JEE Main 2016 Maths Question Paper Set 2 With Solution

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Category: IIT-JEE 2016 Math Paper 2

Question 1:

Let P=

[\begin{matrix} 1 & 0 & 0 \\ 4 & 1 & 0 \\ 16 & 4 & 1 \end{matrix}]

and I be the identity matrix of order 3. If Q=

[q i j]

is a matrix such that

P^{50} - Q = I, t h e n \frac{q_{31} + q_{32}}{q_{21}}

equals

A	52	B	103
C	201	D	205

Correct Answer : B

Category: IIT-JEE 2016 Math Paper 2

Question 2:

L e t b_{i} > 1 f o r i = 1, 2, . . ., 101 .

Suppose

\log_{e} b_{1}, \log_{e} b_{2}, . . ., \log_{e} b_{101}

are in Arithmetic Progression (A.P) with the common difference

\log_{e} 2 .

Suppose

a_{1}, a_{2}, . ., a_{101}

are in A.P. such that

a_{1} = b_{1} a n d a_{51} = b_{51} .

If t=

b_{1} + b_{2} + . . . + b_{51}

and

s = a_{1} + a_{2} + . . . + a_{51}, t h e n

A	$s > t a n d a_{101} > b_{101}$	B	$s > t a n d a_{101} < b_{101}$
C	$s < t a n d a_{101} > b_{101}$	D	$s < t a n d a_{101} < b_{101}$

Correct Answer : B

Category: IIT-JEE 2016 Math Paper 2

Question 3:

The value of

\sum_{k = 1}^{13} \frac{1}{\sin (\frac{π}{4} + \frac{(k - 1) π}{6}) \sin (\frac{π}{4} + \frac{k π}{6})}

is equal to

A	$3 - \sqrt{3}$	B	$2 (3 - \sqrt{3)}$
C	$2 (\sqrt{3} - 1)$	D	$2 (2 + \sqrt{3})$

Correct Answer : C

Category: IIT-JEE 2016 Math Paper 2

Question 4:

The value of

\int_{- \frac{π}{2}}^{\frac{π}{2}} \frac{x^{2} \cos x}{1 + e^{x}} d x i s e q u a l t o

A	$\frac{π^{2}}{4} - 2$	B	$\frac{π^{2}}{4} + 2$
C	$π^{2} - e^{\frac{x}{2}}$	D	$π^{2} + e^{\frac{π}{2}}$

Correct Answer : A

Category: IIT-JEE 2016 Math Paper 2

Question 5:

Area of the region

\{(x, y) \in ℝ^{2} : y \geq \sqrt{|x + 3|}, 5 y \leq x + 9 \leq 15\} i s e q u a l t o

A	$\frac{1}{6}$	B	$\frac{4}{3}$
C	$\frac{3}{2}$	D	$\frac{5}{3}$

Correct Answer : C

Category: IIT-JEE 2016 Math Paper 2

Question 6:

Let P be the image of the point (3,1,7) with respect to the plane x- y+ z=3. Then the equation of the plane passing through P and containing the straight line

\frac{x}{1} = \frac{y}{2} = \frac{z}{1}

A	x+y-3z=0	B	3x+z+0
C	x-4y+7z=0	D	2x-y=0

Correct Answer : C

Category: IIT-JEE 2016 Math Paper 2

Question 7:

Let f(x)=

\lim_{n \to \infty} {(\frac{n^{n} (x + n) (x + \frac{n}{2}) . . . (x + \frac{n}{n})}{n! (x^{2} + n^{2}) (x^{2} + \frac{n^{2}}{4}) . . . (x^{2} + \frac{n^{2}}{n^{2}})})}^{\frac{x}{n}},

for all x > 0. Then

A	f $(\frac{1}{2}) \geq$ f(1)	B	f $(\frac{1}{3}) \leq f (\frac{2}{3})$
C	f'(2) $\leq$ 0	D	$\frac{f' (3)}{f (3)} \geq \frac{f' (2)}{f (2)}$

Correct Answer : B and C

Category: IIT-JEE 2016 Math Paper 2

Question 8:

Let a, b

\in ℝ a n d f : ℝ \to ℝ b e d e f i n e d b y f (x) = a \cos (|x^{3} - x|) + b |x| \sin (|x^{3} + x|) .

