If ${△}_1 = \left|\begin{array}{ccc}\mathrm{x}& \sin \left(\mathrm{\theta }\right)& \cos \left(\mathrm{\theta }\right)\\ - \sin \left(\mathrm{\theta }\right)& - \mathrm{x}& 1\\ \cos \left(\mathrm{\theta }\right)& 1& \mathrm{x}\end{array}\right| \mathrm{and} \left|\begin{array}{ccc}\mathrm{x}& \sin \left(2\mathrm{\theta }\right)& \cos \left(2\mathrm{\theta }\right)\\ - \sin \left(2\mathrm{\theta }\right)& - \mathrm{x}& 1\\ \cos \left(2\mathrm{\theta }\right)& 1& \mathrm{x}\end{array}\right|$, $\mathrm{x} \ne 0$; then for all $\mathrm{\theta } \in \left(0, \frac{\mathrm{\pi }}{2}\right)$ :

• ${△}_1 - {△}_2 = \mathrm{x}\left(\cos \left(2\mathrm{\theta }\right) - \cos \left(4\mathrm{\theta }\right)\right)$

• ${△}_1 + {△}_2 = - 2{\mathrm{x}}^3$

• ${△}_1 + {△}_2 = - 2\left({\mathrm{x}}^3 + \mathrm{x} - 1\right)$

• ${△}_1 - {△}_2 = - 2{\mathrm{x}}^3$

If $$\begin{gathered} \mathrm{f}\left(\mathrm{x}\right) = \left[\frac{\sin \left(\mathrm{p} + 1\right)\mathrm{x} + \sin \left(\mathrm{x}\right)}{\mathrm{x}}, \mathrm{x} < 0 \\ \mathrm{q}, \mathrm{x} = 0 \\ \frac{\sqrt{{\mathrm{x}}^2 + \mathrm{x}} - \sqrt{\mathrm{x}}}{{\mathrm{x}}^{{\displaystyle \raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}}, \mathrm{x} > 0\right] \end{gathered}$$ is continuous at x = 0, then the ordered pair (p, q) is equal to :

• $\left(- \frac{1}{2}, \frac{3}{2}\right)$

• $\left(- \frac{3}{2}, - \frac{1}{2}\right)$

• $\left(\frac{5}{2}, \frac{1}{2}\right)$

• $\left(- \frac{3}{2}, \frac{1}{2}\right)$

Let f(x) = e^x - x and g(x) = x² - x, $\forall \mathrm{x} \in \mathrm{R}$. Then the set of all x $\in$ R, where the function h(x) = (fog)(x) is increasing, is :

• $[- \frac{1}{2}, 0] \cup [1, \infty )$

• $[- 1, - \frac{1}{2}] \cup [\frac{1}{2}, \infty )$

• $[0, \infty )$

• $[0, \frac{1}{2}] \cup [1, \infty )$

If the system of linear equations x + y + z = 5, x + 2y + 2z = 6, x + 3y + $\mathrm{\lambda z} = \mathrm{\mu }, (\mathrm{\lambda }, \mathrm{\mu } \in \mathrm{R})$ has infinitely many solutions, then the value of $\mathrm{\lambda } +\hspace{0.17em}\mathrm{\mu }$ is

• 7

• 10

• 12

• 9

Let f R $\to$ R : be differentiable at c $\in$ R  and f(c) = 0. If g(x) = $\left|\mathrm{f}\left(\mathrm{x}\right)\right|$, then at x = c , g is

• not differentiable

• differentiable if f'(c) = 0

• not differentiable if f'(c) = 0

• differentiable if f'(c) $\ne$ 0

If y = y(x) is the solution of the differential equation $\frac{\mathrm{dy}}{\mathrm{dx}} = \left(\tan \left(\mathrm{x}\right) - \mathrm{y}\right){\sec }^2\left(\mathrm{x}\right), \mathrm{x} \in \left(- \frac{\mathrm{\pi }}{2}, \frac{{\displaystyle \mathrm{\pi }}}{{\displaystyle 2}}\right)$, such that y(0) = 0, then $\mathrm{y}\left(- \frac{\mathrm{\pi }}{4}\right)$ is equal to

• $2 + \frac{1}{\mathrm{e}}$

• e - 2

• $\frac{1}{2} - \mathrm{e}$

• $\frac{1}{\mathrm{e}} - 2$

If the length of perpendicular from point $\left(\mathrm{\beta }, 0, \mathrm{\beta }\right)$, $\left(\mathrm{where} \mathrm{\beta } \ne 0\right)$ to the line $\frac{\mathrm{x}}{1} = \frac{\mathrm{y} - 1}{0} = \frac{\mathrm{z} + 1}{- 1} \mathrm{is} \sqrt{\frac{3}{2}}$, then the value of $\mathrm{\beta }$ is

• - 1

• - 2

• 1

• 2

Assume that each born child is equally likely to be a boy or a girl. If two families have two children each, then the conditional probability that all children are girls given that at least two are girls is :

• $\frac{1}{17}$

• $\frac{1}{10}$

• $\frac{1}{11}$

• $\frac{1}{12}$

If $\int \frac{\mathrm{dx}}{{\left({\mathrm{x}}^2 - 2\mathrm{x} + 10\right)}^2} = \mathrm{A}\left({\tan }^{-1}\left(\frac{\mathrm{x} - 1}{3}\right) + \frac{\mathrm{f}\left(\mathrm{x}\right)}{{\mathrm{x}}^2 - 2\mathrm{x} + 10}\right) + \mathrm{C}$ where C is a constant of integration, then

• $\mathrm{A} = \frac{1}{54}, \mathrm{f}\left(\mathrm{x}\right) = 3\left(\mathrm{x} - 1\right)$

• $\mathrm{A} = \frac{1}{54}, \mathrm{f}\left(\mathrm{x}\right) = 9{\left(\mathrm{x} - 1\right)}^2$

• $\mathrm{A} = \frac{1}{27}, \mathrm{f}\left(\mathrm{x}\right) = 9\left(\mathrm{x} - 1\right)$

• $\mathrm{A} = \frac{1}{81}, \mathrm{f}\left(\mathrm{x}\right) = 3\left(\mathrm{x} - 1\right)$

The value of ${\int }_0^{2\mathrm{\pi }}\left[\sin \left(2\mathrm{x}\right)\left(1 + \cos \left(3\mathrm{x}\right)\right)\right]\mathrm{dx}$, (where [t] denotes Greatest Integer Function)

• $- 2\mathrm{\pi }$

• $\mathrm{\pi }$

• $2\mathrm{\pi }$

• $- \mathrm{\pi }$