21.

The relation R defined on the set of natural numbers as { (a, b) : a differs from b by 3} is given

• {(1, 4), (2, 5), (3, 6), ... }

• {(4, 1), (5, 2), (6, 3), ... }

• {(1, 3), (2, 6), (3, 9), ... }

• None of the above

Let A be any element in a Boolean algebra B, if a + x = 1 and ax = 0, then

• x = 1

• x = 0

• x = a

• x = a'

Dual of (x + y) . (x + 1) = x + x . y + y is

• (x . y) + (x . 0) = x . (x + y) . y

• (x + y) + (x . 1) = x . (x + y) . y

• (x . y) (x . 0) = x . (x + y) . y

• None of these

If A = $\left[\begin{array}{ccc}1& - 5& 7\\ 0& 7& 9\\ 11& 8& 9\end{array}\right]$, then trace of matrix A is

• 17

• 25

• 3

• 12

The value of the determinant $\left|\begin{array}{ccc}\cos \left(\mathrm{\alpha }\right)& - \sin \left(\mathrm{\alpha }\right)& 1\\ \sin \left(\mathrm{\alpha }\right)& \cos \left(\mathrm{\alpha }\right)& 1\\ \cos \left(\mathrm{\alpha } + \mathrm{\beta }\right)& - \sin \left(\mathrm{\alpha } + \mathrm{\beta }\right)& 1\end{array}\right|$ is

• independent of $\mathrm{\alpha }$

• independent of $\mathrm{\beta }$

• independent of $\mathrm{\alpha } \mathrm{and} \mathrm{\beta }$

• None of the above

$$\begin{gathered} \mathrm{If} \mathrm{f}\left(\mathrm{x}\right) = \left\{\mathrm{mx} + 1, \mathrm{x} \le \frac{\mathrm{\pi }}{2} \\ \sin \left(\mathrm{x} + \mathrm{n}\right), \mathrm{x} > \frac{\mathrm{\pi }}{2}\right\} \mathrm{is} \mathrm{continuous} \mathrm{at} \mathrm{x} = \frac{\mathrm{\pi }}{2}, \mathrm{then} \end{gathered}$$

• m = 1, n = 0

• $\mathrm{m} = \frac{\mathrm{n\pi }}{2} + 1$

• $\mathrm{n} = \mathrm{m}\frac{\mathrm{\pi }}{2}$

• $\mathrm{m} = \mathrm{n} = \frac{\mathrm{\pi }}{2}$

The domain of the function f(x) = $\frac{\sqrt{4 - {\mathrm{x}}^2}}{{\sin }^{-1}\left(2 - \mathrm{x}\right)}$ is

• [0, 2]

• [0, 2)

• [1, 2)

• [1, 2]

If x, y and z are all distinct and $\left|\begin{array}{ccc}\mathrm{x}& {\mathrm{x}}^2& 1 + {\mathrm{x}}^3\\ \mathrm{y}& {\mathrm{y}}^2& 1 + {\mathrm{y}}^3\\ \mathrm{z}& \mathrm{z}& 1 + {\mathrm{z}}^3\end{array}\right|$ = 0, then the value of xyz is 

• - 2

• - 1

• - 3

• None of these

$2{\tan }^{-1}\left(\cos \left(\mathrm{x}\right)\right) = {\tan }^{-1}\left(2\csc \left(\mathrm{x}\right)\right)$, then value of x is

• $\frac{3\mathrm{\pi }}{4}$

• $\frac{\mathrm{\pi }}{4}$

• $\frac{\mathrm{\pi }}{3}$

• None of these

If a = i + j + k, b = i + 3j + 5k and c = 7i + 9j + 11k, then the area of parallelogram having diagonals a + b and  b + c is

• 4$\sqrt{6}$ sq units

• $\frac{1}{2}\sqrt{21}$ sq units

• $\frac{\sqrt{6}}{2}$ sq units

• $\sqrt{6}$ sq units