51.

The sum of n terms of the following series 1 + (1 + x) + (1 + x + x²) +... will be

• $\frac{1 - {\mathrm{x}}^{\mathrm{n}}}{1 - \mathrm{x}}$

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

• $\frac{\mathrm{n}\left(1 - \mathrm{x}\right) - \mathrm{x}\left(1 - {\mathrm{x}}^{\mathrm{n}}\right)}{{\left(1 - \mathrm{x}\right)}^2}$

• None of the above

If A₁, A_2; G₁, G₂ and H₁, H₂ be two AM's, GM's and HM's between two quantities, then the value of $\frac{{\mathrm{G}}_1{\mathrm{G}}_2}{{\mathrm{H}}_1{\mathrm{H}}_2}$ is

• $\frac{{\mathrm{A}}_1 + {\mathrm{A}}_2}{{\mathrm{H}}_1 + {\mathrm{H}}_2}$

• $\frac{{\mathrm{A}}_1 - {\mathrm{A}}_2}{{\mathrm{H}}_1 + {\mathrm{H}}_2}$

• $\frac{{\mathrm{A}}_1 + {\mathrm{A}}_2}{{\mathrm{H}}_1 - {\mathrm{H}}_2}$

• $\frac{{\mathrm{A}}_1 - {\mathrm{A}}_2}{{\mathrm{H}}_1 - {\mathrm{H}}_2}$

The most general solutions of the equation

$\sec \left(\mathrm{x}\right) - 1 = \left(\sqrt{2} - 1\right)\tan \left(\mathrm{x}\right)$ are given by

• $\mathrm{n\pi } + \frac{\mathrm{\pi }}{8}$

• $2\mathrm{n\pi }, 2\mathrm{n\pi } + \frac{\mathrm{\pi }}{4}$

• $2\mathrm{n\pi }$

• None of these

The angle between the straight lines

$\mathrm{x} - \mathrm{y}\sqrt{3} = 5 \mathrm{and} \sqrt{3}\mathrm{x} + \mathrm{y} = 7$ is

• 90$°$

• $60°$

• $75°$

• $30°$

The value of k so that x² + y² + kx + 4y + 2 = 0 and 2(x² + y) - 4x - 3y + k = 0 cut orthogonally is

• $\frac{10}{3}$

• $\frac{- 8}{3}$

• $\frac{- 10}{3}$

• $\frac{8}{3}$

An equilateral triangle is inscribed in the parabola y² = 4ax, whose vertex is at the vertex of the parabola. The length of its side is

• 2a$\sqrt{3}$

• 4a$\sqrt{3}$

• $6\mathrm{a}\sqrt{3}$

• 8a$\sqrt{3}$

The period of the function f(x) = $\sin \left(\sin \left(\frac{\mathrm{x}}{5}\right)\right)$ is

• $2\mathrm{\pi }$

• $\frac{2\mathrm{\pi }}{5}$

• $10\mathrm{\pi }$

• $5\mathrm{\pi }$

$\underset{\mathrm{x}\to \infty }{\lim }{\left(1 - \frac{4}{\mathrm{x} - 1}\right)}^{3\mathrm{x} - 1}$ is equal to

• e¹²

• e^{- 12}

• e⁴

• e³

The number of solutions of ${\log }_2\left(\mathrm{x} - 1\right) = 2{\log }_2\left(\mathrm{x} - 3\right)$

• 2

• 1

• 6

• 7

The complex numbers sin(x) + i cos(2x) and cos(x) - i sin(2x) are conjugate to each other for

• x = $\mathrm{n\pi }$

• $\mathrm{x} = \left(\mathrm{n} + \frac{1}{2}\right)\mathrm{\pi }$

• x = 0

• No value of x