JEE Mains 2003

A student is to answer 10 out of 13 questions in an examination such that he must choose at least 4 from the first five questions. The number of choices available to him is (a) 140 (b) 196 (c) 280 (d) 346

Topic:

Permutations and Combinations

Exam:

JEE Mains 2003

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Question 2

denotes the number of combinations of

^{n} C_{r}

n

things taken

r

at a time, then the expression

^{n} C_{r + 1} +^{n} C_{r - 1} + 2 \times^{n} C_{r}

equal to (a)

^{n + 2} C_{r}

^{n + 2} C_{r + 1}

^{n + 1} C_{r}

^{n + 1} C_{r + 1}

Topic:

Permutations and Combinations

Exam:

JEE Mains 2003

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Question 3

The number of integral terms in the expansion of

(3 + 85)^{256}

is (a) 32 (b) 33 (c) 34 (d) 35

Topic:

Binomial Theorem

Exam:

JEE Mains 2003

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Question 4

is positive, the first negative term in the expansion of

x

(1 + x)^{27/5}

is (a) 7th term (b) 5th term (c) 8th term (d) 6 th term

Topic:

Binomial Theorem

Exam:

JEE Mains 2003

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Question 5

The sum of the series

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

upto

\infty

2 lo g_{e} 2

lo g_{e} 2 - 1

lo g_{e} 2

lo g_{e} (\frac{4}{e})

Topic:

Sequences and Series

Exam:

JEE Mains 2003

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Question 6

be a polynomial function of second degree. If

Let

f (x)

f (1) = f (- 1)

a, b, c

are in AP, then

f^{'} (a), f^{'} (b)

are in (a) AP (b) GP (c) HP (d) Arithmetic-Geometric progression

f^{'} (c)

Topic:

Continuity and Differentiability

Exam:

JEE Mains 2003

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Question 7

when added to its inverse gives the minimum value of the sum at

The real number

x

x

equal to (a) 2 (b) 1 (c)

- 1

- 2

Topic:

Application of Derivatives

Exam:

JEE Mains 2003

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Question 8

f (x) = x^{n}

, then the value of

f (1) - \frac{f ^{'} ( 1 )}{1 !} + \frac{f ^{''} ( 1 )}{2 !} - \frac{f ^{'''} ( 1 )}{3 !} + \dots + \frac{( - 1 ) ^{n} f ^{n} ( 1 )}{n !}

is (a)

2^{n}

2^{n - 1}

Topic:

Continuity and Differentiability

Exam:

JEE Mains 2003

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Question 9

lim_{x \to \frac{π}{2}} \frac{[ 1 - t a n ( \frac{x}{2} ) ] ( 1 - s i n x )}{[ 1 + t a n ( \frac{x}{2} ) ] ( π - 2 x ) ^{3}}

\frac{1}{8}

(b) 0 (c)

\frac{1}{32}

\infty

Topic:

Limits and Derivatives

Exam:

JEE Mains 2003

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Question 10

lim_{x \to 0} \frac{l o g ( 3 + x ) - l o g ( 3 - x )}{x} = k

, then the value of

k

is (a) 0 (b)

- 1/3

2/3

- 2/3

Topic:

Limits and Derivatives

Exam:

JEE Mains 2003

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Question 11

exist and are not equal for some

Let

f (a) = g (a) = k

and their

n

th derivatives

f^{n} (a), g^{n} (a)

n

.Further, if

lim_{x \to a} \frac{f ( a ) g ( x ) - f ( a ) - g ( a ) f ( x ) + g ( a )}{g ( x ) - f ( x )} = 4

, then the value of

k

is (a) 4 (b) 2 (c) 1 (d) 0

Topic:

Limits and Derivatives

Exam:

JEE Mains 2003

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Question 12

(a) continuous as well as differentiable for all

f (x) = {x e^{- (\frac{1}{∣ x ∣} + \frac{1}{x})}, 0, x \neq = 0, then f (x) is x = 0

x

(b) continuous for all

x

but not differentiable at

x = 0

x = 0

(d) discontinuous everywhere

Topic:

Continuity and Differentiability

Exam:

JEE Mains 2003

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Question 13

, attains its maximum and minimum at

If the function

f (x) = 2 x^{3} - 9 a x^{2} + 12 a^{2} x + 1

, where

a > 0

p

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

q

respectively, such that

p^{2} = q

, then

a

1/2

Topic:

Application of Derivatives

Exam:

JEE Mains 2003

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Question 14

f (y) = e^{y}, g (y) = y; y > 0

F (t) = \int_{0}^{t} f (t - y) g (y) d y

, then (a)

F (t) = 1 - e^{- t} (1 + t)

F (t) = e^{t} - (1 + t)

F (t) = t e^{t}

F (t) = t e^{- t}

Topic:

Integrals

Exam:

JEE Mains 2003

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Question 15

f (a + b - x) = f (x)

, then

\int_{a}^{b} x f (x) d x

\frac{a + b}{2} \int_{a}^{b} f (b - x) d x

\frac{a + b}{2} \int_{a}^{b} f (x) d x

\frac{b - a}{2} \int_{a}^{b} f (x) d x

\frac{a + b}{2} \int_{a}^{b} f (a + b + x) d x

Topic:

Integrals

Exam:

JEE Mains 2003

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Question 16

The value of

lim_{x \to 0} \frac{\int _{0}^{x^{2}} s e c ^{2} t d t}{x s i n x}

is (a) 3 (b) 2 (c) 1 (d)

- 1

Topic:

Integrals

Exam:

JEE Mains 2003

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Question 17

The value of

I = \int_{0}^{1} x (1 - x)^{n} d x

is (a)

\frac{1}{n + 1}

\frac{1}{n + 2}

\frac{1}{n + 1} - \frac{1}{n + 2}

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

Topic:

Integrals

Exam:

JEE Mains 2003

View 5 solutions

Question 18

lim_{n \to \infty} \frac{[ 1 + 2 ^{4} + 3 ^{4} + \dots + n ^{4} ]}{n ^{5}} - lim_{n \to \infty} \frac{[ 1 + 2 ^{3} + 3 ^{3} + \dots + n ^{3} ]}{n ^{5}}

1/30

(b) 0 (c)

1/4

1/5

Topic:

Integrals

Exam:

JEE Mains 2003

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Question 19

, then one of the possible value of

Let

\frac{d}{d x} F (x) = (\frac{e ^{s i n x}}{x}), x > 0

. If

\int_{1}^{4} \frac{3}{x} e^{s i n x^{3}} d x = F (k) - F (1)

k

, is (a) 15 (b) 16 (c) 63 (d) 64

Topic:

Integrals

Exam:

JEE Mains 2003

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Question 20

f (x)

is a function satisfying

f^{'} (x) = f (x)

with

f (0) = 1

is a function that satisfies

g (x)

f (x) + g (x) = x^{2}

. Then, the value of

\int_{0}^{1} f (x) g (x) d x

, is (a)

e - \frac{e ^{2}}{2} - \frac{5}{2}

e + \frac{e ^{2}}{2} - \frac{3}{2}

e - \frac{e ^{2}}{2} - \frac{3}{2}