JEE Mains 2014

If the coefficients of

x^{3}

x^{4}

in the expansion of

(1 + a x + b x^{2}) (1 - 2 x)^{18}

in powers of

x

are both zero, then

(a, b)

(16, \frac{251}{3})

(14, \frac{251}{3})

(14, \frac{272}{3})

(16, \frac{272}{3})

Topic:

Binomial Theorem

Exam:

JEE Mains 2014

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Question 2

(10)^{9} + 2 (11)^{1} (10)^{8} + 3 (11)^{2} (10)^{7}

+ \dots + 10 (11)^{9} = K (10)^{9}

, then

k

is equal to

$\frac{121}{10}$
$\frac{441}{100}$
100
110

Topic:

Sequences and Series

Exam:

JEE Mains 2014

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Question 3

Three positive numbers form an increasing GP. If the middle term in this GP is doubled, then new numbers are in AP. Then, the common ratio of the GP is

$2 + 3$
$3 + 2$
$2 - 3$
$2 + 3$

Topic:

Sequences and Series

Exam:

JEE Mains 2014

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Question 4

lim_{x \to 0} \frac{s i n ( π c o s ^{2} x )}{x ^{2}}

\frac{π}{2}

(b) 1 (c)

- π

π

Topic:

Limits and Derivatives

Exam:

JEE Mains 2014

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Question 5

f

are differentiable functions in

g

(0, 1)

satisfying

f (0) = 2 = g (1), g (0) = 0

f (1) = 6

, then for some

c \in] 0, 1 [

(a)

2 f^{'} (c) = g^{'} (c)

2 f^{'} (c) = 3 g^{'} (c)

f^{'} (c) = g^{'} (c)

f^{'} (c) = 2 g^{'} (c)

Topic:

Continuity and Differentiability

Exam:

JEE Mains 2014

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Question 6

x = - 1

x = 2

are extreme points of

f (x) = α lo g ∣ x ∣ + β x^{2} + x

, then (a)

α = - 6, β = \frac{1}{2}

α = - 6, β = - \frac{1}{2}

α = 2, β = - \frac{1}{2}

α = 2, β = \frac{1}{2}

Topic:

Application of Derivatives

Exam:

JEE Mains 2014

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Question 7

The integral

\int (1 + x - \frac{1}{x}) e^{x + \frac{1}{x}} . d x

(x - 1) e^{x + \frac{1}{x}} + C

x e^{x + \frac{1}{x}} + C

(x + 1) e^{x + \frac{1}{x}} + C

- x e^{x + \frac{1}{x}} + C

Topic:

Integrals

Exam:

JEE Mains 2014

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Question 8

The integral

\int_{0}^{π} 1 + 4 sin^{2} \frac{x}{2} - 4 sin \frac{x}{2} . d x

π - 4

\frac{2 π}{3} - 4 - 43

43 - 4

43 - 4 - \frac{π}{3}

Topic:

Integrals

Exam:

JEE Mains 2014

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Question 9

lim_{n \to \infty} \sum_{r = 1}^{n} \frac{1}{n} e^{r / n}

e

e - 1

1 - e

e + 1

Topic:

Integrals

Exam:

JEE Mains 2014

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Question 10

Let the population of rabbits surviving at a time

t

be governed by the differential equation

\frac{d p ( t )}{d t} = \frac{1}{2} p (t) - 200

. If

p (0) = 100

, then

p (t)

400 - 300 e^{t /2}

300 - 200 e^{- t /2}

600 - 500 e^{t /2}

400 - 300 e^{- t /2}

Topic:

Differential Equations

Exam:

JEE Mains 2014

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Question 11

Let

a, b, c

be non-zero numbers. If the point of intersection of the lines

d

4 a x + 2 a y + c = 0

lies in the fourth quadrant and is equidistant from the two axes, then

5 b x + 2 b y + d = 0

$2 b c - 3 a d = 0$
$2 b c + 3 a d = 0$
$2 a d - 3 b c = 0$
$3 b c + 2 a d = 0$

Topic:

