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

Let P(x) = a₀ + a₁x² + a₂x² + a₃x⁶ + ... + a_nx²ⁿ be a polynomial in a real variables with 0 < a₀ < a₁ < a₂ < .... < a_n. The function P(x) has

• neither a maxima nor a minima

• only one maxima

• both maxima and minima

• only one minima

The maximum value of f(x) = $2\sin \left(\mathrm{x}\right) + \cos \left(2\mathrm{x}\right), 0 \le \mathrm{x} \le \frac{\mathrm{\pi }}{2}$ occurs at x is equal to

• 0

• $\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$6$}\right.$

• $\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$2$}\right.$

• None of these

The equation of tangent of the curve y = be^-x/a at the point, where the curve meet y-axis is

• bx + ay - ab = 0

• ax + by - ab = 0

• bx - ay - ab = 0

• ax + by - ab = 0

If y = 4x - 5 is a tangent to the curve y² = px³ + q at (2, 3), then

• p = 2, q = - 7

• p = - 2, q = 7

• p = - 2, q = - 7

• p = 2, q = 7