Relations and Functions

If a, b, c and d ∈ R such that a² + b² = 4 and ${\mathrm{c}}^2 + {\mathrm{d}}^2 = 4$and if (a + ib) = (c + id)² (x + iy), then x² + y² is equal to

• 4

• 3

• 2

• 1

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}

$\mathrm{If} \mathrm{f}\left(\mathrm{x}\right) = \left|\mathrm{x} - 1\right| + \left|\mathrm{x} - 2\right| + \left|\mathrm{x} - 3\right|, 2 < \mathrm{x} < 3, \mathrm{then} \mathrm{f} \mathrm{is}$

• an onto function but not one-one

• one-one function but not onto

• a bijection

• neither one-one nor onto

If x = a is a root of multiplicity two of a polynomial equation f(x) = 0, then

• f'(a) = f''(a) = 0

• f''(a) = f(a) = 0

• $\mathrm{f}\text{'}\left(\mathrm{a}\right) \ne 0 \ne \mathrm{f}\text{'}\text{'}\left(\mathrm{a}\right)$

• $\mathrm{f}\left(\mathrm{a}\right) = \mathrm{f}\text{'}\left(\mathrm{a}\right) = 0, \mathrm{f}\text{'}\text{'}\left(\mathrm{a}\right) \ne 0$

Let A = {- 4, - 2, - 1, 0, 3, 5} and f : A $\to$ IR be defined by

$$\begin{gathered} \mathrm{f}\left(\mathrm{x}\right) = \left\{3\mathrm{x} - 1 \mathrm{for} \mathrm{x} > 3 \\ {\mathrm{x}}^2 + 1 \mathrm{for} - 3 \le \mathrm{x} \le 3 \\ 2\mathrm{x} - 3 \mathrm{for} \mathrm{x} < - 3\right\} \\ \mathrm{Then} \mathrm{the} \mathrm{range} \mathrm{of} \mathrm{f} \mathrm{is} \end{gathered}$$

• $\left\{- 11, 5, 2, 1, 10, 14\right\}$

• $\left\{- 11, - 7, 2, 1, 8, 14\right\}$

• $\left\{- 11, 5, 2, 1, 8, 14\right\}$

• $\left\{- 11, - 7, - 5, 1, 10, 14\right\}$

The angle between the tangents drawn from the point (1, 2) to the ellipse 3x² + 2y² = 5 is

• ${\tan }^{-1}\left(\frac{12\sqrt{5}}{5}\right)$

• ${\tan }^{-1}\left(\frac{12\sqrt{5}}{13}\right)$

• $\frac{\mathrm{\pi }}{4}$

• $\frac{\mathrm{\pi }}{2}$

The equation of the circle whose diameter is the common chord of the circles x² + y² + 2x + 2y + 1 = 0 and x² + y² + 4x + 6 y + 4 = 0 is

• $10{\mathrm{x}}^2 + 10{\mathrm{y}}^2 + 14\mathrm{x} + 8\mathrm{y} +\hspace{0.17em}1 = 0$

• $3{\mathrm{x}}^2 + 3{\mathrm{y}}^2 - 3\mathrm{x} + 6\mathrm{y} - 8 = 0$

• $2{\mathrm{x}}^2 +\hspace{0.17em}2{\mathrm{y}}^2 - 2\mathrm{x} + 4\mathrm{y} +\hspace{0.17em}1 = 0$

• $2{\mathrm{x}}^2 + 2{\mathrm{y}}^2 - 2\mathrm{x} + 4\mathrm{y} + 1 = 0$

$\mathrm{If} \frac{5{\mathrm{x}}^2 + 2}{{\mathrm{x}}^3 + \mathrm{x}} = \frac{{\mathrm{A}}_1}{\mathrm{x}} + \frac{{\mathrm{A}}_2\mathrm{x} + {\mathrm{A}}_3}{{\mathrm{x}}^2 + 1}, \mathrm{then} \left({\mathrm{A}}_1, {\mathrm{A}}_2, {\mathrm{A}}_3\right) = ?$

• (0, 2, 3)

• $\left(3, 0, 2\right)$

• $\left(2, 3, 0\right)$

• $\left(2, 0, 3\right)$

Area (in sq. units) of the region outside $\frac{\left|\mathrm{x}\right|}{2} + \frac{\left|\mathrm{y}\right|}{3} = 1$ and inside the ellipse $\frac{{\mathrm{x}}^2}{4} + \frac{{\mathrm{y}}^2}{9} = 1$

• $3\left(\mathrm{\pi } - 2\right)$

• $3\left(4 - \mathrm{\pi }\right)$

• $6\left(\mathrm{\pi } - 2\right)$

• $6\left(4 - \mathrm{\pi }\right)$

$$\begin{gathered} \mathrm{If} \mathrm{R} = \left\{\left(\mathrm{x}, \mathrm{y}\right) :\hspace{0.17em}\mathrm{x}, \mathrm{y} \in \mathrm{Z}, {\mathrm{x}}^2 + 3{\mathrm{y}}^2 \le 8\right\} \mathrm{is} \mathrm{a} \mathrm{relation} \mathrm{on} \mathrm{the} \mathrm{set} \mathrm{of} \mathrm{integers} \mathrm{Z}, \\ \mathrm{then} \mathrm{the} \mathrm{domain} \mathrm{R}{ }^{- 1} \mathrm{is} ; \end{gathered}$$

• $\left\{ - 1, 0, 1\right\}$

• $\left\{ - 2, - 1, 1, 2\right\}$

• $\left\{0, 1\right\}$

• $\left\{ - 2, - 1, 0, 1, 2\right\}$