relations and functions

${\log }_{\mathrm{e}}\left(\frac{1 + 3\mathrm{x}}{1 - 2\mathrm{x}}\right)$ is equal to

• $- 5\mathrm{x} - \frac{5{\mathrm{x}}^2}{2} - \frac{35{\mathrm{x}}^3}{3} - ...$

• $- 5\mathrm{x} + \frac{5{\mathrm{x}}^2}{2} - \frac{35{\mathrm{x}}^3}{3} + ...$

• $5\mathrm{x} - \frac{5{\mathrm{x}}^2}{2} + \frac{35{\mathrm{x}}^3}{3} - ...$

• $5\mathrm{x} + \frac{5{\mathrm{x}}^2}{2} + \frac{35{\mathrm{x}}^3}{3} + ...$

Let $\mathrm{f}\left(\mathrm{x}\right) = \frac{{\mathrm{\alpha x}}^2}{\mathrm{x} + 1}, \mathrm{x} \ne - 1$. The value of $\mathrm{\alpha }$ for which f (a) = a, ($\mathrm{a} \ne 0$) is

• 1 - $\frac{1}{\mathrm{a}}$

• $\frac{1}{\mathrm{a}}$

• $1 + \frac{1}{\mathrm{a}}$

• $\frac{1}{\mathrm{a}} - 1$

For a, b $\in$ R, define $\mathrm{a} * \mathrm{b} = \frac{\mathrm{a}}{\mathrm{a} + \mathrm{b}}$, where a + b $\ne$ 0. If a * b = 5,  then the value of b * a is

• 5

• - 5

• 4

• - 4

Let A = {x, y, z} and B = {a, b, c, d}. Which one of the following is not a relation from A to B ?

• {(x, a), (x, c)}

• {(y, c), (y, d)}

• {(z, a), (z, d)}

• {(z, b), (y, b), (a, d)}

If f(x) = x² - 1 and g(x) = (x + 1)², then (gof) (x) is

• (x + 1)⁴ - 1

• x⁴ - 1

• x⁴

• (x + 1)⁴

If the function f : $[1, \infty ) \to [1, \infty )$ is defined by f(x) = 2^{x(x - 1)}, then f^-1(x) is

• ${\left(\frac{1}{2}\right)}^{\mathrm{x}\left(\mathrm{x} - 1\right)}$

• $\frac{1}{2}\left(1 - \sqrt{1 + 4{\log }_2\left(\mathrm{x}\right)}\right)$

• $\frac{1}{2}\sqrt{1 + 4{\log }_2\left(\mathrm{x}\right)}$

• $\frac{1}{2}\left[1 + \sqrt{1 + 4{\log }_2\left(\mathrm{x}\right)}\right]$

If n(A) = 8 and $\mathrm{n}\left(\mathrm{A} \cap \mathrm{B}\right) = 2$, then $\mathrm{n}\left(\left(\mathrm{A} \cap \mathrm{B}\right)\text{'} \cap \mathrm{A}\right)$ is equal to

• 2

• 4

• 6

• 8

If f(x) = $\sin \left(\mathrm{x}\right) + \cos \left(\mathrm{x}\right)$, $\mathrm{x} \in \left(- \infty , \infty \right)$ and g(x) = x², $\mathrm{x} \in \left(- \infty , \infty \right)$, then (fog)(x) is equal to

• 1

• 0

• ${\sin }^2\left(\mathrm{x}\right) + \cos \left({\mathrm{x}}^2\right)$

• $\sin \left({\mathrm{x}}^2\right) + \cos \left({\mathrm{x}}^2\right)$

If n(A) = 5 and n(B) = 7, then the number of relations on A x B is

• 2³⁵

• 2⁴⁹

• 2²⁵

• $2^{35 \times 35}$

Let $\mathrm{\varphi }\left(\mathrm{x}\right) = \frac{\mathrm{b}\left(\mathrm{x} - \mathrm{a}\right)}{\mathrm{b} - \mathrm{a}} + \frac{\mathrm{a}\left(\mathrm{x} - \mathrm{b}\right)}{\mathrm{a} - \mathrm{b}}$, where x $\in$ R and a and b are fixed real numbers with $\mathrm{a} \ne \mathrm{b}$. Then, $\mathrm{\varphi }\left(\mathrm{a} + \mathrm{b}\right)$ is equal to

• $\mathrm{\varphi }\left(\mathrm{ab}\right)$

• $\mathrm{\varphi }\left(- \mathrm{ab}\right)$

• $\mathrm{\varphi }\left(\mathrm{a}\right) + \mathrm{\varphi }\left(\mathrm{b}\right)$

• $\mathrm{\varphi }\left(\mathrm{a} - \mathrm{b}\right)$