A circle of non-zero radius has origin as its centre. If it passes through the point of intersection of two curves \(y^2 = 4ax\) and \(x^2 = 4ay\), then its equation is

A person walks one lap along a circle at a speed \(v\). Thereafter, he runs one lap along the boundary of the largest square that can be inscribed in the circle at a speed \(3v\). The ratio of the time he walks to the time he runs is

The possible values of \(x\) in the set \(\{1, 5, 13\}\) for which the mean of eight observations \(5, 8, 3x+2, 15, 27, 29, 36, 5x-2\) equals their median are

A certain number of people contributed to a charity. The first person contributed one rupee. The rule for contribution was that the next person would contribute double the amount already raised. If the total money raised for the charity was 2187 rupees, then the number of people…

Positive reals \(x, y\) satisfy \(x \neq y\) and \(\frac{x^2 + y^2}{xy} = k\). If replacing \(x\) by \(x+y\) and \(y\) by \(|x-y|\) leaves the value of \(k\) unchanged, then \(k\) equals

The approximate value of the expression \(2\log_3 3n - \log_3 (n^2 + 1)\) for a sufficiently large \(n\) is

The number of ways in which coins of denominations of rupees 2, 5 and 10 can be combined to make a value of rupees 50 is

The number of values \(a\) can take such that \(x^4 + ax^3 + (3a-4)x^2 + 2(a-1)x - 4\) can be expressed as a product of two quadratic polynomials, \(x^2 + px + 2\) and \(x^2 + qx - 2\), where \(p\) and \(q\) are real, is

If \(a\), \(b\), and \(c\) are three prime numbers such that \(abc = 23(a+b+c)\), then the maximum possible value of \(a+b-c\) is