21.

If f(x) = 2¹⁰⁰x + 1, g(x) = 3¹⁰⁰x + 1,  then the set ofreal numbers x such that f{g(x)} = x is 

• empty

• a singleton

• a finite set with more than one element

• infinite

The limit of $\left\{\frac{1}{\mathrm{x}}\sqrt{1 + \mathrm{x}} - \sqrt{1 + \frac{1}{{\mathrm{x}}^2}}\right\} \mathrm{as} \mathrm{x} \to 0$

• does not exist

• is equal to $\frac{1}{2}$

• is equal to 0

• is equal to 1

The value of

${\cos }^2\left(75°\right) + {\cos }^2\left(45°\right) + {\cos }^2\left(15°\right) - {\cos }^2\left(30°\right) - {\cos }^2\left(60°\right)$ is

• 0

• 1

• $\frac{1}{2}$

• $\frac{1}{4}$

The maximum and minimum values of ${\cos }^6\left(\mathrm{\theta }\right) + {\sin }^6\left(\mathrm{\theta }\right)$  are respectively

• $1 \mathrm{and} \frac{1}{4}$

• 1 and 0

• 2 and 0

• 1 and $\frac{1}{2}$

If z = x + iy, where x and y are real numbers and i = $\sqrt{- 1}$, then the points (x, y) for which $\frac{\mathrm{z} - 1}{\mathrm{z} - \mathrm{i}}$ is real, lie on

• an ellipse

• a circle

•  a parabola

• a straight line

If a, band c are in AP, then the straight line ax + 2by + c = 0 will always pass through a fixed point whose coordinates are

• (1, - 1)

• (- 1, 1)

• (1, - 2)

• (- 2, 1)

The equation 2x² + 5xy - 12y² = 0 represents a

• circle

• pair of non-perpendicular intersecting straight lines

• pair of perpendicular straight lines

• hyperbola

If one end of a diameter of the circle 3x² + 3y² - 9x + 6y + y = 0  is (1, 2), then the other end is

• (2, 1)

• (2, 4)

• (2, - 4)

• (- 4, 2)

The line y = x intersects the hyperbola $\frac{{\mathrm{x}}^2}{9} - \frac{{\mathrm{y}}^2}{25} = 1$ at the points P and Q. The eccentricity of ellipse with PQ as major axis and minor axis of length $\frac{5}{\sqrt{2}}$ is

• $\frac{\sqrt{5}}{3}$

• $\frac{5}{\sqrt{3}}$

• $\frac{5}{9}$

• $\frac{2\sqrt{2}}{9}$

The limit of $\mathrm{xsin}\left({\mathrm{e}}^{\frac{1}{\mathrm{x}}}\right)$ as $\mathrm{x} \to 0$

• is equal to 0

• is equal to 1

• is equal to $\frac{\mathrm{e}}{2}$

• does not exist