Maths IIT-JEE ‘Best Approach’ (MCSIR)

 ln xdx
1. Evaluate 0 x  2x  4
2

2. 8 clay targets are arranged as shown in the figure. In hoe many ways they can be shot
(one at a time) if no target can be shot until the target(s) below it have been shot no fire gets
waste.

3. The sum of all real values of 'm' for which the polynomial
f(x) = {x2 – 2mx – 4(m2 +1)} . {x2 – 4x – 2m (m2 +1)} has exactly three distinct real linear
factors, is :

4. For a polynomial g(x) with integral coefficients, let ng denotes the number of distinct integral
roots of g(x)  k for a real number k.

Now let S be the set of polynomials defined as-

S  {(a n x n  a n 1x n 1  a n 2 x n  2  a n 3 x n 3  ....);a i  R}

For a polynomial f  S, let n f  4 for f (x)  5. Then the value of n f for f (x)  16 is
____?

  1 n 
 1   
If Lim  
n 2 2 ae 2
5. – e n = (a, b  N). Find the minimum value of (a + b).
n   n

 1 b
 1 –  
 n 

6. The range of the function f : R  R

3  2sin x
f x  contains N integers.
1  cos x  1  cos x
Find the value of 10 N.

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Maths IIT-JEE ‘Best Approach’ (MCSIR)
7. If bisector of angle C of an acute ABC cuts the side AB at D and circumcircle of
CE  a  b 
2

ABC in E, then  where K is equal to :

DE Kc 2

Comprehension (Q.8 to Q.10)

Let xi' s, i = 1,2, .... n be a sequence of integers such that xi  [–1,2], i = 1, 2, ...n.
n n
 xi  19 and  x i2  99
i 1 i 1
n
8. The minimum possible value of  x 3i is :
i 1

(A) 18 (B) 21 (C) 19 (D) 13

n
9. The maximum possible value of  x 3i is :
i 1

(A) 133 (B) 121 (C) 143 (D) 165

n
10. For the maximum possible value of  x 3i the number of 2's that the sequence must
i 1
have is :
(A) 18 (B) 19 (C) 20 (D) 21

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Maths IIT-JEE MC SIR

1. All solutions of the equation 4x2 – 40 x + 51 + 40 {x} = 0 lie in the interval ({x} represents fractional
part of x)

 23 83   23 15   83   23 
(A)  ,  (B)  ,  (C)  7,  (D)  ,7 
 10 10   10 2   10   10 
2. f(x) increases in the complete interval :
(A) (–, –1) (–1, 0) (0, 1) (1, ) (B) (–, )
(C) (––1) (–1, 0) (D) (0, 1) (1, )
3
dx
3. Let I n   1  x n  n  1, 2,3..... and Lim I n  I0 say, then which of the following statement(s) is/are
n
0
correct ? (Given : e = 2.71828)
(A) I1 > I0 (B) I2 < I0 (C) I0 + I1 + I2 > 3 (D) I0 > I1 > 2

4. If the matrix A and B are of 3 × 3 and (I –AB) is invertible, then which of the following
statement is/are correct ?
(A) I  BA is not invertible
(B) I  BA is invertible
(C) I  BA has its inverse I  B(I  AB)1 A
(D) I – BA has its inverse I  A(I  BA) 1 B

5. All the 7 digit numbers containing each of the digits 1,2,3,4,5,6,7 exactly once and not divisible by 5 are
arranged in increasing order. Then
(A) 1800th number in the list in 3124567 (B) 1897th number in the list is 4213567
(C) 1994 th number in the list is 4312567 (D) 2001 th number in the list is 4315726

6. f(x) = 0 is a cubic equation with positive and distinct roots,  such that  is H.M. of the roots of
5
 2  i
f'(x) = 0 . If K    where [.] represents greatest integer function, then find   2 .
    iK

  1 x2 
7. The value of   1  x 4  dx is.

8. Every ray of light, emerging from (1, 2), after strikinh at an elliptical curve, whose eccentricity is
2 5
, always passes through (3,6) after reflection. If () is a point on this curve such
2  5  45
that it is at unit distance from origin then 2   is

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Maths IIT-JEE MC SIR
9. Let f(x) = x2 + 10x + 20. Find the number of real solution of the equation f(f(f(f(x)))) = 0

10. Let ABC be equilateral on side BA produced, we choose a point P such that A lies between P and B.
We now denote 'a' as the length of a side of ABC; r1 as the radius of incrircle of PAC; and r2 as the
radius of the excircle of PBC with respect to side BC. Determine the sum (r1 + r2) as function of 'a'
alone.

