If ${\sec }^{-1}\left(\frac{1 + \mathrm{x}}{1 - \mathrm{y}}\right) = \mathrm{a}, \mathrm{then} \frac{\mathrm{dy}}{\mathrm{dx}}$ is

• $\frac{\mathrm{y} - 1}{\mathrm{x} + 1}$

• $\frac{\mathrm{y} + 1}{\mathrm{x} - 1}$

• $\frac{\mathrm{x} - 1}{\mathrm{y} - 1}$

• $\frac{\mathrm{x} - 1}{\mathrm{y} + 1}$

If y = ${\cos }^2\left(\frac{3\mathrm{x}}{2}\right) - {\sin }^2\left(\frac{3\mathrm{x}}{2}\right), \mathrm{then} \frac{{\mathrm{d}}^2\mathrm{y}}{{\mathrm{dx}}^2}$ is

• $- 3\sqrt{1 - {\mathrm{y}}^2}$

• 9y

• - 9y

• $3\sqrt{1 - {\mathrm{y}}^2}$

The point on the curve y² = x the tangent at which makes an angle 45° with X-axis ts

• $\left(\frac{1}{4}, \frac{1}{2}\right)$

• $\left(\frac{1}{2}, \frac{1}{4}\right)$

• $\left(\frac{1}{2}, \frac{- 1}{2}\right)$

• $\left(\frac{1}{2}, \frac{1}{2}\right)$

The length of the subtangent to the curve x²y² = a⁴ at (- a, a)

• $\frac{\mathrm{a}}{2}$

• 2a

• a

• $\frac{\mathrm{a}}{3}$

If $\mathrm{a} = 2\hat{\mathrm{i}} +\hspace{0.17em}3\hat{\mathrm{j}} - \hat{\mathrm{k}}, \mathrm{b} = \hat{\mathrm{i}} +\hspace{0.17em}2\hat{\mathrm{j}} - 5\hat{\mathrm{k}}, \mathrm{c} = 3\hat{\mathrm{i}} + 5\hat{\mathrm{j}} - \hat{\mathrm{k}}$, then a vector perpendicular to a and in the plane containing b and c is

• $- 17\hat{\mathrm{i}} + 21\hat{\mathrm{j}} - \hat{\mathrm{k}}$

• $17\hat{\mathrm{i}} + 21\hat{\mathrm{j}} - 123\hat{\mathrm{k}}$

• $- 17\hat{\mathrm{i}} - 21\hat{\mathrm{j}} + 97\hat{\mathrm{k}}$

• $- 17\hat{\mathrm{i}} - 21\hat{\mathrm{j}} -97 \hat{\mathrm{k}}$

16.

OA and BO are two vectors of magnitudes 5 and 6 respectively. If $\angle$BOA = 60°, then OA · OB is equal to

• 0

• 15

• - 15

• $15\sqrt{3}$

A vector perpendicularto the plane containing the points A (1, - 1, 2), B(2, 0, - 1), C(0, 2, 1) is

• $4\hat{\mathrm{i}} + 8\hat{\mathrm{j}} - 4\hat{\mathrm{k}}$

• $8\hat{\mathrm{i}} + 4\hat{\mathrm{j}} + 4\hat{\mathrm{k}}$

• $3\hat{\mathrm{i}} + \hat{\mathrm{j}} + 2\hat{\mathrm{k}}$

• $\hat{\mathrm{i}} + \hat{\mathrm{j}} - \hat{\mathrm{k}}$

If a and b are vectors such that $\left|\mathrm{a} + \mathrm{b}\right| = \left|\mathrm{a} - \mathrm{b}\right|$, then the angle between a and b is

• 120°

• 60°

• 90°

• 30°

The value of the integral ${\int }_0^{\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\left({\sin }^{100}\left(\mathrm{x}\right) - {\cos }^{100}\left(\mathrm{x}\right)\right)\mathrm{dx}$ is

• $\frac{1}{100}$

• $\frac{100!}{{\left(100\right)}^{100}}$

• $\frac{\mathrm{\pi }}{100}$

• 0

If k${\int }_0^1\mathrm{x} . \mathrm{f}\left(3\mathrm{x}\right)\mathrm{dx} = {\int }_0^3\mathrm{t} . \mathrm{f}\left(\mathrm{t}\right)\mathrm{dt}$, then the value of k is

• 9

• 3

• $\frac{1}{9}$

• $\frac{1}{3}$