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Relations and Functions

Multiple Choice Questions

91.

If $y = 2^{\frac{1}{\log_{x} (8)}}$ , then x is equal to

y
y²
y³
None of these

92.

The domain of the function

$f (x) = \log_{10} (\sqrt{x - 4} + \sqrt{6 - x})$ is

[4, 6]
$(- \infty, 6)$
(2, 3)
None of these

93.

The inverse of the function f(x) = $f (x) = \frac{10^{x} - 10^{- x}}{10^{x} + 10^{- x}}$ is

$\log_{10} (2 - x)$
$\frac{1}{2} \log_{10} (\frac{1 + x}{1 - x})$
$\frac{1}{2} \log_{10} (2 x - 1)$
$\frac{1}{4} \log (\frac{2 x}{2 - x})$

94.
The number of solutions of $\log_{2} (x - 1) = 2 \log_{2} (x - 3)$

2

1

6

7

$Given, \log_{2} (x - 1) = 2 \log_{2} (x - 3) \Rightarrow \log_{2} (x - 1) = \log_{2} {(x - 3)}^{2} \Rightarrow x - 1 = {(x - 3)}^{2} \Rightarrow x - 1 = x^{2} + 9 - 6 x \Rightarrow x^{2} - 7 x + 10 = 0 \Rightarrow (x - 2) (x - 5) = 0 \Rightarrow x = 2, 5$

Since, x = 2 does not satisfis the equation.

So, x = 5 is the only solution.

Hence, number of solution is one.

95.

$If f (x) = \frac{x - 1}{x + 1}$ , then f(2x) is

$\frac{f (x) + 1}{f (x) + 3}$
$\frac{3 f (x) + 1}{f (x) + 3}$
$\frac{f (x) + 3}{f (x) + 1}$
$\frac{f (x) + 3}{3 f (x) + 1}$

96.

∼ (P ∨ q) v ( ∼ p ∧ q) is logically equivalent to

97.

Given, $f (x) = \log (\frac{1 + x}{1 - x})$ and $g (x) = \frac{3 x + x^{3}}{1 + 3 x^{2}}$ , then fog(x) equals

[f(x)]³
- f(x)
3f(x)
None of these

98.

The relation on the set A = {x : $|x| < 3, x \in Z$ } is defined by R = {(x, y) : y = $|x|, x \neq - 1$ } Then, the number of elements in the power set of R is