Let $f(x)$ be a quadratic polynomial. If $f(1) = f(-1)$ and $a, b, c$ are in arithmetic progression,then $f'(a), f'(b), f'(c)$ are in..... progression.

If twice the $11^{th}$ term of an Arithmetic Progression is equal to seven times its $21^{st}$ term,then its $25^{th}$ term is .......

If the average of the first $n$ numbers in the sequence $148, 146, 144, \ldots$ is $125$,then $n$ is equal to

Let $S_n$ denote the sum of the first $n$ terms of an arithmetic progression. If $S_{20} = 790$ and $S_{10} = 145$,then $S_{15} - S_5$ is:

If the roots of the equation $6x^3-11x^2+6x-1=0$ are in harmonic progression,then the roots of $x^3-6x^2+11x-6=0$ will be in

Let $S = \{(a, b, c) \in \mathbb{N} \times \mathbb{N} \times \mathbb{N} : a+b+c=21, a \leq b \leq c\}$ and $T = \{(a, b, c) \in \mathbb{N} \times \mathbb{N} \times \mathbb{N} : a, b, c \text{ are in } AP\}$, where $\mathbb{N}$ is the set of all natural numbers. Then, the number of elements in the set $S \cap T$ is: