In a right-angled triangle ABC, the hypotenuse AC is of length 13 cm. A line drawn connecting the midpoints D and E of sides AB and AC is found to be 6 cm in length. The length of BC is

Let A and B be two sets such that the Cartesian product \(A \times B\) consists of four elements. If two elements of \(A \times B\) are (1, 4) and (4, 1), then

In a bowl containing 60 ml orange juice, 40 ml of water is poured. Thereafter, 100 ml of apple juice is poured to make a fruit punch. Madhu drinks 50 ml of this fruit punch and comments that the proportion of orange juice needs to be higher for better taste. How much orange juice…

The set of all possible values of f(x) for which \((81)^x + (81)^{f(x)} = 3\) is

In a room, there are n persons whose average height is 160 cm. If m more persons, whose average height is 172 cm, enter the room, then the average height of all persons in the room becomes 164 cm. Then m : n is

In a 400-metre race, Ashok beats Bipin and Chandan respectively by 15 seconds and 25 seconds. If Ashok beats Bipin by 150 metres, by how many metres does Bipin beat Chandan in the race?

For \(0 < \theta < \frac{\pi}{4}\), let \(a = ((\sin\theta)^{\sin\theta})(\log_2 \cos\theta)\), \(b = ((\cos\theta)^{\sin\theta})(\log_2 \sin\theta)\), \(c = ((\sin\theta)^{\cos\theta})(\log_2 \cos\theta)\) and \(d = ((\sin\theta)^{\sin\theta})(\log_2 \sin\theta)\). Then, the med…

When the square of the difference of two natural numbers is subtracted from the square of the sum of the same two numbers and the result is divided by four, we get

If \(f(x^2 + f(y)) = xf(x) + y\) for all non-negative integers x and y, then the value of \([f(0)]^2 + f(0)\) equals ________.