The height of a right circular cylinder is decreasing while its diameter is increasing at a rate of $4 \text{ cm/s}$ so as to keep its volume unchanged. The rate of change in its lateral surface area (in $\text{cm}^2/\text{s}$) at the instant when its diameter is $8 \text{ cm}$ and height is $12 \text{ cm}$, is (in $\pi$)

$A$ particle moves according to the law $s=t^{3}-6t^{2}+9t+25$. Find the displacement of the particle when its velocity is zero. (in $\text{ units}$)

The relation between pressure $p$ and volume $V$ is given by $p V^{1/4} = \text{constant}$. If the percentage decrease in volume is $\frac{1}{2} \%$, then the percentage increase in pressure is:

The total revenue in Rupees received from the sale of $x$ units of a product is given by $R(x) = 13x^2 + 26x + 15$. Find the marginal revenue when $x = 7$.

$A$ particle starts moving from rest from a fixed point in a fixed direction. The distance $s$ from the fixed point at a time $t$ is given by $s = t^{2} + at - b + 17$, where $a$ and $b$ are real numbers. If the particle comes to rest after $5 \ s$ at a distance of $s = 25$ units from the fixed point, then the values of $a$ and $b$ are respectively:

$A$ man,$2 \ m$ tall,walks at the rate of $1 \frac{2}{3} \ m/s$ towards a street light which is $5 \frac{1}{3} \ m$ above the ground. At what rate is the tip of his shadow moving? At what rate is the length of the shadow changing when he is $3 \frac{1}{3} \ m$ from the base of the light?