Then f is

A	differentiable at x = 0 if α = 0 and b = 1
B	differentiable at x = 1 if α = 1 and b = 0
C	NOT differentiable at x = 0 if α =1 and b = 0
D	NOT differentiable at x =1 if $α = 1 a n d b = 1$

Correct Answer : A and B

Category: IIT-JEE 2016 Math Paper 2

Question 9:

Let f :

ℝ \to (0, \infty)

and

g : ℝ \to ℝ

be twice differentiable functions such that f" and g" are continuous functions on

ℝ

. Suppose f'(2)=g(2)=0, f" and g'(2)

\neq 0

. If

\lim_{x \to 2} \frac{f (x) g (x)}{f' (x) g' (x)} = 1,

then

A	f has a local minimum at x=2
B	f has a local maximum at x=2
C	f"(2)> f(2)
D	f(x) - f"(x) =0 for at least one x ∈ $ℝ$

Correct Answer : D and A

Category: IIT-JEE 2016 Math Paper 2

Question 10:

Let

f : [- \frac{1}{2}, 2] \to ℝ a n d g : [- \frac{1}{2}, 2] \to ℝ b e f u n c t i o n s d e f i n e d b y f (x) = [x^{2} - 3] a n d

g(x) =

|x| f (x) + |4 x - 7| f (x), w h e r e [y]

denotes the greatest integer less than or equal to y for y ∈

ℝ

. Then

A	f is discontinuous exactly at three points in $[- \frac{1}{2}, 2]$
B	f is discontinuous exactly at four points in $[- \frac{1}{2}, 2]$
C	g is NOT differentiable exactly at four points in $(- \frac{1}{2}, 2)$
D	g is NOT differentiable exactly at five points in $(- \frac{1}{2}, 2)$

Correct Answer : B and C

Category: IIT-JEE 2016 Math Paper 2

Question 11:

Let α, b ∈

ℝ

and

α^{2} + b^{2} \neq 0 .

Suppose

S = \{z \in ℂ : z = \frac{1}{a + i b t}, t \in ℝ, t \neq 0\},

where i =

\sqrt{- 1}

. If z =x +i y and s ∈ S, then (x, y) lies on

A	the circle with radius $\frac{1}{2 α}$ and centre $(\frac{1}{2 α}, 0) f o r α > 0, b \neq 0$
B	the circle with radius $- \frac{1}{2 α}$ and centre $(- \frac{1}{2 α}, 0)$ for $α > 0, b \neq 0$
C	the x-axis for $α \neq 0, b = 0$
D	the y-axis for $α = 0, b \neq 0$

Correct Answer : A, C and D

Category: IIT-JEE 2016 Math Paper 2

Question 12:

Let P be the point on the parabola

y^{2}

=4x which is at the shortest from the center S of the circle

x^{2} + y^{2} - 4 x - 16 y + 64 = 0

. Let Q be the point on the circle dividing the line segment SP internally . Then

A	SP = $2 \sqrt{5}$
B	SQ : QP =( $\sqrt{5} + 1)$ :2
C	the x-intercept of the normal to the parabola at P is 6
D	the slope of the tangent to the circle at Q is $\frac{1}{2}$

Correct Answer : A, C and D

Category: IIT-JEE 2016 Math Paper 2

Question 13:

Let

α, λ, μ \in ℝ .

Consider the system of linear equations

$α x + 2 y = λ 3 x - 2 y = μ$

Which of the following statement(s) is (are) correct?

