If x, y, z are not equal and $\ne 0, \ne 1$, then the value of $\left|\begin{array}{ccc}\log \left(\mathrm{x}\right)& \log \left(\mathrm{y}\right)& \log \left(\mathrm{z}\right)\\ \log \left(2\mathrm{x}\right)& \log \left(2\mathrm{y}\right)& \log \left(2\mathrm{z}\right)\\ \log \left(3\mathrm{x}\right)& \log \left(3\mathrm{y}\right)& \log \left(3\mathrm{z}\right)\end{array}\right|$ is equal to

• $\log \left(\mathrm{xyz}\right)$

• $\log \left(6\mathrm{xyz}\right)$

• 0

• $\log \left(\mathrm{x} + \mathrm{y} + \mathrm{z}\right)$

The function f(x) = [x], where [x] is the greatest integer function, is continuous at

• 1.5

• 4

• 1

• - 2

The slope of the tangent to the curve x = t² + 3t - 8, y = 2t² - 2t - 5 at the point (2, - 1) is

• $\frac{22}{7}$

• $\frac{6}{7}$

• $\frac{7}{6}$

• $\frac{- 6}{7}$

14.

If ${\tan }^{-1}\left({\mathrm{x}}^2 + {\mathrm{y}}^2\right) = \mathrm{\alpha }, \mathrm{then} \frac{\mathrm{dy}}{\mathrm{dx}}$ is equal to

• $\frac{- \mathrm{x}}{\mathrm{y}}$

• xy

• $\frac{\mathrm{x}}{\mathrm{y}}$

• - xy

The equation of the normal to the curve y(1 + x²) = 2 - x, where the tangent crosses X-axis, is

• 5x - y - 10 = 0

• x - 5y - 10 = 0

• 5x + y + 10 = 0

• x + 5y + 10 = 0

The maximum value of ${\left(\frac{1}{\mathrm{x}}\right)}^{\mathrm{x}}$ is

• e

• e^e

• e^1/e

• ${\left(\frac{1}{\mathrm{e}}\right)}^{\mathrm{e}}$

The rate of change of area of a circle with  respect to its radius at r = 2 cm is

• 4

• 2$\mathrm{\pi }$

• 2

• 4$\mathrm{\pi }$

If ${\sin }^{-1}\left(\mathrm{x}\right) + {\sin }^{-1}\left(\mathrm{y}\right) = \frac{\mathrm{\pi }}{2}$, then x² is equal to

• 1 - y²

• y²

• 0

• $\sqrt{1 - \mathrm{y}}$

If A = $\left[\begin{array}{cc}\cos \left(2\mathrm{\theta }\right)& - \sin \left(2\mathrm{\theta }\right)\\ \sin \left(2\mathrm{\theta }\right)& \cos \left(2\mathrm{\theta }\right)\end{array}\right]$ and A + A^T = I, where I is the unit matrix of 2 x 2 and A^T is the transpose of A, then the value of $\mathrm{\theta }$ is equal to

• $\frac{\mathrm{\pi }}{6}$

• $\frac{\mathrm{\pi }}{3}$

• $\mathrm{\pi }$

• $\frac{3\mathrm{\pi }}{2}$

If A = $\left[\begin{array}{cc}3& 1\\ - 1& 2\end{array}\right]$, then A² - 5A is equal to

• I

• - I

• 7I

• - 7I