IPMAT Indore (MCQ) PYQs

If $f(n)= 1 + 2 + 3 +\cdots+(n+1)$ and $g(n)= \sum_{k=1}^{k=n} \dfrac{1}{f(k)}$, then the least value of $n$ for which $g(n)$ exceeds the value $\dfrac{99}{100}$ is:

Given that $1 + \frac{1}{2^2} + \frac{1}{3^2} + \frac{1}{4^2} + ... = \frac{\pi^2}{6}$, the value of $1 + \frac{1}{3^2} + \frac{1}{5^2} + \frac{1}{7^2} + ...$ is

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?