JEE Main 2015 Maths Question Paper Set 2 With Solution

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Category: IIT-JEE 2015 Math Paper 2

Question 1:

For any integer k, let

α_{k} = \cos (\frac{k π}{7}) + i \sin (\frac{k π}{7}), w h e r e i = \sqrt{- 1 .}

The value of the expression

\frac{\sum_{k = 1}^{12} |α_{k + 1} - α_{k}|}{\sum_{k = 1}^{3} |α_{4 k - 1^{-}} α_{4 k - 2}|} i s

Correct Answer : 4

Category: IIT-JEE 2015 Math Paper 2

Question 2:

Suppose that all the terms of an arithmetic progression (A.P.) are natural numbers. If the ratio of the sum of the first seven terms to the sum of the first eleven terms is 6:11 and the seventh term lines in between 130 and 140, then the common difference of this A.P. is

Category: IIT-JEE 2015 Math Paper 2

Question 3:

The coefficient of

x^{9}

in the expansion of

(1 + x) (1 + x^{2}) (1 + x^{3}) . . . . . . (1 + x^{100}) i s

Correct Answer : 8

Category: IIT-JEE 2015 Math Paper 2

Question 4:

Suppose that the foci of the ellipse

\frac{x^{2}}{9} + \frac{y^{2}}{5} = 1 a r e (f_{1}, 0) a n d (f_{2}, 0) w h e r e f_{1} > 0 a n d f_{2} < 0 .

Let

p_{1} a n d p_{2}

be two parabolas with a common vertex at (0,0) and with foci at

(f_{1,} 0) a n d (2 f_{2,} 0),

respectively. Let

T_{1}

be a tangent to

p_{1}

which passes through

(2 f_{2}, 0) a n d T_{2}

be a tangent to

P_{2}

which passes through ((

f_{1}, 0

). If

m_{1}

is the slope of

T_{2}

, then the value of

(\frac{1}{m_{1}^{2}} + m_{2}^{2}) i s

Correct Answer : 5

Category: IIT-JEE 2015 Math Paper 2

Question 5:

Let m and n be two positive integers greater than 1.If

\lim_{α \to 0} (\frac{e^{\cos (α^{n})}}{α^{m}}) = - (\frac{e}{2})

then the value of

\frac{m}{n} i s

Correct Answer : 2

Category: IIT-JEE 2015 Math Paper 2

Question 6:

α = \int_{0}^{1} (e^{9 x + 3 \tan^{- 1}} x) (\frac{12 + 9 x^{2}}{1 + x^{2}}) d x

where

\tan^{- 1} x

takes only principal values,then the value of

(\log_{e} |1 + α| - \frac{3 π}{4}) i s

Correct Answer : 9

Category: IIT-JEE 2015 Math Paper 2

Question 7:

Let f :

ℝ \to ℝ

be a continuous odd function, which vanishes exactly at one point and

f (1) = \frac{1}{2} .

Suppose that F (x) =

\int_{- 1}^{x} f (t) d t f o r a l l x \in [- 1, 2] a n d G (x)

\int_{- 1}^{x} t |f (f (t))| d t

for all

x \in [- 1, 2] . I f \lim_{x \to 1} \frac{F (x)}{G (x)} = \frac{1}{14}, t h e n t h e v a l u e o f f (\frac{1}{2}) i s

Correct Answer : 7

Category: IIT-JEE 2015 Math Paper 2

Question 8:

Suppose that

\vec{p}, \vec{q} a n d \vec{r}

are three non-coplanar vectors in

ℝ^{3}

. Let the components of a vector

\vec{s}

along

\vec{p}, \vec{q} a n d \vec{r}

be 4,3 and 5, respectively . If the components of this vector

\vec{s}

along

(- \vec{p} + \vec{q} + \vec{r}), (\vec{p} - \vec{q} + \vec{r}) a n d (- \vec{p} - \vec{q} + \vec{r})

are x,y and z respectively , then the value of

2 x + y + z

Correct Answer : 9

Category: IIT-JEE 2015 Math Paper 2

Question 9:

Let S be the set of all non-zero real numbers

α

such that the quadratic equation

α x^{2} - x + α = 0

has two distinct real roots

x_{1} a n d x_{2}

satisfying the inquality

|x_{1} - x_{2}| < 1 .

which of the following intervals is (are) a subset (s) of S?

