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

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 ________.

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

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…

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…

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 one of the factors of the number \(3^7\, 2^8\, 17^3\) is randomly chosen, then the probability that the chosen factor will be a perfect square is.

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 value of k for which the following lines x-y-1 = 0 2x+3y-12 = 0 2x-3y+k = 0 are concurrent is