AP ICET 2014

The speed of doing work is assumed to be constant. Hence, direct variation concept is applicable here. 

$$\frac{\left(Time\ to\ complete\ \left(\frac{4}{5}\right)th\ work\right)}{Time\ to\ complete\ \left(\frac{1}{5}\right)th\ work\ }=\frac{\left(1.5\ days\right)}{x\ }$$

Hence, x = (3/8)th day = 3*24/8 hours = 9 hours

In an equilateral triangle, the incenter, orthocenter, median are all coincident. 
Thus, the center of the inscribed circle is also the centroid of the triangle and the radius of the circle is the smaller division of the median which gets cut by the centroid. The centroid divides the median in the ratio of 2:1. Thus, we can say that the height of the triangle will be 3 times the radius of the inscribed circle.

Radius of the circle = sqrt (462*7/22) = $$7\sqrt{\ 3}$$
Height of the triangle = $$21\sqrt{\ 3}$$

Now, Height of triangle = $$\frac{\sqrt{\ 3}}{2}\cdot Side\ of\ triangle$$
Hence, Side of triangle = 42 cm and thus, perimeter = 126 cm

Let the length , breadth, diagonal length be l, b and d respectively.
We know that d = $$\sqrt{\ l^2+b^2}$$

We are given that : 
$$l+\sqrt{\ l^2+b^2}\ =\ 5\cdot b$$
$$\sqrt{\ l^2+b^2}\ =\ 5\cdot b-l$$
Squaring both sides and then putting l*b = 60 (given in question), we get:
$$b^2=25b^2-600$$
$$b=5$$
Hence, l = 12 and difference in length and breadth = 7 

The volume of the clay available remains constant.
Hence, equating the volumes of the cone and the sphere:
$$\frac{1}{3}\pi\ \cdot6^2\cdot24=\frac{4}{3}\pi\ \cdot R^3$$

Solving this, we get R = 6, and hence, diameter = 12. 

Let radius of the sphere be R. 

Hence, according to given data, we get 
$$4\cdot\pi\ \cdot R^2=2\cdot\pi\ \cdot6\cdot12$$
where, the 6 in RHS is the radius of the cylinder and 12 is the height

Solving the above equation, we get R = 6

Number of smaller cubes obtained = (Volume of original cube)/(Volume of each of the smaller cube)
Hence, Number of smaller cubes obtained = (9*9*9)/(3*3*3) = 27 

Now, surface area "s" of the original cube = 6*(9*9) sq. cm
Sum of Surface areas of all the smaller cubes, S = 27 * [6*(3*3)] 

Hence, s:S = 1:3

The cyclicity of 13 is 4. Now 400 is perfectly divisible by 4. Hence units digit of $$13^{400}$$ is equivalent to $$13^{4}$$ which is equal to 1.

Hence, the correct answer is Option D

Given, a* = k denotes that k is the remainder when a is divided by 7.

Dividing 100 by 7 gives remainder 2.

Similarly, 100* = 2

Hence, the correct answer is Option B