JEE Mains · Maths · STD 11 - 4.2 Quadratic equations and inequations
If \(\alpha, \beta\) are the roots of the equation, \(x^2-x-1=0\) and \(S_n=2023 \alpha^n+2024 \beta^n\), then
- A \(2 \mathrm{~S}_{12}=\mathrm{S}_{11}+\mathrm{S}_{10}\)
- B \(\mathrm{S}_{12}=\mathrm{S}_{11}+\mathrm{S}_{10}\)
- C \(2 \mathrm{~S}_{11}=\mathrm{S}_{12}+\mathrm{S}_{10}\)
- D \(\mathrm{S}_{11}=\mathrm{S}_{10}+\mathrm{S}_{12}\)
Answer & Solution
Correct Answer
(B) \(\mathrm{S}_{12}=\mathrm{S}_{11}+\mathrm{S}_{10}\)
Step-by-step Solution
Detailed explanation
\({x}^2-\mathrm{x}-1=0 \) \(\mathrm{~S}_{\mathrm{n}}=2023 \alpha^{\mathrm{n}}+2024 \beta^{\mathrm{n}} \) \( \mathrm{S}_{\mathrm{n}-1}+\mathrm{S}_{\mathrm{n}-2}=2023 \alpha^{\mathrm{n}=1}+2024 \beta^{\mathrm{n}-1}+2023 \alpha^{\mathrm{n}-2}+2024 \beta^{\mathrm{n}-2} \)…
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
- Let \(L_1, L_2\) be the lines passing through the point \(\mathrm{P}(0,1)\) and touching the parabola \(9 x^2+12 x+18 y-14=0\). Let \(Q\) and \(R\) be the points on the lines \(\mathrm{L}_1\) and \(\mathrm{L}_2\) such that the \(\triangle \mathrm{PQR}\) is an isosceles triangle with base \(\mathrm{QR}\). If the slopes of the lines \(Q R\) are \(m_1\) and \(m_2\). then \(16\left(m_1^2+m_2^2\right)\) is equal to ..............JEE Mains 2024 Hard
- Let \(S\, = \,\left\{ {\theta \, \in \,[ - \,2\,\pi ,\,\,2\,\pi ]\, :\,2\,{{\cos }^2}\,\theta \, + \,3\,\sin \,\theta \, = \,0} \right\}\). Then the sum of the elements of \(S\) isJEE Mains 2019 Hard
- If \( X=\begin{bmatrix}x\\ y\\ z\end{bmatrix} \) is a solution of the system of equations \( AX=B \) where \( adj A=\begin{bmatrix}4&2&2\\ -5&0&5\\ 1&-2&3\end{bmatrix} \) and \( B=\begin{bmatrix}4\\ 0\\ 2\end{bmatrix}, \) then \( |x+y+z| \) is equal to:JEE Mains 2026 Easy
- A straight the through a fixed point \((2, 3)\) intersects the coordinate axes at distinct points \(P\) and \(Q.\) If \(O\) is the origin and the rectangle \(OPRQ\) is completed, then the locus of \(R\) is:JEE Mains 2018 Hard
- A line passes through \(A(4,-6,-2)\) and \(B(16,-2,4)\). The point \(\mathrm{P}(\mathrm{a}, \mathrm{b}, \mathrm{c})\) where \(\mathrm{a}, \mathrm{b}, \mathrm{c}\) are non-negative integers, on the line \(\mathrm{AB}\) lies at a distance of 21 units, from the point \(\mathrm{A}\). The distance between the points \(\mathrm{P}(\mathrm{a}, \mathrm{b}, \mathrm{c})\) and \(\mathrm{Q}(4,-12,3)\) is equal to ...........JEE Mains 2024 Medium
- Let \(z_{1}\) and \(z_{2}\) be two complex numbers such that \(\arg \left(\mathrm{z}_{1}-\mathrm{z}_{2}\right)=\frac{\pi}{4}\) and \(\mathrm{z}_{1}, \mathrm{z}_{2}\) satisfy the equation \(|z-3|=\operatorname{Re}(z) .\) Then the imaginary part of \(z_{1}+z_{2}\) is equal to ..... .JEE Mains 2021 Hard
More PYQs from JEE Mains
- The area of the region bounded by the parabola \((y-2)^{2}=(x-1)\), the tangent to it at the point whose ordinate is \(3\) and the \(\mathrm{x}\)-axis is :JEE Mains 2021 Hard
- Let \(\mathrm{f}(\mathrm{x})=\cos \left(2 \tan ^{-1} \sin \left(\cot ^{-1} \sqrt{\frac{1-\mathrm{x}}{\mathrm{x}}}\right)\right)\) \(0<\mathrm{x}<1\). Then :JEE Mains 2021 Hard
- If the mean and variance of eight numbers \(3,7,9,12,13,20, x\) and \(y\) be \(10\) and \(25\) respectively, then \(\mathrm{x} \cdot \mathrm{y}\) is equal toJEE Mains 2020 Hard
- Let \(\mathrm{C}\) be the centroid of the triangle with vertices \((3,-1),(1,3)\) and \((2,4) .\) Let \(P\) be the point of intersection of the lines \(x+3 y-1=0\) and \(3 \mathrm{x}-\mathrm{y}+1=0 .\) Then the line passing through the points \(\mathrm{C}\) and \(\mathrm{P}\) also passes through the pointJEE Mains 2020 Hard
- \(\lim _{n \rightarrow \infty} \frac{\left(1^2-1\right)(n-1)+\left(2^2-2\right)(n-2)+\ldots .+\left((n-1)^2-(n-1)\right) \cdot 1}{\left(1^3+2^3+\ldots .+n^3\right)-\left(1^2+2^2+\ldots . .+n^2\right)}\) is equal to:JEE Mains 2024 Hard
- Let for \(A=\left[\begin{array}{lll}1 & 2 & 3 \\ a & 3 & 1 \\ 1 & 1 & 2\end{array}\right],|A|=2\). If \(|2 \operatorname{adj}(2 \operatorname{adj}(2 A ))|\) \(=32^{ n }\), then \(3 n +\alpha\) is equal toJEE Mains 2023 Hard