The maximum area of a rectangle that can be inscribed in a circle of radius 2 units is

• $8\mathrm{\pi } \mathrm{sq} \mathrm{units}$

• 4 sq units

• 5 sq units

• 8 sq units

If the length of the subtangent at any point to the curve xyⁿ = a is proportional to the abscissa, then n is

• any non-zero real number

• 2

• - 2

• 1

3.

If $\frac{{\left(\mathrm{x} + 1\right)}^2}{{\mathrm{x}}^3 + \mathrm{x}} = \frac{\mathrm{A}}{\mathrm{x}} + \frac{\mathrm{Bx} + \mathrm{C}}{{\mathrm{x}}^2 + 1}$, then ${\sin }^{-1}\left(\mathrm{A}\right) + {\tan }^{-1}\left(\mathrm{B}\right) + {\sec }^{-1}\left(\mathrm{C}\right)$ is equal to

• $\frac{\mathrm{\pi }}{2}$

• $\frac{\mathrm{\pi }}{6}$

• 0

• $\frac{5\mathrm{\pi }}{6}$

$\underset{\mathrm{x}\to 0}{\lim }\frac{{\log }_{\mathrm{e}}\left(1 +\hspace{0.17em}\mathrm{x}\right)}{3^{\mathrm{x}} - 1}$ is equal to

• ${\log }_{\mathrm{e}}\left(3\right)$

• 0

• ${\log }_3\left(\mathrm{e}\right)$

• 1

If f(x) = $$\begin{gathered} \left\{\mathrm{x}, \mathrm{if} \mathrm{x} \mathrm{is} \mathrm{irrational} \\ 0, \mathrm{if} \mathrm{x} \mathrm{is} \mathrm{rational}\right\} \end{gathered}$$, then f is

• continuous everywhere

• discontinuous everywhere

• continuous only at x = 0

• continuous at all rational numbers

If A and B are square matnces of order n such that A² - B² = (A - B) (A + B), then which of the following will be true?

• Either A or B is zero matrix

• A = B

• AB = BA

• Either A or B is identity matrix

If A = $\left[\begin{array}{cc}\mathrm{\alpha }& 2\\ 2& \mathrm{\alpha }\end{array}\right] \mathrm{and} \left|{\mathrm{A}}^3\right| = 125$, then $\mathrm{\alpha }$ is equal to

• $\pm 1$

• $\pm 2$

• $\pm 3$

• $\pm 5$

If $\mathrm{A} = \left|\begin{array}{ccc}\mathrm{x}& 1& 1\\ 1& \mathrm{x}& 1\\ 1& 1& \mathrm{x}\end{array}\right| \mathrm{and} \mathrm{B} = \left|\begin{array}{cc}\mathrm{x}& 1\\ 1& \mathrm{x}\end{array}\right|$, then $\frac{\mathrm{dA}}{\mathrm{dx}}$ is equal to

• 3B + 1

• 3B

• - 3B

• 1 - 3B

If the determinant of the adjoint of a (real) matrix of order 3 is 25, then the determinant of the inverse of the matrix is

• 0.2

• $\pm 5$

• $\frac{1}{\sqrt[5]{625}}$

• $\pm 0.2$

If the matrix $\left[\begin{array}{cc}2& 3\\ 5& - 1\end{array}\right] = \mathrm{A} + \mathrm{B}$, where A is symmetric and B is skew - symmetric, then B is equal to

• $\left[\begin{array}{cc}2& 4\\ 4& - 1\end{array}\right]$

• $\left[\begin{array}{cc}0& - 2\\ 2& 0\end{array}\right]$

• $\left[\begin{array}{cc}0& 1\\ - 1& 0\end{array}\right]$

• $\left[\begin{array}{cc}0& - 1\\ 1& 0\end{array}\right]$