1.

Let f(x) = $$\begin{gathered} \left\{5^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\mathrm{x}$}\right.}, \mathrm{x} < 0 \\ \mathrm{\lambda }\left[\mathrm{x}\right], \mathrm{x} \ge 0\right\} \mathrm{and} \mathrm{\lambda } \in \mathrm{R} \end{gathered}$$, then at x = 0

• f is discontinuous

• f is continuous only, if $\mathrm{\lambda } = 0$

• f is continuous only, whatever $\mathrm{\lambda }$ maybe

• None of the above

Which one of the following is not true?

• Matrix addition is commutative

• Matrix addition is associative

• Matrix multiplication is commutative

• Matrix multiplication is associative

If matrix $\left[\begin{array}{ccc}0& 1& - 2\\ - 1& 0& 3\\ \mathrm{\lambda }& - 3& 0\end{array}\right]$ is singular, then $\mathrm{\lambda }$ equal to

• - 2

• - 1

• 1

• 2

If A = $\left[\begin{array}{cc}2\mathrm{x}& 0\\ \mathrm{x}& \mathrm{x}\end{array}\right]$ and A^-1 = $\left[\begin{array}{cc}- 1& 0\\ - 1& 2\end{array}\right]$, then x equals

• 2

• $- \frac{1}{2}$

• S

• $\frac{1}{2}$

The determinant $\left|\begin{array}{ccc}\mathrm{a}& \mathrm{b}& \mathrm{a\alpha } + \mathrm{b}\\ \mathrm{b}& \mathrm{c}& \mathrm{b\alpha } + \mathrm{c}\\ \mathrm{a\alpha } + \mathrm{b}& \mathrm{b\alpha } + \mathrm{c}& 0\end{array}\right| = 0$, if a, b, c are in

• AP

• GP

• HP

• None of these

If ${\tan }^{-1}\left(\mathrm{x} - 1\right) + {\tan }^{-1}\left(\mathrm{x}\right) + {\tan }^{-1}\left(\mathrm{x} + 1\right) = {\tan }^{-1}\left(3\mathrm{x}\right)$, then x is

• $\pm \frac{1}{2}$

• $0, \frac{1}{2}$

• $0, - \frac{1}{2}$

• $0, \pm \frac{1}{2}$

The minimum value of ${\mathrm{x}}^2 + \frac{1}{1 + {\mathrm{x}}^2}$ is at

• x = 0

• x = 1

• x = 4

• x = 3

Let f(x) = $\frac{1}{2} - \tan \left(\frac{\mathrm{\pi x}}{2}\right)$, - 1 < x < 1 and g(x) = $\sqrt{3 + 4\mathrm{x} - 4{\mathrm{x}}^2}$, then dom(f + g) is given by

• $\left[\frac{1}{2}, 1\right]$

• $[\frac{1}{2}, - 1)$

• $[ - \frac{1}{2}, 1)$

• $\left[- \frac{1}{2}, - 1\right]$

If $\mathrm{a} \ne \mathrm{b} \ne \mathrm{c}$, the value of x which satisfies the equation $\left|\begin{array}{ccc}0& \mathrm{x} - \mathrm{a}& \mathrm{x} - \mathrm{b}\\ \mathrm{x} + \mathrm{a}& 0& \mathrm{x} - \mathrm{c}\\ \mathrm{x} + \mathrm{b}& \mathrm{x} + \mathrm{c}& 0\end{array}\right|$ = 0, is

• x = 0

• x = a

• x = b

• 1

For how many values of x in the closed interval [- 4, - 1], is the matrix $\left[\begin{array}{ccc}3& - 1 + \mathrm{x}& 2\\ 3& - 1& \mathrm{x} + 2\\ \mathrm{x} + 3& - 1& 2\end{array}\right]$ singular ?

• 2

• 0

• 3

• 1