If the roots of x^3+a x^2+b x+c=0 are in arit | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(A) Let the roots of $x^3+a x^2+b x+c=0$ be $\alpha-1, \alpha, \alpha+1$ as the roots are in $AP$ with common difference $1$.Sum of roots $= (\alpha-1

Vedclass · Accepted Answer

$9 c=a(b-2)$

Vedclass · Answer

(A) Let the roots of $x^3+a x^2+b x+c=0$ be $\alpha-1, \alpha, \alpha+1$ as the roots are in $AP$ with common difference $1$.Sum of roots $= (\alpha-1) + \alpha + (\alpha+1) = 3\alpha = -a \Rightarrow \alpha = -\frac{a}{3} \quad \dots(i)$Sum of product of roots taken two at a time $= (\alpha-1)\alpha + \alpha(\alpha+1) + (\alpha-1)(\alpha+1) = b$$\Rightarrow \alpha^2 - \alpha + \alpha^2 + \alpha + \alpha^2 - 1 = b$ $\Rightarrow 3\alpha^2 - 1 = b \quad \dots(ii)$Product of roots $= (\alpha-1)\alpha(\alpha+1) = \alpha(\alpha^2-1) = -c \quad \dots(iii)$Substitute $\alpha = -\frac{a}{3}$ into $(ii)$: $3(-\frac{a}{3})^2 - 1 = b$ $\Rightarrow \frac{a^2}{3} - 1 = b$ $\Rightarrow a^2 = 3(b+1)$.Substitute $\alpha = -\frac{a}{3}$ into $(iii)$: $(-\frac{a}{3})((-\frac{a}{3})^2 - 1) = -c$$\Rightarrow \frac{a}{3}(\frac{a^2}{9} - 1) = c$ $\Rightarrow \frac{a}{3}(\frac{3(b+1)}{9} - 1) = c$$\Rightarrow \frac{a}{3}(\frac{b+1}{3} - 1) = c$ $\Rightarrow \frac{a}{3}(\frac{b-2}{3}) = c$ $\Rightarrow 9c = a(b-2)$.

If the roots of $x^3+a x^2+b x+c=0$ are in arithmetic progression with common difference $1$,then

Similar Questions

The series of positive multiples of $3$ is divided into sets: $\{3\}, \{6, 9, 12\}, \{15, 18, 21, 24, 27\}, \ldots$. Then the sum of the elements in the $11^{\text{th}}$ set is equal to $................$

If the roots of the equation $x^3+3px^2+3qx-8=0$ are in an arithmetic progression,then $2p^3-3pq=$

If $\frac{3 + 5 + 7 + \dots \text{ to } n \text{ terms}}{5 + 8 + 11 + \dots \text{ to } 10 \text{ terms}} = 7$,then the value of $n$ is

If the $p^{th}$ term of an arithmetic progression is $q$ and the $q^{th}$ term is $p$,then its $n^{th}$ term is:

If for an arithmetic progression $S_{2n} = 2S_n$,then $S_{3n} / S_n = \dots$

Vedclass Test Series

Exam Paper Generator

Online Exam Module