Rate of Change

Q. Two trains running in opposite directions cross a man standing on the platform in $27$ seconds and $17$ seconds respectively and they cross each other in $23$ seconds. The ratio of their speeds is

1. $1:2$
2. $3:2$
3. $3:4$
4. $4:3$

Q. If $x=2\cos t-\cos 2t$ and $y=2\sin t-\sin 2t,$ then $\frac{dy}{dx}$ is equal to

1. $\tan t$
2. $-\tan \frac{t}{2}$
3. $\tan \left(\frac{3t}{2}\right)$
4. $-\tan \left(\frac{3t}{2}\right)$

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

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

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. $\frac{1}{18\pi }$

Q. If the normal to the ellipse $3x^2+4y^2=12$ at a point $P$ on it is parallel to the line, $2x+y=4$ and the tangent to the ellipse at $P$ passes through $Q(4,4)$ then $PQ$ is equal to

1. $\frac{5\sqrt{5}}{2}$ unit
2. $\frac{\sqrt{157}}{2}$ unit
3. $\frac{\sqrt{61}}{2}$ unit
4. $\frac{\sqrt{221}}{2}$ unit

Q.

If the volume of a spherical balloon is increasing at the rate $900{\mathrm{cm}}^3/\sec$, then the rate of change of radius of balloon instant when radius is $15$ cm [in cm/sec]

1. $\frac{22}{7}$

2. $22$

3. $\frac{7}{22}$

4. None of these

Q.

Suppose oil spills from a ruptured tanker and spreads in a circular pattern. If the radius of the oil spill increases at a constant rate of $1\mathrm{m}/\mathrm{s}$, how fast is the area of the spill increasing when the radius is $37\mathrm{m}$?

Q.

The diameter of a tennis ball is $6.58cm$. What is the volume?

Q.

A tank with rectangular base and rectangular sides, open at the top is to be constructed so that its depth is 2m and volume is $8m^3$. If building of tank costs Rs 70 per sq metre for the base and Rs 45 per sq metre for sides. What is the cost of least expensive tank ?

Q.

The population of a city is $20,000.$ Find the population of the city after three years if the population increase by $5\%$ every year .

Q. The sides of an equilateral triangle are increasing at the rate of 4 cm/sec. The rate at which its area is increasing, when the side is 14 cm is

1. $42c{m}^2/sec$
2. $10\sqrt{3}c{m}^2/sec$
3. $14c{m}^2/sec$
4. $28\sqrt{3}c{m}^2/sec$

Q.

The radius of an air bubble is increasing at the rate of $\frac{1}{2}$ cm/s. At what rate is the volume of the bubble increasing when the radius is 1 cm?

Q. The total cost $C\left(x\right)$ associated with the production of $x$ units of an item is given by $C\left(x\right)=0.005x^3-0.02x^2+30x+5000$.
Find the marginal cost when $3$ units are produced, where by marginal cost we mean the instantaneous rate of change of total cost at any level of output.

Q.

A balloon, which always remains spherical, has a variable diameter $\frac{3}{2}(2x+1)$ . Find the rate of change of its volume with respect to x.

Q.

The radius of a circle is increasing at a constant rate of $0.2$ meters per second. what is the rate increase in the area of the circle at the instant when the circumference of the circle is $20\pi$ meters?

Q. A conical tent is to accommodate 11 persons. Each person must have 4 sq.m of the space on the ground and 20 cubic metre of air to breath. Find the height of the cone.

Q.

A particle moves in a straight line, so that $s=\sqrt{t}$ then its acceleration is proportional to

1. $velocity$

2. ${\left(velocity\right)}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}$

3. ${\left(velocity\right)}^3$

4. ${\left(velocity\right)}^2$

Q. The total cost $C\left(x\right)$ in Rupees, associated with the production of $x$ units of an item is given by
$C\left(x\right)=0.005x^3-0.02x^2+30x+5000$
Find the marginal cost when $3$ units are produced, where by marginal cost we mean the instantaneous rate of change of total cost at any level of output.

Q. Which of the following functions in the interval's mentioned are one-one functions.

1. $\sin x,x\in (0,2\pi )$
2. $\sin x,x\in (\frac{\pi }{2},\frac{3\pi }{2})$
3. $\tan x,x\in (0,\pi )-\left\{\frac{\pi }{2}\right\}$
4. $\tan x,x\in (\frac{\pi }{2},\frac{3\pi }{2})$

Q. On the ellipse $4x^2+9y^2=1$, the points at which the tangents are parallel to the line $8x=9y$ are

1. $(\frac{2}{5},\frac{1}{5})$
2. $(-\frac{2}{5},\frac{1}{5})$
3. $(-\frac{2}{5},-\frac{1}{5})$
4. $(\frac{2}{5},-\frac{1}{5})$

Q. An aeroplane flying at a constant speed, parallel to the horizontal ground, $\sqrt{3}km$ above it, is observed at an elevation of ${60}^{\circ }$ from a point on the ground. If, after five seconds, its elevation from the same point, is ${30}^{\circ }$, then the speed (in $km/hr$) of the aeroplane, is :

1. $1500$
2. $1440$
3. $750$
4. $720$

Q. A tank with rectangular base and rectangular sides, open at the top is to be constructed so that its depth is $2\text{m}$ and volume is $8{\text{m}}^3.$ If building of tank costs Rs. $70$ per square metre for the base and Rs. $45$ per square metre for the sides, what is the cost of least expensive tank ?

Q. If the law of motion in a straight line is $s=\frac{1}{2}vt$, then acceleration is

1. Constant
2. Proportional to t
3. Proportional to s
4. Proportional to v

Q. The approximate change in volume $V$ of a cube of side $x$ meters caused by increasing the side by $2$% is:

1. $0.03x^3$ cubic meter
2. $0.04x^3$ cubic meter
3. $0.06x^3$ cubic meter
4. $0.08x^3$ cubic meter

Q. The temperature at $12$ noon was ${10}^{o}C$ above zero. If it decreases at the rate of $2^{o}C$ per hour until midnight, at what time would the temperature be $8^{o}C$ below zero? What would be the temperature at midnight?

1. ${14}^{o}C$
2. ${-14}^{o}C$
3. ${16}^{o}C$
4. ${-16}^{o}C$

Q.

What Is MC and AC?

Q.

The equation of motion of a particle moving along a straight line is $s=2t^3–9t^2+12t$, where the units of $s$and $t$ are cm and sec. The acceleration of the particle will be zero after

1. $\frac{3}{2}sec$

2. $\frac{2}{3}sec$

3. $\frac{1}{2}sec$

4. Never

Q.

If the area of a circle increases at a uniform rate, then prove that perimeter varies inversely as the radius.

Q. The number of point(s) of discontinuity of function $f\left(x\right)=\left[\frac{6x}{\pi }\right]\cos \left[\frac{3x}{\pi }\right]$ in the interval $(\frac{\pi }{10},\frac{11\pi }{10})$ is
(where [.] is greatest integer function)

Q. The radius of a circle is increasing at the rate of 0.7 cm/s. What is the rate of increase of its circumference?