Bodhee Prep-CAT Online Preparation

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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