The value of the expression 1 . (2 - w)(2 - w²) + 2 . (3 - w)(3 - w²) + ... + (n - 1)(n - w²), where w is an imaginary cube root of unity is

• ${\left\{\frac{\mathrm{n}\left(\mathrm{n} + 1\right)}{2}\right\}}^2$

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

• ${\left\{\frac{\mathrm{n}\left(\mathrm{n} + 1\right)}{2}\right\}}^2 + \mathrm{n}$

• None of these

${∆}_1 = \left|\begin{array}{ccc}\mathrm{x}& \mathrm{b}& \mathrm{b}\\ \mathrm{a}& \mathrm{x}& \mathrm{b}\\ \mathrm{a}& \mathrm{a}& \mathrm{x}\end{array}\right| \mathrm{and} {∆}_2 = \left|\begin{array}{cc}\mathrm{x}& \mathrm{b}\\ \mathrm{a}& \mathrm{x}\end{array}\right|$ are the given determinants, then

• ${∆}_1 = 3{\left({∆}_2\right)}^2$

• $\frac{\mathrm{d}}{\mathrm{dx}}\left({∆}_1\right) = 3{∆}_2$

• $\frac{\mathrm{d}}{\mathrm{dx}}\left({∆}_1\right) = 3{\left({∆}_2\right)}^2$

• ${∆}_1 = 3{{∆}_2}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}$

If A and B are square matrices of the same order and A is non-singular, then for a positive integer n, (A^-1BA)ⁿ is equal to

• A^-nBⁿAⁿ

• AⁿBⁿA^-n

• A^-1BⁿA

• n(A^-1BA)

4.

The function f(x) = $$\begin{gathered} \left\{\raisebox{1ex}{${\mathrm{x}}^2$}\!\left/ \!\raisebox{-1ex}{$\mathrm{a}$}\right., 0 \le \mathrm{x} < 1 \\ \mathrm{a}, 1 \le \mathrm{x} < \sqrt{2} \\ \frac{2{\mathrm{b}}^2 - 4\mathrm{b}}{{\mathrm{x}}^2}, \sqrt{2} \le \mathrm{x} < \infty \right\} \end{gathered}$$ is continuous for $0 \le \mathrm{x} < \infty$,then the most suitable values of a and b are

• a = 1, b = - 1

• a = - 1, b = 1 + $\sqrt{2}$

• a = - 1, b = 1

• None of the above

If f(x) = $$\begin{gathered} \left\{1, \mathrm{x} < 0 \\ 1 + \sin \left(\mathrm{x}\right), 0 \le \mathrm{x} < \frac{\mathrm{\pi }}{2}\right\} \end{gathered}$$,then at x = 0 the derivative f'(x) is

• 1

• 0

• infinite

• not defined

Let P(x) = a₀ + a₁x² + a₂x² + a₃x⁶ + ... + a_nx²ⁿ be a polynomial in a real variables with 0 < a₀ < a₁ < a₂ < .... < a_n. The function P(x) has

• neither a maxima nor a minima

• only one maxima

• both maxima and minima

• only one minima

If ${\left({\tan }^{-1}\left(\mathrm{x}\right)\right)}^2 + {\left({\cot }^{-1}\left(\mathrm{x}\right)\right)}^2 = \frac{5{\mathrm{\pi }}^2}{8}$, then x is equal to

• - 1

• 1

• 0

• None of these

$\frac{{\mathrm{\alpha }}^3}{2}{\csc }^2\left(\frac{1}{2}{\tan }^{-1}\left(\frac{\mathrm{\alpha }}{\mathrm{\beta }}\right)\right) + \frac{{\displaystyle {\mathrm{\beta }}^3}}{{\displaystyle 2}}{\sec }^2\left(\frac{{\displaystyle 1}}{{\displaystyle 2}}\right)\left(\frac{1}{2}{\tan }^{-1}\left(\frac{{\displaystyle \mathrm{\beta }}}{{\displaystyle \mathrm{\alpha }}}\right)\right)$ is equal to

• $\left(\mathrm{\alpha } - \mathrm{\beta }\right)\left({\mathrm{\alpha }}^2 + {\mathrm{\beta }}^2\right)$

• $\left(\mathrm{\alpha } + \mathrm{\beta }\right)\left({\mathrm{\alpha }}^2 - {\mathrm{\beta }}^2\right)$

• $\left(\mathrm{\alpha } + \mathrm{\beta }\right)\left({\mathrm{\alpha }}^2 + {\mathrm{\beta }}^2\right)$

• None of the above

$\int \frac{\mathrm{dx}}{9 + 16{\sin }^2\left(\mathrm{x}\right)}$ is equal to

• $\frac{1}{3}{\tan }^{-1}\left(\frac{3\tan \left(\mathrm{x}\right)}{5}\right) + \mathrm{c}$

• $\frac{1}{5}{\tan }^{-1}\left(\frac{\tan \left(\mathrm{x}\right)}{15}\right) + \mathrm{c}$

• $\frac{1}{15}{\tan }^{-1}\left(\frac{\tan \left(\mathrm{x}\right)}{5}\right) + \mathrm{c}$

• $\frac{1}{15}{\tan }^{-1}\left(\frac{5\tan \left(\mathrm{x}\right)}{3}\right) + \mathrm{c}$

$\int \frac{{\mathrm{x}}^2\mathrm{dx}}{{\left(\mathrm{xsin}\left(\mathrm{x}\right) + \cos \left(\mathrm{x}\right)\right)}^2}$ is equal to

• $\frac{\sin \left(\mathrm{x}\right) + \cos \left(\mathrm{x}\right)}{\mathrm{xsin}\left(\mathrm{x}\right) + \cos \left(\mathrm{x}\right)} + \mathrm{c}$

• $\frac{x\sin \left(\mathrm{x}\right) - \cos \left(\mathrm{x}\right)}{\mathrm{xsin}\left(\mathrm{x}\right) + \cos \left(\mathrm{x}\right)} + \mathrm{c}$

• $\frac{\sin \left(\mathrm{x}\right) - \mathrm{xcos}\left(\mathrm{x}\right)}{\mathrm{xsin}\left(\mathrm{x}\right) + \cos \left(\mathrm{x}\right)} + \mathrm{c}$

• None of these