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Q. The smallest value of $\mathrm{k},$ for which both the roots of the equation, $\mathrm{x}^{2}-8 \mathrm{kx}+16\left(\mathrm{k}^{2}-\mathrm{k}+1\right)=0$ are real, distinct and have values at least $4,$ is

[JEE 2009, 4 (–1)]

Sol. 2

Q. Let p and q be real numbers such that $p \neq 0, p^{3} \neq q$ and $p^{3} \neq-q .$ If $\alpha$ and $\beta$ are nonzero complex numbers satisfying $\alpha+\beta=-p$ and $\alpha^{3}+\beta^{3}=q$, then a quadratic equation having $\frac{\alpha}{\beta}$ and $\frac{\beta}{\alpha}$ as its roots is

(A) $\left(p^{3}+q\right) x^{2}-\left(p^{3}+2 q\right) x+\left(p^{3}+q\right)=0$

(B) $\left(\mathrm{p}^{3}+\mathrm{q}\right) \mathrm{x}^{2}-\left(\mathrm{p}^{3}-2 \mathrm{q}\right) \mathrm{x}+\left(\mathrm{p}^{3}+\mathrm{q}\right)=0$

(C) $\left(\mathrm{p}^{3}-\mathrm{q}\right) \mathrm{x}^{2}-\left(5 \mathrm{p}^{3}-2 \mathrm{q}\right) \mathrm{x}+\left(\mathrm{p}^{3}-\mathrm{q}\right)=0$

(D) $\left(p^{3}-q\right) x^{2}-\left(5 p^{3}+2 q\right) x+\left(p^{3}-q\right)=0$

[JEE 2010, 3]

Sol. (B)

Q. Let $\alpha$ and $\beta$ be the roots of $\mathrm{x}^{2}-6 \mathrm{x}-2=0,$ with $\alpha>\beta .$ If $\mathrm{a}_{\mathrm{n}}=\alpha^{\mathrm{n}}-\beta^{\mathrm{n}}$ for $\mathrm{n} \geq 1,$ then the value of $\frac{\mathrm{a}_{10}-2 \mathrm{a}_{8}}{2 \mathrm{a}_{9}}$ is

(A) 1              (B) 2             (C) 3             (D) 4

[JEE 2011]

Sol. (C)

Q. A value of b for which the equations

$\mathrm{x}^{2}+\mathrm{bx}-1=0$

$\mathrm{x}^{2}+\mathrm{x}+\mathrm{b}=0$

have one root in common is –

(A) $-\sqrt{2}$

(B) $-i \sqrt{3}$

(C) $\mathrm{i} \sqrt{5}$

(D) $\sqrt{2}$

[JEE 2011]

Sol. (B)

Q. Let S be the set of all non-zero numbers a such that the quadratic equation $\alpha \mathrm{x}^{2}$ – x + a = 0 has two distinct real roots $\mathrm{x}_{1}$ and $\mathrm{x}_{2}$ satisfying the inequality $\left|x_{1}-x_{2}\right|<1$. Which of the following intervals is(are) a subset(s) of S ?

(A) $\left(-\frac{1}{2},-\frac{1}{\sqrt{5}}\right)$

(B) $\left(-\frac{1}{\sqrt{5}}, 0\right)$

(C) $\left(0, \frac{1}{\sqrt{5}}\right)$

(D) $\left(\frac{1}{\sqrt{5}}, \frac{1}{2}\right)$

[JEE 2015, 4M, –0M]

Sol. (A,D)

Q. Let $-\frac{\pi}{6}<\theta<-\frac{\pi}{12}$. Suppose $\alpha_{1}$ and $\beta_{1}$ are the roots of the equation $\mathrm{x}^{2}-2 \mathrm{xsec} \theta+1=0$ and $\alpha_{2}$ and $\beta_{2}$ are the roots of the equation $\mathrm{x}^{2}+2 \mathrm{xtan} \theta-1=0$. If $\alpha_{1}>\beta_{1}$ and $\alpha_{2}>\beta_{2},$ then $\alpha_{1}+\beta_{2}$ equals

(A) $2(\sec \theta-\tan \theta)$

(B) $2 \sec \theta$

(C) $-2 \tan \theta$

(D) 0

Sol. (C)

PARAGRAPH 2

Let p,q be integers and let $\alpha, \beta$ be the roots of the equation, $x^{2}-x-1=0,$ where $\alpha \neq \beta .$ For

$n=0,1,2, \ldots . .,$ let $a_{n}=p \alpha^{n}+q \beta^{n}$.

FACT : If a and b are rational numbers and $a+b \sqrt{5}=0,$ then $a=0=b$.

Q. If $a_{4}=28,$ then $p+2 q=$

(A) 14 (B) 7 (C) 12 (D) 21

Sol. (C)

Q. $\mathrm{a}_{12}=$

(A) $2 \mathrm{a}_{11}+\mathrm{a}_{10}$

(B) $a_{11}-a_{10}$

(C) $\mathrm{a}_{11}+\mathrm{a}_{10}$

(D) $a_{11}+2 a_{10}$