JEE Mains · Maths · STD 11 - 4.2 Quadratic equations and inequations
If \(\alpha \) and \( \beta \) are roots of the equation, \({x^2} - 4\sqrt 2\,kx + 2\,{e^{4\ln \,k}} - 1 = 0\) for some \(k\), and \({\alpha ^2} + {\beta ^2} = 66\), then \({\alpha ^3} + {\beta ^3}\) is equal to
- A \(248\sqrt 2 \)
- B \(280\sqrt 2 \)
- C \(-32\sqrt 2 \)
- D \(-280\sqrt 2 \)
Answer & Solution
Correct Answer
(D) \(-280\sqrt 2 \)
Step-by-step Solution
Detailed explanation
\(x^{2}-4 \sqrt{2} k x+2 e^{4 \ln k}-1=0\) or, \(x^{2}-4 \sqrt{2} k x+2 k^{4}-1=0\) \(\alpha+\beta=4 \sqrt{2} k\) and \(\alpha . \beta=2 k^{4}-1\) Squaring both sides, we get \((\alpha+\beta)^{2}=(4 \sqrt{2} k)^{2}\) \(\Rightarrow \alpha^{2}+\beta^{2}+2 \alpha \beta=32 k^{2}\)…
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 A be the focus of the parabola \(y^{2}=8x.\) Let the line \(y=mx+c\) intersect the parabola at two distinct points B and C. If the centroid of the triangle ABC is \((\frac{7}{3},\frac{4}{3})\) , then \((BC)^{2}\) is equal to:JEE Mains 2026 Medium
- The largest \(\mathrm{n} \in \mathrm{N}\) such that \(3^{\mathrm{n}}\) divides 50 ! is:JEE Mains 2025 Easy
- Let \( (2\alpha, \alpha) \) be the largest interval in which the function \( f(t)=\frac{|t+1|}{t^{2}}, t<0 \), is strictly decreasing. Then the local maximum value of the function \( g(x)=2\log_{e}(x-2)+\alpha x^{2}+4x-\alpha, x>2 \), isJEE Mains 2026 Medium
- If \(\mathrm{y}=\mathrm{y}(\mathrm{x})\) is the solution of the differential equation, \(\mathrm{e}^{\mathrm{y}}\left(\frac{\mathrm{dy}}{\mathrm{dx}}-1\right)=\mathrm{e}^{\mathrm{x}}\) such that \(\mathrm{y}(0)=0,\) then \(\mathrm{y}(1)\) is equal toJEE Mains 2020 Hard
- Let \(A(2,3,5)\) and \(C(-3,4,-2)\) be opposite vertices of a parallelogram \(A B C D\) if the diagonal \(\overrightarrow{B D}=\hat{i}+2 \hat{j}+3 \hat{k}\) then the area of the parallelogram is equal toJEE Mains 2024 Medium
- The constant term in the expansion of \(\left(2 x+\frac{1}{x^7}+3 x^2\right)^5 \text { is }........\).JEE Mains 2023 Hard
More PYQs from JEE Mains
- If the line \(x-1=0\), is a directrix of the hyperbola \(kx ^{2}- y ^{2}=6\), then the hyperbola passes through the point.JEE Mains 2022 Medium
- Consider \(10\) observation \(\mathrm{x}_1, \mathrm{x}_2, \ldots, \mathrm{x}_{10}\). such that \(\sum_{i=1}^{10}\left(x_i-\alpha\right)=2\) and \(\sum_{i=1}^{10}\left(x_i-\beta\right)^2=40\), where \(\alpha, \beta\) are positive integers. Let the mean and the variance of the observations be \(\frac{6}{5}\) and \(\frac{84}{25}\) respectively. The \(\frac{\beta}{\alpha}\) is equal to :JEE Mains 2024 Hard
- Let \(O \) be the vertex and \(Q\) be any point on the parabola \({x^2} = 8y\) .If the point \(P\) divides the line segment \(OQ\) internally in the ratio \( 1:3\) , then locus of \(P\) is :JEE Mains 2015 Hard
- Let \(\vec{a}=2 \hat{i}+3 \hat{j}+4 \hat{k}, \vec{b}=2 \hat{i}-2 \hat{j}-2 \hat{k}\) and \(\overrightarrow{ c }=-\hat{ i }+4 \hat{ j }+3 \hat{ k }\). If \(\overrightarrow{ d }\) is a vector perpendicular to both \(\vec{b}\) and \(\overrightarrow{ c }\) and \(\overrightarrow{ a } \cdot \overrightarrow{ d }=18\), Then \(|\overrightarrow{ a } \times \overrightarrow{ d }|^2\) is equal to \(..........\).JEE Mains 2023 Hard
- If \(x \phi(x)=\int_{5}^{x}\left(3 t^{2}-2 \phi^{\prime}(t)\right) d t, x\,>\,-2\), and \(\phi(0)=4\) then \(\phi(2)\) is .... .JEE Mains 2021 Hard
- Let \(M =\left[\begin{array}{cc}0 & -\alpha \\ \alpha & 0\end{array}\right]\), where \(\alpha\) is a non-zero real number an \(N =\sum\limits_{ k =1}^{49} M ^{2 k }\). If \(\left( I - M ^{2}\right) N =-2 I\), then the positive integral value of \(\alpha\) isJEE Mains 2022 Hard