JEE Mains · Maths · STD 11 - 4.1 complex nubers
Let \(\alpha, \beta\) be the roots of the quadratic equation \(x^2+\sqrt{6} x+3=0\). Then \(\frac{\alpha^{23}+\beta^{23}+\alpha^{14}+\beta^{14}}{\alpha^{15}+\beta^{15}+\alpha^{10}+\beta^{10}}\) is equal to
- A \(729\)
- B \(72\)
- C \(81\)
- D \(9\)
Answer & Solution
Correct Answer
(C) \(81\)
Step-by-step Solution
Detailed explanation
\(\alpha, \beta=\frac{-\sqrt{6} \pm \sqrt{6-12}}{2}=\frac{-\sqrt{6} \pm \sqrt{6} i }{2}\) \(=\sqrt{3} e ^{ \pm \frac{3 \pi i }{4}}\) Required expression…
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 \(\alpha\) and \(\beta\) be the coefficients of \(x^{4}\) and \(x^{2}\) respectively in the expansion of \((\mathrm{x}+\sqrt{\mathrm{x}^{2}-1})^{6}+(\mathrm{x}-\sqrt{\mathrm{x}^{2}-1})^{6}\), thenJEE Mains 2020 Hard
- If the system of equations
\(\begin{aligned} & (\lambda-1) x+(\lambda-4) y+\lambda z=5 \\ & \lambda x+(\lambda-1) y+(\lambda-4) z=7 \\ & (\lambda+1) x+(\lambda+2) y-(\lambda+2) z=9\end{aligned}\)
has infinitely many solutions, then \(\lambda^2+\lambda\) is equal toJEE Mains 2025 Easy - Let \(f : R \rightarrow R\) be continuous function satisfying \(f ( x )+ f ( x + k )= n\), for all \(x \in R\) where \(k >0\) and \(n\)is a positive integer. If \(I _{1}=\int\limits_{0}^{4 n k} f ( x ) dx\) and \(I _{2}=\int\limits_{- k }^{3 k } f ( x ) dx\), thenJEE Mains 2022 Hard
- The vertices B and C of a triangle ABC lie on the line \(\frac{x}{1}=\frac{1-y}{-2}=\frac{z-2}{3}\). The coordinates of A and B are (1, 6, 3) and \((4,9, \alpha)\) respectively and C is at a distance of 10 units from B. The area (in sq. units) of \( \Delta ABC \) is:JEE Mains 2026 Medium
- The integral \(\int {x\,{{\cos }^{ - 1}}\,\left( {\frac{{1 - {x^2}}}{{1 + {x^2}}}} \right)dx} \,\left( {x > 0} \right)\) is equal toJEE Mains 2014 Hard
- lf Rolle's theorem holds for the function \(f(x) =2x^3 + bx^2 + cx, x \in [-1, 1],\) at the point \(x = \frac {1}{2},\) then \(2b+ c\) equalsJEE Mains 2015 Medium
More PYQs from JEE Mains
- If \((\sqrt{3}+\mathrm{i})^{100}=2^{99}(\mathrm{p}+\mathrm{i} \mathrm{q})\), then \(\mathrm{p}\) and \(\mathrm{q}\) are roots of the equation :JEE Mains 2021 Hard
- If \(y=\frac{(\sqrt{x}+1)\left(x^2-\sqrt{x}\right)}{x \sqrt{x}+x+\sqrt{x}}+\frac{1}{15}\left(3 \cos ^2 x-5\right) \cos ^3 x\), then \(96 y^{\prime}\left(\frac{\pi}{6}\right)\) is equal to :JEE Mains 2024 Hard
- Let \([\mathrm{x}]\) denote the greatest integer less than or equal to \(\mathrm{x}\). Then, the values of \(x \in R\) satisfying the equation \(\left[e^{x}\right]^{2}+\left[e^{x}+1\right]-3=0\) lie in the interval:JEE Mains 2021 Hard
- The area (in sq. units) of the smaller portion enclosed between the curves, \(x^2 + y^2 = 4\) and \(y^2 =3x\), isJEE Mains 2017 Hard
- Let the line L: \(\frac{ x -1}{2}=\frac{ y +1}{-1}=\frac{ z -3}{1}\) intersect the plane \(2 x+y+3 z=16\) at the point \(P.\) Let the point \(Q\) be the foot of perpendicular from the point \(R(1,-1,-3)\) on the line \(L\). If \(\alpha\) is the area of triangle \(PQR.\) then \(\alpha^2\) is equal to \(...........\).JEE Mains 2023 Hard
- Let \(\vec{a}=\alpha \hat{i}+\hat{j}+\beta \hat{k}\) and \(\vec{b}=3 \hat{i}-5 \hat{j}+4 \hat{k}\) be two vectors, such that \(\vec{a} \times \vec{b}=-\hat{i}+9 \hat{i}+12 k\). Then the projection of \(\vec{b}-2 \vec{a}\) on \(\vec{b}+\vec{a}\) is equal to.JEE Mains 2022 Hard