JEE Mains 2011

is the complex cube root of unity and matrix

ω \neq = 1

H = [ω 0 0 ω]

, then

H^{70}

is equal to

$H$
$0$
$- H$
$H^{2}$

Topic:

Matrices

Exam:

JEE Mains 2011

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Question 2

Statement I. The number of ways of distributing 10 identical balls in 4 distinct boxes such that no box is empty, is

^{9} C_{3}

. Statement II. The number of ways of choosing any 3 places from 9 different places is

^{9} C_{3}

. (a) Statement I is true, Statement II is true; Statement II is not a correct explanation of Statement I (b) Statement I is true, Statement II is false (c) Statement I is false, Statement II is true (d) Statement I is true, Statement II is true; Statement II is a correct explanation of Statement I

Topic:

Permutations and Combinations

Exam:

JEE Mains 2011

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Question 3

There are 10 points in a plane, out of these 6 are collinear. If

N

is the number of triangles formed by joining these points, then (a)

N > 190

N \leq 100

100 < N \leq 140

140 < N \leq 190

Topic:

Permutations and Combinations

Exam:

JEE Mains 2011

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Question 4

Statement I. For each natural number

n, (n + 1)^{7} - n^{7} - 1

is dizvisible by 7 . Statement II. For each natural number

n

n^{7} - n

is divisible by 7 . (a) Statement I is false, Statement II is true (b) Statement I is true, Statement II is true; Statement II is correct explanation of Statement I (c) Statement I is true, Statement II is true; Statement II is not a correct explanation of Statement I (d) Statement I is true, Statement II is false

Topic:

Principle of Mathematical Induction

Exam:

JEE Mains 2011

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Question 5

The coefficient of

x^{7}

in the expansion of

(1 - x - x^{2} + x^{3})^{6}

is (a)

- 132

- 144

Topic:

Binomial Theorem

Exam:

JEE Mains 2011

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Question 6

in each of the first three months of his service. In each of the subsequent months, his saving increases by Rs.

A man saves Rs.

200

40

more than the saving of immediately previous month. His total saving from the start of service will be Rs. 11040 after

19 months
20 months
21 months
18 months

Topic:

Sequences and Series

Exam:

JEE Mains 2011

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Question 7

a_{n}

be the

n

th term of an AP. If

\sum_{r = 1}^{100} a_{2 r} = α

, then the common difference of the AP is-

\sum_{r = 1}^{100} a_{2 r - 1} = β

$\frac{α - β}{200}$
$α - β$
$\frac{α - β}{100}$
$β - α$

Topic:

Sequences and Series

Exam:

JEE Mains 2011

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Question 8

lim_{x \to 2} (\frac{1 - { c o s 2 ( x - 2 )}}{x - 2})

is equal to (a)

2

- 2

\frac{1}{2}

(d) Does not exist

Topic:

Limits and Derivatives

Exam:

JEE Mains 2011

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Question 9

\frac{d ^{2} x}{d y ^{2}}

is equal to (a)

- (\frac{d ^{2} y}{d x ^{2}})^{- 1} . (\frac{d y}{d x})^{- 3}

(\frac{d ^{2} y}{d x ^{2}}) . (\frac{d y}{d x})^{- 2}

- (\frac{d ^{2} y}{d x ^{2}}) . (\frac{d y}{d x})^{- 3}

(\frac{d ^{2} y}{d x ^{2}})^{- 1}

Topic:

Continuity and Differentiability

Exam:

JEE Mains 2011

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Question 10

The values of

p

q

for which the function

f (x) = ⎩ ⎨ ⎧ \frac{s i n ( p + 1 ) x + s i n x}{x} q, \frac{x + x ^{2} - x}{x ^{3/2}},, x < 0 x = 0 x > 0

is continuous for all

x

R

, are (a)

p = \frac{5}{2}, q = \frac{1}{2}

p = - \frac{3}{2}, q = \frac{1}{2}

p = \frac{1}{2}, q = \frac{3}{2}

p = \frac{1}{2}, q = - \frac{3}{2}

Topic:

Continuity and Differentiability

Exam:

JEE Mains 2011

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Question 11

For

x \in (0, \frac{5 π}{2})

, define

f (x) = \int_{0}^{x} t sin t d t

. Then,

f

has (a) local minimum at

π

2 π

(b) local minimum at

π

and local maximum at

2 π

π

and local minimum at

2 π

(d) local maximum at

π

2 π

Topic:

Application of Derivatives

Exam:

JEE Mains 2011

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Question 12

is equal to (a) 3 (b) 0 (c) 1 (d) 2

f : R \to [0, \infty)

be such that

lim_{x \to 5} f (x)

exists and

lim_{x \to 5} \frac{[ f ( x ) ] ^{2} - 9}{∣ x - 5∣} = 0

. Then,

lim_{x \to 5} f (x)

Topic:

Limits and Derivatives

Exam:

JEE Mains 2011

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Question 13

(a) Statement I is false, Statement II is true (b) Statement I is true, Statement II is true; Statement II is correct explanation of Statement I (c) Statement I is true, Statement II is true; Statement II is not a correct explanation of Statement I (d) Statement I is true, Statement II is false

f

be a function defined by

f (x) = {\frac{t a n x}{x}, 1, x \neq = 0 x = 0

Statement I.

x = 0

is point of minima of

f

. Statement II.

f^{'} (0) = 0

Topic:

Application of Derivatives

Exam:

JEE Mains 2011

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Question 14

If function

f (x)

is differentiable at

x = a

, then

lim_{x \to a} \frac{x ^{2} f ( a ) - a ^{2} f ( x )}{x - a}

is equal to (a)

2 a f (a) + a^{2} f^{'} (a)

- a^{2} f^{'} (a)

a f (a) - a^{2} f^{'} (a)

2 a f (a) - a^{2} f^{'} (a)

Topic:

Limits and Derivatives

Exam:

JEE Mains 2011

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Question 15

as the product of two real functions

Define

f (x)

f_{1} (x) = x, x \in I R

f_{2} (x) = {sin \frac{1}{x}, 0 if x \neq = 0, if x = 0

as follows

f (x) = {f_{1} (x) \cdot f_{2} (x) 0, if x \neq = 0, if x = 0

Statement I.

F (x)

is continuous on

I R

. Statement II.

f_{1} (x)

. (a) Statement I is false, Statement II is true (b) Statement I is true, Statement II is true; Statement II is correct explanation of Statement I. (c) Statement I is true, Statement II is true; Statement II is not a correct explanation of Statement I (d) Statement I is true, Statement II is false

f_{2} (x)

are continuous on

I R

Topic:

Relations and Functions

Exam:

JEE Mains 2011

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Question 16

The value of

\int_{0}^{1} \frac{8 l o g ( 1 + x )}{1 + x ^{2}} d x

is (a)

\frac{π}{8} lo g 2

\frac{π}{2} lo g 2

lo g 2

π lo g 2

Topic:

Integrals

Exam:

JEE Mains 2011

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Question 17

Let [ ] denotes the greatest integer function, then the value of

\int_{0}^{1.5} x [x^{2}] d x

is (a)

\frac{5}{4}

(b) 0 (c)

\frac{3}{2}

\frac{3}{4}

Topic:

Integrals

Exam:

JEE Mains 2011

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Question 18

\frac{d y}{d x} = y + 3 > 0

is equal to (a) 5 (b) 13 (c)

y (0) = 2

, then

y (lo g 2)

- 2

(d) 7

Topic:

Differential Equations

Exam:

JEE Mains 2011

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Question 19

be the purchase value of an equipment and

I

V (t)

be the value after it has been used for

t

years. The value

V (t)

depreciates at a rate given by differential equation

\frac{d V ( t )}{d t} = - k (T - t)

, where

k > 0

is a constant and

T

is the total life in years of the equipment. Then, the scrap value

V (T)

of the equipment is (a)

I - \frac{k}{2} T^{2}

I - \frac{k ( T - t ) ^{2}}{2}

e^{- k T}

T^{2} - \frac{1}{k}

Topic:

Differential Equations

Exam:

JEE Mains 2011

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Question 20

The equation of the circle passing through the point

(1, 0)