A	$(- \frac{1}{2}, - \frac{1}{\sqrt{5}})$	B	$(- \frac{1}{\sqrt{5}}, 0)$
C	$(0, \frac{1}{\sqrt{5}})$	D	$(\frac{1}{\sqrt{5}}, \frac{1}{2})$

Correct Answer : D and A

Category: IIT-JEE 2015 Math Paper 2

Question 10:

α = 3 \sin^{- 1} (\frac{6}{11}) a n d β = \cos^{- 1} (\frac{4}{9}),

where the inverse trigonometric function take only the principal values, then the correct option (s) is (are)

A	$\cos β > 0$	B	$\sin β < 0$
C	$\cos (α + β) > 0$	D	$\cos α < 0$

Correct Answer : B, C and D

Category: IIT-JEE 2015 Math Paper 2

Question 11:

Let

E_{1} a n d E_{2}

be two ellipses whose centers are at the origin.The major axes of

E_{1} a n d E_{2}

lie along the x-axis and the y-axis, respectively. Let S be the circle

x^{2} + {(y - 1)}^{2} = 2 .

The straight line x+y=3 touches the curves S,

E_{1} a n d E_{2}

at P,Q and R, respectively .suppose that

P Q = P R = \frac{2 \sqrt{2}}{3} .

e_{1} a n d e_{2}

are the eccentricities of

E_{1} a n d E_{2},

respectively, than the correct expression(s) is (are)

A	$e_{1}^{2} + e_{2}^{2} = \frac{43}{40}$	B	$e_{1} e_{2} = \frac{\sqrt{7}}{2 \sqrt{10}}$
C	$\|e_{1}^{2} - e_{2}^{2}\| = \frac{5}{8}$	D	$e_{1} e_{2} = \frac{\sqrt{3}}{4}$

Correct Answer : A and B

Category: IIT-JEE 2015 Math Paper 2

Question 12:

Consider the hyperbola

H : x^{2} - y^{2} = 1

and a circle S with center

N (x_{2}, 0) .

Suppose that H and S touch each other at a point P

(x_{1}, y_{1})

with

x_{1} > 1 a n d y_{1} > 0 .

The common tangent to H and S at P intersects the x-axis at point M.If (I,m) is centroid of the triangle

∆ P M N,

then the correct expression (s) is (are)

A	$\frac{d l}{d x_{1}} = 1 - \frac{1}{3 x_{1}^{2}} f o r x_{1} > 1$	B	$\frac{d m}{d x_{1}} = \frac{x_{1}}{3 (\sqrt{x_{1}^{2}} - 1)} f o r x_{1} > 1$
C	$\frac{d l}{d x_{1}} = 1 + \frac{1}{3 x_{1}^{2}} f o r x_{1} > 1$	D	$\frac{d m}{d y_{1}} = \frac{1}{3} f o r y_{1} > 0$

Correct Answer : A, B and D

Category: IIT-JEE 2015 Math Paper 2

Question 13:

The option(s) with the values of

α

and L that satisfy the following equation is (are )

\frac{\int_{0}^{4 π} e^{t} (\sin^{6} α t + \cos^{4} α t) d t}{\int_{0}^{π} e^{t} (\sin^{6} α t + \cos^{4} α t) d t} = L ?

A	$α = 2, L = \frac{e^{4 π} - 1}{e^{π} - 1}$	B	$α = 2, L = \frac{e^{4 π} + 1}{e^{π} + 1}$
C	$α = 4, L = \frac{e^{4 π} - 1}{e^{π} - 1}$	D	$α = 4, L = \frac{e^{4 π} + 1}{e^{π} + 1}$

Correct Answer : A and C

Category: IIT-JEE 2015 Math Paper 2

Question 14:

Let f,

g : [- 1, 2] \to ℝ

be continuous functions which are twice differential on the interval (-1,2). Let the values of f and g at the points-1,0 and 2 be as given in the following table:

	x=-1	x=0	x=2
f(x)	3	6	0
g(x)	0	1	-1

In each of intervals (-1,0) and (0,2) the function (f-3g)'' never vanishes.Then the correct statement (s) is 9(are)

A	f'(x)-3g'9(x)=0 has exactly three solutions in (-1,0)∪(0,2)	B	f'(x)-3g'9(x)=0 has exactly three solutions in (-1,0)
C	f'(x)-3g'9(x)=0 has exactly three solutions in(0,2)	D	f'(x)-3g'9(x)=0 has exactly three solutions in (-1,0) and exactly two solutions in (0,2)

