21.

The radius of the sphere x² + y² + z² = 12x + 4y + 3z is

• 13/2

• 13

• 26

• 52

$\underset{\mathrm{x}\to \infty }{\lim }{\left(\frac{\mathrm{x} + 5}{\mathrm{x} + 2}\right)}^{\mathrm{x} + 3}$ equals

• e

• e²

• e³

• e⁵

If $\mathrm{p} \Rightarrow \left(~ \mathrm{p} \vee \mathrm{q}\right)$ is false, the truth value of p and q are respectively

• F, T

• F, F

• T, F

• T, T

The roots of (x - a)(x - a - 1) + (x - a - 1)(x - a - 2) + (x - a)(x - a - 2) = 0, $\mathrm{a} \in \mathrm{R}$ are always

• equal

• imaginary

• real and distinct

• rational and equal

Let f(x) = x² + ax + b, where a, b $\in$ R. If f(x) = 0 has all its roots imaginary, then the roots of f(x) + f'(x) + f''(x) = 0 are

• real and distinct

• imaginary

• real and distinct

• rational and equal

If f(x) = 2x⁴ - 13x² + ax + b is divisible by x² - 3x + 2, then (a, b) is equal to

• (- 9, - 2)

• (6, 4)

• (9, 2)

• (2, 9)

If x, y, z are all positive and are the pth , qth and rth terms of a geometric progression respectively, then the value of determinant $\left|\begin{array}{ccc}\log \left(\mathrm{x}\right)& \mathrm{p}& 1\\ \log \left(\mathrm{y}\right)& \mathrm{q}& 1\\ \log \left(\mathrm{z}\right)& \mathrm{r}& 1\end{array}\right|$ equals

• log(xyz)

• (p - 1)(q - 1)(r - 1)

• pqr

• 0

${\cos }^{-1}\left(\frac{- 1}{2}\right) - 2{\sin }^{-1}\left(\frac{1}{2}\right) + 3{\cos }^{-1}\left(\frac{- 1}{\sqrt{2}}\right) - 4{\tan }^{-1}\left(- 1\right)$ equals

• $\frac{19\mathrm{\pi }}{12}$

• $\frac{35\mathrm{\pi }}{12}$

• $\frac{47\mathrm{\pi }}{12}$

• $\frac{43\mathrm{\pi }}{12}$

If f : R $\to$ R is defined by

$$\begin{gathered} \mathrm{f}\left(\mathrm{x}\right) = \left\{\frac{2\sin \left(\mathrm{x}\right) - \sin \left(2\mathrm{x}\right)}{2\mathrm{xcos}\left(\mathrm{x}\right)}, \mathrm{if} \mathrm{x}\ne 0 \\ \mathrm{a}, \mathrm{if} \mathrm{x} = 0, \mathrm{if} \mathrm{x} = 0\right\} \end{gathered}$$

then the value of a so that f is continuous at 0 is

• 2

• 1

• - 1

• 0

$\mathrm{x} = {\cos }^{-1}\left(\frac{1}{\sqrt{1 + {\mathrm{t}}^2}}\right), \mathrm{y} = {\sin }^{-1}\left(\frac{1}{\sqrt{1 + {\mathrm{t}}^2}}\right) \Rightarrow \frac{\mathrm{dy}}{\mathrm{dx}}$ is equal to

• 0

• tan(t)

• 1

• sin(t) cos(t)