Conic Sections

Exam:

JEE Mains 2014

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Question 12

The locus of the foot of perpendicular drawn from the centre of the ellipse

x^{2} + 3 y^{2} = 6

on any tangent to it, is (a)

(x^{2} - y^{2})^{2} = 6 x^{2} + 2 y^{2}

(x^{2} - y^{2})^{2} = 6 x^{2} - 2 y^{2}

(x^{2} + y^{2})^{2} = 6 x^{2} + 2 y^{2}

(x^{2} + y^{2})^{2} = 6 x^{2} - 2 y^{2}

Topic:

Conic Sections

Exam:

JEE Mains 2014

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Question 13

The slope of the line touching both the parabolas

y^{2} = 4 x

x^{2} = - 32

is,

$\frac{1}{2}$
$\frac{3}{2}$
$\frac{1}{8}$
$\frac{2}{3}$

Topic:

Conic Sections

Exam:

JEE Mains 2014

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Question 14

The image of the line

\frac{x - 1}{3} = \frac{y - 3}{1} = \frac{z - 4}{- 5}

in the plane

2 x - y + z + 3 = 0

is the line (a)

\frac{x + 3}{3} = \frac{y - 5}{1} = \frac{z - 2}{- 5}

\frac{x + 3}{- 3} = \frac{y - 5}{- 1} = \frac{z + 2}{5}

\frac{x - 3}{3} + \frac{y + 5}{1} = \frac{z - 2}{- 5}

\frac{x - 3}{- 3} = \frac{y + 5}{- 1} = \frac{z - 2}{5}

Topic:

Three Dimensional Geometry

Exam:

JEE Mains 2014

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Question 15

Distance between two parallel planes

2 x + y + 2 z = 8

4 x + 2 y + 4 z + 5 = 0

is (a)

\frac{3}{2}

\frac{5}{2}

\frac{7}{2}

\frac{9}{2}

Topic:

Three Dimensional Geometry

Exam:

JEE Mains 2014

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Question 16

Let

A

B

be two events such that

P (\overline{A \cup B}) = \frac{1}{6}, P (A \cap B) = \frac{1}{4}

stands for the complement of the event

P (\overset{ˉ}{A}) = \frac{1}{4}

, where

\overset{ˉ}{A}

A

. Then, the events

A

are (a) independent but not equally likely (b) independent and equally likely (c) mutually exclusive and independent (d) equally likely but not independent

B

Topic:

Probability

Exam:

JEE Mains 2014

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Question 17

The variance of first 50 even natural numbers is (a)

\frac{833}{4}

(b) 833 (c) 437 (d)

\frac{437}{4}

Topic:

Statistics

Exam:

JEE Mains 2014

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Question 18

f_{k} (x) = \frac{1}{k} (sin^{k} x + cos^{k} x)

, where

x \in R

k \geq 1

, then

f_{4} (x) - f_{6} (x)

\frac{1}{6}

\frac{1}{3}

\frac{1}{4}

\frac{1}{12}

Topic:

Trigonometric Functions

Exam:

JEE Mains 2014

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Question 19

Let

α

β

be the roots of equation

p x^{2} + q x + r = 0, p \neq = 0

. If

p, q

r

are in

AP

\frac{1}{α} + \frac{1}{β} = 4

, then the value of

∣ α - β ∣

is (a)

\frac{61}{9}

\frac{2 17}{9}

\frac{34}{9}

\frac{2 13}{9}

Topic:

Complex Number and Quadratic Equations

Exam:

JEE Mains 2014

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Question 20

X = (4^{n} - 3 n - 1 : n \in N)

is the set of natural numbers, then

Y = {9 (n - 1) : n \in N}

; where

N

X \cup Y

N

Y - X

X