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1

### WB JEE 2009

The domain of definition of the function $$f(x) = \sqrt {1 + {{\log }_e}(1 - x)}$$ is

A
$$- \infty < x \le 0$$
B
$$- \infty < x \le {{e - 1} \over e}$$
C
$$- \infty < x \le 1$$
D
$$x \ge 1 - e$$

## Explanation

$$f(x) = \sqrt {1 + {{\log }_e}(1 - x)}$$

Value of f(x) is real when

$$1 + {\log _e}(1 - x) \ge 0$$ and $$1 - x > 0$$

$$\Rightarrow {\log _e}(1 - x) \ge - 1$$ and $$x < 1$$

$$\Rightarrow {\log _e}(1 - x) \ge {\log _e}{e^{ - 1}}$$ and $$x < 1$$

$$\Rightarrow 1 - x \ge {1 \over e}$$ and $$x < 1 \Rightarrow x \le {{e - 1} \over e}$$ and $$x < 1$$

or, $$x \le {{e - 1} \over e}$$

2

### WB JEE 2009

A mapping from IN to IN is defined as follows:

$$f:IN \to IN$$

$$f(n) = {(n + 5)^2},\,n \in IN$$

(IN is the set of natural numbers). Then

A
f is not one-to-one
B
f is onto
C
f is both one-to-one and onto
D
f is one-to-one but not onto

## Explanation

$$f(n) = {(n + 5)^2}$$

Let n = n1, n2 $$\in$$ N such that n1 $$\ne$$ n2

$$\Rightarrow {n_1} + 5 \ne {n_2} + 5 \Rightarrow {({n_1} + 5)^2} \ne {({n_2} + 5)^2} \Rightarrow f({n_1}) \ne f({n_2})$$

$$\therefore$$ f is one-one

$$\because$$ n $$\in$$ N i.e., n = 1, 2, 3, .....

$$\therefore$$ f(1) = 36, f(2) = 49, .....

$$\therefore$$ Range = {f(1), f(2), f(3), .....}

= {36, 49, .....} $$\ne$$ N (codomain)

$$\therefore$$ f is one to one but not onto.

3

### WB JEE 2009

For any two sets A and B, A $$-$$ (A $$-$$ B) equals

A
B
B
A $$-$$ B
C
A $$\cap$$ B
D
Ac $$\cap$$ Bc

## Explanation

A $$-$$ (A $$-$$ B)

= A $$\cap$$ (A $$\cap$$ Bc)c = A $$\cap$$ (Ac $$\cup$$ B)

= $$\phi$$ $$\cup$$ (A $$\cap$$ B) = A $$\cap$$ B

4

### WB JEE 2008

Let A = {1, 2, 3} and B = {2, 3, 4}, then which of the following relations is a function from A to B?

A
{(1, 2), (2, 3), (3, 4), (2, 2)}
B
{(1, 2), (2, 3), (1, 3)}
C
{(1, 2), (2, 3), (3, 3)}
D
{(1, 2), (2, 3), (3, 4)}

## Explanation

If a relation is function from set A to set B, then first element of ordered pairs in relation should not be repeated and second element of ordered pairs should be in set B. So Relation {(1, 3), (2, 3), (3, 3)} is a function from A = {1, 2, 3} to B = {2, 3, 4}.

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