JEE Main 2015 Maths Question Paper Set 1 With Solution

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Category: IIT-JEE 2015 Math Paper 1

Question 1:

The number of distinct solutions of the equation

\frac{5}{4} \cos^{2} 2 x + \cos^{4} x + \sin^{4} x \cos^{6} x + \sin^{6} x = 2

in the interval

[0, 2 π} is

Correct Answer : 9

Category: IIT-JEE 2015 Math Paper 1

Question 2:

Let the curve C be the mirror image of the parabola

y^{2} = 4 x

with respect to the line x+y+4=0. If A and B are the points of intersection of C with the line y=-5, then the distance between A and B is

Correct Answer : 4

Category: IIT-JEE 2015 Math Paper 1

Question 3:

The minimum number of times a fair coin needs to be tossed so that the probability of getting at least two heads is at least 0.96,is

Correct Answer : 8

Category: IIT-JEE 2015 Math Paper 1

Question 4:

Let n be the number of ways in which 5 boys and 5 girls can stand in a queue in such a way that all the girls stand consecutively in the queue. Let m be the numbers of ways in which 5 boys and 5 girls can stand in a queue in such a way that exactly four girls stand consecutively in the queue .then the value of

\frac{m}{n} i s

Correct Answer : 5

Category: IIT-JEE 2015 Math Paper 1

Question 5:

If the normals of the parabola

y^{2} = 4 x

drawn at the end points of its latus rectum are tangents to the circle

{(x - 3)}^{2} + {(y + 2)}^{2} = r^{2},

then the value of

r^{2}

Correct Answer : 2

Category: IIT-JEE 2015 Math Paper 1

Question 6:

Let f:

ℝ \to ℝ

be a function defined by f(x)=

\{\begin{array}{l} [x] \\ 0, \end{array},

x \leq 2 x > 2

., where [x] is the greatest integer less than or equal to x .If

I = \int_{- 1}^{2} \frac{x f (x^{2})}{2 + f (x + 1)}

dx, then the value of (4I-1)is

Correct Answer : 0

Category: IIT-JEE 2015 Math Paper 1

Question 7:

A cylindrical container is to made from certain solid material with the following constraints:it has a fixed inner volume of V

m m^{3}

, has a 2mm thick solid wall is open at the top. The bottom of the container is a solid circular disc of thickness 2 mm and is of radius equal to the outer radius of the container. If the volume of the material used to make the container is minimum when the inner radius of the container is 10mm, then the value of

\frac{V}{250 π} i s

Correct Answer : 4

Category: IIT-JEE 2015 Math Paper 1

Question 8:

Let F (x)=

\int_{x}^{x^{2} + \frac{π}{6}}

\cos^{2}

tdt for all

x \in ℝ a n d f : [0, \frac{1}{2}] \to [0, \infty)

be a continuos function . for

α \in [0, \frac{1}{2}], i f F' (a) + 2 i s

the area of the region bounded by x=0,y=0,y=f(x) and x=a, then f a(0) is

Correct Answer : 3

Category: IIT-JEE 2015 Math Paper 1

Question 9:

Let X and Y be two arbitrary 3x3, non-zero,skew-symmetric matrices and Z be an arbitrary 3x3, non-zero, symmetric matrix. Then which of the following matrices is (are) skew-symmetric?

A	$Y^{3} Z^{4} - Z^{4} Y^{3}$	B	$X^{44} + Y^{44}$
C	$X^{4} Z^{3} - Z^{3} X^{4}$	D	$X^{23} + Y^{23}$

Correct Answer : C and D

Category: IIT-JEE 2015 Math Paper 1

Question 10:

which of the following values of

α

satisfy the equation

|\begin{matrix} {(1 + a)}^{2} & {(1 + 2 α)}^{2} & {(1 + 3 α)}^{2} \\ {(2 + α)}^{2} & {(2 + 2 α)}^{2} & {(2 + 3 α)}^{2} \\ {(3 + α)}^{2} & {(3 + 2 α)}^{2} & {(3 + 3 α)}^{2} \end{matrix}| = 648 α ?

A	-4	B	9
C	-9	D	4

Correct Answer : B and C

Category: IIT-JEE 2015 Math Paper 1

Question 11:

ℝ^{3},

consider the planes

p_{1} : y = 0

and

p_{2} : x + z = 1 . L e t p_{3}

be a plane,different from

p_{1} a n d p_{2}

which passes through the intersection of

p_{1} a n d p_{2}

.If the distance of the point (0,1,0) from

p^{3}

is 1 and the distance of a (

α, β, γ

)from

p_{3}

is 2, then which of the following relations is (are) true?

