11.

In a triangle ABC, if $\tan \left(\frac{\mathrm{A}}{2}\right) = \frac{5}{6} \mathrm{and} \tan \left(\frac{\mathrm{B}}{2}\right) = \frac{20}{37}$ , then a + c is equal to

• b

• 2b

• 3b

• 4b

Equation of the pair of straight lines bisecting the angles between the lines represented by ax² + hxy + by² = 0

• $\frac{{\mathrm{x}}^2 - {\mathrm{y}}^2}{\mathrm{a} - \mathrm{b}} = \frac{2\mathrm{xy}}{\mathrm{h}}$

• $\frac{{\mathrm{x}}^2 + {\mathrm{y}}^2}{\mathrm{a} + \mathrm{b}} = \frac{2\mathrm{xy}}{\mathrm{h}}$

• $\frac{{\mathrm{x}}^2 - {\mathrm{y}}^2}{\mathrm{a} - \mathrm{b}} = \frac{\mathrm{xy}}{\mathrm{h}}$

• None of these

The equation of circle which touches the axes and the line and whose centre lies inthe first quadrant is x² + y² - 2cx - 2cy + c² = 0. Then, c is equal to

• 1

• 2

• 3

• 6

Angle between any two diagonals ofa cube is

• $\frac{\mathrm{\pi }}{3}$

• ${\cos }^{-1}\left(\frac{1}{3}\right)$

• ${\cos }^{-1}\left(\frac{1}{\sqrt{3}}\right)$

• None of the above

If a₁, a₂, a₃ ..., a_n are in AP, where a_i > 0 for all i. Find the sum of series $\frac{1}{\sqrt{{\mathrm{a}}_1} + \sqrt{{\mathrm{a}}_2}} + \frac{1}{\sqrt{{\mathrm{a}}_2} + \sqrt{{\mathrm{a}}_3}} + \frac{1}{\sqrt{{\mathrm{a}}_3} + \sqrt{{\mathrm{a}}_4}} + ... + \frac{1}{\sqrt{{\mathrm{a}}_{\mathrm{n} - 1}} + \sqrt{{\mathrm{a}}_{\mathrm{n}}}}$

• $\frac{\mathrm{n} + 1}{\sqrt{{\mathrm{a}}_1} + \sqrt{{\mathrm{a}}_{\mathrm{n}}}}$

• $\frac{\mathrm{n} - 1}{\sqrt{{\mathrm{a}}_1} - \sqrt{{\mathrm{a}}_{\mathrm{n}}}}$

• $\frac{\mathrm{n} + 1}{\sqrt{{\mathrm{a}}_1} + \sqrt{{\mathrm{a}}_{\mathrm{n}}}}$

• $\frac{\mathrm{n} - 1}{\sqrt{{\mathrm{a}}_1} + \sqrt{{\mathrm{a}}_{\mathrm{n}}}}$

$\mathrm{y} = {\tan }^{-1}\left(\frac{\sqrt{\left(1 + {\mathrm{x}}^2\right)} + \sqrt{\left(1 - {\mathrm{x}}^2\right)}}{\sqrt{\left(1 + {\mathrm{x}}^2\right)} - \sqrt{\left(1 - {\mathrm{x}}^2\right)}}\right), \mathrm{then} \frac{\mathrm{dy}}{\mathrm{dx}}$ is equal to

• $\frac{1}{\sqrt{\left(1 + {\mathrm{x}}^2\right)}}$

• $- \frac{1}{2}$

• $- \frac{\mathrm{x}}{\sqrt{\left(1 - {\mathrm{x}}^4\right)}}$

• $\frac{\mathrm{x}\sqrt{1 + {\mathrm{x}}^2}}{1 - {\mathrm{x}}^4}$

If $\underset{\mathrm{x}\to \mathrm{a}}{\lim }\frac{{\mathrm{a}}^{\mathrm{x}} - {\mathrm{x}}^{\mathrm{a}}}{{\mathrm{x}}^{\mathrm{x}} - {\mathrm{a}}^{\mathrm{a}}} = - 1$, then

• a = 1

• a = 0

• a = e

• $\mathrm{a} = \frac{1}{\mathrm{e}}$

If four digits are taken fromthe digits 1, 2, 3, 4, 5, 6, 7. The probability that the sum of digits is less than 12, is

• $\frac{3}{35}$

• $\frac{4}{35}$

• $\frac{2}{35}$

• $\frac{1}{35}$

The probability of happening exactly one of the two events A and B is

• $\mathrm{P}\left(\mathrm{A}\right) + \mathrm{P}\left(\mathrm{B}\right) - 2\mathrm{P}\left(\mathrm{A} \cap \mathrm{B}\right)$

• $\mathrm{P}\left(\mathrm{A}\right) + \mathrm{P}\left(\mathrm{B}\right) - \mathrm{P}\left(\mathrm{A} \cap \mathrm{B}\right)$

• P(A) - P(B)

• None of the above

If the equation (a² + 4a + 3)x² + (a² - a - 2).x + a(a + 1) = 0 has more than two roots, then values of a is

• 0

• 1

• - 1

• None of the above