x
f ( x) is maximum,
(where f (x) is odd function)
(D) The values of m for which f x
Space for rough work
x
2
sin m x
is an even function, are
m
1
2
3.
3
4.
4
SECTION V
Subjective or Numerical Type
The answer to these questions would lie between
0 to 9999. For any answer all four bubbles must be
filled, for example if you plan to answer as 16 then fill
0016 and if you plan to answer 0 then fill 0000 in the
grid provided in answer sheet. Any incomplete filling
will be considered as incorrect filling.
Illustration: If you want to fill 2379 as your answer
then it will be filled as
21.
Let N =
be a 6 digit number (all digit repeated) and N is divisible by 924 and let ,
0 , then product of all possible values of is
be the roots of the equation x 2 11x
________ .
22.
Let
2 1 x3
100
100
i 0
___________ .
Space for rough work
50
ai x i
cos
2
a 2i
x i , if
i 0
2 k , then the value of k is