$\mathrm{Evaluate} \sum _{\mathrm{k} = 1}^6\left(\sin \left(\frac{2\mathrm{k\pi }}{7}\right) - \mathrm{icos}\left(\frac{2\mathrm{k\pi }}{7}\right)\right)$

• 2i

• - i

• i

• - 2i

If a, b, c are in HP, then the value of $\frac{\mathrm{b} +\hspace{0.17em}\mathrm{a}}{\mathrm{b} - \mathrm{a}} + \frac{\mathrm{b} + \mathrm{c}}{\mathrm{b} - \mathrm{c}}$ is

• 0

• 1

• 2

• 3

If x² - 4x + log_1/2(a) = 0 does not have two distinct real roots, then maximum value of a is

• $- \frac{1}{4}$

• $\frac{1}{16}$

• $\frac{1}{4}$

• None of these

A polygon has 44 diagonals. Find the number of sides.

• 8

• 10

• 11

• 13

The coefficient of x⁴ in the expansion of log (1 + 3x + 2x²) is

• $\frac{16}{3}$

• $- \frac{16}{3}$

• $\frac{17}{4}$

• $- \frac{17}{4}$

In a $∆$ABC, if cot(A) cot(B) cot(C) > 0, then the triangle is

• acute angled

• right angled

• obtuse angled

• does not exist

If ${\left(\frac{1 + \sin \left(\mathrm{\theta }\right) - \cos \left(\mathrm{\theta }\right)}{ 1 + \sin \left(\mathrm{\theta }\right) + \cos \left(\mathrm{\theta }\right)}\right)}^2 = \mathrm{\lambda }\left(\frac{1 - \cos \left(\mathrm{\theta }\right)}{1 +\hspace{0.17em}\cos \left(\mathrm{\theta }\right)}\right)$, then $\mathrm{\lambda }$ equals

• - 1

• 1

• 2

• - 2

If the sides of a $∆$ABC are in AP and a is the smallest side, then cos(A) equals

• $\frac{3\mathrm{c} - 4\mathrm{b}}{2\mathrm{c}}$

• $\frac{3\mathrm{c} - 4\mathrm{b}}{2\mathrm{b}}$

• $\frac{4\mathrm{c} - 3\mathrm{b}}{2\mathrm{c}}$

• $\frac{4\mathrm{c} - 3\mathrm{b}}{2\mathrm{b}}$

The number of common tangents that can be drawn to the circles x² + y² - 4x - 6y - 3 = 0 and x² + y² + 2x + 2y + 1 = 0 is

• 1

• 2

• 3

• 4

10.

If two circles (x - 1)² + (y - 3)² = r² and x² + y² - 8x + 2y + 8 = 0 intersect in two distinct points, then

• 2 < r < 8

• r < 2

• r = 2

• r > 2