If a, b, and c are all positive integers, with 4a > b, then which of the following conditions is BOTH NECESSARY AND SUFFICIENT for the expression ∛[(3)^a (21)^(3a−b) (49)^(2b+c)] to be a positive integer?

In a computer game, each move requires pressing a button. When the button is pressed for the first time, as a move, the computer randomly chooses a cell from a 4 × 4 grid of sixteen cells and puts an "X" mark on that cell. When the button is pressed subsequently, the computer ran…

A solid trophy, consisting of two parts, has been designed in the following manner: the bottom part is a frustum of a cone with the bottom radius 30 cm, the top radius 20 cm, and height 40 cm, while the top part is a hemisphere with radius 20 cm. Moreover, the flat surface of the…

A straight line L1 has the equation y = k(x - 1), where k is some real number. The straight line L1 intersects another straight line L2 at the point (5,8). If L2 has a slope of 1, which of the following is definitely FALSE?

An industrial robot manufacturing company is tasked to design humanoid robots to be used in warehouses where the robots need to pick items from a stack of shelves. The height of the topmost shelf from the ground is 7 feet. To operate, the robot has to move on a track, running par…

Ramesh bought a mobile from a local store. He paid 1/6 of the price via UPI and 1/3 of the price via cash. He agreed to pay the balance amount a year later. While paying back the balance amount, Ramesh paid 10% interest on the balance amount. If the interest paid was Rs. 6000, wh…

There are five dustbins along a circular path at different places. Ramesh takes multiple rounds of the path every morning, always at the same speed. He noticed that it took him a different number of steps to walk between any two consecutive dustbins. Ramesh also noticed that star…

Adu and Amu have bought two pieces of land on the Moon from an e-store. Both the pieces of land have the same perimeters, but Adu's piece of land is in the shape of a square, while Amu's piece of land is in the shape of a circle. The ratio of the areas of Adu's piece of land to A…

For how many distinct real values of x does the equation below hold true? (Consider a > 0.) x²·logₐ(16)/logₐ(32) − logₐ(64)/logₐ(32) − x = 0