21.

$$\begin{gathered} \mathrm{If} \mathrm{the} \mathrm{curves} \frac{{\mathrm{x}}^2}{{\mathrm{a}}^2} + \frac{{\mathrm{y}}^2}{{\mathrm{b}}^2} = 1 \mathrm{and} \frac{{\mathrm{x}}^2}{25} + \frac{{\mathrm{y}}^2}{16} = 1 \\ \mathrm{cut} \mathrm{each} \mathrm{other} \mathrm{orthogonally}, \mathrm{then} {\mathrm{a}}^2 - {\mathrm{b}}^2 = ? \end{gathered}$$

• 9

• 400

• 75

• 41

The value of c in the Lagrange's mean value theorem for f(x) = $\sqrt{\mathrm{x} - 2}$ in the interval [2, 6] is

• $\frac{9}{2}$

• $\frac{5}{2}$

• 3

• 4

Te area (m sq units) of the region bounded by x = -1, x = 2, y = x² + 1 and y = 2x - 2 is

• 10

• 7

• 8

• 9

If R is the set of all real numbers and $\mathrm{f} : \mathrm{R} - \left\{2\right\} \to \mathrm{R} \mathrm{is} \mathrm{defined} \mathrm{by} \mathrm{f}\left(\mathrm{x}\right) = \frac{2 +\hspace{0.17em}\mathrm{x}}{2 - \mathrm{x}} \mathrm{for} \mathrm{x} \in \mathrm{R} - \left\{2\right\}$

• R - {- 2}

• R

• R - {1}

• R - {- 1}

Let Q be the set of all rational numbers in [0, 1]and f: [0, 1] $\to$ [0,1] be defined by

$$\begin{gathered} \mathrm{f}\left(\mathrm{x}\right) = \left\{\mathrm{x}, \mathrm{for} \mathrm{x} \in \mathrm{Q} \\ 1 - \mathrm{x} \mathrm{for} \mathrm{x}\notin \mathrm{Q}\right\} \\ \mathrm{Then}, \mathrm{the} \mathrm{set} \mathrm{S} = \left\{\mathrm{x} \in \left[0, 21\right] : \left(\mathrm{fof}\right)\left(\mathrm{x}\right)\right\} = ? \end{gathered}$$

• $[0, 1]$

• - Q

• $[0, 1]\; -\; Q$

• (0, 1)

$\sum _{\mathrm{k} = 1}^{2\mathrm{n} + 1}{\left(- 1\right)}^{\mathrm{k} - 1} . {\mathrm{k}}^2 = ?$

• (n - 1)(2n - 1)

• (n + 1)(2n + 1)

• (n + 1)(2n - 1)

• (n - 1)(2n + 1)

If in the angles of a triangle are in the ratio1 : 1 : 4, then the ratio of the perimeter of the triangle to its largest side is

• $\sqrt{2} + 2 : \sqrt{3}$

• 3 : 2

• $\sqrt{3} + 2 . \sqrt{2}$

• $2 + \sqrt{3} : \sqrt{3}$

If in a $∆$ABC, r₁ = 2, r₂ = 3 and 7a = 6, then ae quals to

• 4

• 1

• 2

• 3

The number of solutions for ${\mathrm{z}}^3 + \overline{\mathrm{z}} = 0$ , is

• 5

• 1

• 3

• 2

The least positive integer n for which (1 + i)ⁿ = (1 - i)ⁿ , is

• 8

• 2

• 4

• 6