One ticket is selected at random from 100 tickets numbered 00, 01, 02,..., 98, 99. If x₁ and x₂ denote the sum and product of the digits on the tickets, then P(x₁ = 9/x₂ = 0) is equal to

• $\frac{1}{50}$

• $\frac{2}{19}$

• $\frac{19}{100}$

• None of these

The shortest distance from the plane 12x + y + 3z = 327 to the sphere x² + y² + z² + 4x - 2y - 6z = 155 is

• 13

• 26

• 39

• $41\frac{4}{13}$

13.

If $\mathrm{\alpha } + \mathrm{\beta } + \mathrm{\gamma } = \mathrm{a\delta } \mathrm{and} \mathrm{\beta } + \mathrm{\gamma } + \mathrm{\delta } = \mathrm{b\alpha }, \mathrm{\alpha } \mathrm{and} \mathrm{\delta }$ are non-collinear, then $\mathrm{\alpha } + \mathrm{\beta } + \mathrm{\gamma } + \mathrm{\delta }$ equals

• 0

• $\mathrm{a\alpha }$

• b$\mathrm{\delta }$

• (a + b)$\mathrm{\gamma }$

The value of $\underset{\mathrm{x}\to \infty }{\lim }\frac{{\left(2^{{\mathrm{x}}^{\mathrm{n}}}\right)}^{{\displaystyle {\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\mathrm{e}$}\right.}^{\mathrm{x}} }}- {\left(3^{{\mathrm{x}}^{\mathrm{n}}}\right)}^{{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\mathrm{e}$}\right.}^{\mathrm{x}} }}{{\mathrm{x}}^{\mathrm{n}}}$ (where, x $\in$ N) is

• 0

• $\mathrm{n} \log \left(\mathrm{n}\left(\frac{2}{3}\right)\right)$

• $\log \left(\left(\frac{2}{3}\right)\right)$

• not defined

If A = $\left[\begin{array}{ccc}0& 1& - 1\\ 2& 1& 3\\ 3& 2& 1\end{array}\right]$, then (A(adj A)A^{- 1}) is equal to

• $2\left[\begin{array}{ccc}3& 0& 0\\ 0& 3& 0\\ 0& 0& 3\end{array}\right]$

• $\left[\begin{array}{ccc}0& 1/6& - 1/6\\ 2/6& 1/6& 3/6\\ 3/6& 2/6& 1/6\end{array}\right]$

• $\left[\begin{array}{ccc}- 6& 0& 0\\ 0& - 6& 0\\ 0& 0& - 6\end{array}\right]$

• None of these

Let A = $\left[\begin{array}{ccc}1& - 1& 1\\ 2& 1& - 3\\ 3& 2& 1\end{array}\right]$ and 10B = $\left[\begin{array}{ccc}4& 2& 2\\ - 5& 0& \mathrm{\alpha }\\ 1& - 2& 3\end{array}\right]$. If B is the inverse of A, then $\mathrm{\alpha }$ is

• 5

• - 2

• 1

• - 1

If A and B are two matrices such that rank of A = m and rank of B = n, then

• rank (AB) $\ge$ rank (B)

• rank (AB) $\ge$ rank (A)

• rank (AB) $\le$ min (rank A, rank B)

• rank(AB) = mn

Let a, b and c be positive real numbers. The following system of equations in x, y and z,

$$\begin{gathered} \frac{{\mathrm{x}}^2}{{\mathrm{a}}^2} + \frac{{\mathrm{y}}^2}{{\mathrm{b}}^2} - \frac{{\mathrm{z}}^2}{{\mathrm{c}}^2} = 1, \frac{{\displaystyle {\mathrm{x}}^2}}{{\displaystyle {\mathrm{a}}^2}} - \frac{{\displaystyle {\mathrm{y}}^2}}{{\displaystyle {\mathrm{b}}^2}} + \frac{{\displaystyle {\mathrm{z}}^2}}{{\displaystyle {\mathrm{c}}^2}} = 1 \\ \mathrm{and} \frac{{\displaystyle - {\mathrm{x}}^2}}{{\displaystyle {\mathrm{a}}^2}} + \frac{{\displaystyle {\mathrm{y}}^2}}{{\displaystyle {\mathrm{b}}^2}} + \frac{{\displaystyle {\mathrm{z}}^2}}{{\displaystyle {\mathrm{c}}^2}} = 1 \mathrm{has} \end{gathered}$$

• finitely many solutions

• no solution

• unique solution

• infinitely many solutions

Range of f(x) = ${\sin }^{-1}\left(\mathrm{x}\right) + {\tan }^{-1}\left(\mathrm{x}\right) + {\sec }^{-1}\left(\mathrm{x}\right)$ is

• $\left[\frac{\mathrm{\pi }}{4}, \frac{{\displaystyle 5\mathrm{\pi }}}{{\displaystyle 4}}\right]$

• $\left\{\frac{\mathrm{\pi }}{4}, \frac{3\mathrm{\pi }}{4}\right\}$

• $\left(\frac{\mathrm{\pi }}{4}, \frac{7\mathrm{\pi }}{4}\right)$

• None of these

If $\frac{1}{2}{\sin }^{-1}\left[\frac{3\sin \left(2\mathrm{\theta }\right)}{5 + 4\cos \left(2\mathrm{\theta }\right)}\right] = {\tan }^{-1}\left(x\right)$, then x is equal to

• 3$\cot \left(\mathrm{\theta }\right)$

• 3$\tan \left(\mathrm{\theta }\right)$

• $\tan \left(3\mathrm{\theta }\right)$

• $\frac{1}{3}\tan \left(\mathrm{\theta }\right)$