JEE Mains · Maths · STD 11 - 4.2 Quadratic equations and inequations
Let \(\alpha, \beta ; \alpha>\beta\), be the roots of the equation \(x^2-\sqrt{2} x-\sqrt{3}=0\). Let \(P_n=\alpha^n-\beta^n, n \in N\). Then \((11 \sqrt{3}-10 \sqrt{2}) \mathrm{P}_{10}+(11 \sqrt{2}+10) \mathrm{P}_{11}-11 \mathrm{P}_{12}\) is equal to :
- A \(10 \sqrt{2} \mathrm{P}_9\)
- B \(10 \sqrt{3} \mathrm{P}_9\)
- C \(11 \sqrt{2} \mathrm{P}_9\)
- D \(11 \sqrt{3} \mathrm{P}_9\)
Answer & Solution
Correct Answer
(B) \(10 \sqrt{3} \mathrm{P}_9\)
Step-by-step Solution
Detailed explanation
\( x^2-\sqrt{2 x}-\sqrt{3}=0\left\langle_\beta^\alpha\right. \) \( \alpha^{n+2}-\sqrt{2} \alpha^{n+1}-\sqrt{3} \alpha^n=0 \) \( \text { and } \beta^{n+2}-\sqrt{2} \beta^{n+1}-\sqrt{3} \beta^n=0\) Subtracting…
See the Complete Solution
Get step-by-step explanations for this and 2.5 Lakh+ more JEE, NEET & CET questions.
- Unlock all solutions
- Practice the full chapter
- Track accuracy across PYQs
4.8 rated on Google Play · 14,000+ reviews
More questions from Maths
- If \(\lim _{x \rightarrow 0} \frac{a x^2 e^x-b \log _e(1+x)+c x e^{-x}}{x^2 \sin x}=1\), then \(16\left(a^2+b^2+c^2\right)\) is equal to ...........JEE Mains 2024 Hard
- If the mirror image of the point \(\mathrm{P}(3,4,9)\) in the line \(\frac{x-1}{3}=\frac{y+1}{2}=\frac{z-2}{1}\) is \((\alpha, \beta, \gamma)\), then \(14(\alpha+\beta+\gamma)\) is :JEE Mains 2024 Hard
- For \(x \in R\) , \(f\left( x \right) = \left| {\log 2 - \sin x} \right|\) and \(g\left( x \right) = f\left( {f\left( x \right)} \right)\) then .. .JEE Mains 2016 Hard
- The derivative of \({\tan ^{ - 1}}\left( {\frac{{\sin \,x - \cos \,x}}{{\sin \,x + \cos \,x}}} \right)\), with respect to \(\frac{x}{2}\), where \(\left( {x \in \left( {0,\frac{\pi }{2}} \right)} \right)\) isJEE Mains 2019 Hard
- Let the mirror image of a circle \(c_{1}: x^{2}+y^{2}-2 x-\) \(6 y+\alpha=0\) in line \(y=x+1\) be \(c_{2}: 5 x^{2}+5 y^{2}+10 g x\) \(+10 f y +38=0\). If \(r\) is the radius of circle \(c _{2}\), then \(\alpha+6 r^{2}\) is equal to\(.....\)JEE Mains 2022 Hard
- If the equation of the plane passing through the point \((1,1,2)\) and perpendicular to the line \(x-3 y+2 z-1=04 x-y+z\) is \(Ax + By + Cz =1\), then \(140( C - B + A )\) is equal to \(.........\).JEE Mains 2023 Hard
More PYQs from JEE Mains
- The value of \(\sum_{n=1}^{100} \int_{n-1}^{n} e^{x-[x]} d x,\) where \([x]\) is the greatest integer \(\leq x ,\) isJEE Mains 2021 Medium
- Let the tangents at the points \(A (4,-11)\) and \(B (8,-5)\) on the circle \(x^2+y^2-3 x+10 y-15=0\), intersect at the point \(C\). Then the radius of the circle, whose centre is \(C\) and the line joining \(A\) and \(B\) is its tangent, is equal toJEE Mains 2023 Hard
- The area (in sq. units) of the region \(\left\{(\mathrm{x}, \mathrm{y}) \in \mathrm{R}^{2} | 4 \mathrm{x}^{2} \leq \mathrm{y} \leq 8 \mathrm{x}+12\right)\) isJEE Mains 2020 Hard
- For \(\alpha, \beta \in \mathrm{R}\) and a natural number \(\mathrm{n}\), let \(A_r=\left|\begin{array}{ccc}r & 1 & \frac{n^2}{2}+\alpha \\ 2 r & 2 & n^2-\beta \\ 3 r-2 & 3 & \frac{n(3 n-1)}{2}\end{array}\right|\). Then \(2 A_{10}-A_8\)JEE Mains 2024 Hard
- The lengths of the sides of a triangle are \(10+ x ^{2}\), \(10+ x ^{2}\) and \(20-2 x ^{2}\). If for \(x = k\), the area of the triangle is maximum, then \(3 K ^{2}\) is equal toJEE Mains 2022 Hard
- Let \(A=\{1,2,3, \ldots .7\}\) and let \(P(1)\) denote the power set of \(A\). If the number of functions \(\mathrm{f}: \mathrm{A} \rightarrow \mathrm{P}(\mathrm{A})\) such that \(a \in \mathrm{f}(\mathrm{a}), \forall \mathrm{a} \in \mathrm{A}\) is \(\mathrm{m}^{\mathrm{n}}, \mathrm{m}\) and \(\mathrm{n} \in \mathrm{N}\) and \(\mathrm{m}\) is least, then \(\mathrm{m}+\mathrm{n}\) is equal to ...........JEE Mains 2024 Hard