The slope of the tangent to the curve x = 3t² + 1, y = t³ - 1 at x = 1 is

• $\frac{1}{2}$

• 0

• - 2

• $\infty$

The rate ofchange of the surface area ofthe sphere of radius r when the radius is increasing at the rate of 2 cm/sec is proportional to

• $\frac{1}{{\mathrm{r}}^2}$

• $\frac{1}{\mathrm{r}}$

• r²

• r

For the curve xy = c², the subnormal at any point varies as

• x³

• x²

• y³

• $\infty$

Let the function $\mathrm{f} : \mathrm{R} \to \mathrm{R}$ be defined by f(x) = 2x + cos(x), then f

• has maximum at x = 0

• has minimum at x = $\mathrm{\pi }$

• is an increasing function

• is a decreasing

The equation to the tangent to the curve y = be^{- x/a} at the point where it crosses the Y-axis is

• ax + by = 1

• $\frac{\mathrm{x}}{\mathrm{a}} - \frac{\mathrm{y}}{\mathrm{b}} = 1$

• $\frac{\mathrm{x}}{\mathrm{a}} + \frac{\mathrm{y}}{\mathrm{b}} = 1$

• ax - by = 1

The height of the cylinder of maximum volume inscribed in a sphere of radius 'a' is

• $\frac{3\mathrm{a}}{2}$

• $\frac{\sqrt{2}\mathrm{a}}{3}$

• $\frac{\mathrm{a}}{\sqrt{3}}$

• $\frac{2\mathrm{a}}{\sqrt{3}}$

The perimeter of a sector is P. The area of the sectoris maximum when its radius is

• $\frac{1}{\sqrt{\mathrm{P}}}$

• $\frac{\mathrm{P}}{2}$

• $\frac{\mathrm{P}}{4}$

• $\sqrt{\mathrm{P}}$

The function f(x) = x³ - 3x is

• increasing on $\left(- \infty , - 1\right) \cup [1, \infty ) \mathrm{and} \mathrm{decreasing} \mathrm{on} (- 1, 1)$

• $\mathrm{decreasing} \mathrm{on} \left(- \infty , - 1\right) \cup [1, \infty ) \mathrm{and} \mathrm{increasing} \mathrm{on} (- 1, 1)$

• $\mathrm{increasing} \mathrm{on} (0, \infty ) \mathrm{and} \mathrm{decreasing} \mathrm{on} (- \infty , 0)$

• $\mathrm{decreasing} \mathrm{on} (0, \infty ) \mathrm{and} \mathrm{increasing} \mathrm{on} (- \infty , 0)$

669.

A population p(t) of 1000 bacteria introduced into nutrient medium grows according to the relation p(t) = 1000 + $\frac{1000\mathrm{t}}{100 +\hspace{0.17em}{\mathrm{t}}^2}$. The maximum 100+t size of this bacterial population is

• 1100

• 1250

• 1050

• 5250

If ST and SN are the lengths of the subtangent and the subnormal at the point $\mathrm{\theta } = \frac{\mathrm{\pi }}{2}$ on the curve x = a$\left(\mathrm{\theta } + \sin \left(\mathrm{\theta }\right)\right)$, y = a$\left(1 - \cos \left(\mathrm{\theta }\right)\right)$, $\mathrm{a} \ne 1$, then

• ST = SN

• ST = 2SN

• ST² = aSN³

• ST³ = aSN