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4-2.Quadratic Equations and Inequations
normal
If $\alpha ,\beta,\gamma$ are the roots of equation $x^3 + 2x -5 = 0$ and if equation $x^3 + bx^2 + cx + d = 0$ has roots $2 \alpha + 1, 2 \beta + 1, 2 \gamma + 1$ , then value of $|b + c + d|$ is (where $b,c,d$ are coprime)-
A
$41$
B
$39$
C
$40$
D
$43$
Solution
Let $2 \alpha+1=x \Rightarrow \alpha=\frac{x-1}{2}$
$\therefore $ required equation is $\left(\frac{\mathrm{x}-1}{2}\right)^{3}+\mathrm{x}-1-5=0$
$\Rightarrow x^{3}-3 x^{2}+3 x-1+8 x-98=0$
$\Rightarrow x^{3}-3 x^{2}+11 x-49=0$
$\therefore \mathrm{b}=-3, \mathrm{c}=11, \mathrm{d}=-49$
$\Rightarrow|b+c+d|=41$.
Standard 11
Mathematics
