AP ICET 16th May 2016

If A completes 60% of work in 15 days alone, then he can do entire work alone in 25 days. 
Now, it is given that A and B complete 40% of work together in 5 days.
Hence, A and B together can do entire work in 100/8 i.e. 12.5 days

Now, let B be able to do entire work alone in x days
Hence, going by the standard formula:
$$\frac{100}{8}=\frac{25x}{25+x}$$ ==> x = 25 days

Let the length of the train be L meters and its speed be S m/s

By given info, we can say that:
L = 15*S and L+100 = 25*S 
Solving both equations we get L = 150 meters

Time for which train was travelling = 140/70 = 2 hours

Time for which the train needed to cover 80 km = 80/60 = 4/3 hours
Hence, remaining time = 2 - (4/3) = 2/3 hour 
Speed for the remaining 60 km = 60/(2/3) = 90 kmph

The angle subtended by the sector is directly proportional to the area it commands. 
Hence, we can write

$$\frac{2\pi}{\frac{22}{7}\cdot14\cdot14}=\frac{?}{\frac{154}{3}}$$ 

This gives us the ? value = Pi/6

Let the angles be x, 2x, 3x 
x+2x+3x =180 ++> x = 30 degrees 

Hence, it is a 30-60-90 triangle. 
The length of the smallest side i.e. the length of the side opposite the 30 degree angle = 10 cm
Hence, the largest side = Hypotenuse = 2*10 = 20 cm 

Volume of ice cream in each cone = $$\frac{1}{10}\cdot\frac{22}{7}\cdot36\cdot15\ =\ \frac{33\cdot36}{7}$$

The ice cream is filled in the cone plus a surplus hemispherical lobe at top of it. 
Hence, 
$$\ \frac{33\cdot36}{7}=\frac{1}{3}\cdot\frac{22}{7}\cdot r^2\cdot4r\ +\ \frac{2}{3}\cdot\frac{22}{7}\cdot r^3$$

Solving this, we easily get r = 3 

Given, the circus tent is in the form of a cylinder surmounted by a cone.

The circus is in the shape of cone placed over cylinder. Radius of both will be equal.

Radius of the cylinder = 30 feet

$$\Rightarrow$$ Radius of the cone = 30 feet

Height of the cylinder = 10 feet

Total height of the tent = 20 feet

$$\Rightarrow$$ Height of the cone = 20 - 10 = 10 feet

$$\therefore\ $$Volume of the tent = Volume of the cylinder + Volume of the cone

$$=\pi\ \left(30\right)^2\left(10\right)+\frac{1}{3}\pi\left(30\right)^2\left(10\right)$$

$$=\frac{4}{3}\pi\left(30\right)^2\left(10\right)$$

$$=\frac{4}{3}\pi\times9000$$

$$=12000\pi$$ cubic feet

Hence, the correct answer is Option C

Given,

$$a = x^{b - c}$$

$$\Rightarrow$$  $$a^a=x^{ab-ac}$$ ........(1)

$$b = x^{c - a}$$

$$\Rightarrow$$  $$b^b=x^{bc-ba}$$ ........(2)

$$c = x^{a - b}$$

$$\Rightarrow$$  $$c^c=x^{ca-cb}$$ ........(3)

Now multiplying the equations (1), (2), (3)

$$a^a\ b^b\ c^c=x^{ab-ac+bc-ba+ca-cb}=x^0=1$$

Hence, the correct answer is Option B

Now 13b7 is divisible by 11. Hence (b+1)-10 =11m (divisibility rule for 11)

As we can see, this is only possible when m=0 ( do remember that "b" has to be a single-digit)

Hence b+1-10 or b=9  Hence the number is 1397. Subtracting 984 from it we get 413. Hence a=1

So (a+b)^2=100