31.

Let f(x) be differentiable on the interval (0, $\infty$) such that f(1) =1 and $\underset{\mathrm{t}\to \infty }{\lim }\frac{{\mathrm{t}}^2\mathrm{f}\left(\mathrm{x}\right) - {\mathrm{x}}^2\mathrm{f}\left(\mathrm{t}\right)}{\mathrm{t} - \mathrm{x}} = 1$ for each x > 0. Then, f(x) is equal to

• $\frac{1}{3\mathrm{x}} + \frac{2}{3}{\mathrm{x}}^2$

• $- \frac{\mathrm{x}}{3} + \frac{4{\mathrm{x}}^2}{3}$

• $- \frac{1}{\mathrm{x}}$

• $- \frac{1}{\mathrm{x}} + \frac{2}{{\mathrm{x}}^2}$

If A = $\left[\begin{array}{cc}4& 11\\ 2& 6\end{array}\right]$, then A^-1 is equal to

• $\left[\begin{array}{cc}- 1& \frac{- 11}{2}\\ 3& 2\end{array}\right]$

• $\left[\begin{array}{cc}3& \frac{- 11}{2}\\ - 1& 2\end{array}\right]$

• $\left[\begin{array}{cc}3& 2\\ - 1& \frac{- 11}{2}\end{array}\right]$

• None of these

Let $\mathrm{f}\left(\mathrm{x}\right) = \left|\begin{array}{ccc}\sin \left(3\mathrm{x}\right)& 1& 2{\left(\cos \left(\frac{3\mathrm{x}}{2}\right) + \sin \left(\frac{3\mathrm{x}}{2}\right)\right)}^2\\ \cos \left(3\mathrm{x}\right)& - 1& 2{\cos }^2\left(\frac{3\mathrm{x}}{2}\right) - {\sin }^2\left(\frac{3\mathrm{x}}{2}\right)\\ \tan \left(3\mathrm{x}\right)& 4& 1 + 2\tan \left(3\mathrm{x}\right)\end{array}\right|$. Then, the value of f'(x) at x = (2n + 1) $\mathrm{\pi }$, n $\in \mathrm{I}$ (the set of integers) is equal to

• (- 1)ⁿ

• (- 1)^{n + 1}

• 3

• 9

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

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

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

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

• None of these

The number of points, where f(x) = [sin(x) + cos(x)] (where [] denotes the greatest integerfunction) and x $\in$ (0, 2) is not continuous, is

• 3

• 4

• 5

• 6

The value of ${\cot }^{-1}\left(\frac{\sqrt{1 + \sin \left(\mathrm{x}\right)} + \sqrt{1 - \sin \left(\mathrm{x}\right)}}{\sqrt{1 + \sin \left(\mathrm{x}\right)} - \sqrt{1 - \sin \left(\mathrm{x}\right)}}\right)$ is equal to

• $\frac{\mathrm{x}}{3}$

• $\frac{\mathrm{x}}{4}$

• 1

• $\frac{\mathrm{x}}{2}$

The value of $\left|\begin{array}{ccc}1& \mathrm{a}& \mathrm{b} + \mathrm{c}\\ 1& \mathrm{b}& \mathrm{c}\hspace{0.17em}+ \mathrm{a}\\ 1& \mathrm{c}& \mathrm{a} +\hspace{0.17em}\mathrm{b}\end{array}\right|$ is

• 0

• a + b + c

• abc

• 1

The angle between planes 2x - y + z = 6 and x + y + 2z = 8 is

• 30°

• 60°

• ${\cos }^{-1}\left(\sqrt{\frac{3}{2}}\right)$

• ${\sin }^{-1}\sqrt{\left(\frac{3}{2}\right)}$

According to Simpson's rule, the value of ${\int }_1^7\frac{\mathrm{dx}}{\mathrm{x}}$ is

• 1.358

• 1.958

• 1.625

• 1.458

Equation of a plane passing through (- 1, 1, 1) and (1, - 1, 1) and perpendicular to x + 2y + 2z = 5 is

• 2x + 3y - 3z + 3 = 0

• x + y + 3z - 5 = 0

• 2x+ 2y - 3z + 3 = 0

• x + y + z - 3 = 0