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

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

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

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

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

The range of the function f(x) = $\frac{{\mathrm{x}}^2 + 8}{{\mathrm{x}}^2 + 4}, \mathrm{x} \in \mathrm{R}$ is

• $\left[- 1, \frac{3}{2}\right]$

• (1, 2]

• (1, 2)

• [1, 2]

If n(A) = 1000, n(B) = 500 and if n(A $\cap$ B) $\ge$1 and  $\mathrm{n}\left(\mathrm{A} \cup \mathrm{B}\right) = \mathrm{p}$, then

• $500 \le \mathrm{p} \le 1000$

• $1001 \le \mathrm{p} \le 1498$

• $1000 \le \mathrm{p} \le 1498$

• $1000 \le \mathrm{p} \le 1499$

The domain of the function $\mathrm{f}\left(\mathrm{x}\right) = {\sin }^{-1}\left(\frac{\mathrm{x} + 5}{2}\right)$ is

• [- 1, 1]

• [2, 3]

• [3, 7]

• [- 7, - 3]

If f(x) = x + 1 and g(x) = 2x, then f{g(x)} is equal to

• 2(x + 1)

• 2x(x + 1)

• x

• 2x + 1

The domain of the function f(x) = $\frac{{\log }_2\left(\mathrm{x} +\hspace{0.17em}3\right)}{{\mathrm{x}}^2 + 3\mathrm{x} +\hspace{0.17em}2}$ is

• R - {- 1, - 2}

• R - {- 1, - 2, 0}

• (- 3, - 1) ∪ (- 1, $\infty$)

• (- 3, $\infty$) - {- 1, - 2}

227.

If * is defined by a * b = a - b² and $\oplus$ is defined by a $\oplus$ b = a² + b, where a and b are integers, then (3 $\oplus$ 4) * 5 is equal to

• 164

• 38

• - 12

• - 28

The image of the interval [- 1, 3] under the mapping f : R $\to$ R given by f(x) = 4x³ - 12x is

• [8, 72]

• [0, 72]

• [- 8, 72]

• [0, 8]

If the operation $\oplus$ is defined by a $\oplus$ b = a² + b² for all real numbers a and b, then (2 $\oplus$ 3) $\oplus$ 4 is equal to

• 120

• 185

• 175

• 129

The domain of the function f(x) = $\sqrt{7 - 3\mathrm{x}} + {\log }_{\mathrm{e}}\left(\mathrm{x}\right)$ is

• $0 < \mathrm{x} < \infty$

• $\frac{7}{3} \le \mathrm{x} < \infty$

• $0 < \mathrm{x} \le \frac{7}{3}$

• $- \infty < \mathrm{x} < 0$