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1

WB JEE 2010

The function $$f(x) = \sec \left[ {\log \left( {x + \sqrt {1 + {x^2}} } \right)} \right]$$ is

A
odd
B
even
C
neither odd nor even
D
constant

Explanation

$$\because$$ $$f(x) = \sec \left[ {\log \left( {x + \sqrt {1 + {x^2}} } \right)} \right]$$

$$\therefore$$ $$f(x) = \sec [\log \{ - x + \sqrt {1 + {{( - x)}^2}} \} ]$$

$$= \sec \left[ {\log {{(\sqrt {1 + {x^2}} - x)(\sqrt {1 + {x^2}} + x)} \over {(\sqrt {1 + {x^2}} + x)}}} \right]$$

$$= \sec \left[ {\log \left( {{{1 + {x^2} - {x^2}} \over {x + \sqrt {1 + {x^2}} }}} \right)} \right] = \sec \log {(x + \sqrt {1 + {x^2}} )^{ - 1}}$$

$$= \sec \{ - \log (x + \sqrt {1 + {x^2}} )\} = \sec [\log (x + \sqrt {1 + {x^2}} )\} = f(x)$$

$$\therefore$$ f(x) is an even function.

2

WB JEE 2010

If A = {1, 2, 3, 4}, B = {1, 2, 3, 4, 5, 6} are two sets, and function f : A $$\to$$ B is defined by f(x) = x + 2 $$\forall$$ x$$\in$$ A, then the function f is

A
bijective
B
onto
C
one-one
D
many-one

Explanation

$$\therefore$$ f(x) = x + 2

Here, f is one-one only. f is not onto. [$$\because$$ 1, 2 have no pre-image]

3

WB JEE 2010

Let R be the set of real numbers and the mapping f : R $$\to$$ R and g : R $$\to$$ R be defined by f(x) = 5 $$-$$ x2 and g(x) = 3x $$-$$ 4, then the value of (fog)($$-$$1) is

A
$$-$$ 44
B
$$-$$ 54
C
$$-$$ 32
D
$$-$$ 64

Explanation

$$fog( - 1) = f\{ g( - 1)\} = f\{ 3( - 1) - 4\}$$

$$= f( - 7) = 5 - {( - 7)^2} = - 44$$

4

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

Questions Asked from Sets and Relations

On those following papers in MCQ (Single Correct Answer)
Number in Brackets after Paper Indicates No. of Questions
WB JEE 2022 (3)
WB JEE 2021 (2)
WB JEE 2020 (3)
WB JEE 2019 (3)

Joint Entrance Examination

JEE Main JEE Advanced WB JEE