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Q.1 Let A, B, C be n  n real matrices and are pairwise commutative and ABC  On and if
  det  A3  B3  C3  .det  A  B  C  then

A)   0
B)   0
C)   0

D)    ,   0

Q.2 P(x) & Q(x) are two quadratic expressions with leading coefficients being one, such
that P  x  Q  x   Q  P  x   x  R then.
A) Q(0) = 0
B) P(1) = 1

C) Q  x   P  x   Q '  0 

D) P(1) = 2

Q.3 In ABC , a is the arithmetic mean and b, c are two geometric mean of two positive
 sin12 B  sin12 C 
distinct numbers, then the value of 44   can be equal to
  sin 2A  sin 2B  sin 2C  
4
 
1
A)
2
1
B)
4
C) 2
D) 3

Q.4 If , ,  are the roots of x3  x 2  2x  1  0, then the value of  is equal to______

 2
 2
2   2

Where   2   2
 2
2
2 2   2 2

99

 10  n
Q.5 If P  n 1
99
, then  P  is {where [.] is GIF}

n 1
10  n
Q.6 The value of a, if a and b are integers such the x 2  x  1 is a factor of ax17  bx16  1 is
P
P, then the value of .
329
k

2 r n
Cr k 1
Cr 1
Q.7 The value of r 1
k
(where n  k ) is equal to _____

r 0
n
Cr n  k  r 1
Cn 1

1
Let I   1  x 50  x100 dx and
99
Q.8
0

1
J   1  x 50 
100
x100 dx
0

I m  m  n 1
If is equal to where m and n are co-prime then find the value of  .
J n  20 

Q.9 Let M n be the n  n matrix with entries as follow

for 1  i  n mi,i  10 ; for 1  i  n  1 mi 1,i  mi,i 1  3 all other enteries in M n are


1
zero, Let D n be the determinant of matrix M n then  8D
n 1 1
can be represented as
n

p
, where p and q are relatively prime integer, the value of p + q is A, then value of
q
A
is.
73

Q.10 The value of mm  

m
 m  1 r  1 mr 1 is equal to _____
r 1 r m Cr
 2 2
dy y
  ydx and f 1  2 then
dx x 1  ydx is equal to
Q.1 If y is a function of x satisfies
1
12
A)
7  4 ln 2
12
B)
7  8ln 2
12
C)
7  8ln 2
D) none of these

Q.2 If W  24.32.5 then number of factors of W2 which are less than W but are not factors
of W is:
A) 36
B) 132
C) 66
D) 38

 8sin x cos x  f  x  f  x  dx   , then which of the options are correct.

2
Q.3 Let
0
 /2
4
A)  f  x  dx  3
0

B) f(x) is periodic function with period 2

C) f(x) is even function

D)  f  x  dx  0
0

1
Q.4 a, b, c are three unit vectors such that a  b  b  c  c  a  . If a  b  pa  qb  rc
3
where p, q, r are scalars then
1
A) p 2 
10
1
B) p2 
15
1
C) q 2 
15
 r2
D)  16
q2

 /4
 100 100   
2  sec x  cos ec  x  4   dx is equal to
100
Q.5
0   
ln 1 2 

A)  2  eu  e u  du
99

 /4

2
101
B) sec101 x dx
0

ln  2 1 
 4  eu  e u  du
98
C)
0

 /4

2
101
D) sec100 x dx
0

Q.6 Let, a, b and c be positive real numbers such that a + b + c = 1 then which of the
following is/are true
A) a 2  b2  c2  a a bbcc

B) a 2  b2  c2  a a bbcc

C) a 2  b2  c2  1 18 abc

D) a 2  b2  c2  1 18abc

Q.7 A triangle ABC is such that a circle passing through vertex C, centroid G touches side
AB at B. If AB = 6, BC = 4 then the length of median through A is equal to
13
A)
2
B) 42

C) 2 42
D) none of these

 /4
 cos x  sin x  1 2  
 esec x    k, then 3 ln(2|k|) is equal to_____
2
Q.8  dx e
0  2 cos x  sin 2x  2
 2
Let a complex number z = x + iy satisfies equation z  16 z  3z2  3z  9  0 . If a
4 2
Q.9
and b are the maximum and minimum value of |z| then ab is equal to______

