For what values of the parameter ' $a$ ' does the function $f(x)=x^3+3(a-7) x^2+3\left(a^2-9\right) x-1$ have a positive point of maximum.

The slope of the curve $2 y^2=a x^2+b$ at $(1,-1)$ is -1 . Then, the value of $b$ is

A cylindrical tank of radius $10 \mathrm{~m}$ is being filled with wheat at the rate of $200 \pi$ cubic metre per hour. Then, the depth of the wheat is increasing at the rate of

Water is being filled at the rate of $1 \mathrm{~cm}^3 / \mathrm{s}$ in a right circular conical vessel (vertex downwards) of height $35 \mathrm{~cm}$ and diameter $14 \mathrm{~cm}$. When the height of the water levels is $10 \mathrm{~cm}$, the rate (in $\mathrm{cm}^2 / \mathrm{sec}$) at which the wet conical surface area of the vessel increases is

A running track of 440 ft is to be laid out enclosing a football field, the shape of which is a rectangle with a semi-circle at each end. If the area of the rectangular portion is to be maximum, then the lengths of its side are

A spherical balloon is filled with 4500$\pi$ cubic meters of helium gas. If a leak in the balloon causes the gas to escape at the rate of 72$\pi$ cubic meters per minute then the rate (in meters per minute) at which the radius of the balloon decreases 49 min after the leakage began is

The slope of the tangent to the curve x = t² + 3t $-$ 8, y = 2t² $-$ 2t $-$ 5 at the point t = 2 is

Let $f(x) = {a_0} + {a_1}{x^2} + {a_2}{x^4} + {a_3}{x^6} + ... + {a_n}{x^{2n}}$ be a polynomial in a real variable x with $0 < {a_1} < {a_2} < {a_3} < .... < {a_n}$, the function f(x) has