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Previous Years JEE Advanced Questions
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. 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)]
Ans. 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]
Ans. (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]
Ans. (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]
Ans. (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]
Ans. (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)]
Ans. (C)
Q. If $a_{4}=28,$ then $p+2 q=$
(A) 14 (B) 7 (C) 12 (D) 21
[JEE(Advanced 2017, 3(–1)]
Ans. (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) $a_{11}+2 a_{10}$
[JEE(Advanced 2017, 3(–1)]
Ans. (C)
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July 25, 2021, 10:59 a.m.
Sir one doubt if e = q/eo = [E] = rsinΦdΦ = q/eo = 4πr*2[e] = r = 14631m = 1/4π(14631)*8.85×10*-9f/m = 42.00479
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Vinit jangir
June 12, 2021, 2:29 p.m.
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Keshav
May 11, 2021, 9:31 p.m.
Last one paragraph is best. But i am disappointed because there was only 7 or 8 question 😖.
But thanks for that