The values of a and b for which the function y = alog_e(x ) + bx² + x, has extremum at the points x₁ = 1 and x₂ = 2 are

• $\mathrm{a} = \frac{2}{3}, \mathrm{b} = - \frac{1}{6}$

• $\mathrm{a} = - \frac{2}{3}, \mathrm{b} = - \frac{1}{6}$

• $\mathrm{a} = - \frac{2}{3}, \mathrm{b} = \frac{1}{6}$

• $\mathrm{a} = - \frac{1}{3}, \mathrm{b} = - \frac{1}{6}$

The value of maxima of ${\left(\frac{1}{\mathrm{x}}\right)}^{\mathrm{x}}$ is

• ${\left(\frac{1}{\mathrm{e}}\right)}^{\mathrm{e}}$

• e^e

• e

• e^1/e

193.

A point particle moves along a straight line such that x = $\sqrt{\mathrm{t}}$, where t is time. Then, ratio of acceleration to cube of the velocity is

• - 1

• - 0.5

• - 3

• - 2

The tangents to curve y = x³ - 2x² + x - 2 which are parallel to straight line y = x, are

• $\mathrm{x} + \mathrm{y} = 2 \mathrm{and} \mathrm{x} - \mathrm{y} = \frac{86}{27}$

• $\mathrm{x} - \mathrm{y} = 2 \mathrm{and} \mathrm{x} - \mathrm{y} = \frac{86}{27}$

• $\mathrm{x} - \mathrm{y} = 2 \mathrm{and} \mathrm{x} + \mathrm{y} = \frac{86}{27}$

• $\mathrm{x} + \mathrm{y} = 2 \mathrm{and} \mathrm{x} + \mathrm{y} = \frac{86}{27}$

If two sides of a triangle are given, then the area of the triangle will be maximum, if. the angle between the given sides is

• $\frac{\mathrm{\pi }}{3}$

• $\frac{\mathrm{\pi }}{4}$

• $\frac{\mathrm{\pi }}{6}$

• $\frac{\mathrm{\pi }}{2}$

If f(x) = $\frac{80}{3{\mathrm{x}}^4 + 8{\mathrm{x}}^3 - 18{\mathrm{x}}^2 + 60}$, then the points of local maxima for the function f(x) are

• 1, 3

• - 3, 1

• - 1, 3

• - 1, - 3

The adjacent sides of a rectangle with given parameter as 200 cm and enclosing minimum area are

• 20 cm and 80 cm

•  50 cm and 50 cm40 cm and 60 cm

• 50 cm and 50 cm

• 30 cm and 70 cm

The altitude of the right circular cone of maximum volume that can be inscribed in a sphere of radius r is

• $\frac{\mathrm{r}}{2}$

• $\frac{\mathrm{r}}{3}$

• $\frac{3\mathrm{r}}{4}$

• $\frac{4\mathrm{r}}{3}$

Let f(x) = x(x - 1)², the point at which f(x) assumes maximum and minimum are respectively

• $\frac{1}{3}, 1$

• $1, \frac{1}{3}$

• 3, 1

• None of these

Rectangles are inscribed ina circle of radius r. The dimensions of the rectangle which has the maximum area, are

• r, r

• 2r, 2r

• $\sqrt{2}\mathrm{r}, \sqrt{2}\mathrm{r}$

• None of the above