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:

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?

There are numbers $a_1, a_2, a_3, \ldots, a_n$ each of them being $+1$ or $-1$. If it is known that $a_1 a_2 + a_2 a_3 + a_3 a_4 + \ldots a_{n-1} a_n + a_n a_1 = 0$ then