Consider a right-angled triangle ABC, right angled at B. Two circles, each of radius r, are drawn inside the triangle in such a way that one of them touches AB and BC, while the other one touches AC and BC. The two circles also touch each other (see the image below). If AB = 18 cm and BC = 24 cm, then find the value of r. [Figure: see page 11 of source PDF]

Consider a 4-digit number of the form abbb, i.e., the first digit is a (a > 0) and the last three digits are all b. Which of the following conditions is both NECESSARY and SUFFICIENT to ensure that the 4-digit number is divisible by a?

A king has distributed all his rare jewels in three boxes. The first box contains 1/3 of the rare jewels, while the second box contains k/5 of the rare jewels, for some positive integer value of k. The third box contains 66 rare jewels. How many rare jewels does the king have?

Consider the system of two linear equations as follows: 3x + 21y + p = 0; and qx + ry – 7 = 0, where p, q, and r are real numbers. Which of the following statements DEFINITELY CONTRADICTS the fact that the lines represented by the two equations are coinciding?

From the passage: Aman has come to the market with Rs. 100. If he buys 5 kilograms of ca...

Aman decides to buy only onion, in whatever maximum quantity possible (in positive integer kilogram), with the money he has come to the market with. How much money will he be left with after the purchase?

A local restaurant has 16 vegetarian items and 9 non-vegetarian items in their menu. Some items contain gluten, while the rest are gluten-free. One evening, Rohit and his friends went to the restaurant. They planned to choose two different vegetarian items and three different non…

The roots of the polynomial P(x) = 2x³ − 11x² + 17x + 6 are the radii of three concentric circles. The ratio of their area, when arranged from the largest to the smallest, is:

From the passage: 41 applicants have been shortlisted for interviews for some data analy...

What BEST can be concluded about the value of x?

The least common multiple of a number and 990 is 6930. The greatest common divisor of that number and 550 is 110. What is the sum of the digits of the least possible value of that number?