If the magnitude of the coefficient of x⁷ in the expansion of ${\left({\mathrm{ax}}^2 + \frac{1}{\mathrm{bx}}\right)}^8$, where a, b are positive numbers, is equal to the magnitude of the coefficient of x^-7 in the expansion of ${\left({\mathrm{ax}}^2 - \frac{1}{\mathrm{bx}}\right)}^8$, then a and b are connected by the relation

• ab = 1

• ab = 2

• a²b = 1

• ab² = 2

The mapping f: N $\to$ N given by f(n) = 1 + n², n $\in$ N where N is the set of natural numbers, is

• one - to - one and onto

• onto but not one - to - one

• one - to - one but not onto

• neither one - to - one nor onto

193.

The range of the function

$\mathrm{f}\left(\mathrm{x}\right) = \tan \left(\sqrt{\frac{{\mathrm{\pi }}^2}{9} - {\mathrm{x}}^2}\right)$ is

• [0, $\sqrt{3}$]

• (0, $\sqrt{3}$)

• $[0, \sqrt{3})$

• $(0, \sqrt{3}]$

If N is a set of natural numbers, then under binary operation a · b = a + b, (N, ·) is

• quasi-group

• semi-group

• monoid

• group

If $\mathrm{f} : \mathrm{R} \to \mathrm{R}$ be such that f(1) = 3 and f'(1) = 6. Then $\underset{\mathrm{x}\to 0}{\lim }{\left\{\frac{\mathrm{f}\left(1 + \mathrm{x}\right)}{\mathrm{f}\left(1\right)}\right\}}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\mathrm{x}$}\right.}$ equals to

• 1

• e^1/2

• e²

• e³

The domain of the function f(x) = $\frac{1}{{\log }_{10}\left(1 - \mathrm{x}\right)} + \sqrt{\mathrm{x} +\hspace{0.17em}2}$ is

• $\left[- 3, - 2.5 \left[\cap \right] - 2.5, - 2\right]$

• $\left[- 2, 0 \left[\cup \right] 0, 1\right]$

• [0, 1]

• None of the above

The relation R defined on set A = $\left\{\mathrm{x} : \left|\mathrm{x}\right| < 3, \mathrm{x} \in \mathrm{I}\right\}$ by $\mathrm{R} = \left\{\left(\mathrm{x}, \mathrm{y}\right) : \mathrm{y} = \left|\mathrm{x}\right|\right\}$ is

• {(- 2, 2), (- 1, 1), (0, 0), (1, 1), (2, 2)}

• {(- 2, - 2), (- 2, 2), (-1, 1), (0, 0), (1, - 2), (1, 2), (2, - 1), (2, - 2)}

• {(0, 0), (1, 1), (2, 2)}

• None of the above

The domain of the function f(x) = $\frac{\sqrt{4 - {\mathrm{x}}^2}}{{\sin }^{-1}\left(2 - \mathrm{x}\right)}$ is

• [0, 2]

• [0, 2)

• [1, 2)

• [1, 2]

The roots of (x - a)(x - a - 1) + (x - a - 1)(x - a - 2) + (x - a)(x - a - 2) = 0, $\mathrm{a} \in \mathrm{R}$ are always

• equal

• imaginary

• real and distinct

• rational and equal

Let f(x) = x² + ax + b, where a, b $\in$ R. If f(x) = 0 has all its roots imaginary, then the roots of f(x) + f'(x) + f''(x) = 0 are

• real and distinct

• imaginary

• real and distinct

• rational and equal