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

Gita starts from point A and walks 1000 m east. She then walks 800 m north, followed by 640 m west and 512 m south, reaching point B. After this, she continues moving in the same cyclic order: east, north, west, south, with each successive movement 20% shorter than the previous o…

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

The equation \(2^x - x^2 = 0\) has

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…

In a class, 25% of all students read news from the Internet. Moreover, 45% of all students read news from printed newspaper. Further, 20% of all students read news from both the Internet and printed newspaper and they do not play video games. It is also known that 30% of the stud…

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

If \(x\) is a real number such that \(\max(\min(x, 2-x), x-4, 2x-8) = \pi - 3\), then the number of possible values of \(x\) is

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