(B) Let the roots of the equation $x^2 - 30x + p = 0$ be $\alpha$ and $\alpha^2$.From the relation between roots and coefficients,the sum of roots is

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(B) Let the roots of the equation $x^2 - 30x + p = 0$ be $\alpha$ and $\alpha^2$.From the relation between roots and coefficients,the sum of roots is $\alpha + \alpha^2 = 30$ and the product of roots is $\alpha \cdot \alpha^2 = \alpha^3 = p$.Solving $\alpha^2 + \alpha - 30 = 0$:$(\alpha + 6)(\alpha - 5) = 0$,which gives $\alpha = -6$ or $\alpha = 5$.If $\alpha = 5$,then $p = \alpha^3 = 5^3 = 125$.If $\alpha = -6$,then $p = \alpha^3 = (-6)^3 = -216$.Thus,the values of $p$ are $125$ and $-216$.

Class 11 Mathematics 4-2.Quadratic Equations and Inequations Relation between roots and coefficients

Medium

The values of $p$ for which one root of the equation $x^2 - 30x + p = 0$ is the square of the other are:

A
$125$ only
B
$125$ and $-216$
C
$125$ and $215$
D
$216$ only

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4-2.Quadratic Equations and Inequations

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Relation between roots and coefficients

If $\sin 2\theta$ and $\cos 2\theta$ are solutions of $x^2 + ax - c = 0$,then

MediumTS EAMCET 2023

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$\alpha, \beta, \gamma$ are the roots of the equation $x^3-10 x^2+7 x+8=0$. Match the following and choose the correct answer.
$A. \alpha + \beta + \gamma$ $(1) -\frac{43}{4}$
$B. \alpha^2 + \beta^2 + \gamma^2$ $(2) -\frac{7}{8}$
$C. \frac{1}{\alpha} + \frac{1}{\beta} + \frac{1}{\gamma}$ $(3) 86$
$D. \frac{\alpha}{\beta \gamma} + \frac{\beta}{\gamma \alpha} + \frac{\gamma}{\alpha \beta}$ $(4) 0$
$(5) 10$

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For the quadratic equation $2x^2 - 2(p - 2)x - p - 1 = 0$,what should be the value of $p$ such that the sum of the squares of its roots is minimized?

Medium

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If the sum of the squares of the roots of the equation $x^2-(a-2)x-(a+1)=0$ is least for an appropriate value of the variable parameter $a$, then the value of $a$ will be

MediumWBJEE 2025

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If $\alpha, \beta, \gamma$ are the roots of the equation $x^3 + x + 1 = 0$,then the value of $\alpha^3 \beta^3 \gamma^3$ is:

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$A. \alpha + \beta + \gamma$	$(1) -\frac{43}{4}$
$B. \alpha^2 + \beta^2 + \gamma^2$	$(2) -\frac{7}{8}$
$C. \frac{1}{\alpha} + \frac{1}{\beta} + \frac{1}{\gamma}$	$(3) 86$
$D. \frac{\alpha}{\beta \gamma} + \frac{\beta}{\gamma \alpha} + \frac{\gamma}{\alpha \beta}$	$(4) 0$
	$(5) 10$

The values of $p$ for which one root of the equation $x^2 - 30x + p = 0$ is the square of the other are:

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