Question

The polynomial $P(x)=x^3+a{x}^2+bx+c$ has the property that the mean of its roots, the product of its roots and the sum of its coefficient are all equal. If the $y$-intercept of the graph of $y=P(x)$ is 2 , then 4
If sum of first 10 terms of an A.P. is $c+10$ and sum of odd numbered terms lying in first 10 terms i s $b+17,$ then a is equal to

• A. sum of first 5 terms
• B. sum of first 10 terms
• C. sum of first 15 terms
• D. sum of first 20 terms

Answer 1

Answer: (A)

Let A.P. be $a_1,a_2,a_3,\dots \dots$ with $d$ as common difference
$\because \frac{10}{2}\left(2a_1+9d\right)=c+10\Rightarrow 2a_1+9d=\frac{12}{5}$......(1)
and $\frac{5}{2}\left(2a_1+8d\right)=b+17\Rightarrow 2a_1+8d=\frac{12}{5}$.....(2)
From (1) and (2) $\Rightarrow d=0$
$\therefore a_1=\frac{6}{5}$
$\therefore$ all terms are equal to $\frac{6}{5}$
$\therefore$ Sum of first five terms $=\frac{6}{5}\times 5=6.$