31.

If $\mathrm{x} = {\mathrm{acos}}^3\left(\mathrm{\theta }\right), \mathrm{y} = {\mathrm{asin}}^3\left(\mathrm{\theta }\right), \mathrm{then} \sqrt{1 + {\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)}^2}$ is equal to

• ${\tan }^2\left(\mathrm{\theta }\right)$

• $\sec \left(\mathrm{\theta }\right)$

• ${\sec }^2\left(\mathrm{\theta }\right)$

• $\left|\sec \left(\mathrm{\theta }\right)\right|$

The difference of the numbers (1100110011)₂ and (1101001011)₂ in binary system is

• (100000)₂

• (101010)₂

• (11000)₂

• (10111)₂

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

• 2

• 1

• 0

• 3

If $\mathrm{\theta }$ is the angle between vectors p = ai + bj + ck and q = bi + cj + ak, then $\mathrm{\theta }$ lies in

• $\left[0, \frac{\mathrm{\pi }}{2}\right]$

• $\left[\frac{\mathrm{\pi }}{2}, \mathrm{\pi }\right]$

• $\left[\frac{\mathrm{\pi }}{2}, \frac{2\mathrm{\pi }}{3}\right]$

• $\left[0, \frac{2\mathrm{\pi }}{3}\right]$

If a line makes angles $\mathrm{\alpha }, \mathrm{\beta }, \mathrm{\gamma } \mathrm{and} \mathrm{\delta }$ with the four diagonals of a cube, then${\cos }^2\left(\mathrm{\alpha }\right) + {\cos }^2\left(\mathrm{\beta }\right) + {\cos }^2\left(\mathrm{\gamma }\right) +\hspace{0.17em}{\cos }^2\left(\mathrm{\delta }\right)$ is

• 1

• 2

• 2/3

• 4/3

The value of ${\int }_0^1{\sin }^{-1}\left(\mathrm{x}\right)\mathrm{dx}$ is

• $\frac{\mathrm{\pi }}{2}$

• $\frac{\mathrm{\pi }}{2} - 1$

• $\frac{\mathrm{\pi }}{2} + 1$

• 0

The solution of the differential equation $\mathrm{x}\frac{\mathrm{dy}}{\mathrm{dx}} = \mathrm{y} + \mathrm{xtan}\left(\frac{\mathrm{y}}{\mathrm{x}}\right)$ is

• $\sin \left(\frac{\mathrm{x}}{\mathrm{y}}\right) = \mathrm{x} + \mathrm{C}$

• $\sin \left(\frac{\mathrm{y}}{\mathrm{x}}\right) = \mathrm{Cx}$

• $\sin \left(\frac{\mathrm{x}}{\mathrm{y}}\right) = \mathrm{Cy}$

• $\sin \left(\frac{\mathrm{y}}{\mathrm{x}}\right) = \mathrm{Cy}$

If a, b and c are non-coplanar vectors and $\mathrm{p} = \frac{\mathrm{b} \times \mathrm{c}}{\left[\mathrm{a} \mathrm{b} \mathrm{c}\right]}$, q = $\mathrm{q} = \frac{\mathrm{c} \times \mathrm{a}}{\left[\mathrm{a} \mathrm{b} \mathrm{c}\right]}$ and $\mathrm{r} = \frac{\mathrm{a} \times \mathrm{b}}{\left[\mathrm{a} \mathrm{b} \mathrm{c}\right]}$, then a . p + b . q + c . r is equal to

• 3

• - 3

• 0

• None of the above

The angle between the straight lines $\frac{\mathrm{x} + 1}{2} = \frac{\mathrm{y} - 2}{5} = \frac{\mathrm{z} + 3}{4}$ and $\frac{\mathrm{x} - 1}{1} = \frac{\mathrm{y} + 2}{2} = \frac{\mathrm{z} - 3}{- 3}$ is

• $45°$

• $30°$

• $60°$

• $90°$

The ratio in which the join of (1, - 2,3) and (4, 2, - 1) is divided by the XOY plane is

• 1 : 3

• 3 : 1

• - 1 : 3

• None of these