Average Rate of Change

Q.

A person whose height is $6\mathrm{feet}$ is walking away from the base of a street light along a straight path at a rate of $4$ feet per second.

If the height of a street light is $15\mathrm{feet}$, what is the rate at which a person’s shadow is lengthening?

Q. Two buses start from a bus terminal with a speed of $20$ km/hr at interval of $10$ minutes. What is the speed of a man coming from the opposite direction towards the bus terminal if he meets the buses at interval of $8$ minutes?

1. $3$ km/hr
2. $4$ km/hr
3. $5$ km/hr
4. $7$ km/hr

Q.

Derivative of

${\tan }^{-1}x$

Q.

If $\alpha +\beta =\frac{\pi }{2}$ and $\beta +\gamma =\alpha$, then the value of $\tan \alpha$ is

1. $\tan \beta +\tan \gamma$

2. $2(\tan \beta +\tan \gamma )$

3. $\tan \beta +2\tan \gamma$

4. $2\tan \beta +\tan \gamma$

Q. Two trains of equal length are running on parallel lines in the same direction at $46$ km/hr and $36$ km/hr. The faster train passes the slower train in $36$ seconds. The length of each train is

1. $50$ m
2. $72$ m
3. $80$ m
4. $100$ m

Q.

Prove that $\left(\frac{\tan A}{1-cotA}\right)+\left(\frac{cotA}{1-\tan A}\right)=1+\tan A+cotA$

Q.

If the radius of a circle is increasing at a uniform rate of $2\mathrm{cm}{\mathrm{s}}^{-1}$. The area of increasing the area of a circle, at the instant when the radius is $20\mathrm{cm}$, is

1. $70\mathrm{\pi }{\mathrm{cm}}^2{\mathrm{s}}^{-1}$

2. $70{\mathrm{cm}}^2{\mathrm{s}}^{-1}$

3. $80\mathrm{\pi }{\mathrm{cm}}^2{\mathrm{s}}^{-1}$

4. $80{\mathrm{cm}}^2{\mathrm{s}}^{-1}$

Q.

The diagonal of a square is changing at a rate of $0.5cm/sec$. Then the rate of change of area, when the area is $400c{m}^2$, is equal to

1. $20\sqrt{2}c{m}^2/sec$

2. $10\sqrt{2}c{m}^2/sec$

3. $\frac{1}{10\sqrt{2}}c{m}^2/sec$

4. $\left[10\sqrt{2}\right]c{m}^2/sec$

5. $5\sqrt{2}c{m}^2/sec$

Q.

The approximate value of square root of 25.2 is

1. 5.01

2. 5.03

3. 5.02

4. 5.04

Q.

How do you simplify the expression $1-se{c}^{2}x$?

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?

Q. A person reaches his destination $40$ minutes late if his speed is $3$ km/hr and reaches $30$ minutes before time if his speeed is $4$ km/hr. Then the distance of his destination from his starting point is

1. $14$ km
2. $12$ km
3. $7$ km
4. $3$ km

Q.

The lateral surface area of cube is $256m^2$. Find its volume.

Q.

A square piece of tin of side 18 cm is to be made into a box without top, by cutting-off square from each corner and folding up the flaps of the box. What should be the side of the square to be cut off so that the volume of the box is maximum possible?

Q. Two vertical poles of heights, $20$ m and $80$ m stand apart on a horizontal plane. The height (in metres) of the point of intersection of the lines joining the top of each pole to the foot of the other, from this horizontal plane is :

1. $16$
2. $18$
3. $12$
4. $15$

Q. The value of $\sqrt{0.6}$ is (approximate upto three decimal)

1. $0.735$
2. $0.775$
3. $0.755$
4. $0.745$

Q. A passenger train runs at the rate of $80$ km/h. It starts from the station, $6$ hours after a goods train leaves the station. The passenger train overtakes the goods train after $4$ hours. The speed of goods train is

1. $32$ km/h
2. $45$ km/h
3. $50$ km/h
4. $64$ km/h

Q.

A and B take 30 days to finish a work, B and C take 24 days to finish that particular work, C and A take 20 days to finish the same work. All of them can finish the work in 10 days and If B and C leave the job, then how many days does A take to finish the same work?

Q. Radius of a sphere is increasing at the rate of $2\text{cm/s}$. The rate at which its volume increasing when its radius is $10\text{cm}$ is ______

Q.

If an open box with a square base is to be made of a given quantity of cardboard of area $c^2,$ then show that the maximum volume of the box is $\frac{c^3}{6\sqrt{3}}$ cu units.

Q. The edge of a cube is increasing at the rate of 5cm/sec.How fast is the volume of the cube increasing when the edge is 12cm long

1. $432c{m}^3/sec$
2. $2160c{m}^3/sec$
3. $180c{m}^3/sec$
4. $Noneofthese$

Q.

Show that the semi-vertical angle of the right circular cone of given surface area and maximum volume is $si{n}^{-1}\left(\frac{1}{3}\right)$

Q.

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 and at what rate is the length of the shadow changing when he is $3\frac{1}{3}$ m from the base of the light?

Q. Let $y=f\left(x\right)=2x^2-3x+2$. The differential of $y$ when $x$ changes from $2$ to $1.99$ is

1. $-0.05$
2. $0.07$
3. $0.01$
4. $0.18$

Q.

Find the average of $4.2,3.8$ and $7.6$

Q.

The radius of a circle is increasing at the rate of $0.1cm/s$. When the radius of the circle is $5cm$, the rate of change of its area is

1. $-\pi c{m}^2/s$

2. $10\pi c{m}^2/s$

3. $0.1\pi c{m}^2/s$

4. $5\pi c{m}^2/s$

5. $\pi c{m}^2/s$

Q.

$\int \frac{\left(4e^x+6e^{-x}\right)}{\left(9e^x-4e^{-x}\right)}dx=$

1. $\frac{3}{2}x+\left(\frac{35}{35}\right)\log \left|9e^{2x}-4\right|+C$

2. $\frac{3}{2}x+\left(\frac{35}{36}\right)\log \left|9e^{2x}-4\right|+C$

3. $-\frac{3}{2}x+\left(\frac{35}{36}\right)\log \left|9e^{2x}-4\right|+C$

4. $-\frac{5}{2}x+\left(\frac{35}{36}\right)\log \left|9e^{2x}-4\right|+C$

5. $\frac{5}{2}x+\left(\frac{35}{66}\right)\log \left|9e^{2x}-4\right|+C$

Q. Walking $\frac{3}{4}$ of his normal speed, Ravi is $16$ minutes late in reaching his office. Then the usual time taken by him to cover the distance between his home and office is

1. $48$ minutes
2. $60$ minutes
3. $42$ minutes
4. $62$ minutes

Q.

A particle moves along $X-$axis in such a way that its coordinate $\left(X\right)$ varies with time $\left(T\right)$ according to the expression, $x=2-5t+6t^2$. Then the initial velocity of the particle is

Q.

What percentage is $2\mathrm{minutes}24\mathrm{seconds}$ of an hour?