If f : $\mathrm{R} \to \mathrm{R}$ be the signum function and g : $\mathrm{R} \to \mathrm{R}$ be the greatest integer function, then $\sin \left\{\mathrm{\pi }\left(\left(\mathrm{fog}\right)\left(\frac{1}{2}\right)\right)\right\}$ is equal to

• 1

• $\frac{\sqrt{3}}{2}$

• 0

• $\frac{1}{\sqrt{2}}$

If A and B are mutually exclusive events, then P(A/B) is equal to

• 0

• 1

• $\frac{\mathrm{P}\left(\mathrm{A} \cap \mathrm{B}\right)}{\mathrm{P}\left(\mathrm{A}\right)}$

• $\frac{\mathrm{P}\left(\mathrm{A} \cap \mathrm{B}\right)}{\mathrm{P}\left(\mathrm{B}\right)}$

If both the coefficients of regression between x and y are 0.8 and 0.2, then the coefficient of correlation between them will be

• 0.4

• 0.6

• 0.3

• 0.5

If the angle between two lines of regression is $\mathrm{\theta }$, then the value of $\mathrm{\theta }$ will be

• ${\tan }^{-1}\left[\frac{{\mathrm{b}}_{\mathrm{yx}} - {\displaystyle \frac{1}{{\mathrm{b}}_{\mathrm{xy}}}}}{1 + {\displaystyle \frac{{\mathrm{b}}_{\mathrm{xy}}}{{\mathrm{b}}_{\mathrm{yx}}}}}\right]$

• ${\tan }^{-1}\left[\frac{{\mathrm{b}}_{\mathrm{yx}} - {\displaystyle {\mathrm{b}}_{\mathrm{xy}} - 1}}{{\mathrm{b}}_{\mathrm{xy}} + {\mathrm{b}}_{\mathrm{yx}}}\right]$

• ${\tan }^{-1}\left[\frac{{\mathrm{b}}_{\mathrm{xy}} - {\displaystyle \frac{1}{{\mathrm{b}}_{\mathrm{yx}}}}}{1 + {\displaystyle \frac{{\mathrm{b}}_{\mathrm{xy}}}{{\mathrm{b}}_{\mathrm{yx}}}}}\right]$

• ${\tan }^{-1}\left[\frac{{\mathrm{b}}_{\mathrm{yx}} - {\displaystyle {\mathrm{b}}_{\mathrm{xy}}}}{1 + {\mathrm{b}}_{\mathrm{xy}} . {\mathrm{b}}_{\mathrm{yx}}}\right]$

Which of the following function is inverse of itself

• $\mathrm{f}\left(\mathrm{x}\right) = \frac{1 - \mathrm{x}}{1 + \mathrm{x}}$

• g(x) = 5^log(x)

• h(x) = 2^{x(x - 1)}

• None of the above

Let f : R $\to$ R be a differentiable function and f(1) = 4. Then, the value of $\underset{\mathrm{x}\to 1}{\lim }{\int }_4^{\mathrm{f}\left(\mathrm{x}\right)}\frac{2\mathrm{t}}{\mathrm{x} - 1}\mathrm{dt}$ is

• 8f'(1)

• 4f'(1)

• 2f'(1)

• f'(1)

If f''(x) = - f(x) and g(x) = f'(x) anf F(x) = ${\left[\mathrm{f}\left(\frac{\mathrm{x}}{2}\right)\right]}^2 + {\left[\left(\frac{\mathrm{x}}{2}\right)\right]}^2$ and given that F(5) = 5, then the value of F(10) is

• 15

• 0

• 5

• 10

28.

A function g defined for all real x > 0 satisfies g(1) = 1, g'(x²) = x³ for all x > 0, then value of g(4) is

• $\frac{13}{3}$

• 3

• $\frac{67}{5}$

• None of these

The value of ${\sin }^{-1}\left(\frac{4}{5}\right) + 2{\tan }^{-1}\left(\frac{2}{3}\right)$ is

• $\frac{\mathrm{\pi }}{3}$

• $\frac{\mathrm{\pi }}{2}$

• $\frac{\mathrm{\pi }}{6}$

• None of these

A particle moving on a curve has the position at a time t is given by x = f'(t)sin(t) + f'(t)cos(t), y = f'(t)cos(t) - f'(t)sin(t), where f is a twice differentiable function. Then, the velocity of the particle at time t is

• f'(t) + f''(t)

• f'(t) - f''(t)

• f'(t) + f'''(t)

• f'(t) - f''(t)