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 median value in the sequence a, b, c, d 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

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 ________.

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 curve represented by the equation \(\frac{x^2}{\sin\sqrt{2}-\sin\sqrt{3}}+\frac{y^2}{\cos\sqrt{2}-\cos\sqrt{3}}=1\)

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

The number of four-digit integers which are greater than 1000 and divisible by both 2 and 3, but not by 5, 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

Ayesha is standing atop a vertical tower 200m high and observes a car moving away from the tower on a straight horizontal road from the foot of the tower. At 11:00 AM, she observes the angle of depression of the car to be 45°. At 11:02 AM, she observes the angle of depression of…