Q.10 Let ABC be right angle triangle, B  90 . The median through A and C are y = 3x +
1 and y = x + 1 respectively. If AC = 8 and then area of triangle ABC is_____
Q.1 Let S1 be the locus of the point ‘P’ which moves such that OP = 1 unit. The plane
x + y + z = 1 cuts the curve S1 in another curve S2 . Find the volume of solid formed by
joining the point O with every point on the curve S2 . (O is origin)

2 3 C) 4 4 3
A) 3 B) D)
27 2 7

Q.2 Let f : R → R and g : R → R be two functions,

f ( x ) = x  x 2 − 1 & g ( x ) = x +  x 2 − 1 then (where [.] denotes G.I.F.)
A) f is discontinuous exactly at five points in [-1, 2]
B) g is discontinuous exactly at five points in [-1, 2]
1
2

 ( f ( x ) − g ( x ) ) dx = 2
1
C)
1
−
2

1
2
D)  ( f ( x ) − g ( x ) ) dx = 1
1
−
2

Q.3 Two triangles ABC& DEF are non congruent but have 5 elements equal (out of 3
side lengths & 3 angles). Which of the following statements are true?
A) Side length of given triangles are in G.P
B) it is possible to construct ABC with side lengths 1 & 9

C) It is possible to construct DEF with side lengths 1 & 2

D) It is not possible to have triangle ABC such that its sides are in A.P.

Q.4 If A = 1, 2,3, 4 , B = 1, 2,3, 4,5, 6 and f : A → B is an injective mapping satisfying
f ( i )  i, then number of such mappings are :
A) 182 B) 181 C) 183 D) none of these

Q.5 For the curve sin x + sin y = 1 lying in the first quadrant there exists a constant  for
d2 y
which lim x  2 = L (not zero), then 2 =
x →0 dx

+ he−1/h 

 dx −  x 2e − x dx
2 − x2 2
xe
0 0
Q.6 Value of lim+ is equal to :
h →0 he −1/h
 (
A)  1 − 2 e− ) (
B) 2 1 − 2 e− )
2 2

C)  (1 −  ) e − D) 2e−
2

x k + 2 2k
1 n
Q.7 The value of  lim
0
n →
k =0 k!
dx is :

A) e2 − 1 B) 2

e2 − 1 e2 − 1
C) D)
2 4

If r th , ( r + 2 ) , ( r + 6 ) terms of an increasing A.P are last three terms of a G.P. and

th th
Q.8
there lies exactly 2p + 1 number of terms of that G.P in the interval (p, 2018p)
( p  N ) then maximum value of p is____

Q.9 If a, b, c are real numbers such that a + b + c = 0 , then value of

( ab ) ( bc ) ( ac )
2 2 2

+ + is
(a 2
− bc )( b2 − ca ) (b 2
− ca )( c2 − ab ) (a 2
− bc )( c 2 − ab )

Q.10 The number of solution of trigonometric equation sin 5x = 16sin 5 x in  0, 2 is

Q.1

 4  
 n3 −1 
Let L =  1 − 2  ; M =  3  and N = 

(1 + n ) , −1 2

−1
n =3  n  n =2  n + 1  n =1 1 + 2n

Then find the value of L−1 + M−1 + N−1.

Q.2 A ball moving around the circle x 2 + y2 − 2x − 4y − 20 = 0 in anticlockwise direction

leaves it tangentially at P ( −2, − 2 ) . After getting reflected from a straight ‘L’ it passes
5
through the centre of circle. If distance of P from line L isand 2 be the angle
2
between incident ray and reflected say then 10cot 2 cos  is equal to______


Q.3 Let in ABC, A is and D is point on BC such that AD is altitude. If E, F and I
2
AI
are incentre of ABD, ADC and ABC respectively, then the value of is equal
EF
to____

x 2 y2
Q.4 A variable chord of the hyperbola − = 1 , subtends a right angle at the centre of
4 8
the hyperbola. If this chord touches a fixed circle concentric with the hyperbola then
the square of radius of the fixed circle is equal to_____

n −1 2k ( k −1) 2 
L
Let lim  e
i i
Q.5 n
−e n
= L , then is equal to______
n → 2
k =0

Q.6 The arithmetic mean of all integral values of real parameter  for which

z − (  2 − 7 + 13 + i ) = 1 and arg z  is satisfied for at least one z is equal to_____
2
 /2  /2  /2
Suppose I1 =  cos ( sin x ) dx ; I =  cos ( 2 sin x ) and I =  cos ( sin x ) dx, then
2 2
Q.7 2 3
0 0 0

I1 + I 2 + I3 =

Q.8 Let A and B are two square idempotent matrices such that AB + BA, AB – BA are
null matrices, then the sum of squares of the possible values det (A – B) is

