If α, β are roots of equation x^{2}+5 √{2} x+10=0, α,>,β and P

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4-2.Quadratic Equations and Inequations

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If $\alpha, \beta$ are roots of the equation $x^{2}+5 \sqrt{2} x+10=0, \alpha\,>\,\beta$ and $P_{n}=\alpha^{n}-\beta^{n}$ for each positive integer $\mathrm{n}$, then the value of $\left(\frac{P_{17} P_{20}+5 \sqrt{2} P_{11} P_{19}}{P_{18} P_{19}+5 \sqrt{2} P_{18}^{2}}\right)$ is equal to $....$

$4$

$3$

$2$

$1$

(JEE MAIN-2021)

Solution

$x^{2}+5 \sqrt{2} x+10=0$

$\& p_{n}=\alpha^{n}-\beta^{n} \text { (Given) }$

$\text { Now } \frac{P_{17} P_{20}+5 \sqrt{2} p_{11} P_{19}}{P_{18} P_{19}+5 \sqrt{2} P_{18}^{2}}=\frac{P_{17}\left(P_{20} 5 \sqrt{2} P_{19}\right)}{P_{18}\left(P_{19}+5 \sqrt{2 P}_{18}\right)}$

$\frac{P_{17}\left(\alpha^{20}-\beta^{20}+5 \sqrt{2}\left(\alpha^{19}-\beta^{19}\right)\right)}{P_{18}\left(\alpha^{19}-\beta^{19}+5 \sqrt{2}\left(\alpha^{18}-\beta^{18}\right)\right)}$

$\frac{P_{17}\left(\alpha^{19}(\alpha+5 \sqrt{2})-\beta^{19}(\beta+5 \sqrt{2})\right)}{P_{18}\left(\alpha^{18}(\alpha+5 \sqrt{2})-\beta^{18}(\beta+5 \sqrt{2})\right)}$

$\text { Since } \alpha+5 \sqrt{2}=-10 / \alpha \ldots \ldots \ldots \ldots . .(1)$

$\&\, \beta+5 \sqrt{2}=-10 / \beta \ldots \ldots \ldots \ldots(2)$

Now put there values in above expression

$=-\frac{10 P_{1} P_{18}}{-10 P_{18} P_{17}}=1$

Standard 11

Mathematics

Let $\mathrm{S}=\left\{x \in R:(\sqrt{3}+\sqrt{2})^x+(\sqrt{3}-\sqrt{2})^x=10\right\}$. Then the number of elements in $\mathrm{S}$ is :

hard

(JEE MAIN-2024)

If $x+\frac{1}{x}=a, x^2+\frac{1}{x^3}=b$, then $x^3+\frac{1}{x^2}$ is

hard

(KVPY-2011)

Below are four equations in $x$. Assume that $0 < r < 4$. Which of the following equations has the largest solution for $x$ ?

normal

(KVPY-2011)

Let $a$ ,$b$, $c$ , $d$ , $e$ be five numbers satisfying the system of equations

$2a + b + c + d + e = 6$
$a + 2b + c + d + e = 12$
$a + b + 2c + d + e = 24$
$a + b + c + 2d + e = 48$
$a + b + c + d + 2e = 96$ ,

then $|c|$ is equal to

normal

Consider the cubic equation $x^3+c x^2+b x+c=0$ where $a, b, c$ are real numbers. Which of the following statements is correct?

normal

(KVPY-2011)

If $\alpha, \beta$ are roots of the equation $x^{2}+5 \sqrt{2} x+10=0, \alpha\,>\,\beta$ and $P_{n}=\alpha^{n}-\beta^{n}$ for each positive integer $\mathrm{n}$, then the value of $\left(\frac{P_{17} P_{20}+5 \sqrt{2} P_{11} P_{19}}{P_{18} P_{19}+5 \sqrt{2} P_{18}^{2}}\right)$ is equal to $....$

Solution

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Let $a$ ,$b$, $c$ , $d$ , $e$ be five numbers satisfying the system of equations

$2a + b + c + d + e = 6$
$a + 2b + c + d + e = 12$
$a + b + 2c + d + e = 24$
$a + b + c + 2d + e = 48$
$a + b + c + d + 2e = 96$ ,

then $|c|$ is equal to