11.

$\mathrm{For} \mathrm{real} \mathrm{value} \mathrm{of} \mathrm{x}, \mathrm{the} \mathrm{rane} \mathrm{of} \frac{{\mathrm{x}}^2 + 2\mathrm{x} +\hspace{0.17em}1}{{\mathrm{x}}^2 + 2\mathrm{x} - 1} \mathrm{is}$

• $\left(- \infty , 0\right) \cup \left(1, \infty \right)$

• $\left[\frac{1}{2}, 2\right]$

• $\left(- \infty , \frac{- 2}{9}\right) \cup \left(1, \infty \right)$

• $\left(- \infty , - 6\right) \cup \left(- 2, \infty \right)$

T_m denotes the number of triangles that canbe formed with the vertices of a regular polygon of m sides. If T_{m + 3} - T_m = 15, then m is equal to

• 3

• 6

• 9

• 12

If $\left|\mathrm{x}\right|$ < 1, then the coefficient of x⁵ in the expansion of $\frac{3\mathrm{x}}{\left(\mathrm{x} - 2\right)\left(\mathrm{x} + 1\right)}$ is

• $\frac{33}{32}$

• $\frac{- 33}{32}$

• $\frac{31}{32}$

• $\frac{- 31}{32}$

If the coefficients of x⁹, x1⁰ and x¹¹ in the expansion of (1 + x)ⁿ are in arithmetic progression, then n² - 41n is equal to

• 399

• 298

• - 398

• 198

If s$\sin \left(\mathrm{\theta }\right) + \cos \left(\mathrm{\theta }\right) = \mathrm{p}$ and $\tan \left(\mathrm{\theta }\right) + \cot \left(\mathrm{\theta }\right) = \mathrm{q}$, then q(p² - 1) is equal to

• $\frac{1}{2}$

• 2

• 1

• 3

$\tan \left(\frac{\mathrm{\pi }}{5}\right) + 2\tan \left(\frac{2\mathrm{\pi }}{5}\right) + 4\cot \left(\frac{4\mathrm{\pi }}{5}\right)$ is equal to

• $\cot \left(\frac{\mathrm{\pi }}{5}\right)$

• $\cot \left(\frac{2\mathrm{\pi }}{5}\right)$

• $\cot \left(\frac{3\mathrm{\pi }}{5}\right)$

• $\cot \left(\frac{4\mathrm{\pi }}{5}\right)$

If sin(A) + sin(B) + sin(C) = 0 and cos(A) + cos(B) + cos(C) = 0, then cos (A + B) + cos (B + C) + cos(C + A) is equal to

• cos(A + B + C)

• 2

• 1

• 0

If $\tan \left(\mathrm{\theta }\right) . \tan \left(120° - \mathrm{\theta }\right)\tan \left(120° + \mathrm{\theta }\right) = \frac{1}{\sqrt{3}}$, then $\mathrm{\theta }$ is equal to

• $\frac{\mathrm{n\pi }}{3} + \frac{\mathrm{\pi }}{18}, \mathrm{n} \in \mathrm{Z}$

• $\frac{\mathrm{n\pi }}{3} + \frac{\mathrm{n\pi }}{12}, \mathrm{n} \in \mathrm{Z}$

• $\frac{\mathrm{n\pi }}{12} + \frac{\mathrm{\pi }}{12}, \mathrm{n} \in \mathrm{Z}$

• $\frac{\mathrm{n\pi }}{3} + \frac{\mathrm{\pi }}{6}, \mathrm{n} \in \mathrm{Z}$

If in a $∆\mathrm{ABC}$, r₁ = 2r₂ = 3r₃, then the perimeter of the triangle is equal to

• 3a

• 3b

• 3c

• 3(a + b + c)

The arithmetic mean of the observations 10, 8, 5, a, b is 6 and their variance is 6.8, then ab is equal to

• 6

• 4

• 3

• 12