JEE Mains 2008

matrix with real entries. Let

Let

A

2 \times 2

I

be the

2 \times 2

identity matrix. Denote by

tr (A)

, the sum of diagonal entries of

A

. Assume that

A^{2} = I

. Statement I. If

A \neq = I

A \neq = - I

det (A) = - 1

. Statement II. If

A \neq = I

A \neq = - I

. (a) Statement I is false, Statement II is true (b) Statement I is true, Statement II is true; Statement II is a 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

tr (A) \neq = 0

Topic:

Matrices

Exam:

JEE Mains 2008

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Question 2

be a square matrix all of whose entries are integers. Then, which one of the following is true? (a) If

Let

A

det (A) = \pm 1

A^{- 1}

need not exist (b) If

det (A) = \pm 1

exists but all its entries are not necessarily integers (c) If

A^{- 1}

det (A) \neq = \pm 1

exists and all its entries are non-integers (d) If

A^{- 1}

det (A) = \pm 1

exists and all its entries are integers

A^{- 1}

Topic:

Matrices

Exam:

JEE Mains 2008

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Question 3

In a shop, there are five types of ice-creams available. A child buys six ice-creams. Statement I. The number of different ways the child can buy the six ice creams is

^{10} C_{5}

. Statement II. The number of different ways the child can buy the six ice creams is equal to the number of different ways of arranging 6 A's and 4 B's in a row. (a) Statement I is false, Statement II is true (b) Statement I is true, Statement II is true; Statement II is a 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:

Permutations and Combinations

Exam:

JEE Mains 2008

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Question 4

How many different words can be formed by jumbling the letters in the word 'MISSISSIPPI' in which no two

S

are adjacent? (a)

7 \cdot^{6} C_{4} \cdot^{8} C_{4}

8 \cdot^{6} C_{4} \cdot^{7} C_{4}

6 \cdot 7 \cdot^{8} C_{4}

6 \cdot 8 \cdot^{7} C_{4}

Topic:

Permutations and Combinations

Exam:

JEE Mains 2008

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Question 5

Statement I.For every natural number

n \geq 2

\frac{1}{1} + \frac{1}{2} + \dots + \frac{1}{n} > n

Statement II. For every natural number

n \geq 2, n (n + 1) < n + 1

Topic:

Principle of Mathematical Induction

Exam:

JEE Mains 2008

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Question 6

is false, Statement II is true (b) Statement I is true, Statement II is true; Statement II is a 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

Statement I.

\sum_{r = 0}^{n} (r + 1) \cdot^{n} C_{r} = (n + 2) 2^{n - 1}

Statement II.

\sum_{r = 0}^{n} (r + 1)^{n} C_{r} \cdot x^{r}

= (1 + x)^{n} + n x (1 + x)^{n - 1}

(a) Statement

I

Topic:

Binomial Theorem

Exam:

JEE Mains 2008

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Question 7

The first two terms of a geometric progression add upto 12 . The sum of the third and the fourth terms is 48 . If the terms of the geometric progression are alternately positive and negative, then the first term is

4
$- 4$
$- 12$
12

Topic:

Sequences and Series

Exam:

JEE Mains 2008

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Question 8

Then, which one of the following is true? (a)

Let

f (x) = {(x - 1) sin \frac{1}{x - 1}, 0, if x \neq = 1 if x = 1

f

is differentiable at

x = 1

but not at

x = 0

is neither differentiable at

f

x = 0

nor at

x = 1

f

is differentiable at

x = 0

and at

x = 1

f

is differentiable at

x = 0

but not at

x = 1

Topic:

Continuity and Differentiability

Exam:

JEE Mains 2008

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Question 9

How many real solutions does the equation

x^{7} + 14 x^{5} + 16 x^{3} + 30 x - 560 = 0

have? (a) 5 (b) 7 (c) 1 (d) 3

Topic:

Limits and Derivatives

Exam:

