If $\left|\begin{array}{ccc}\mathrm{x} +\hspace{0.17em}1& \mathrm{x} + 2& \mathrm{x} + \mathrm{a}\\ \mathrm{x} + 2& \mathrm{x} + 3& \mathrm{x} + \mathrm{b}\\ \mathrm{x} + 3& \mathrm{x} + 4& \mathrm{x} +\hspace{0.17em}\mathrm{c}\end{array}\right|$ = 0, then a, b, c are

• in GP

• in HP

• equal

• 9

The value of $\left|\begin{array}{ccc}1& {\log }_{\mathrm{x}}\left(\mathrm{y}\right)& {\log }_{\mathrm{x}}\left(\mathrm{z}\right)\\ {\log }_{\mathrm{y}}\left(\mathrm{x}\right)& 1& {\log }_{\mathrm{y}}\left(\mathrm{z}\right)\\ {\log }_{\mathrm{z}}\left(\mathrm{x}\right)& {\log }_{\mathrm{z}}\left(\mathrm{y}\right)& 1\end{array}\right|$ is equal to

• 0

• 1

• xyz

• log(xyz)

If A = $\left[\begin{array}{ccc}2& 1& 0\\ 0& 2& 1\\ 1& 0& 2\end{array}\right]$, then $\left|\mathrm{adj}\right|$ is equal to

• 0

• 9

• $\frac{1}{9}$

• 81

If A and B are square matrices ofthe same order such that (A + B)(A - B) = A² - B², then (ABA^-1)² is equal to

• B²

• I

• A²B²

• A²

15.

On the set of integers Z, define f : Z $\to$ Z as f(n) = $$\begin{gathered} \left\{\frac{\mathrm{n}}{2}, \mathrm{n} \mathrm{is} \mathrm{even} \\ 0, \mathrm{n} \mathrm{is} \mathrm{odd}\right\} \end{gathered}$$, then 'f' is

• injective but not surjective

• neither injective nor surjective

• surjective but not injective

• bijective

${\cot }^{-1}\left(2 . 1^2\right) + {\cot }^{-1}\left(2 . 2^2\right) + {\cot }^{-1}\left(2 . 3^2\right) + ...$ upto $\infty$ is equal to

• $\frac{\mathrm{\pi }}{4}$

• $\frac{\mathrm{\pi }}{3}$

• $\frac{\mathrm{\pi }}{2}$

• $\frac{\mathrm{\pi }}{5}$

If 'x' takes negative permissible value, then sin^-1(x) is equal to

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

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

   

• $\mathrm{\pi } - {\cos }^{-1}\left(\sqrt{1 - {\mathrm{x}}^2}\right)$

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

$\int {\mathrm{e}}^{{\tan }^{-1}\left(\mathrm{x}\right)}\left(1 + \frac{\mathrm{x}}{1 +\hspace{0.17em}{\mathrm{x}}^2}\right)\mathrm{dx}$ is equal to

• ${\mathrm{xe}}^{{\tan }^{-1}\left(\mathrm{x}\right)} +\hspace{0.17em}\mathrm{c}$

• ${\mathrm{e}}^{{\tan }^{-1}\left(\mathrm{x}\right)} +\hspace{0.17em}\mathrm{c}$

• $\frac{1}{2}{\mathrm{e}}^{{\tan }^{-1}\left(\mathrm{x}\right)} +\hspace{0.17em}\mathrm{c}$

• $\frac{1}{2}{\mathrm{xe}}^{{\tan }^{-1}\left(\mathrm{x}\right)} +\hspace{0.17em}\mathrm{c}$

$\int \csc \left(\mathrm{x} - \mathrm{a}\right)\csc \left(\mathrm{x}\right)\mathrm{dx}$ is equal to

• $\frac{- 1}{\sin \left(\mathrm{a}\right)}\log \left|\sin \left(\mathrm{x}\right)\csc \left(\mathrm{x} - \mathrm{a}\right)\right| +\hspace{0.17em}\mathrm{c}$

• $\frac{- 1}{\sin \left(\mathrm{a}\right)}\log \left[\sin \left(\mathrm{x} - \mathrm{a}\right)\sin \left(\mathrm{x}\right)\right] +\hspace{0.17em}\mathrm{c}$

• $\frac{1}{\sin \left(\mathrm{a}\right)}\log \left[\sin \left(\mathrm{x}\right)\csc \left(\mathrm{x} - \mathrm{a}\right)\right] +\hspace{0.17em}\mathrm{c}$

• $\frac{1}{\sin \left(\mathrm{a}\right)}\log \left[\sin \left(\mathrm{x} - \mathrm{a}\right)\sin \left(x\right)\right] +\hspace{0.17em}\mathrm{c}$

If f(x) = ${\int }_{- 1}^{\mathrm{x}}\left|\mathrm{t}\right|\mathrm{dt}$, then for any x $\ge$ 0, f(x) is equal to

• 1 - x²

• $\frac{1}{2}\left(1 +\hspace{0.17em}{\mathrm{x}}^2\right)$

• 1 + x²

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