JEE Main 2014 Maths Question Paper Set 2 With Solution

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Category: IIT-JEE 2014 Math Paper 2

Question 1:

The function 𝑦𝑦=𝑓𝑓(𝑥𝑥) is the solution of the differential equation

\frac{d y}{d x} + \frac{x y}{x^{2} - 1} = \frac{x^{4} + 2 x}{\sqrt{1 - x^{2}}}

in (−1,1) satisfying 𝑓𝑓(0)= 0.

\int_{\frac{\sqrt{3}}{2}}^{\frac{\sqrt{3}}{2}} f (x) d x

Then

A	$\frac{π}{3} - \frac{\sqrt{3}}{2}$	B	$\frac{π}{3} - \frac{\sqrt{3}}{4}$
C	$\frac{π}{6} - \frac{\sqrt{3}}{4}$	D	$\frac{π}{6} - \frac{\sqrt{3}}{2}$

Correct Answer : B

Category: IIT-JEE 2014 Math Paper 2

Question 2:

The following integral

\int_{\frac{π}{4}}^{\frac{π}{2}} {(2 \cos e c x)}^{17} d x

is equal to

A	$\int_{0}^{\log (1 + \sqrt{2})} {(e^{u} + e^{- u})}^{16} d u$	B	$\int_{0}^{\log (1 + \sqrt{2})} {(e^{u} + e^{- u})}^{17} d u$
C	$\int_{0}^{\log (1 + \sqrt{2})} {(e^{u} - e^{- u})}^{17} d u$	D	$\int_{0}^{\log (1 + \sqrt{2})} 2 {(e^{u} + e^{- u})}^{16} d u$

Correct Answer : A

Category: IIT-JEE 2014 Math Paper 2

Question 3:

Coefficient of

x^{11}

in the expansion of

{(1 + x)}^{2} {(1 + x^{3})}^{4} {(1 + x^{3})}^{7} {(1 + x^{4})}^{12} i s

A	1051	B	1106
C	1113	D	1120

Correct Answer : C

Category: IIT-JEE 2014 Math Paper 2

Question 4:

Let 𝑓𝑓:[0,2]→ℝ be a function which is continuous on [0,2] and is differentiable on (0,2) with 𝑓𝑓(0)= 1. Let

F (x) = \int_{0}^{x^{2}} f (\sqrt{t}) d t

for 𝑥𝑥∈[0,2]. If 𝐹𝐹′(𝑥𝑥)= 𝑓𝑓′(𝑥𝑥) for all 𝑥𝑥 ∈(0,2), then 𝐹𝐹(2) equals

A	$e^{2} - 1$	B	$e^{4} - 1$
C	$e - 1$	D	$e^{4}$

Correct Answer : B

Category: IIT-JEE 2014 Math Paper 2

Question 5:

The common tangents to the circle

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

and the parabola

y^{2} = 8 x

touch the circle at the points 𝑃𝑃,𝑄𝑄 and the parabola at the points 𝑅,𝑆. Then the area of the quadrilateral 𝑃QRS is

A	3	B	6
C	9	D	15

Correct Answer : D

Category: IIT-JEE 2014 Math Paper 2

Question 6:

For 𝑥𝑥∈(0,𝜋𝜋), the equation sin𝑥𝑥+2sin2𝑥𝑥−sin3𝑥𝑥=3 has

A	infinitely many solutions	B	three solutions
C	one solution	D	no solution

Correct Answer : D

Category: IIT-JEE 2014 Math Paper 2

Question 7:

In a triangle the sum of two sides is 𝑥 and the product of the same two sides is 𝑦. If

x^{2} - c^{2} = y

where 𝑐𝑐 is the third side of the triangle, then the ratio of the in-radius to the circum-radius of the triangle is

A	$\frac{3 y}{z x (x + c)}$	B	$\frac{3 y}{z c (x + c)}$
C	$\frac{3 y}{4 x (x + c)}$	D	$\frac{3 y}{4 c (x + c)}$

Correct Answer : B

Category: IIT-JEE 2014 Math Paper 2

Question 8:

Six cards and six envelopes are numbered 1,2,3,4,5,6 and cards are to be placed in envelopes so that each envelope contains exactly one card and no card is placed in the envelope bearing the same number and moreover the card numbered 1 is always placed in envelope numbered 2. Then the number of ways it can be done is

A	264	B	265
C	53	D	67

Correct Answer : C

Category: IIT-JEE 2014 Math Paper 2

Question 9:

