21.

The equation of sphere concentric with the sphere x² + y² + z² - 4x - 6y - 8z - 5 = 0 and which passes through the origin, is

• x² + y² + z² - 4x - 6y - 8z = 0

• x² + y² + z² - 6y - 8z = 0

• x² + y² + z² = 0

• x² + y² + z² - 4x - 6y - 8z - 6 = 0

If the lines $\frac{\mathrm{x} - 1}{2} = \frac{\mathrm{y} +\hspace{0.17em}1}{3} = \frac{\mathrm{z} - 1}{4}$ and $\frac{\mathrm{x} - 3}{2} = \frac{\mathrm{y} -\hspace{0.17em}\mathrm{k}}{3} = \frac{\mathrm{z}}{1}$ intersect, then the value of k, is

• $\frac{3}{2}$

• $\frac{9}{2}$

• $- \frac{2}{9}$

• - $\frac{3}{2}$

The two curves y = 3 and y = 5 intersect at an angle

• ${\tan }^{-1}\left(\frac{\log \left(3\right) - \log \left(5\right)}{1 + \log \left(3\right)\log \left(5\right)}\right)$

• ${\tan }^{-1}\left(\frac{\log \left(3\right) + \log \left(5\right)}{1 - \log \left(3\right)\log \left(5\right)}\right)$

• ${\tan }^{-1}\left(\frac{\log \left(3\right) + \log \left(5\right)}{1 + \log \left(3\right)\log \left(5\right)}\right)$

• ${\tan }^{-1}\left(\frac{\log \left(3\right) - \log \left(5\right)}{1 - \log \left(3\right)\log \left(5\right)}\right)$

The equation ${\mathrm{\lambda x}}^2 + 4\mathrm{xy} + {\mathrm{y}}^2 + \mathrm{\lambda x} + 3\mathrm{y} + 2 = 0$ represents parabola, if $\lambda$ is

• 0

• 1

• 2

• 4

If two circles 2x² + 2y² - 3x + 6y + k = 0 and x² + y² - 4x + 10y + 16 = 0 cut orthagobally then, value of k is

• 41

• 14

• 4

• 1

If B is a non-singular matrix and A is a square matrix, then det (B^-1AB) is equal to

• det(A^-1)

• det(B^-1)

• det(A)

• det(B)

The principal value of ${\sin }^{-1}\left\{\sin \left(\frac{5\mathrm{\pi }}{6}\right)\right\}$ is

• $\frac{\mathrm{\pi }}{6}$

• $\frac{5\mathrm{\pi }}{6}$

•  $\frac{7\mathrm{\pi }}{6}$

• None of these

Range of the function y = ${\sin }^{-1}\left(\frac{{\mathrm{x}}^2}{1 + {\mathrm{x}}^2}\right)$, is

• $\left(0, \frac{\mathrm{\pi }}{2}\right)$

• $[0, \frac{\mathrm{\pi }}{2})$

• $(0, \frac{\mathrm{\pi }}{2}]$

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

If y = $\sqrt{\mathrm{x} + \sqrt{\mathrm{y} + \sqrt{\mathrm{x} + \sqrt{\mathrm{y} + ... \infty }}}}$, then $\frac{\mathrm{dy}}{\mathrm{dx}}$ is equal to

• $\frac{\mathrm{y} + \mathrm{x}}{{\mathrm{y}}^2 - 2\mathrm{x}}$

• $\frac{{\mathrm{y}}^3 - \mathrm{x}}{2{\mathrm{y}}^2 - 2\mathrm{xy} - 1}$

• $\frac{{\mathrm{y}}^3 + \mathrm{x}}{2{\mathrm{y}}^2 - \mathrm{x}}$

• None of these

If f(x), g(x) and h(x) are three polynomials of degree 2 and $∆\left(\mathrm{x}\right) = \left|\begin{array}{ccc}\mathrm{f}\left(\mathrm{x}\right)& \mathrm{g}\left(\mathrm{x}\right)& \mathrm{h}\left(\mathrm{x}\right)\\ \mathrm{f}\text{'}\left(\mathrm{x}\right)& \mathrm{g}\text{'}\left(\mathrm{x}\right)& \mathrm{h}\text{'}\left(\mathrm{x}\right)\\ \mathrm{f}\text{'}\text{'}\left(\mathrm{x}\right)& \mathrm{g}\text{'}\text{'}\left(\mathrm{x}\right)& \mathrm{h}\text{'}\text{'}\left(\mathrm{x}\right)\end{array}\right|$, then $∆\left(\mathrm{x}\right)$ is a polynomials of degree

• 2

• 3

• 0

• atmost 3