JEE Mains 2008

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Question 10

I = \int_{0}^{1} \frac{s i n x}{x} d x

. Then, which one of the following is true? (a)

J = \int_{0}^{1} \frac{c o s x}{x} d x

I > \frac{2}{3}

J < 2

I > \frac{2}{3}

J > 2

I < \frac{2}{3}

J < 2

I < \frac{2}{3}

J > 2

Topic:

Integrals

Exam:

JEE Mains 2008

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Question 11

The value of

2 \int \frac{s i n x d x}{s i n ( x - \frac{π}{4} )}

x - lo g ∣ ∣ cos (x - \frac{π}{4}) ∣ ∣ + C

x + lo g ∣ ∣ cos (x - \frac{π}{4}) ∣ ∣ + C

x - lo g ∣ ∣ sin (x - \frac{π}{4}) ∣ ∣ + C

x + lo g ∣ ∣ sin (x - \frac{π}{4}) ∣ ∣ + C

Topic:

Integrals

Exam:

JEE Mains 2008

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Question 12

The differential equation of the family of circles with fixed radius 5 units and centre on the line

y = 2

(x - 2)^{2} y^{'2} = 25 - (y - 2)^{2}

(x - 2) y^{'2} = 25 - (y - 2)^{2}

(y - 2) y^{'2} = 25 - (y - 2)^{2}

(y - 2)^{2} y^{'2} = 25 - (y - 2)^{2}

Topic:

Differential Equations

Exam:

JEE Mains 2008

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Question 13

The solution of the differential equation

\frac{d y}{d x} = \frac{x + y}{x}

satisfying the condition

y (1) = 1

y = x lo g x + x

y = lo g x + x

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

y = x e^{(x - 1)}

Topic:

Differential Equations

Exam:

JEE Mains 2008

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Question 14

The point diametrically opposite to the point

P (1, 0)

on the circle

x^{2} + y^{2} + 2 x + 4 y - 3 = 0

$(3, 4)$
$(3, - 4)$
$(- 3, 4)$
$(- 3, - 4)$

Topic:

Conic Sections

Exam:

JEE Mains 2008

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Question 15

The line passing through the points

(5, 1, a)

(3, b, 1)

crosses the

yz

-plane at the point

(0, \frac{17}{2}, - \frac{13}{2})

. Then, (a)

a = 8, b = 2

a = 2, b = 8

a = 4, b = 6

a = 6, b = 4

Topic:

Three Dimensional Geometry

Exam:

JEE Mains 2008

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Question 16

If the straight lines

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

intersect at a point, then the integer

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

k

is equal to (a)

- 2

- 5

Topic:

Three Dimensional Geometry

Exam:

JEE Mains 2008

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Question 17

lies in the plane of the vectors

The vector

a = α \hat{i} + 2 \hat{j} + β \hat{k}

b = \hat{i} + \hat{j}

and bisects the angle between

c = \hat{j} + \hat{k}

b

. Then, which one of the following gives possible values of

c

α

β

$α = 1, β = 1$
$α = 2, β = 2$
$α = 1, β = 2$
$α = 2, β = 1$

Topic:

Vector Algebra

Exam:

JEE Mains 2008

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Question 18

The non-zero vectors

a, b

c

are related by

a = 8 b

. Then, the angle between

c = - 7 b

a

c

$π$
0
$\frac{π}{4}$
$\frac{π}{2}$

Topic:

Vector Algebra

Exam:

JEE Mains 2008

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Question 19

It is given that the events

A

B

are such that

P (A) = \frac{1}{4}, P (A / B) = \frac{1}{2}

P (B / A) = \frac{2}{3}

. Then,

P (B)

is equal to (a)

\frac{1}{2}

\frac{1}{6}

\frac{1}{3}

\frac{2}{3}

Topic:

Probability

Exam:

JEE Mains 2008

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Question 20

be the event that the number obtained is greater than 3 . Let

A die is thrown. Let

A

B

be the event that the number obtained is less than 5 . Then,

P (A \cup B)

\frac{2}{5}