Divide 10 into two parts such that the sum of double of the first and the square of the second is minimum

• (6, 4)

• (7, 3)

• (8, 2)

• (9, 1)

If 2x + y + $\mathrm{\lambda }$ = 0 is normal to the parabola y² = 8x, then $\mathrm{\lambda }$, is

• - 24

• 8

• - 16

• 24

If the line lx + my + n = 0 is tangent to the parabola y² = 4ax, then

• mn = al²

• lm = an²

• ln = am²

• None of the above

f : R $\to$ R, then f(x) = $\mathrm{x}\left|\mathrm{x}\right|$ will be

• many-one-onto

• one-one-onto

• many-one-into

• one-one-into

5.

If f(x) = $$\begin{gathered} \left\{\frac{\log \left(1 +\hspace{0.17em}2\mathrm{ax}\right) - \log \left(1 - \mathrm{bx}\right)}{\mathrm{x}}, \mathrm{x} \ne 0 \\ \mathrm{k}, \mathrm{x} = 0\right\} \end{gathered}$$ is contonuous at x = 0, then value of k is

• b + a

• b - 2a

• 2a - b

• 2a + b

If $\mathrm{f}\left(\mathrm{x}\right) = \left|\mathrm{x} - 3\right|$, then f'(3) is

• - 1

• 1

• 0

• does not exist

All the points on the curve y² = 4a$\left|\mathrm{x} + \mathrm{asin}\left(\raisebox{1ex}{$\mathrm{x}$}\!\left/ \!\raisebox{-1ex}{$\mathrm{a}$}\right.\right)\right|$, where the tangent is parallel to the axis of x are lies on

• circle

• parabola

• straight line

• None of these

The length of normal at any point to the curve $\mathrm{y} = \mathrm{c} \cosh \left(\frac{\mathrm{x}}{\mathrm{c}}\right)$ is

• fixed

• $\frac{{\mathrm{y}}^2}{{\mathrm{c}}^2}$

• $\frac{{\mathrm{y}}^2}{\mathrm{c}}$

• $\frac{\mathrm{y}}{{\mathrm{c}}^2}$

The height of right circular cylinder of maximum volume inscribed in a sphere of diameter 2a is

• $2\sqrt{3}\mathrm{a}$

• $\sqrt{3}\mathrm{a}$

• $\frac{2\mathrm{a}}{\sqrt{3}}$

• $\frac{\mathrm{a}}{\sqrt{3}}$

If $$\begin{gathered} \mathrm{f}\left(\mathrm{x}\right) = \left\{\mathrm{xsin}\left(\frac{1}{\mathrm{x}}\right), \mathrm{x} \ne 0 \\ 0 , \mathrm{x} = 0\right\} \end{gathered}$$, then at x = 0 the function f(x) is

• continuous

• differentiable

• continuous but not differentiable

• None of the above