60

 60
C j sin ( 2 jx ) 
4
, then find the value of  J ( x ) dx =
j= 0
Q.9 Let J (x) = 60


j= 0
60
C j cos ( 2 jx ) 0
 
cos 
Q.10 Let I =  d , then which of following is/are correct statement(s)
0
10 − 6cos 

A) I < 1 B) I > 0 C) I < 0 D) I > 1

Q.1 Let A, B, C, D be the vertices of a regular tetrahedron each of whose edges measure 1
meter. A bug, starting from vertex, A, observes, the following rules, at each vertex it
chooses one of the three edges meeting at that vertex, each edge being equally likely to
n
be chosen, and crawls along that edge to the vertex at its opposite end. Let p = be
729
the probability that the bug is at vertex A when it has crawled exactly 7 meters. The
z
value of n is z, then the value of is.
91

Q.2 The z6 + z3 + 1 = 0 has complex roots with argument between 90 and 180 in the
p
complex plane, determine the degree measure of  is p, then the value of
40

Q.3 The number of integer values of k in the closed interval [-500, 500] for which the
equation log(kx) = 2log(x + 2) has exactly one real solution.

1 1
4
1 
2 2
Let J =   − x 2  dx and K =  x 4 (1 − x ) dx then
4
Q.4
0 
4 0

J B) J – K = 0
A) =2
K
1
1 4 1
C) J =  x (1 − x ) dx D) K =
4

20 1260

Q.5 Let C be a variable point in the first quadrant on the line x + y = 1 which intersects X
and Y axes at A and B respectively. D and E are foot of perpendiculars from C on X
and Y axes respectively. Let P be a point on the line segment joining D and E.For all
its possible position, P will satisfy the inequality:
1 1
A) xy  B) xy 
16 16
1 1
C) xy  D) xy 
32 32

Q.6 The ordered triplet ( , ,  ) satisfies following equations

x + y + 3z = 1, x − y + z = 1, ax + ( a + b ) y + 6z = 1, ( a, b  R ) If 2 + 2 +  2 is minimum,
then minimum value of a 2 + b2 is:
Q.7 ( )
f : R → R f ( x ) = x 4 − x 2 + 1 . The number of values of x for which f f ( f ( x ) )  x 8
is.

ln (sin x + 3cos x + 2 + 1)
Q.8 For a function f ( x ) = (where {.} denote the fractional
sin x + 1cos x + 1
 −   + 
part of x), then f ( 0 ) + f   − f   is equal to________
−

 2   2 

Q.9 The line x + 2y + c = 0 intersects the circle x 2 + y2 − 4 = 0 at two distinct points P and
Q. Another line 12x – 6y = 41 intersects the circle x 2 + y2 − 4x − 2y + 1 = 0 at two
distinct points R and S. Find the value of c for which the four points P, Q, R and S are
concyclic.

 4034   4034  2  4034  2017  4034 

Q.10 If the expression   + 3  + 3   + ..... + 3   is divisible by
 0   2   4   4034 
2 n then maximum value of n equals ( n  N )

A) 2016 B) 2018 C) 4033 D) 2017

 1 1 1  2  1 1 1 1 
Q.1 If lim  2 + 2 + .... + 2  = and S = lim  3 3 + 3 3 + 3 3 + ..... + ,
n →  1 .2
 n  6 n 3 . ( n + 1) 
n → 1 3
2  2 .3 3 .4
then S + 2 is equal to
A) A non-integral number B) A rational number
C) A transcendental number D) An integer

Q.2 Identify the statement(s) which is/are incorrect?

( ) (
A) a  a  a  b  = a  b a
 )(
2
) ( a, b are non -zero vectors.)

B) If a, b, c are non-zero, noncoplanar vectors and v.a = v.b = v.c = 0 then v must be a null
vector.
C) If a and b lie in a plane normal to the plane containing the vectors c and d then
( a  b )  ( c  d ) = 0 ( a, b, c, d Are non -zero vectors.)
D) If a, b, c and a ', b ',c' are reciprocal system of vectors then a.b '+ b.c '+ c.a ' = 3

Q.3 f : N → A and g : N → B are two onto functions such that f ( n ) =  n sec 2   and
g ( n ) =  n cos ec 2  , for some  . If sec2  is irrational, then (where [.] denotes the greatest
integer function ; N denotes set of natural numbers)
A) A  B = 

