If A and B have n elements in common, then the numberofelements common to A x B and B x A is

• n

• 2n

• n²

• 0

If f(x) = $\left|\begin{array}{ccc}\sin \left(\mathrm{x}\right)& \cos \left(\mathrm{x}\right)& \tan \left(\mathrm{x}\right)\\ {\mathrm{x}}^3& {\mathrm{x}}^2& \mathrm{x}\\ 2\mathrm{x}& 1& \mathrm{x}\end{array}\right|$, then $\underset{\mathrm{x}\to 0}{\lim }\frac{\mathrm{f}\left(\mathrm{x}\right)}{{\mathrm{x}}^2}$ is equal to

• 0

• 3

• 2

• 1

Inverse of a diagonal non-singular matrix is

• scalar matrix

• skew symmetric matrix

• zero matrix

• diagonal matrix

If ax⁴ + bx³ + cx² + dx + e = $\left|\begin{array}{ccc}{\mathrm{x}}^3 + 3\mathrm{x}& \mathrm{x} - 1& \mathrm{x} +\hspace{0.17em}3\\ \mathrm{x} + 1& - 2\mathrm{x}& \mathrm{x} - 4\\ \mathrm{x} - 3& \mathrm{x} + 4& 3\mathrm{x}\end{array}\right|$, then e is equal to

• 1

• 0

• 2

• - 1

Which of the following is false ?

• (N, *) is a group
  
  

• (N, +) is a semi-group

• (Z, +) is a group

• Set of even integers is a group under usual addition

6.

$2{\cos }^{-1}\left(\mathrm{x}\right) = {\sin }^{-1}\left(2\mathrm{x}\sqrt{1 - {\mathrm{x}}^2}\right)$ is valid for all values of x satisfying

• $- 1 \le \mathrm{x} \le 1$

• $0 \le \mathrm{x} \le 1$

• $\frac{1}{\sqrt{2}} \le \mathrm{x} \le 1$

• $0 \le \mathrm{x} \le \frac{1}{\sqrt{2}}$

If ${\mathrm{msin}}^{-1}\left(\mathrm{x}\right) = {\log }_{\mathrm{e}}\left(\mathrm{y}\right), \mathrm{then} \left(1 - {\mathrm{x}}^2\right)\mathrm{y}\text{'}\text{'} - \mathrm{xy}\text{'}$ is equal to

• m²y

• - m²y

• 2y

• - 2y

If y = ${\log }_{\mathrm{e}}\left[1 + \mathrm{x} + {\mathrm{x}}^2 + .. \infty \right], \mathrm{then} \frac{\mathrm{dy}}{\mathrm{dx}}$ is equal to

• $\frac{1}{{\left(1 +\hspace{0.17em}\mathrm{x}\right)}^2}$

• $\frac{1}{{\left(1 -\hspace{0.17em}\mathrm{x}\right)}^2}$

• $\frac{- 1}{{\left(1 +\hspace{0.17em}\mathrm{x}\right)}^2}$

• $\frac{- 1}{{\left(1 -\hspace{0.17em}\mathrm{x}\right)}^2}$

Length of the subtangent at (x₁, y₁) on xⁿy^m = a^{m + n}, m, n > 0, is

• $\frac{\mathrm{n}}{\mathrm{m}}{\mathrm{x}}_1$

• $\frac{\mathrm{m}}{\mathrm{n}}\left|{\mathrm{x}}_1\right|$

• $\frac{\mathrm{n}}{\mathrm{m}}\left|{\mathrm{y}}_1\right|$

• $\frac{\mathrm{n}}{\mathrm{m}}\left|{\mathrm{x}}_1\right|$

If a ball is thrown vertically upwards and the height 's' reached in time 't' is given by s = 22t - 11t², then the total distance travelled by the ball is

• 44 units

• 33 units

• 11 units

• 22 units