The minimum value of $\left(\mathrm{x} - \mathrm{\alpha }\right)\left(\mathrm{x} - \mathrm{\beta }\right)$ is

• 0

• $\mathrm{\alpha \beta }$

• $\frac{1}{4}{\left(\mathrm{\alpha } - \mathrm{\beta }\right)}^2$

• $- \frac{1}{4}{\left(\mathrm{\alpha } - \mathrm{\beta }\right)}^2$

The equation of tangent to the curve 6y = 7 - x³ at (1, 1) is

• 2x + y = 3

• x + 2y = 3

• x + y = 1

• x + y + 2 = 0

The maximum value of xy subject to x + y = 7 is

• 10

• 12

• $\frac{49}{4}$

• $\frac{55}{4}$

If u = ${\mathrm{e}}^{{\mathrm{x}}^2 - {\mathrm{y}}^2}$, then

• xu_x = yu_y

• yu_x = xy_y

• yu_x + xu_y = 0

• x²u_y + y²u_x = 0

If u = ${\mathrm{xy}}^2{\tan }^{-1}\left(\frac{\mathrm{y}}{\mathrm{x}}\right)$, then $\mathrm{x}\frac{\partial \mathrm{u}}{\partial \mathrm{x}} + \mathrm{y}\frac{\partial \mathrm{u}}{\partial \mathrm{y}}$ is equal to

• 2u

• u

• 3u

• $\frac{1}{3}\mathrm{u}$

The family of curves in which the sub-tangent at any point to any curve is double the abscissa is given by

• x = Cy²

• y = Cx²

• x² = Cy²

• y² = Cx²

67.

Let Z denote the set of integers define f : $\mathrm{Z} \to \mathrm{Z}$ by f(x) = $$\begin{gathered} \left\{\frac{\mathrm{x}}{2}, \mathrm{x} \mathrm{is} \mathrm{even} \\ 0, \mathrm{x} \mathrm{is} \mathrm{odd}\right\} \end{gathered}$$, then f is

• onto but not one-one

• one-one but not onto

• one-one and onto

• neither one-one nor onto

Let f : $\mathrm{R} \to \mathrm{R}$ be defined by

$$\begin{gathered} \mathrm{f}\left(\mathrm{x}\right) =\left\{ \mathrm{x} +\hspace{0.17em}2, \mathrm{x} \le - 1 \\ {\mathrm{x}}^2, - 1 < \mathrm{x} < 1 \\ 2 - \mathrm{x}, \mathrm{x} \ge 1\right\} \end{gathered}$$

Then, the value of f(- 1.75) + f(0.5) + f(1.5) is

• 0

• 1

• 2

• - 1

Two functions f : $\mathrm{R} \to \mathrm{R}$, g : $\mathrm{R} \to \mathrm{R}$ are defined as follows

$$\begin{gathered} \mathrm{f}\left(\mathrm{x}\right) = \left\{0, \mathrm{x} \mathrm{is} \mathrm{rational} \\ 1, \mathrm{x} \mathrm{is} \mathrm{irational}\right\} \\ \mathrm{g}\left(\mathrm{x}\right) = \left\{- 1, \mathrm{x} \mathrm{is} \mathrm{rational} \\ 0, \mathrm{x} \mathrm{is} \mathrm{irrational}\right\} \end{gathered}$$

Then, (fog) $\left(\mathrm{\pi }\right)$ + (gof) (e) is equal to

• 0

• - 1

• 2

• 1

If $\mathrm{A} = \left[\begin{array}{cc}- 2& 2\\ - 3& 2\end{array}\right], \mathrm{B} = \left[\begin{array}{cc}0& - 1\\ 1& 0\end{array}\right]$, then (B^-1A^-1) is equal to

• $\left[\begin{array}{cc}2& 2\\ 2& 3\end{array}\right]$

• $\left[\begin{array}{cc}3& - 2\\ 2& 2\end{array}\right]$

• $\frac{1}{10}\left[\begin{array}{cc}2& 2\\ 2& 3\end{array}\right]$

• $\frac{1}{10}\left[\begin{array}{cc}3& 2\\ - 2& 2\end{array}\right]$