The minimum value of $2\cos \left(\mathrm{\theta }\right) + \frac{1}{\sin \left(\mathrm{\theta }\right)} + \sqrt{2}\tan \left(\mathrm{\theta }\right)$ in the interval $\left(0, \frac{\mathrm{\pi }}{2}\right)$ is :

• $2 + \sqrt{2}$

• $3\sqrt{2}$

• $2\sqrt{3}$

• $3 + \sqrt{2}$

12.

Let x₁ and x₂ be solutions of the equation ${\sin }^{-1}\left({\mathrm{x}}^2 - 3\mathrm{x} + \frac{5}{2}\right) = \frac{\mathrm{\pi }}{6}$. Then, the value of x₁² + x₂² is :

• 4

• 5

• $\frac{5}{2}$

• 6

If the derivative of the function f(x) is every where continuous and is given by

$$\begin{gathered} \mathrm{f}\left(\mathrm{x}\right) = \left\{{\mathrm{bx}}^2 + \mathrm{ax} + 4; \mathrm{x} \ge - 1 \\ {\mathrm{ax}}^2 + \mathrm{b}; \mathrm{x} < - 1\right\} \end{gathered}$$, then :

• a = 2, b = - 3

• a = 3, b = 2

• a = - 2, b = - 3

• a = - 3, b = - 2

If f(x + y) = f(x)f(y) for all real x and y, f(6) = 3 and f'(0) = 10, then f'(6) is :

• 30

• 13

• 10

• 0

The value of determinant, $\left|\begin{array}{ccc}\sqrt{13} + \sqrt{3}& 2\sqrt{5}& \sqrt{5}\\ \sqrt{15} + \sqrt{26}& 5& \sqrt{10}\\ 3 + \sqrt{65}& \sqrt{15}& 5\end{array}\right|$ is :

• $5\left(\sqrt{6} - 5\right)$

• $5\sqrt{3}\left(\sqrt{6} - 5\right)$

• $\sqrt{5}\left(\sqrt{6} - \sqrt{3}\right)$

• $\sqrt{2}\left(\sqrt{7} - \sqrt{5}\right)$

Derivative of ${\sec }^{-1}\left(\frac{1}{1 - 2{\mathrm{x}}^2}\right)$ w . r. t ${\sin }^{-1}\left(3\mathrm{x} - 4{\mathrm{x}}^3\right)$ is :

• $\frac{1}{4}$

• $\frac{3}{2}$

• 1

• $\frac{2}{3}$

Let D = {1, 2, 35, 6, 10, 15, 30}. Define the operattons '+', ' . ' and ' ' ' on D as follows a + b = LCM(a, b), a . b = GCD(a, b) and a' = $\frac{30}{\mathrm{a}}$ Then (15' + 6) · 10 1s equal to :

• 1

• 2

• 3

• 5

$\left|\begin{array}{ccc}\mathrm{a} +\hspace{0.17em}\mathrm{x}& \mathrm{b}& \mathrm{c}\\ \mathrm{a}& \mathrm{b} +\hspace{0.17em}\mathrm{y}& \mathrm{c}\\ \mathrm{a}& \mathrm{b}& \mathrm{c} +\hspace{0.17em}\mathrm{z}\end{array}\right|$ is equal to :

• $\mathrm{abc}\left(1 + \frac{\mathrm{x}}{\mathrm{a}} + \frac{\mathrm{y}}{\mathrm{b}} + \frac{\mathrm{z}}{\mathrm{c}}\right)$

• $\mathrm{abc}\left(1 + \frac{\mathrm{a}}{\mathrm{x}} + \frac{\mathrm{b}}{\mathrm{y}} + \frac{\mathrm{c}}{\mathrm{z}}\right)$

• $\mathrm{xyz}\left(1 + \frac{\mathrm{a}}{\mathrm{x}} + \frac{\mathrm{b}}{\mathrm{y}} + \frac{\mathrm{c}}{\mathrm{z}}\right)$

• $\mathrm{xyz}\left(1 + \frac{\mathrm{x}}{\mathrm{a}} + \frac{\mathrm{y}}{\mathrm{b}} + \frac{\mathrm{z}}{\mathrm{c}}\right)$

The solution of the differential equation $\mathrm{x}\frac{\mathrm{dy}}{\mathrm{dx}} + 2\mathrm{y} = {\mathrm{x}}^2$ is :

• $\mathrm{y} = \frac{{\mathrm{x}}^2 + \mathrm{c}}{4{\mathrm{x}}^2}$

• $\mathrm{y} = \frac{{\mathrm{x}}^2}{4} + \mathrm{c}$

• $\mathrm{y} = \frac{{\mathrm{x}}^2 + \mathrm{c}}{{\mathrm{x}}^2}$

• $\mathrm{y} = \frac{{\mathrm{x}}^4 + \mathrm{c}}{4{\mathrm{x}}^2}$

y = - A cos(5x) + B sin(5x) satisfies the differential equation :

• $\frac{{\mathrm{d}}^2\mathrm{y}}{{\mathrm{dx}}^2} + 10\frac{\mathrm{dy}}{\mathrm{dx}} + 25\mathrm{y} = 0$

• $\frac{{\mathrm{d}}^2\mathrm{y}}{{\mathrm{dx}}^2} - 10\frac{\mathrm{dy}}{\mathrm{dx}} + 25\mathrm{y} = 0$

• $\frac{{\mathrm{d}}^2\mathrm{y}}{{\mathrm{dx}}^2} - 25\mathrm{y} = 0$

• $\frac{{\mathrm{d}}^2\mathrm{y}}{{\mathrm{dx}}^2} + 25\mathrm{y} = 0$