If x > 0, x^y = e^{x - y}, then $\frac{\mathrm{dy}}{\mathrm{dx}}$ is equal to

• $\frac{1}{{\left(1 + \log \left(\mathrm{x}\right)\right)}^2}$

• $\frac{\log \left(\mathrm{x}\right)}{{\left(1 + \log \left(\mathrm{x}\right)\right)}^2}$

• ${\left(\frac{\log \left(\mathrm{x}\right)}{{\left(1 + \log \left(\mathrm{x}\right)\right)}^2}\right)}^2$

• $\frac{{\left(\log \left(\mathrm{x}\right)\right)}^2}{1 + \log \left(\mathrm{x}\right)}$

A particle moves along the curve y = x² + 2x. Then, the point on the curve such that x and y coordinates of the particle change with same rate is

• (1, 3)

• $\left(\frac{1}{2}, \frac{5}{2}\right)$

• $\left(- \frac{1}{2}, - \frac{3}{4}\right)$

• (- 1, - 1)

53.

The function f: $\mathrm{R} \to \mathrm{R}$ is defined by f(x)=3^{- x}. Observe the following statements
of it
I. f is one-one
II. f is onto
III. f is a decreasing function
Out of these, true statement are

• Only I, II

• Only II, III

• Only I, III

• I, II, III

The value of $\left|\begin{array}{ccc}1990& 1991& 1992\\ 1991& 1992& 1993\\ 1992& 1993& 1994\end{array}\right|$ is

• 1992

• 1993

• 1994

• 0

The rank of $\left[\begin{array}{ccc}1& - 1& 1\\ 1& 1& - 1\\ - 1& 1& 1\end{array}\right]$ is

• 0

• 1

• 2

• 3

$\mathrm{If} {\sin }^{-1}\left(\mathrm{x}\right) + {\sin }^{-1}\left(1 - \mathrm{x}\right) = {\cos }^{-1}\left(\mathrm{x}\right), \mathrm{then} \mathrm{x} \in \mathrm{to}$

• {1, 0}

• {- 1, 1}

• $\left\{0, \frac{1}{2}\right\}$

• {2, 0}

$\mathrm{If} \mathrm{y} = {\sin }^{-1} \mathrm{x}, \mathrm{then} \left(1 - {\mathrm{x}}^2\right)\frac{{\mathrm{dy}}^2}{{\mathrm{dx}}^2} \mathrm{is} \mathrm{equal} \mathrm{to}$

• $- \mathrm{x} \frac{\mathrm{dy}}{\mathrm{dx}}$

• 0

• $\mathrm{x}\frac{\mathrm{dy}}{\mathrm{dx}}$

• $\mathrm{x}{\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)}^2$

A point is moving on y = 4 - 2x². The x-coordinate of the point is decreasing at the rate of 5 units/second. Then, the rate at which y coordinate of the point is changing when the point is at (1, 2) is

• 5 units/s

• 10 units/s

• 15 units/s

• 20 units/s

$\mathrm{f}\left(\mathrm{x}, \mathrm{y}\right) = 2{\left(\mathrm{x} - \mathrm{y}\right)}^2 - {\mathrm{x}}^4 - {\mathrm{y}}^4$ ${\left|\left({\mathrm{f}}_{\mathrm{xx}}{\mathrm{f}}_{\mathrm{yy}} - {\mathrm{f}}_{\mathrm{xy}}^2\right)\right|}_{\left(0, 0\right)}$

• 32

• 16

• 0

• - 1

$$\begin{gathered} \mathrm{If} \hat{\mathrm{i}} + 2\hat{\mathrm{j}} + 3\hat{\mathrm{k}}, 3\hat{\mathrm{i}} + 2\hat{\mathrm{j}} + \hat{\mathrm{k}} \mathrm{are} \mathrm{sides} \mathrm{of} \mathrm{a} \mathrm{parallelogram}, \mathrm{then} \mathrm{a} \\ \mathrm{unit} \mathrm{vector} \mathrm{is} \mathrm{parallel} \mathrm{to} \mathrm{one} \mathrm{of} \mathrm{the} \mathrm{diagonals} \mathrm{of} \mathrm{the} \mathrm{parallelogram} \mathrm{is} \end{gathered}$$

• $\frac{\hat{\mathrm{i}} + \hat{\mathrm{j}} + \hat{\mathrm{k}}}{\sqrt{3}}$

• $\frac{\hat{\mathrm{i}} + \hat{\mathrm{j}} - \hat{\mathrm{k}}}{\sqrt{3}}$

• $\frac{\hat{\mathrm{i}} - \hat{\mathrm{j}} + \hat{\mathrm{k}}}{\sqrt{3}}$

• $\frac{- \hat{\mathrm{i}} + \hat{\mathrm{j}} + \hat{\mathrm{k}}}{\sqrt{3}}$