JEE Main 2016 Maths Question Paper Set 1 With Solution

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Category: IIT-JEE 2016 Math Paper 1

Question 1:

Let -

\frac{π}{6} < θ < - \frac{π}{12} .

suppose

α_{1} a n d β_{1}

are the roots of the equation

x^{2}

-2x

s e c θ + 1 = 0

and

α_{2} a n d β_{2}

are the roots of the equation

x^{2}

+2x tan

θ

-1=0. If

α_{1} > β_{1} a n d α_{2} > β_{2},

then

α_{1} + β_{2}

equals

A	2(sec $θ$ -tan $θ$ )	B	2secθ
C	-2tanθ	D	0

Correct Answer : C

Category: IIT-JEE 2016 Math Paper 1

Question 2:

A debate club consists of 6 girls and 4 boys. A team of members is to selected from this club including the selection of a captain (from among these 4 members) for the team. If the team has include at most one boy, then the number of ways of selecting the team is

A	380	B	320
C	260	D	95

Correct Answer : A

Category: IIT-JEE 2016 Math Paper 1

Question 3:

Let

S = \{x \in (- π, π) : x \neq 0, \pm \frac{π}{2}\} .

The sum of all distinct solutions of the equation

\sqrt{3}

sec x + cosec x + 2(tan x-cot x) =0 in the set S is equal to

A	$- \frac{7 π}{9}$	B	$- \frac{2 π}{9}$
C	0	D	$\frac{5 π}{9}$

Correct Answer : C

Category: IIT-JEE 2016 Math Paper 1

Question 4:

A computer producing factory has only two plants

T_{1}

and

T_{2}

. Plants

T_{1}

produces 20% and plant

T_{2}

produces 80% of the total computers produced. 7% of computer produced in the factory turn out to be defective. It is known that P(Computer turns out to be defective given that it is produced in plant

T_{1}

) =10P (Computer turns out to be defective given that it is produced in plant

T_{2}

) where P(E) denotes the probability of an event E. A computer produced in the factory is randomly selected and it does not turn out to be defective. Then the probability that it is produced in plant

T_{2}

A	$\frac{36}{73}$	B	$\frac{47}{79}$
C	$\frac{78}{93}$	D	$\frac{75}{83}$

Correct Answer : C

Category: IIT-JEE 2016 Math Paper 1

Question 5:

The least value of α ε

ℝ

for which

4 a x^{2} + \frac{1}{x} \geq 1,

for all x > 0, is

A	$\frac{1}{64}$	B	$\frac{1}{32}$
C	$\frac{1}{27}$	D	$\frac{1}{25}$

Correct Answer : C

Category: IIT-JEE 2016 Math Paper 1

Question 6:

consider a pyramid OPQRS located in the first octant (x

\geq 0, y \geq 0, z \geq 0

) with O as origin, and OP and OR along the x-axis and the y-axis, respectively. The base OPQR of the pyramid is a square with OP = 3. The point S is directly above the mid-point T of diagonal OQ such that TS=3. Then

A	the acute angle between OQ and OS is $\frac{π}{3}$	B	the equation of the plane containing the triangle OQS is x-y=0
C	the length of the perpendicular from P to the plane containing the triangle OQS is $\frac{3}{\sqrt{2}}$	D	the perpendicular distance from O to the straight line containing RS is $\sqrt{\frac{15}{2}}$

Correct Answer : B,C and D

Category: IIT-JEE 2016 Math Paper 1

Question 7:

Let f :(0,

\infty

)

\overset{}{\to}

ℝ

be a differentiable function such that f '(x)=

\frac{f (x)}{x}

for all x ε(0,∞) and f(1)

\neq

1. Then

A	$\lim_{x \to 0 +}$ f' $(\frac{1}{x}) = 1$	B	$\lim_{x \to 0 +}$ x f $(\frac{1}{x})$ =2
C	$\lim_{x \to 0 +} x^{2}$ f'(x)=0	D	\| f(x) \| $\leq$ 2 for all x ∈ (0,2)

Correct Answer : A

Category: IIT-JEE 2016 Math Paper 1

Question 8:

Let P=

[\begin{matrix} 3 & - 1 & - 2 \\ 2 & 0 & α \\ 3 & - 5 & 0 \end{matrix}]

, where

α \in ℝ .

Suppose Q=[

q_{i j}

] is a matrix such that PQ=k I, where

k \in ℝ, k \neq 0

and I is the identity matrix of order 3. If

q_{23}

\frac{k^{}}{2}

and det(Q)=

\frac{k^{2}}{2}

, then

A	α=0,k=8	B	4 $α$ -k+8=0
C	det(P adj(Q))= $2^{9}$	D	det(Q adj(P))= $2^{13}$

Correct Answer : B and C

Category: IIT-JEE 2016 Math Paper 1

Question 9:

In a triangle XYZ, let x,y,z be the lengths of sides opposite to the angles X,Y,Z respectively, and 2s=x+y+z. If

