If A = $\left[\begin{array}{cc}3& 2\\ 1& 1\end{array}\right]$, then A² + xA + yI = 0 for (x, y) is

• (- 4, 1)

• (- 1, 3)

• (4, - 1)

• (1, 3)

The constant term of the polynomial $\left|\begin{array}{ccc}\mathrm{x} + 3& \mathrm{x}& \mathrm{x} + 2\\ \mathrm{x}& \mathrm{x} + 1& \mathrm{x} - 1\\ \mathrm{x} + 2& 2\mathrm{x}& 3\mathrm{x} + 1\end{array}\right|$ is

• 0

• 2

• - 1

• 1

If a > b > 0, ${\sec }^{-1}\left(\frac{\mathrm{a} + \mathrm{b}}{\mathrm{a} - \mathrm{b}}\right) = 2{\sin }^{-1}\left(\mathrm{x}\right)$, then x is

• $- \sqrt{\frac{\mathrm{b}}{\mathrm{a} +\hspace{0.17em}\mathrm{b}}}$

• $\sqrt{\frac{\mathrm{b}}{\mathrm{a} +\hspace{0.17em}\mathrm{b}}}$

• $- \sqrt{\frac{\mathrm{a}}{\mathrm{a} +\hspace{0.17em}\mathrm{b}}}$

• $\sqrt{\frac{\mathrm{a}}{\mathrm{a} +\hspace{0.17em}\mathrm{b}}}$

If $\mathrm{x} \ne \mathrm{n\pi }, \mathrm{x} \ne \left(2\mathrm{n} + 1\right)\frac{\mathrm{\pi }}{2}, \mathrm{n} \in \mathrm{Z}$, then $\frac{{\sin }^{-1}\left(\cos \left(\mathrm{x}\right)\right) + {\cos }^{-1}\left(\sin \left(\mathrm{x}\right)\right)}{{\tan }^{-1}\left(\cot \left(\mathrm{x}\right)\right) + {\cot }^{-1}\left(\tan \left(\mathrm{x}\right)\right)}$ is

• $\frac{\mathrm{\pi }}{2}$

• $\frac{\mathrm{\pi }}{6}$

• $\frac{\mathrm{\pi }}{4}$

• $\frac{\mathrm{\pi }}{3}$

The function f(x) = [x], where [x] denotes the greatest integer not greater than x , is

• continuous for all non-integral values of x

• continuous only at positive integral values of x

• continuous for all real values of x

• continuous only at rational values of x

If A is a 3 x 3 non-singular matrix and if $\left|\mathrm{A}\right| = 3, \mathrm{then} \left|{\left(2\mathrm{A}\right)}^{- 1}\right|$ is

• 24

• 3

• $\frac{1}{3}$

• $\frac{1}{24}$

The inverse of 2010 in the group Q^* of all positive rational under the binary operation * defined by a * b = $\frac{\mathrm{ab}}{2010}, \forall \mathrm{a}, \mathrm{b} \in {\mathrm{Q}}^{+}$ is

• 2009

• 2011

• 1

• 2010

8.

If the three function f(x), g(x) and h(x) are such that h(x) = f(x) g(x) and f'(x) g'(x) = c where c is constant, then

$\frac{\mathrm{f}\text{'}\text{'}\left(\mathrm{x}\right)}{\mathrm{f}\left(\mathrm{x}\right)} + \frac{\mathrm{g}\text{'}\text{'}\left(\mathrm{x}\right)}{\mathrm{g}\left(\mathrm{x}\right)} + \frac{2\mathrm{c}}{\mathrm{f}\left(\mathrm{x}\right) . \mathrm{g}\left(\mathrm{x}\right)}$ is equal to

• h'(x) . h''(x)

• $\frac{\mathrm{h}\left(\mathrm{x}\right)}{\mathrm{h}\text{'}\text{'}\left(\mathrm{x}\right)}$

• $\frac{\mathrm{h}\text{'}\text{'}\left(\mathrm{x}\right)}{\mathrm{h}\left(\mathrm{x}\right)}$

• $\frac{\mathrm{h}\left(\mathrm{x}\right)}{\mathrm{h}\text{'}\left(\mathrm{x}\right)}$

The derivative of e^ax cos(bx) with respect x is re^ax cos(bx) ${\tan }^{-1}\left(\frac{\mathrm{b}}{\mathrm{a}}\right)$ when a>0,b>0, then a value of r, is

• $\sqrt{{\mathrm{a}}^2 + {\mathrm{b}}^2}$

• $\frac{1}{\sqrt{\mathrm{ab}}}$

• ab

• a + b

In $∆\mathrm{ABC}$, if a = 2, B = ${\tan }^{-1}\left(\frac{1}{2}\right)$ and C = ${\tan }^{-1}\left(\frac{1}{3}\right)$, then (A, b) equals

• $\frac{3\mathrm{\pi }}{4}, \frac{2}{\sqrt{5}}$

• $\frac{\mathrm{\pi }}{4}, \frac{2\sqrt{2}}{\sqrt{5}}$

• $\frac{3\mathrm{\pi }}{4}, \frac{2\sqrt{2}}{\sqrt{5}}$

• $\frac{\mathrm{\pi }}{4}, \frac{2}{\sqrt{5}}$