A	If α =-3, then the system has infinitely many solutions for all values of λ and μ
B	If $α \neq - 3$ , then the system has a unique solution for all values of λ and μ
C	If λ + $μ$ =0, then the system has infinitely many solutions for $α = - 3$
D	If $λ + μ \neq 0,$ then the system has no solution for $α = - 3$

Correct Answer : B, C and D

Category: IIT-JEE 2016 Math Paper 2

Question 14:

Let

\hat{u} = u_{1} \hat{i} + u_{3} \hat{k}

be a unit vector in

ℝ^{3}

and

\hat{w}

\frac{1}{\sqrt{6}}

(iˆ +jˆ+ 2kˆ). Given that there exists a vector

\vec{v}

ℝ^{3}

such that

|\hat{u} x \vec{v}|

=1 and

\hat{w}

(\hat{u} x \vec{v})

=1. Which of the following statement(s) is (are) correct?

A	There is exactly one choice for such $\vec{v}$
B	There are infinitely many choices for such $\vec{v}$
C	If $\hat{u}$ lies in the xy-plane then $\|u_{1}\| = \|u_{2}\|$
D	If $\hat{u}$ lies in the xz-plane then $2 \|u_{1}\| = \|u_{3}\|$

Correct Answer : B and C

Category: IIT-JEE 2016 Math Paper 2

Question 15:

Football teams

T_{1} a n d T_{2}

have to play two games against each other. It is assumed that the outcomes of the two games are independent. The probabilities of

T_{1}

winning, drawing and losing a game against

T_{2} a r e \frac{1}{2}, \frac{1}{6} a n d \frac{1}{3},

respectively. Each team gets 3 points for a draw and 0 point for a loss in a game. Let X and Y denote the total points scored by teams

T_{1}

and

T_{2},

respectively, after two games. P(X > Y) is

A	$\frac{1}{4}$	B	$\frac{5}{12}$
C	$\frac{1}{2}$	D	$\frac{7}{12}$

Correct Answer : B

Category: IIT-JEE 2016 Math Paper 2

Question 16:

Football teams

T_{1} a n d T_{2}

have to play two games against each other. It is assumed that the outcomes of the two games are independent. The probabilities of

T_{1}

winning, drawing and losing a game against

T_{2} a r e \frac{1}{2}, \frac{1}{6} a n d \frac{1}{3},

respectively. Each team gets 3 points for a draw and 0 point for a loss in a game. Let X and Y denote the total points scored by teams

T_{1}

and

T_{2},

respectively, after two games. P(X =Y) is

A	$\frac{11}{36}$	B	$\frac{1}{3}$
C	$\frac{13}{36}$	D	$\frac{1}{2}$

Correct Answer : C

Category: IIT-JEE 2016 Math Paper 2

Question 17:

Let

F_{1} (x_{1}, 0) a n d F_{2} (x_{2}, 0), f o r x_{1} < 0,

be the foci of the ellipse

\frac{x^{2}}{9} + \frac{y^{2}}{8} = 1

. Suppose a parabola having vertex at the origin and focus at

F_{2}

intersects the ellipse at point M in the first quadrant and at pint N in the fourth quadrant. The orthocentre of the triangle

F_{1} M N

A	$(- \frac{9}{10}, 0)$	B	$(\frac{2}{3}, 0)$
C	$(\frac{9}{10}, 0)$	D	$(\frac{2}{3}, \sqrt{6})$

Correct Answer : A

Category: IIT-JEE 2016 Math Paper 2

Question 18:

Let

F_{1} (x_{1}, 0) a n d F_{2} (x_{2}, 0), f o r x_{1} < 0,

be the foci of the ellipse

\frac{x^{2}}{9} + \frac{y^{2}}{8} = 1

. Suppose a parabola having vertex at the origin and focus at

F_{2}

intersects the ellipse at point M in the first quadrant and at pint N in the fourth quadrant. If the tangents to the ellipse at M and N meet at R and the normal to the parabola at M meets the x-axis at Q, then the ratio of area of the triangle MQR to area of the quadrilateral

M F_{1} N F_{2}

A	3 : 4	B	4 : 5
C	5 : 8	D	2 : 3

Correct Answer : C

JEE Main 2016 Maths Question Paper Set 2 With Solution

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