(B) Given that $\alpha$ and $\beta$ are the roots of $a x^2+b x+c=0$. $\alpha+\beta = -\frac{b}{a}$ and $\alpha \beta = \frac{c}{a}$. Let the roots of

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(B) Given that $\alpha$ and $\beta$ are the roots of $a x^2+b x+c=0$. $\alpha+\beta = -\frac{b}{a}$ and $\alpha \beta = \frac{c}{a}$. Let the roots of $p x^2+q x+r=0$ be $\gamma = \frac{1-\alpha}{\alpha} = \frac{1}{\alpha}-1$ and $\delta = \frac{1-\beta}{\beta} = \frac{1}{\beta}-1$. Then $\alpha = \frac{1}{1+\gamma}$ and $\beta = \frac{1}{1+\delta}$. Since $\alpha$ is a root of $a x^2+b x+c=0$,we have $a(\frac{1}{1+x})^2 + b(\frac{1}{1+x}) + c = 0$. $a + b(1+x) + c(1+x)^2 = 0$. $a + b + bx + c(1 + 2x + x^2) = 0$. $c x^2 + (b+2c)x + (a+b+c) = 0$. Comparing this with $p x^2+q x+r=0$,we get $r = a+b+c$.

Class 11 Mathematics 2 Complex Numbers and Quadratic Equations: Quadratic Equations Relation between roots and coefficients

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If $\alpha$ and $\beta$ are the roots of the equation $a x^2+b x+c=0$ and if $p x^2+q x+r=0$ has roots $\frac{1-\alpha}{\alpha}$ and $\frac{1-\beta}{\beta}$,then $r$ is equal to

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A
$a+2 b$
B
$a+b+c$
C
$a b+b c+c a$
D
$a b c$

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2 Complex Numbers and Quadratic Equations: Quadratic Equations

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Relation between roots and coefficients

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If the ratio of the roots of the equations $x^2 + bx + c = 0$ and $x^2 + qx + r = 0$ is the same,then:

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Let $f(x) = x^3 - 2x + 2$. If real numbers $a, b, c$ are such that $|f(a)| + |f(b)| + |f(c)| = 0$,then the value of $f^2(a^2 + \frac{2}{a}) + f^2(b^2 + \frac{2}{b}) - f^2(c^2 + \frac{2}{c})$ is equal to:

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Let $\alpha$ and $\beta$ be the roots of the quadratic equation $a x^2+b x+c=0$. Match the conditions in List-$I$ with the corresponding relations in List-$II$.
List-$I$ List-$II$
$(i) \alpha = \beta$ $(A) (ac^2)^{1/3} + (a^2c)^{1/3} + b = 0$
$(ii) \alpha = 2\beta$ $(B) 2b^2 = 9ac$
$(iii) \alpha = 3\beta$ $(C) b^2 = 6ac$
$(iv) \alpha = \beta^2$ $(D) 3b^2 = 16ac$
$(E) b^2 = 4ac$
$(F) (ac^2)^{1/3} + (a^2c)^{1/3} = b$

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If $\alpha, \beta, \gamma$ are the roots of $x^3+p x^2+q x+r=0$,then the value of $(1+\alpha^2)(1+\beta^2)(1+\gamma^2)$ is

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Let $\alpha$ and $\beta$ be two distinct roots of the equation $x^3 + 3x^2 - 1 = 0$. The equation which has $(\alpha \beta)$ as its root is equal to

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List-$I$	List-$II$
$(i) \alpha = \beta$	$(A) (ac^2)^{1/3} + (a^2c)^{1/3} + b = 0$
$(ii) \alpha = 2\beta$	$(B) 2b^2 = 9ac$
$(iii) \alpha = 3\beta$	$(C) b^2 = 6ac$
$(iv) \alpha = \beta^2$	$(D) 3b^2 = 16ac$
	$(E) b^2 = 4ac$
	$(F) (ac^2)^{1/3} + (a^2c)^{1/3} = b$

If $\alpha$ and $\beta$ are the roots of the equation $a x^2+b x+c=0$ and if $p x^2+q x+r=0$ has roots $\frac{1-\alpha}{\alpha}$ and $\frac{1-\beta}{\beta}$,then $r$ is equal to

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If the ratio of the roots of the equations $x^2 + bx + c = 0$ and $x^2 + qx + r = 0$ is the same,then:

Let $f(x) = x^3 - 2x + 2$. If real numbers $a, b, c$ are such that $|f(a)| + |f(b)| + |f(c)| = 0$,then the value of $f^2(a^2 + \frac{2}{a}) + f^2(b^2 + \frac{2}{b}) - f^2(c^2 + \frac{2}{c})$ is equal to:

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