B) A  B = N

C) f and g are bijective

D) A  B is non-empty finite set

 
Q.4 If f (  ) = sin  + cos  + tan  + cot  − sec  − cos ec;    0,  , then
 2

A) f (  )  1 B) f ( )  2

C) f (  )  2 D) f (  )  1


Q.5 Let , ,  are roots of x3 + px + r = 0 and , , 2 are the roots of equation `
2
x3 + qx + r = 0 then
3 2 B) p = q
q  r 
A)   +   = 0
7 6
2 D) None of these
C) ( pq ) = 5r 3
Q.6 Let S2 = 0 be the mirror image of S1 : x 2 + y2 − 4x − 6y + 12 = 0 w.r.t the line
L1 :104 x + (104 + 10 ) y + (104 + 20 ) = 0 . Let L 2 : 211 x + ( 211 + 212 ) y + ( 211 + 213 ) = 0 be a
line. The equation of the line passing through the point of intersection of the line L2 = 0 with
the radical axis of S1 = 0,S2 = 0 and making equal intercepts with the coordinate axes, is

A) x − y − 3 = 0 B) x + y + 1 = 0

C) 2x − 2y + 1 = 0 D) 2x + 2y + 3 = 0

Q.7 If z  C , which satisfies 3 z − 12 = 5 z − 8i and z − 4 = z − 8 absolute difference of

imaginary values of z?

Q.8 3 distinct integers are selected at random from 1, 2, 3……,20. If probability that their sum is
a
divisible by 5, is (a, b are co-prime). Find b−a ?
b

Q.9 If the maximum value of x 4 − 7x 2 − 4x + 20 − x 4 + 9x 2 + 16 is P, find the value of [P]

(where [.] denotes GIF)

Q.10 If a,b,c  N and a 2 bc + 4abc + 4bc = 81 . Find the value of a + b + c such that 2a + b + c + 4
assumes its least value_______
1. Number of ordered triplets (p, q, r), where p, q, r,  [1, 100] such that 2 p  3q  5r is multiple of 4 is.
(A) 5× 106 (B) 5 × 104 (C) 5 × 105 (D) 5 × 103

2. Let  be two real numbers satisfying the following relation 2 +  2 = 5, 3(5 +  5) = 11 (3 +  3)
then
(A) Possible value of  is 2
(B) Possible value of  is  3
(C) Possible value of  is 3
(D) Possible value of  is  2

3. f(x) = –1 + kx + k either touches or does not intersect the curve f(x) = nx, then minimum value of k 
1 1   1 
(A)  ,  (B) (e, e2) (C)  ,e  (D) None of these
e e   e 

4. Let x1, x2, x3, x4 be the roots of x4 + ax3 + bx2 + cx + d = 0 if x1 + x2 = x3 + x4 and a, b, c, d  R then

(A) If a = 2 then the value of b – c is 1.

(B) If a = –2 then the value of b + c = 1
2c
(C) x1x 2  x 3 x 4 
a
 1
(D) If b + c = 1 and a  –2 then for real values of ‘a’, c   ,
 4 

5. In a ABC, AB  3, BC  4, CA  5 : P is any point inside the ABC , such the

PAB  PBC  PCA  
16
(A) Ratio of area of PBC & PAB is
25
8
(B) Ratio of area of PBC & PAB is
15
25
(C) cot  
12
2
(D) cot  
5

6. Let A = [aij]n ×n be a matrix such that aij = i 2–j then find the value of lim (trace An)1/n (where n  N).
n 

7. Let f(x) = x2 + 6x + c  x  R, where c is some real number. For what value of c does f(f (x)) = 0 have
exactly 3 distinct real roots.
11  13 11  2 13 11  2 13 11 – 13
(A) (B) (C) (D)
2 2 2 2
8. Let a, b, c be the real numbers such that a + b + c = 2 and ab + bc + ca = 1, where a  b  c, then
The complete set of values of ‘a’, ‘b’ and ‘c’ are (l, m), (p, q) and (r, s) respectively then
l + m + p + q + r + s is equal

(A) 2 (B) 3 (C) 4 (D) 5

9. The two points on the curve y = x4 – 2x2 – x that have a common tangent line are (1,  1) and
(2,  2), then |1| + | 1| + |2| + | 2| =

10. In an acute triangle ABC, altitudes from the vertices A, B and C meet the opposite sides at the points D,
E and F respectively. If the radius of the circumcircle of AFE, BFD, CED and ABC be respectively
R1, R2 R3 and R. Then the maximum value of R1 + R2 + R3 is.
3R 2R 4R 3R
(A) (B) (C) (D)
8 3 3 2