JEE Mains 2015

A

be two sets containing four and two elements respectively. Then, the number of subsets of the set

B

A \times B

, each having at least three elements are (a) 219 (b) 256 (c) 275 (d) 510

Topic:

Permutations and Combinations

Exam:

JEE Mains 2015

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Question 2

is the AM of two distinct real numbers

m

l

n (l, n > 1)

G_{1}, G_{2}

are three geometric means between

G_{3}

l

n

, then

G_{1}^{4} + 2 G_{2}^{4} + G_{3}^{4}

equals

$4 l^{2} mn$
$4 l m^{2} n$
$4 m n^{2}$
$4 l^{2} m^{2} n^{2}$

Topic:

Sequences and Series

Exam:

JEE Mains 2015

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Question 3

The sum of first 9 terms of the series

\frac{1 ^{3}}{1} + \frac{1 ^{3} + 2 ^{3}}{1 + 3} + \frac{1 ^{3} + 2 ^{3} + 3 ^{3}}{1 + 3 + 5} + \dots

Topic:

Sequences and Series

Exam:

JEE Mains 2015

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Question 4

is equal to (a) 4 (b) 3 (c) 2 (d)

lim_{x \to 0} \frac{( 1 - c o s 2 x ) ( 3 + c o s x )}{x t a n 4 x}

\frac{1}{2}

Topic:

Limits and Derivatives

Exam:

JEE Mains 2015

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Question 5

differentiable, then the value of

If the function

g (x) = {k x + 1 m x + 2, 0 \leq x \leq 3, 3 < x \leq 5

k + m

is (a) 2 (b)

\frac{16}{5}

\frac{10}{3}

(d) 4

Topic:

Continuity and Differentiability

Exam:

JEE Mains 2015

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Question 6

be a polynomial of degree four having extreme values at

f (x)

x = 1

x = 2

. If

lim_{x \to 0} [1 + \frac{f ( x )}{x ^{2}}] = 3

, then

f (2)

is equal to (a)

- 8

- 4

Topic:

Limits and Derivatives

Exam:

JEE Mains 2015

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Question 7

The integral

\int \frac{d x}{x ^{2} ( x ^{4} + 1 ) ^{\frac{3}{4}}}

equals (a)

(\frac{x ^{4} + 1}{x ^{4}})^{1/4} + c

(x^{4} + 1)^{1/4} + c

- (x^{4} + 1)^{1/4} + c

- (\frac{x ^{4} + 1}{x ^{4}})^{1/4} + c

Topic:

Integrals

Exam:

JEE Mains 2015

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Question 8

is equal to (a) 2 (b) 4 (c) 1 (d) 6

The integral

\int_{2}^{4} \frac{l o g x ^{2}}{l o g x ^{2} + l o g ( 36 - 12 x + x ^{2} )} d x

Topic:

Integrals

Exam:

JEE Mains 2015

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Question 9

The area (in sq. units) of the region described by

{(x, y) : y^{2} \leq 2 x

y \geq 4 x - 1}

\frac{7}{32}

\frac{5}{64}

\frac{15}{64}

\frac{9}{32}

Topic:

Application of Integrals

Exam:

JEE Mains 2015

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Question 10

be the solution of the differential equation

y (x)

(x lo g x) \frac{d y}{d x} + y = 2 x lo g x, (x \geq 1)

. Then,

y (e)

is equal to (a)

e

(b) 0 (c) 2 (d)

2 e

Topic:

Differential Equations

Exam:

JEE Mains 2015

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Question 11

The number of common tangents to the circles

x^{2} + y^{2} - 4 x - 6 y - 12 = 0

x^{2} + y^{2} + 6 x + 18 y + 26 = 0

Topic:

Conic Sections

Exam:

JEE Mains 2015

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Question 12

The distance of the point

(1, 0, 2)

from the point of intersection of the line

\frac{x - 2}{3} = \frac{y + 1}{4} = \frac{z - 2}{12}

and the plane

x - y + z = 16

214

(b) 8 (c)

321

(d) 13

Topic:

Three Dimensional Geometry

Exam:

JEE Mains 2015

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Question 13

The equation of the plane containing the line

2 x - 5 y + z = 3, x + y + 4 z = 5

and parallel to the plane

x + 3 y + 6 z = 1

2 x + 6 y + 12 z = 13

x + 3 y + 6 z = - 7

x + 3 y + 6 z = 7

2 x + 6 y + 12 z = - 13

Topic:

Three Dimensional Geometry

Exam:

JEE Mains 2015

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Question 14

The angle between the lines whose direction cosines satisfy the equation

l + m + n = 0

l^{2} = m^{2} + n^{2}

\frac{π}{3}

\frac{π}{4}

\frac{π}{6}

\frac{π}{2}

Topic:

Three Dimensional Geometry

Exam:

JEE Mains 2015

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Question 15

be three non-zero vectors such that no two of them are collinear and

Let a, b and

c

(a \times b) \times c = \frac{1}{3} ∣ b ∥ c ∣ a

. If

θ

is the angle between vectors band c, then a value of

sin θ

$\frac{2 2}{3}$
$\frac{- 2}{3}$
$\frac{2}{3}$
$\frac{- 2 3}{3}$

Topic:

Vector Algebra

Exam:

JEE Mains 2015

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Question 16

If 12 identical balls are to be placed in 3 identical boxes, then the probability that one of the boxes contains exactly 3 balls, is (a)

\frac{55}{3} (\frac{2}{3})^{11}

55 (\frac{2}{3})^{10}

220 (\frac{1}{3})^{12}

22 (\frac{1}{3})^{11}

Topic:

Probability

Exam:

JEE Mains 2015

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Question 17

The mean of the data set comprising of 16 observations is 16 . If one of the observation valued 16 is deleted and three new observations valued 3, 4 and 5 are added to the data, then the mean of the resultant data is (a)

16.8

16.0

15.8

14.0

Topic:

Statistics

Exam:

JEE Mains 2015

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Question 18

tan^{- 1} y = tan^{- 1} x + tan^{- 1} (\frac{2 x}{1 - x ^{2}})

, where

∣ x ∣ < \frac{1}{3}

. Then, a value of

y

\frac{3 x - x ^{3}}{1 - 3 x ^{2}}

\frac{3 x + x ^{3}}{1 - 3 x ^{2}}

\frac{3 x - x ^{3}}{1 + 3 x ^{2}}

\frac{3 x + x ^{3}}{1 + 3 x ^{2}}

Topic:

Inverse Trigonometric Functions

Exam:

JEE Mains 2015

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Question 19

The negation of

\sim s \lor (\sim r \land s)

is equivalent to (a)

s \land \sim r

s \land (r \land \sim s)

s \lor (r \lor \sim s)

s \land r

Topic:

Mathematical Reasoning

Exam:

JEE Mains 2015

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Question 20