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*Simulator***Previous Years JEE Advanced Questions**

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)]**
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]**
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]**
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]**
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]****Download eSaral App for Video Lectures, Complete Revision, Study Material and much more...**

**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

**[JEE(Advanced)-2016, 3(–1)]****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

**[JEE(Advanced 2017, 3(–1)]**
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) $a_{11}+2 a_{10}$

**[JEE(Advanced 2017, 3(–1)]**
Awesome questions

Yes

Really helpful and the solutions are really good and very concise.

Mostly we can use theory of equation in solving these questions. Some objects are misprinted in questions, so kindly revert them to understanding text.

Really good

Really very helpful

pretty good and nice questions

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