Three boys and two girls stand in a queue. The probability, that the number of boys ahead of every girl is at least one more than the number of girls ahead of her, is

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

Correct Answer : A

Category: IIT-JEE 2014 Math Paper 2

Question 10:

The quadratic equation 𝑝𝑝(𝑥𝑥)= 0 with real coefficients has purely imaginary roots. Then the equation

p (p (x)) = 0

has

A	only purely imaginary roots	B	all real roots
C	two real and two purely imaginary roots	D	neither real nor purely imaginary roots

Correct Answer : D

Category: IIT-JEE 2014 Math Paper 2

Question 11:

Let 𝑎,𝑟,𝑠,𝑡 be nonzero real numbers. Let 𝑃(𝑎

t^{2}

,2𝑎t ), 𝑄, 𝑅(

a r^{2}, 2 a r

) and 𝑆(

a s^{2}, 2 a s

) be distinct points on the parabola

y^{2} = 4 a x

. Suppose that 𝑃Q is the focal chord and lines 𝑄Rand 𝑃K are parallel, where 𝐾𝐾 is the point (2𝑎,0). The value of 𝑟𝑟 is Choices:

A	$- \frac{1}{t}$	B	$\frac{t^{2} + 1}{t}$
C	$\frac{1}{t}$	D	$\frac{t^{2} - 1}{t}$

Correct Answer : D

Category: IIT-JEE 2014 Math Paper 2

Question 12:

If 𝑠 =1, then the tangent at 𝑃 and the normal at 𝑆 to the parabola meet at a point whose ordinate is

A	$\frac{{(t^{2} + 1)}^{2}}{2 t^{3}}$	B	$\frac{a {(t^{2} + 1)}^{2}}{2 t^{3}}$
C	$\frac{a {(t^{2} + 1)}^{2}}{t^{3}}$	D	$\frac{a {(t^{2} + 2)}^{2}}{t^{3}}$

Correct Answer : B

Category: IIT-JEE 2014 Math Paper 2

Question 13:

Given that for each 𝑎𝑎∈(0,1),

\lim_{h \to 0^{+}} \int_{h}^{1 - h} t^{- a} {(1 - t)}^{a - 1} d t

exists. Let this limit be 𝑔 (𝑎𝑎). In addition, it is given that the function 𝑔 (𝑎𝑎) is differentiable on (0,1). 53. The value of 𝑔

(\frac{1}{2}) i s

A	$π$	B	2 $π$
C	$\frac{π}{2}$	D	$\frac{π}{4}$

Correct Answer : A

Category: IIT-JEE 2014 Math Paper 2

Question 14:

Given that for each 𝑎𝑎∈(0,1),

\lim_{h \to 0^{+}} \int_{h}^{1 - h} t^{- a} {(1 - t)}^{a - 1} d t

exists. Let this limit be 𝑔 (𝑎𝑎). In addition, it is given that the function 𝑔 (𝑎𝑎) is differentiable on (0,1). The value of 𝑔 ′

(\frac{1}{2}) i s

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

Correct Answer : D

Category: IIT-JEE 2014 Math Paper 2

Question 15:

Box 1 contains three cards bearing numbers 1,2,3; box 2 contains five cards bearing numbers 1,2,3,4,5; and box 3 contains seven cards bearing numbers 1,2,3,4,5,6,7. A card is drawn from each of the boxes. Let 𝑥𝑖 be the number on the card drawn from the

i^{t h}

box, 𝑖=1,2,3. The probability that

x_{1} + x_{2} + x_{3}

is odd, is

A	$\frac{29}{105}$	B	$\frac{53}{105}$
C	$\frac{57}{105}$	D	$\frac{1}{2}$

Correct Answer : B

Category: IIT-JEE 2014 Math Paper 2

Question 16:

i^{t h}

box, 𝑖=1,2,3. The probability that

x_{1,} x_{2}, x_{3}

are in an arithmetic progression, is

A	$\frac{9}{105}$	B	$\frac{10}{105}$
C	$\frac{11}{105}$	D	$\frac{7}{105}$

Correct Answer : C

Category: IIT-JEE 2014 Math Paper 2

Question 17:

Let

Z_{K} = \cos (\frac{2 k π}{10}) + \sin (\frac{2 k π}{10}); k = 1, 2, . . . 9 .

