If :$\mathrm{R}\to \mathrm{R}$  defined by

$$\begin{gathered} \mathrm{f} \left(\mathrm{x}\right) = {\mathrm{a}}^2{\cos }^2\mathrm{x} + {\mathrm{b}}^2{\sin }^2\mathrm{x}, \mathrm{x} \le 0 \\ = {\mathrm{e}}^{\mathrm{ax} + \mathrm{b}}, \mathrm{x} >0 \\ \mathrm{is} \mathrm{continuous} \mathrm{function}, \mathrm{then} \end{gathered}$$

• $\mathrm{b} = 2\log \left(\left|\begin{array}{c}\mathrm{a}\end{array}\right|\right)$

• $2\mathrm{b} = \log \left(\left|\begin{array}{c}\mathrm{a}\end{array}\right|\right)$

• $\mathrm{b} = \log \left(\left|\begin{array}{c}2\mathrm{a}\end{array}\right|\right)$

• ${\mathrm{b}}^2 = \log \left(\left|\begin{array}{c}\mathrm{a}\end{array}\right|\right)$

Let f(x) = e^x, g(x) = sin ^{- 1}x and h(x) = f(g(x)), then

$\frac{\mathrm{h}\text{'}\left(\mathrm{x}\right)}{\mathrm{h}\left(\mathrm{x}\right)} \mathrm{is} \mathrm{equal} \mathrm{to}$

• ${\sin }^{-1}\left(\mathrm{x}\right)$

• $\frac{1}{\sqrt{1 - {\mathrm{x}}^2}}$

• $\frac{1}{\sqrt{1 - \mathrm{x}}}$

• ${\mathrm{e}}^{{\sin }^{-1}\left(\mathrm{x}\right)}$

1013.

$\mathrm{If} \mathrm{f}\left(\mathrm{x}\right) = \sqrt{\mathrm{ax}} + \frac{{\mathrm{a}}^2}{\sqrt{\mathrm{ax}}}, \mathrm{then} \mathrm{f}\text{'}\left(\mathrm{a}\right) \mathrm{is} \mathrm{equal} \mathrm{to}$

• 0

• - 1

• 1

• a

If y = ae^x + be^-x + c, where a, b, c are parameters, then x²y" + xy' is equal to 

• 0

• y

• $\mathrm{y}\text{'}$

• $\mathrm{y}\text{'}\text{'}$

If y = acos(log (x)) + bsin (log(x)), where a, b are parameters, then xy" + xy' is equal to

• y

• - y

• 2y

• - 2y

The two curves x = y², xy = a² cut orthogonally at a point, then a² is equal to

• $\frac{1}{3}$

• $\frac{1}{2}$

• 1

• 2

$$\begin{gathered} \mathrm{If} \mathrm{f}\left(\mathrm{x}\right) = \left\{\frac{\sqrt{1 + \mathrm{kx}} - \sqrt{1 - \mathrm{kx}}}{\mathrm{x}}, \mathrm{for} - 1 \le \mathrm{x}< 0 \\ 2{\mathrm{x}}^2 + 3\mathrm{x} + 2, \mathrm{for} 0 \le \mathrm{x} \le 1\right\} \\ \mathrm{is} \mathrm{continuous} \mathrm{at} \mathrm{x} = 0, \mathrm{then} \mathrm{k} \mathrm{is} \mathrm{equal} \mathrm{to} \end{gathered}$$

• - 1

• - 2

• - 3

• - 4

$$\begin{gathered} \mathrm{If} \mathrm{f}\left(\mathrm{x}\right) =\left\{\frac{\mathrm{x} - 1}{2{\mathrm{x}}^2 - 7\mathrm{x} + 5}, \mathrm{for} \mathrm{x} \ne 1 \\ - \frac{1}{3}, \mathrm{for} \mathrm{x} = 1\right\}, \mathrm{then} \mathrm{f}\text{'}\left(1\right) \mathrm{is} \mathrm{equal} \mathrm{to} : \end{gathered}$$

• $- \frac{1}{9}$

• $- \frac{2}{9}$

• $- \frac{1}{3}$

• $\frac{1}{3}$

$\mathrm{If} \mathrm{f}\left(\mathrm{x}\right) = \frac{\mathrm{x}}{1 + \left|\mathrm{x}\right|}, \mathrm{for} \mathrm{x} \in \mathrm{R}, \mathrm{then} \mathrm{f}\text{'}\left(0\right) \mathrm{is} \mathrm{equal} \mathrm{to} :$

• 0

• 1

• 2

• 3

If u(x, y) = ylog(x) + xlog(y), then

u_xu_y - u_xlog(x) - u_ylog(y) + log(x)log(y) is equal to

• 0

• - 1

• 1

• 2