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

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

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

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

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

A spherical balloon is being inflated at the rate of 35 cc/min. The rate of increases of the surface area ofthe balloon when its diameter is 14cm, is

• 7 sq cm/min

• 10 sq cm/min

• 17.5 sq cm/min

• 28 sq cm/min

If the curve y = 2x³ + ax² + bx + c passes through the origin and the tangents drawn to it at x = - 1 and x = 2 are parallel to the x-axis, then the values of a, b and c are respectively :

• 12, - 3 and 0

• - 3, - 12 and 0

• - 3, 12 and 0

• 3, - 12 and 0

A circular sector of perimeter 60 m with maximum area is to be constructed. The radius of the circular arc in metre must be :

• 20

• 5

• 15

• 10

The tangent and the normal drawn to the curve y = x² - x + 4at P(1, 4) cut the x-axis at A and B respectively. If the length of the subtangent drawn to the curve at P is equal to the length of the subnormal, then the area of the triangle PAB in sq unit is :

• 4

• 32

• 8

• 16

The range in which y = - x² + 6x - 3 is increasing is

• x < 3

• x > 3

• 7 < x < 8

• 5 < x < 6

240.

OA and OB are two roads enclosmng an angle of 120°. X and Y start from 'O' at the same time. X travels along OA with a speed of 4 km/h and Y travels along OB with a speed of 3 km/h.The rate at which the shortest distance between X and Y is increasing after 1 h is

• $\sqrt{37} \mathrm{km} /\mathrm{h}$

• 37 km/h

• 13 km/h

• $\sqrt{13} \mathrm{km}/\mathrm{h}$