The coefficient of x⁵ in the expansion of (a + x²)⁵(1 + x)⁴ is

• 30

• 60

• 40

• None of these

$\frac{{\displaystyle \frac{1}{2!} + \frac{1}{4!} + \frac{1}{6!} + ...}}{1 + {\displaystyle \frac{1}{3!}} + {\displaystyle \frac{1}{5!}} + ...}$ equals

• e + 1

• $\frac{\mathrm{e} - 1}{\mathrm{e} + 1}$

• e - 1

• None of these

The sum of the series

$\left(\frac{\mathrm{a} - \mathrm{b}}{\mathrm{a}}\right) + \frac{1}{2}{\left(\frac{\mathrm{a} - \mathrm{b}}{\mathrm{a}}\right)}^2 + \frac{1}{3}{\left(\frac{\mathrm{a} - \mathrm{b}}{\mathrm{a}}\right)}^3 + ... \infty$ is

• ${\log }_{\mathrm{e}}\left(\frac{\mathrm{a}}{\mathrm{b}}\right)$

• ${\log }_{\mathrm{e}}\left(\frac{\mathrm{a} - \mathrm{b}}{\mathrm{a}}\right)$

• ${\log }_{\mathrm{e}}\left(\frac{\mathrm{b}}{\mathrm{a}}\right)$

• None of these

The domain of the function

$\mathrm{f}\left(\mathrm{x}\right) = {\log }_{10}\left(\sqrt{\mathrm{x} - 4} + \sqrt{6 - \mathrm{x}}\right)$ is

• [4, 6]

• $\left(- \infty , 6\right)$

• (2, 3)

• None of these

The inverse of the function f(x) = $\mathrm{f}\left(\mathrm{x}\right) = \frac{{10}^{\mathrm{x}} - {10}^{- \mathrm{x}}}{{10}^{\mathrm{x}} + {10}^{- \mathrm{x}}}$ is

• ${\log }_{10}\left(2 - \mathrm{x}\right)$

• $\frac{1}{2}{\log }_{10}\left(\frac{1 +\hspace{0.17em}\mathrm{x}}{1 - \mathrm{x}}\right)$

• $\frac{1}{2}{\log }_{10}\left(2\mathrm{x} - 1\right)$

• $\frac{1}{4}\log \left(\frac{2\mathrm{x}}{2 - \mathrm{x}}\right)$

36.

The value of $\underset{\mathrm{x}\to \infty }{\lim }\sqrt{{\mathrm{a}}^2{\mathrm{x}}^2 + \mathrm{ax} + 1} - \sqrt{{\mathrm{a}}^2{\mathrm{x}}^2 + 1}$ is

• $\frac{1}{2}$

• 1

• 2

• None of these

The curve represented by

$\mathrm{x} = 3\left(\cos \left(\mathrm{t}\right) + \sin \left(\mathrm{t}\right)\right), \mathrm{y} = 4\left(\cos \left(\mathrm{t}\right) - \sin \left(\mathrm{t}\right)\right)$ is

• ellipse

• parabola

• hyperbola

• circle

The equation of the bisector of the acute angle between the lines 3x - 4y + 7 = 0 and 12x + 5y - 2 = 0 is

• 99x - 27y - 81 = 0

• 11x - 3y + 9 = 0

• 21x + 77y - 101 = 0

• 21x + 77y + 101 = 0

If 5x - 12y + 10 = 0 and 12y - 5x + 16 = 0 are two tangents to a circle, then the radius of the circle is

• 1

• 2

• 4

• 6

If the coefficient of rth and (r + 1)th terms in the expansion of (3 + 7x)²⁹ are equal, then r equals

• 15

• 21

• 14

• None of these