A	$2 α + β + 2 γ + 2 = 0$	B	$2 α - β + 2 γ + 4 = 0$
C	$2 α + β - 2 γ - 10 = 0$	D	$2 α - β + 2 γ - 8 = 0$

Correct Answer : B and D

Category: IIT-JEE 2015 Math Paper 1

Question 12:

ℝ^{3}

,let L be a straight line passing through the origin .suppose that all the points on L are at a constant distance from the two planes

p_{1} : x + 2 y - z + 1 = 0 a n d P_{2} : 2 x - y + z - 1 = 0 .

Let M be the locus of the locus of the feet of theperpendiculars drawn from the points on L to the plane. which of the following points lie(s) on M?

A	$(0, - \frac{5}{6}, - \frac{2}{3})$	B	$(- \frac{1}{6}, - \frac{1}{3}, \frac{1}{6})$
C	$(- \frac{5}{6}, 0, \frac{1}{6})$	D	$(- \frac{1}{3}, 0, \frac{2}{3})$

Correct Answer : A and B

Category: IIT-JEE 2015 Math Paper 1

Question 13:

Let P and Q be distinct points on the parabola

y^{2} = 2 x

such that a circle with PQ as diameter passess through the vertax O Of the parabola .If P lies in the first quadrant and the area of the triangle

∆ O P Q i s 3 \sqrt{2},

then which of the following is (are) the coordinates of P?

A	$(4, 2 \sqrt{2})$	B	$(9, 3 \sqrt{2})$
C	$(\frac{1}{4}, \frac{1}{\sqrt{2}})$	D	$(1, \sqrt{2})$

Correct Answer : A and D

Category: IIT-JEE 2015 Math Paper 1

Question 14:

Let y(x) be a solution of the differential equation (

(1 + e^{x}) y' + y e^{x} = 1

.If y(0)=2, then which of the following statements is (are) true?

A	y(-4)=0	B	y(-2)=0
C	y(x) has a critical point in the interval (-1,0)	D	y(x) has no critical point in the interval (-1,0)

Correct Answer : A and C

Category: IIT-JEE 2015 Math Paper 1

Question 15:

consider the family of all circles whose centres lie on the straight line y=x. If this family of circles is represented by the differential represented by the differential equation py''+Qy'+1=0,where P,Q are functions of x,y and y'(here y'=

\frac{d y}{d x}, y'' = \frac{d^{2} y}{d x^{2}}),

then which of the following statements is (are) true ?

A	P=y+x	B	P=y-x
C	$P + Q = 1 - x + y + {(y')}^{2}$	D	p-Q=x+y-y'- ${(y')}^{2}$

Correct Answer : B and C

Category: IIT-JEE 2015 Math Paper 1

Question 16:

Let g:

ℝ \to ℝ

be a differential function with g(0)=0,g'(0)=0 and g'(1)≠0.Let

f (x) \{\begin{array}{l} x \\ |x| \end{array} g (x), x \neq 0 x = 0

and h(x)

= e^{|x|}

for all

x \in ℝ .

Let

(f \circ h) (x)

denote f(h(x)) and

(h \circ f)

(x) denote h(f(x)) then which of the following is (are) true ?

A	f is differentiable at x=0	B	h is differential at x=0
C	$f \circ h$ is differential at x =0	D	$h \circ f$ is differential at x=0

Correct Answer : A and D

Category: IIT-JEE 2015 Math Paper 1

Question 17:

Let f(x)=sin

(\frac{π}{6} \sin (\frac{π}{2} \sin x))

for all x

\in ℝ a n d g (x) = \frac{π}{2}

sin x for all

x \in ℝ

.Let (f

\circ g) (x)

denote f(g(x)) and (g

\circ

f)(x).Then which of the following is (are) true ?

A	Range of f is $[- \frac{1}{2}, \frac{1}{2}]$	B	$R a n g o o f f \circ g i s [- \frac{1}{2}, \frac{1}{2}]$
C	$\lim_{x \to 0} \frac{f (x)}{g (x)} = \frac{π}{6}$	D	There is an $x \in ℝ s u c h t h a t (g \circ f) (x) = 1$

Correct Answer : A,B and C

Category: IIT-JEE 2015 Math Paper 1

Question 18:

Let

∆ P Q R

be a triangle.

