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# Quadratic equation assignement

Q1. If $\alpha$ and $\beta$ be two roots of the equation $x^{2}-64 x+256=0$. Then the value of $\left(\frac{\alpha^{3}}{\beta^{5}}\right)^{\frac{1}{8}}+\left(\frac{\beta^{3}}{\alpha^{5}}\right)^{\frac{1}{8}}$ is

Q2. If roots of this equation $a(b-c) x^{2}+b(c-a) x+c(a-b)=0$, (where $\left.a, b, c \in Q\right)$ are equal then a, b, and c are in

(1) AP

(2) GP

(3) HP

(4) none of these

Q3. If quadratic $f(x)=a x^{2}+b x+c,(a, b, c \in R)$ has non-real roots and $(4 a-2 b+c)<0$ then correct statement is/are:
(i) $(a+b+c)<0$
(ii) $a.c<0$
(iii) $a x^{2}+b x+c<0$ for every real $^{\prime} x^{\prime}$.
Q4. A quadratic function $f(x)$ satisfies $f(x) \geq 0$ for all real $x$. If $f(2)=0$ and $f(4)=12$ then the value of $f(6)$ is:
Q5. If the roots of $2 x^{2}+(4 m+1) x+2(2 m-1)=0$ are reciprocals of each other, $\mathrm{m}=$
Q6. Find the equation whose roots are twice the roots of the equation $3 x^{2}-7 x+4=0$
(A) $3 x^{2}-14 x+8=0$
(B) $3 x^{2}+14 x+16=0$
(C) $3 x^{2}+14 x-16=0$
(D) $3 x^{2}-14 x+16=0$
Q7. If one real root of the quadratic equation $81{x}^{2}+{kx}+256=0$ is cube of the other root, then a value of k is:
(A) -81
(B) 100
(C) 144
(D) -300

Q8. If both the roots of the quadratic equation $x^{2}-m x+4=0$ are real and distinct and they lie in the interval [1,5], then m lies in the interval :
(A) (4,5)
(B) (3,4)
(C) (5,6)
(D) (-5,-4)

Q9. The number of all possible positive integral values of $\alpha$ for which the roots of the quadratic equation, $6 x^{2}-11 x+\alpha=0$ are rational numbers is:

Q10. The least positive value of ‘a’ for which the equation, $2 \mathrm{x}^{2}+(\mathrm{a}-10) \mathrm{x}+\frac{33}{2}=2 \mathrm{a}$ has real roots is

Q11. Let $a, b \in R, a \neq 0$ be such that the equation, $a x^{2}-2 b x+5=0$ has a repeated root $\alpha$, which is also a root of the equation, $x^{2}-2 b x-10=0$. If $\beta$ is the other root of this equation, then $\alpha^{2}+\beta^{2}$ is equal to

Q12. If $\alpha$ and $\beta$ are the roots of the equation $x^{2}+p x+2=0$ and $\frac{1}{\alpha}$ and $\frac{1}{\beta}$ are the roots of the equation $2 \mathrm{x}^{2}+2 \mathrm{qx}+1=0$, then $\left(\alpha-\frac{1}{\alpha}\right)\left(\beta-\frac{1}{\beta}\right)\left(\alpha+\frac{1}{\beta}\right)\left(\beta+\frac{1}{\alpha}\right)$ is equal to :

(A) $\frac{9}{4}\left(9+\mathrm{p}^{2}\right)$
(B) $\frac{9}{4}\left(9+q^{2}\right)$
(C) $\frac{9}{4}\left(9-\mathrm{p}^{2}\right)$
(D) $\frac{9}{4}\left(9-q^{2}\right)$

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