\frac{s - x}{4} = \frac{s - y}{3} = \frac{s - z}{2}

and area of incircle of the triangle XYZ is

\frac{8 π}{3},

then

A	area of the triangle XYZ is $6 \sqrt{6}$
B	the radius of circumcircle of the triangle XYZ is $\frac{35}{6} \sqrt{6 .}$
C	$\sin \frac{X}{2} \sin \frac{Y}{2} \sin \frac{Z}{2} = \frac{4}{35}$
D	$\sin^{2} (\frac{X + Y}{2}) = \frac{3}{5}$

Correct Answer : A, C and D

Category: IIT-JEE 2016 Math Paper 1

Question 10:

A solution curve of the differential equation (

x^{2} + x y + 4 x + 2 y + 4) \frac{d y}{d x} - y^{2} = 0, x > 0,

passes through the point (1,3). Then the solution curve

A	intersects y=x+2 exactly at one point
B	intersects y=x+2 exactly at two points
C	intersects $y = {(x + 2)}^{2}$
D	does NOT intersect $y = {(x + 3)}^{2}$

Correct Answer : A and D

Category: IIT-JEE 2016 Math Paper 1

Question 11:

Let f:

ℝ \overset{}{\to ℝ}

, g:

ℝ \overset{}{\to ℝ}

and h:

ℝ \overset{}{\to ℝ}

be differentiable functions such that f(x)=

x^{3} + 3 x + 2,

g(f(x))=x and h (g(g(x)))=x for x for all x ∈

ℝ

. Then

A	g'(2)= $\frac{1}{15}$	B	h'(1)=666
C	h(0)=16	D	h(g(3))=36

Correct Answer : B and C

Category: IIT-JEE 2016 Math Paper 1

Question 12:

The circle

C_{1} : x^{2} + y^{2} = 3,

with centre at O, intersects the parabola

x^{2} = 2 y

at the point P in the firts quadrant. Let the tangent to the tangent to the circle

C_{1}

at P touches other two circles

C_{2}

and

C_{3}

R_{2}

and

R_{3}

, respectively. Suppose

C_{2}

and

C_{3}

have eual radii

2 \sqrt{3}

and centres

Q_{2}

and

Q_{3,}

respectively. If

Q_{2}

and

Q_{3}

lie on the y-axis, then

A	$Q_{2} Q_{3} = 12$	B	$R_{2} R_{3} = 4 \sqrt{6}$
C	area of the tringle $O R_{2} R_{3}$ is $6 \sqrt{2}$	D	area of the triangle $P Q_{2} Q_{3}$ is $4 \sqrt{2}$

Correct Answer : A,B and C

Category: IIT-JEE 2016 Math Paper 1

Question 13:

Let RS be the diameter of the circle

x^{2} + y^{2} = 1

, where S is the point (1,0). Let P be a variable point (other than R and S) on the circle and tangents to the S and P meet at the point Q. The normal to the circle at P intersects a line drawn through Q parallel to RS at point E. Then the locus of E passes through the point(s)

A	$(\frac{1}{3}, \frac{1}{\sqrt{3}})$	B	$(\frac{1}{4}, \frac{1}{2})$
C	$(\frac{1}{3}, \frac{1}{\sqrt{3}})$	D	$(\frac{1}{4}, - \frac{1}{2})$

Correct Answer : A and C

Category: IIT-JEE 2016 Math Paper 1

Question 14:

The total number of distinct x ∈

ℝ

for which

|\begin{matrix} x & x^{2} & 1 + x^{3} \\ 2 x & 4 x^{2} & 1 + 8 x^{3} \\ 3 x & 9 x^{2} & 1 + 27 x^{3} \end{matrix}|

=10 is

Correct Answer : 2

Category: IIT-JEE 2016 Math Paper 1

Question 15:

Let m be the smallest positive integer such that the coefficient of

x^{2}

in the expansion of

{(1 + x)}^{2} + {(1 + x)}^{3} + . . . + {(1 + x)}^{49} + {(1 + m x)}^{50} i s (3 n + 1) C_{3}^{51}

for some positive integer n. Then the value of n is

Correct Answer : 5

Category: IIT-JEE 2016 Math Paper 1

Question 16:

The total number of distinct x ∈ [0,1] for which

\int_{0}^{x} \frac{t^{2}}{1 + t^{4}} d t = 2 x - 1 i s

Correct Answer : 1

Category: IIT-JEE 2016 Math Paper 1

Question 17:

Let α, β ∈

ℝ

be such that

\lim_{x \to 0} \frac{x^{2} \sin (β x)}{α x - \sin x} = 1

Then 6 (

α + β

) equals

Correct Answer : 7

Category: IIT-JEE 2016 Math Paper 1

Question 18:

Let z=

\frac{- 1 + \sqrt{3 i}}{2}

, where i=

\sqrt{- 1}

, and r,s∈ {1,2,3}. Let P

= [\begin{matrix} {(- z)}^{r} z^{2 s} \\ z^{2 s} z^{r} \end{matrix}]

and I be the identity matrix of order 2. Then the total number of pairs (r,s) for which

P^{2} = -

I is

Correct Answer : 1

JEE Main 2016 Maths Question Paper Set 1 With Solution

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