Correct Answer : B and C

Category: IIT-JEE 2015 Math Paper 2

Question 15:

Let f(x)=7

\tan^{8} x + 7 \tan^{6} x - 3 \tan^{4} x - 3 \tan^{2} x f o r a l l x \in (- \frac{π}{2}, \frac{π}{2})

.Then the correct expression (s) is (are)

A	$\int_{0}^{π / 4} x f (x) d x = \frac{1}{2}$	B	$\int_{0}^{π / 4} x f (x) d x = 0$
C	$\int_{0}^{π / 4} x f (x) d x = \frac{1}{6}$	D	$\int_{0}^{π / 4} x f (x) d x = 1$

Correct Answer : A and B

Category: IIT-JEE 2015 Math Paper 2

Question 16:

Let f'(x)=

\frac{192 x^{3}}{2 + \sin^{4} πx}

for all x

\in ℝ w i t h f (\frac{1}{2}) = 0 .

If m

\leq \int_{1 / 2}^{1} (x) d x \leq M

,then the possible values of m and M are

A	m=13,M=24	B	$m = \frac{1}{4}, M = \frac{1}{2}$
C	m=-11,M=0	D	$m = 1, M = 12$

Correct Answer : D

Category: IIT-JEE 2015 Math Paper 2

Question 17:

Let

n_{1} a n d n_{2}

be the number of red and black balls,respectively,in box I. Let

n_{3} a n d n_{4}

be the number of red and black balls,respectively ,in box II. One of two boxes, box I and Box II, was selected at random and a ball was drawn randomly out of this box. The ball was found to be red.If the probability that this red ball was drawn from box II is

\frac{1}{3},

then the correct option (s) with the possible value of

n_{1}, n_{2}, n_{3} a n d n_{4}

is 9(are)

A	$n_{1} = 3, n_{2} = 3, n_{3} = 5, n_{4} = 15,$	B	$n_{1} = 3, n_{2} = 6, n_{3} = 10, n_{4} = 50,$
C	$n_{1} = 8, n_{2} = 6, n_{3} = 5, n_{4} = 20,$	D	$n_{1} = 6, n_{2} = 12, n_{3} = 5, n_{4} = 20$

Correct Answer : A and B

Category: IIT-JEE 2015 Math Paper 2

Question 18:

Let

n_{1} a n d n_{2}

be the number of red and black balls,respectively,in box I. Let

n_{3} a n d n_{4}

be the number of red and black balls,respectively ,in box II. A ball is drawn at random from box I and transferred to box II. If the probability of drawing a red ball from box I, after this transfer, is

\frac{1}{3},

then the correct option (s) with the possible values of

n_{1} a n d n_{2}

is a(are)

A	$n_{1} = 4 a n d n_{2} = 6$	B	$n_{1} = 2 a n d n_{2} = 3$
C	$n_{1} = 10 a n d n_{2} = 20$	D	$n_{1} = 3 a n d n_{2} = 6$

Correct Answer : C and D

Category: IIT-JEE 2015 Math Paper 2

Question 19:

Let F :

ℝ \to ℝ

be a thrice differential function.suppose that F(1)=0, F(3)=-4 and F'(x)

< 0 f o r a l l x \in (1 / 2, 3) . L e t f (x) = x F (x) f o r a l l x \in ℝ

\int_{1}^{3} x^{2} F' (x) d x = - 12 a n d \int_{1}^{3} x^{3} F'' (x) d x = 40

,then the correct expression (s) is (are)

A	$9 f' (3) + f' (1) - 32 = 0$	B	$\int_{1}^{3} f (x) d x = 12$
C	$9 f' (3) + f' (1) + 32 = 0$	D	$\int_{1}^{3} f (x) d x = - 12$

Correct Answer : C and D

Category: IIT-JEE 2015 Math Paper 2

Question 20:

Let F :

ℝ \to ℝ

be a thrice differential function.suppose that F(1)=0, F(3)=-4 and F'(x

< 0 f o r a l l x \in (1 / 2, 3) . L e t f (x) = x F (x) f o r a l l x \in ℝ

The correct statements (s) is (are)

A	f'(1)<0	B	f(2) <0
C	$f (x) \neq 0 f o r a n y x \in (1, 3)$	D	$f' (x) = 0 f o r s o m e x \in (1, 3)$

Correct Answer : A,B and C

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JEE Main 2015 Maths Question Paper Set 2 With Solution

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