LIST I	LIST II
P For each $z_{k}$ there exists a $z_{j}$ such that $z_{k} \cdot z_{j}$	1. True
Q. There exists a 𝑘 ∈ {1,2,…,9} such that $z_{1} \cdot z = z_{k}$ has no solution 𝑧𝑧 in the set of complex numbers.	2. False
R $\frac{\|1 - z_{1} \| \| 1 - z_{2}\| \cdot \cdot \|1 - z_{9}\|}{}$ equals	3. 1
S $1 - \sum_{K = 1}^{9} \cos (\frac{2 k π}{10}) e q u a l s$	4. 2

P Q R S

A	1 2 4 3	B	2 1 3 4
C	1 2 3 4	D	2 1 4 3

Correct Answer : C

Category: IIT-JEE 2014 Math Paper 2

Question 18:

LIST I	LIST II
P The number of polynomials $f (x)$ with non-negative integer coefficients of degree $\leq 2$ , satisfying 𝑓(0)=0 and $\int_{0}^{1} (x) d = 1, i s$	1. 8
Q The number of points in the interval $[- \sqrt{13,} \sqrt{13}]$ at which 𝑓(𝑥)= $\sin {(x)}^{2} + \cos (x^{2})$ attains its maximum value, is	2. 2
R $\int_{- 2}^{2} \frac{3 x^{2}}{(1 + e^{x})} d x e q u a l s$	3. 4
S $\frac{(\int_{\frac{1}{2}}^{\frac{1}{2}} \cos 2 x \log (\frac{1 + x}{1 - x}) d x)}{(\int_{0}^{\frac{1}{2}} \cos 2 x \log (\frac{1 + x}{1 - x}) d x)} e q u a l$	4. 0

P Q R S

A	3 2 4 1	B	2 3 4 1
C	3 2 1 4	D	2 3 1 4

Correct Answer : D

Category: IIT-JEE 2014 Math Paper 2

Question 19:

LIST I	LIST II
P Let y(x)=cos $(3 \cos^{- 1} x), x \in [- 1, 1], x \neq \pm \frac{\sqrt{3}}{2}$ Then $\frac{1}{y (x)} \{(x^{2} - 1)\} \frac{d^{2} y (x)}{d x^{2}} + x \begin{array}{r} d y (x) \\ d x \end{array}\} e q u a l s$	1. 1
Q Let $A_{1} A_{2} . . . A n (n > 2)$ be the vertices of a regular polygon of 𝑛𝑛 sides with its centre at the origin.Let $\vec{a_{k}}$ be the position vector of the point $A_{k}, K = 1, 2 . . ., n . I f \|\sum_{k = 1}^{n - 1} (\vec{a_{k}} \times \vec{a_{k + 1}})\| \|=\| \|\sum_{k = 1}^{n - 1} (\vec{a_{k}} \cdot \vec{a_{k + 1}})\|$ the minimum value of 𝑛𝑛 is	2. 2
R If the normal from the point 𝑃𝑃(ℎ,1) on the ellipse $\frac{x^{2}}{6} + \frac{y^{2}}{3} = 1$ is perpendicular to the line 𝑥𝑥+𝑦𝑦=8, then the value of ℎ is	3. 8
S Number of positive solutions satisfying the equation $\tan^{- 1} (\frac{1}{2 x + 1}) + \tan^{- 1} (\frac{1}{4 x + 1}) = \tan^{- 1} (\frac{2}{x^{2}}) i s$	4. 9

PQRS

A	4 3 2 1	B	2 4 3 1
C	4 3 1 2	D	2 4 1 3

Correct Answer : A

Category: IIT-JEE 2014 Math Paper 2

Question 20:

Let

f_{1}

:ℝ→ℝ ,

f_{2}

:[0,∞)→ℝ,

f_{3}

:ℝ→ℝ and

f_{4}

:ℝ→[0,∞) be defined by

f_{1} (x) = \{\begin{cases} |x| & i f x < 0, \\ e^{x} & i f x \geq 0; \end{cases}

f_{2} (x) = x^{2}; f_{3} (x) = \{\begin{cases} \sin x i f x < 0, \\ u i f x \geq 0 \end{cases} a n d f_{4} (x) = \{\begin{cases} f_{2} (f_{1} (x)) \\ f_{2} (f_{1} (x)) - 1 \end{cases} \frac{i f x < 0,}{i f x \geq 0 .}

LIST I	LIST II
P $f_{4} i s$	1. onto but not one-one
Q $f_{3} i s$	2. neither continuous nor one-one
R $f_{2} ° f_{1} i s$	3. differentiable but not one-one
S $f_{2} i s$	4. continuous and one-one

PQRS

A	3142	B	1342
C	3124	D	1324

Correct Answer : D

JEE Main 2014 Maths Question Paper Set 2 With Solution

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