L e t \vec{α} = \vec{Q R,} \vec{b} = \vec{R P} a n d \vec{c} = \vec{P Q}

,If

|\vec{α}| = 12, |\vec{b}| = 4 \sqrt{3} a n d \vec{b} . \vec{c} = 24,

then which off the following is (are) true?

A	${\|\vec{C}\|}^{2} - \|\vec{α}\| = 12 2$	B	${\|\vec{C}\|}^{2} + \|\vec{α}\| = 30 2$
C	$\| \vec{α} x \vec{b} + \vec{c} x \vec{α} \| = 48 \sqrt{3}$	D	$\vec{α} . \vec{b} = - 72$

Correct Answer : A,C and D

Category: IIT-JEE 2015 Math Paper 1

Question 19:

Column I	Column II
(A)In $ℝ^{2},$ if the magnitude of the projection vector of the vector α $\hat{i}$ $+ β \hat{j} o n \sqrt{3 \hat{i}} + \hat{j} i s \sqrt{3} a n d i f α = 2 + \sqrt{3 β,}$ then possible value (s) of $\|α\|$ is (arezz0	(P) 1
(B)Let a and b real numbers such that the function $f (x) = \{\begin{cases} - 3 α x^{2} - 2, x < 1 \\ b x + α^{2}, x \geq 1 \end{cases}$ is differentiable for all $x \in ℝ$ . Then possible value (s) of $α$ is (are)	(Q) 2
(C)Let $ω \neq 1$ be a complex cube root of unity. If ${(2 + 3 ω - 3 ω^{2})}^{4 n + 3} + {(- 3 + 2 ω + 3 ω^{2})}^{4 n + 3} = 0,$ then possible value (s) of n is (are)	(R) 3
(D)Let the harmonic mean of two positive real numbers $α$ and b 4.If q is a positive real number such that $α$ ,5,q,b is an arithmetic progression ,then the value (s) of $\|q - α\|$ is (are)	(S) 4
	(T) 5

A	A-Q, B-R, C-S, D-P
B	A-P, B-Q, C-S, D-T
C	A-R, B-Q, C-S, D-P
D	A-Q, B- R, C-T, D-S

Correct Answer : B

Category: IIT-JEE 2015 Math Paper 1

Question 20:

Column I	ColumnII
(A)In a triangle $∆ X Y Z, L e t α$ ,b and c be the lengths of the sides opposite to the angles X,Y and Z, respectively.If 2 $(α^{2} - b^{2}) = c^{2} a n d λ = \frac{\sin (x - y)}{\sin z},$ then possible values of n for which cos $(n πλ) = 0$ is (are)	(P)1
(B)In a triangle $∆ X Y Z,$ let a,b and c the lengths of the sides opposite to the angles X,Yand Z, respectively .if 1+cos 2X-2cos2Y=2sinXsinY,then possible value (s)of $\frac{α}{b}$ is (are)	(Q)2
(C)In $ℝ^{2}, l e t \sqrt{3} \hat{i} + \hat{j}, \hat{i} + \sqrt{3 \hat{j}} a n d β \hat{i} + (1 - β) \hat{j}$ be the position vectors of X, Y and Z with respect to the origin O, respectively. If the distance of Z from the bisector of the acute angle of $\vec{O X} w i t h \vec{O Y} i s \frac{3}{\sqrt{2}},$ then possible value (s) of $\|β\|$ is (are)	(R)3
(D)suppose that F $(α)$ denotes the area of the region bounded by $x = 0, x = 2, y^{2} = 4 x a n d y = \|α x - 1\| + \|α x - 2\| + α x, w h e r e α \in \{0, 1\} .$ Then the value (s) of F $(α) + \frac{8}{3} \sqrt{2,} w h e n α = 0 a n d α = 1,$ is (are)	(S)5
	(T)6

A	A-S, B-P, C-Q, D-S
B	A-T, B-P, C-Q, D-R
C	A-R, B-Q, C-S, D-P
D	A-Q, B- R, C-T, D-S

Correct Answer : A

A	${\|\vec{C}\|}^{2} - \|\vec{α}\| = 12 2$	B	${\|\vec{C}\|}^{2} + \|\vec{α}\| = 30 2$
C	$\| \vec{α} x \vec{b} + \vec{c} x \vec{α} \| = 48 \sqrt{3}$	D	$\vec{α} . \vec{b} = - 72$

JEE Main 2015 Maths Question Paper Set 1 With Solution

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