1. Tardigrade
  2. Question
  3. Mathematics
  4. The polynomial P ( x )= x 3+ ax 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

Question Error Report

Thank you for reporting, we will resolve it shortly

Back to Question

Q. The polynomial $P ( x )= x ^3+ ax ^2+ bx + c$ 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

Sequences and Series

Solution:

Let A.P. be $a_1, a_2, a_3, \ldots \ldots$ with $d$ as common difference
$\because \frac{10}{2}\left(2 a _1+9 d \right)= c +10 \Rightarrow 2 a _1+9 d =\frac{12}{5}$......(1)
and $\frac{5}{2}\left(2 a_1+8 d\right)=b+17 \Rightarrow 2 a_1+8 d=\frac{12}{5}$.....(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 . $