21.

The area included between the parabolas y² = 4x and x² = 4y is

• $\frac{8}{3}$ sq unit

• 8 sq unit

• $\frac{16}{3}$ sq unit

• 12 sq unit

The locus of the centres of the circles which touch both the axes is given by

• x² - y² = 0

• x² + y² = 0

• x² - y² = 1

• x² + y² = 1

The sum of the series (1 + 2) + (1 + 2 + 2²) + (1 + 2 + 2² + 2³) + ...upto n terms is

• 2^{n + 2} - n - 4

• 2(2ⁿ - 1) - n

• 2^{n + 1} - n

• 2^{n + 1} - 1

The 5th term of the series $\frac{10}{9}, \frac{1}{3}\sqrt{\frac{20}{3}}, \frac{2}{3}, ...$ is

• $\frac{1}{3}$

• 1

• $\frac{2}{5}$

• $\sqrt{\frac{2}{3}}$

The equation $\sqrt{3}\sin \left(\mathrm{x}\right) + \cos \left(\mathrm{x}\right) = 4$ has

• infinitely many solutions

• no solution

• two solutions

• only one solution

The value of

$\tan \left(\mathrm{\alpha }\right) + 2\tan \left(2\mathrm{\alpha }\right) + 4\tan \left(4\mathrm{\alpha }\right) + ... + 2^{\mathrm{n} - 1}\tan \left(2^{\mathrm{n} - 1}\mathrm{\alpha }\right) + 2^{\mathrm{n}}\cot \left(2^{\mathrm{n}}\mathrm{\alpha }\right)$ is

• $\cot \left(2^{\mathrm{n}}\mathrm{\alpha }\right)$

• $2^{\mathrm{n}}\tan \left(2^{\mathrm{n}}\mathrm{\alpha }\right)$

• 0

• $\cot \left(\mathrm{\alpha }\right)$

Out of 8 given points, 3 are collinear. How many different straight lines can be drawn by joining any two points from those 8 points ?

• 26

• 28

• 27

• 25

How many odd numbers of six significant digits can be formed with the digits 0, 1, 2, 5, 6, 7 when no digit is repeated ?

• 120

• 96

• 360

• 288

Let $\mathrm{\alpha }, \mathrm{\beta }$ be the roots of x² - 2x$\cos \left(\mathrm{\varphi }\right)$ + 1 = 0, then the equation whose roots are ${\mathrm{\alpha }}^{\mathrm{n}}, {\mathrm{\beta }}^{\mathrm{n}}$ is

• ${\mathrm{x}}^2 - 2\mathrm{xcos}\left(\mathrm{n\varphi }\right) - 1$ = 0

• ${\mathrm{x}}^2 - 2\mathrm{xcos}\left(\mathrm{n\varphi }\right) + 1 = 0$

• ${\mathrm{x}}^2 - 2\mathrm{xsin}\left(\mathrm{n\varphi }\right) + 1 = 0$

• ${\mathrm{x}}^2 + 2\mathrm{xsin}\left(\mathrm{n\varphi }\right) - 1 = 0$

The latusrectum of an ellipse is equal to one-half of its minor axis. The eccentricity of the ellipse is

• $\frac{1}{\sqrt{6}}$

• $\frac{\sqrt{3}}{2}$

• $\frac{\sqrt{3}}{4}$

• $\frac{1}{2}$