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

674.

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

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

• $\sqrt{13} \mathrm{km}/\mathrm{h}$

The point on the curve y² = x the tangent at which makes an angle 45° with X-axis ts

• $\left(\frac{1}{4}, \frac{1}{2}\right)$

• $\left(\frac{1}{2}, \frac{1}{4}\right)$

• $\left(\frac{1}{2}, \frac{- 1}{2}\right)$

• $\left(\frac{1}{2}, \frac{1}{2}\right)$

The length of the subtangent to the curve x²y² = a⁴ at (- a, a)

• $\frac{\mathrm{a}}{2}$

• 2a

• a

• $\frac{\mathrm{a}}{3}$

A stone is thrown vertically upwards and the height x ft reached by the stone in t seconds is given by x = 80 t - 16t² The stone reaches the maximum height in

• 2 s

• 2.5 s

• 3 s

• 1.5 s

The maximum value of $\frac{\log \left(\mathrm{x}\right)}{\mathrm{x}} \mathrm{in} \left(2, \infty \right)$ is

• 1

• $\frac{2}{\mathrm{e}}$

• e

• $\frac{1}{\mathrm{e}}$