IPMAT Indore (MCQ) PYQs

If $\frac{1}{1^2} + \frac{1}{2^2} + \frac{1}{3^2} + \ldots$ up to $\infty = \frac{\pi^2}{6}$, then the value of $\frac{1}{1^2} + \frac{1}{3^2} + \frac{1}{5^2} + \ldots$ up to $\infty$ is

A person standing at the centre of an open ground first walks 32 meters towards the east, takes a right turn and walks 16 meters, takes another right turn and walks 8 meters, and so on. How far will the person be from the original starting point after an infinite number of such walks in this pattern?

Let $a_{1}, a_{2}, a_{3}$ be three distinct real numbers in geometric progression. If the equations $a_{1} x ^ 2 + 2a_{2}x + a_{3} = 0$ and $b_{1} x ^ 2 + 2b_{2}x + b_{3} = 0$ have a common root, then which of the following is necessarily true?