Rate of Change

Q.

A man of height 2m walks directly away from a lamp of height 5m, on a level road at 3 m/s. The rate at which the length of his shadow is increasing is

1. 1m/s

2. 2m/s

3. 3m/s

4. 4m/s

Q.

For what values of x is the rate of increase of $x^3-5x^2+5x+8$ is twice rate of increase of x ?

1. $-3,-\frac{1}{3}$

2. $-3,\frac{1}{3}$

3. $3,-\frac{1}{3}$

4. $3,\frac{1}{3}$

Q. Find equation of pair of tangents drawn from point $(6,4)$ to the ellipse $\frac{x^2}{25}+\frac{y^2}{9}=1$

1. $(\frac{x^2}{25}+\frac{y^2}{9}-1)={(\frac{6x}{25}+\frac{4y}{9}-1)}^2$
2. $(\frac{x^2}{25}+\frac{y^2}{9}-1)\left(\frac{499}{225}\right)={(\frac{6x}{25}+\frac{4y}{9}-1)}^2$
3. None of the above
4. $(\frac{x^2}{25}+\frac{y^2}{9}-1)\left(\frac{499}{225}\right)=(\frac{6x}{25}+\frac{4y}{9}-1)$

Q. The radius of a right circular cylinder increases at the rate of $0.1\text{cm/min}$, and the height decreases at the rate of $0.2\text{cm/min}$. The rate of change of the volume of the cylinder, in ${\text{cm}}^3/\text{min}$, when the radius is $2\text{cm}$ and the height is $3\text{cm}$ is
(The negative sign(-) indicates that volume decreases)

1. $-\frac{2\pi }{5}$
2. $\frac{8\pi }{5}$
3. $-\frac{3\pi }{5}$
4. $\frac{2\pi }{5}$

Q. $f\left(x\right)=4+5x^2+6x^4$ has

1. only one minimum
2. neither maximum nor minimum
3. only one maximum
4. two local minimum

Q. For the function $f\left(t\right)=\frac{e^t}{1+e^t},$

1. $f\left(t\right)$ has a minimum at $t=\frac{1}{e}$
2. $f\left(t\right)$ has a minimum at $t=1$
3. $f\left(t\right)$ has a maximum at $t=e$
4. $f\left(t\right)$ has no extreme value

Q. A spherical iron ball of $10\text{cm}$ radius is coated with a layer of ice of uniform thickness that melts at the rate of $50{\text{cm}}^3/\text{min}$. When the thickness of ice is $5\text{cm}$, then the rate (in $\text{cm/min}$) at which the thickness of ice decreases, is :

1. $\frac{5}{6\pi }$
2. $\frac{1}{54\pi }$
3. $\frac{1}{36\pi }$
4. $\frac{1}{18\pi }$