If $\mathrm{P} = \frac{1}{2}{\sin }^2\left(\mathrm{\theta }\right) + \frac{1}{3}{\cos }^2\left(\mathrm{\theta }\right)$ then

• $\frac{1}{3} \le \mathrm{P} \le \frac{1}{2}$

• $\mathrm{P} \ge \frac{1}{2}$

• $2 \le \mathrm{P} \le 3$

• $- \frac{\sqrt{13}}{6} \le \mathrm{P} \le \frac{\sqrt{13}}{6}$

A positive acute angle is divided into two parts whose tangents are $\frac{1}{2}$ and $\frac{1}{3}$. Then, the angle is

• $\frac{\mathrm{\pi }}{4}$

• $\frac{\mathrm{\pi }}{5}$

• $\frac{\mathrm{\pi }}{3}$

• $\frac{\mathrm{\pi }}{6}$

The smallest value of $5\cos \left(\mathrm{\theta }\right) + 12$ is

• 5

• 12

• 7

• 17

Product of any r consecutive natural numbers is always divisible by

• r!

• (r + 4)!

• (r + 1)!

• (r + 2)!

45.

If C₀, C₁, C₂, ..., C_n denote the coefficients in the expansion of (1 + x)ⁿ, then the value of C₁ + 2C₂ + 3C₃ + ... + nC_n is

• n . 2^{n - 1}

• (n + 1)2^{n - 1}

• (n + 1)2ⁿ

• (n + 2)2^{n - 1}

A polygon has 44 diagonals. The number of its sides is

• 10

• 11

• 12

• 13

If $\mathrm{\alpha }, \mathrm{\beta }$ be the roots of x² - a(x - 1) + b = 0, then the value of $\frac{1}{{\mathrm{\alpha }}^2 - \mathrm{a\alpha }} + \frac{{\displaystyle 1}}{{\displaystyle {\mathrm{\beta }}^2 - \mathrm{a\beta }}} + \frac{2}{\mathrm{a} +\hspace{0.17em}\mathrm{b}}$ is

• $\frac{4}{\mathrm{a} + \mathrm{b}}$

• $\frac{1}{\mathrm{a} + \mathrm{b}}$

• 0

• - 1

The angle between the lines joining the foci of an ellipse to one particular extremity of the minor axis is 90°. The eccentricity of the ellipse is

• 1/8

• 1/√3

• √(2/3)

• √(1/2)

The sum of all real roots of the equation ${\left|\mathrm{x} - 2\right|}^2 + \left|\mathrm{x} - 2\right| - 2 = 0$ is

• 7

• 4

• 1

• 5

For each n $\in$ N, 2³ⁿ - 1is divisible by

• 7

• 8

• 6

• 16