JEE Mains 2007

D = ∣ ∣ 111 1 1 + x 1 11 1 + y ∣ ∣

for

x \neq = 0, y \neq = 0

, then

D

divisible by neither $x$ nor $y$
divisible by both $x$ and $y$
divisible by $x$ but not $y$
divisible by $y$ but not $x$

Topic:

Determinants

Exam:

JEE Mains 2007

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Question 2

is to be partitioned into three sets

The set

S = {1, 2, 3, \dots, 12}

A, B

and

C

of equal size.Thus,

A \cup B \cup C = S

A \cap B = B \cap C = A \cap C = ϕ

. The number of ways to partition

S

12! /3! (4!)^{3}

12! /3! (3!)^{4}

12! / (4!)^{3}

12! / (3!)^{4}

Topic:

Permutations and Combinations

Exam:

JEE Mains 2007

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Question 3

In the binomial expansion of

(a - b)^{n}, n \geq 5

, the sum of 5 th and 6 th terms is zero, then

a / b

is equal to (a)

\frac{5}{n - 4}

\frac{6}{n - 5}

\frac{n - 5}{6}

\frac{n - 4}{5}

Topic:

Binomial Theorem

Exam:

JEE Mains 2007

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Question 4

The sum of the series

^{20} C_{0} -^{20} C_{1} +^{20} C_{2} -^{20} C_{3} + \dots +^{20} C_{10}

-^{20} C_{10}

\frac{1}{2}^{20} C_{10}

^{20} C_{10}

Topic:

Binomial Theorem

Exam:

JEE Mains 2007

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Question 5

In a geometric progression consisting of positive terms, each term equals the sum of the next two terms. Then, the common ratio of this progression is equal to

$\frac{1}{2} (1 - 5)$
$\frac{1}{2} 5$
$5$
$\frac{1}{2} (5 - 1)$

Topic:

Sequences and Series

Exam:

JEE Mains 2007

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Question 6

are positive real numbers such that

p

and

q

p^{2} + q^{2} = 1

, then the maximum value of

(p + q)

2
$\frac{1}{2}$
$\frac{1}{2}$
$2$

Topic:

Sequences and Series

Exam:

JEE Mains 2007

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Question 7

are positive real numbers such that

p

and

q

p^{2} + q^{2} = 1

, then the maximum value of

(p + q)

2
$\frac{1}{2}$
$\frac{1}{2}$
$2$

Topic:

Sequences and Series

Exam:

JEE Mains 2007

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Question 8

Easy

The sum of the series

\frac{1}{2 !} - \frac{1}{3 !} + \frac{1}{4 !} - \dots

upto infinity is

$e^{- 2}$
$e^{- 1}$
$e^{- \frac{1}{2}}$
$e^{\frac{1}{2}}$

Topic:

Sequences and Series

Exam:

JEE Mains 2007

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Question 9

from the origin is twice the abscissa of

The normal to a curve at

P (x, y)

meets the

X

-axis at

G

. If the distance of

G

P

, then the curve is a (a) ellipse (b) parabola (c) circle (d) hyperbola

Topic:

Application of Derivatives

Exam:

JEE Mains 2007

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Question 10

for which the conclusion of mean value theorem holds for the function

A value of

C

f (x) = lo g_{e} x

on the interval

[1, 3]

2 lo g_{3} e

\frac{1}{2} lo g_{e} 3

lo g_{3} e

lo g_{e} 3

Topic:

Application of Derivatives

Exam:

JEE Mains 2007

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Question 11

is an increasing function in (a)

The function

f (x) = tan^{- 1} (sin x + cos x)

(\frac{π}{4}, \frac{π}{2})

(- \frac{π}{2}, \frac{π}{4})

(0, \frac{π}{2})

(- \frac{π}{2}, \frac{π}{2})

Topic:

Application of Derivatives

Exam:

JEE Mains 2007

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Question 12

can be made continuous at

The function

f : R / {0} \to R

given by

f (x) = \frac{1}{x} - \frac{2}{e ^{2 x} - 1}

x = 0

by defining

f (0)

as (a) 2 (b)

- 1

Topic:

Continuity and Differentiability

Exam:

JEE Mains 2007

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Question 13

F (x) = f (x) + f (\frac{1}{x})

, where

f (x) = \int_{1}^{x} \frac{l o g t}{1 + t} d t

. Then,

F (e)

is equal to (a)

\frac{1}{2}

(b) 0 (c) 1 (d) 2

Topic:

Integrals

Exam:

JEE Mains 2007

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Question 14

The solution for

x

of the equation

\int_{2}^{x} \frac{d t}{t t ^{2} - 1} = \frac{π}{2}

- 2

π

\frac{3}{2}

22

Topic:

Integrals

Exam:

JEE Mains 2007

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Question 15

\int \frac{d x}{c o s x + 3 s i n x}

is equal to (a)

\frac{1}{2} lo g tan (\frac{x}{2} + \frac{π}{12}) + C

\frac{1}{2} lo g tan (\frac{x}{2} - \frac{π}{12}) + C

lo g tan (\frac{x}{2} + \frac{π}{12}) + C

lo g tan (\frac{x}{2} - \frac{π}{12}) + C

Topic:

Integrals

Exam:

JEE Mains 2007

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Question 16

The differential equation of all circles passing through the origin and having their centres on the

X

-axis is (a)

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

x^{2} = y^{2} + 3 x y \frac{d y}{d x}

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

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

Topic:

Differential Equations

Exam:

JEE Mains 2007

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Question 17

Consider a family of circles which are passing through the point

(- 1, 1)

and are tangent to

X

-axis. If

(h, k)

are the coordinates of the centre of the circles, then the set of values of

k

is given by the interval (a)

0 < k < \frac{1}{2}

k \geq \frac{1}{2}

- \frac{1}{2} \leq k \leq \frac{1}{2}

k \leq \frac{1}{2}

Topic:

Conic Sections

Exam:

JEE Mains 2007

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Question 18

, which of the following remains constant when

For the hyperbola

\frac{x ^{2}}{c o s ^{2} α} - \frac{y ^{2}}{s i n ^{2} α} = 1

α

varies? (a) Eccentricity (b) Directrix (c) Abscissae of vertices (d) Abscissae of foci

Topic:

Conic Sections

Exam:

JEE Mains 2007

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Question 19

The equation of a tangent to the parabola

y^{2} = 8 x

y = x + 2

. The point on this line from which the other tangent to the parabola is perpendicular to the given tangent is (a)

(- 1, 1)

(0, 2)

(2, 4)

(- 2, 0)

Topic:

Conic Sections

Exam:

JEE Mains 2007

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Question 20

is a bisector of the angle between the lines

If one of the line of

m y^{2} + (1 - m^{2})

x y - m x^{2} = 0

x y = 0

, then

m

- \frac{1}{2}

- 2