JEE Mains · Maths · STD 11 - 4.2 Quadratic equations and inequations
Let \(\alpha, \beta\) be the roots of the equation \(x^2-a x-b=0\) with \(\operatorname{Im}(\alpha) \lt \operatorname{Im}(\beta)\). Let \(P_n=\alpha^n-\beta^n\). If \(\mathrm{P}_3=-5 \sqrt{7} i, \mathrm{P}_4=-3 \sqrt{7} i, \mathrm{P}_5=11 \sqrt{7} i\) and \(\mathrm{P}_6=45 \sqrt{7} i\), then \(\left|\alpha^4+\beta^4\right|\) is equal to __________.
- A 30
- B 31
- C 32
- D 33
Answer & Solution
Correct Answer
(B) 31
Step-by-step Solution
Detailed explanation
\(\begin{aligned} & \alpha+\beta=\mathrm{a} \quad \alpha \beta=-\mathrm{b} \\ & \mathrm{P}_6=\mathrm{aP}_5+\mathrm{bP}_4 \\ & 45 \sqrt{7} \mathrm{i}=\mathrm{a} \times 11 \sqrt{7} \mathrm{i}+\mathrm{b}(-3 \sqrt{7}) \mathrm{i} \\ & 45=11 \mathrm{a}-3 \mathrm{~b} \end{aligned}\)…
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 \(\vec{a}=\hat{i}+\hat{j}+\hat{k}, \vec{b}=3 \hat{i}+2 \hat{j}-\hat{k}, \vec{c}=\lambda \hat{j}+\mu \hat{k}\) and \(\hat{d}\) be a unit vector such that \(\overrightarrow{\mathrm{a}} \times \hat{\mathrm{d}}=\overrightarrow{\mathrm{b}} \times \hat{\mathrm{d}}\) and \(\overrightarrow{\mathrm{c}} \cdot \hat{\mathrm{d}}=1\), If \(\vec{c}\) is perpendicular to \(\vec{a}\), then \(|3 \lambda \hat{d}+\mu \overrightarrow{\mathrm{c}}|^2\) is equal to _______ .JEE Mains 2025 Medium
- Let \(\alpha\) and \(\beta\) be the roots of the equation \(\mathrm{x}^{2}-\mathrm{x}-1=0 .\) If \(\mathrm{p}_{\mathrm{k}}=(\alpha)^{\mathrm{k}}+(\beta)^{\mathrm{k}}, \mathrm{k} \geq 1,\) then which one of the following statements is not true?JEE Mains 2020 Hard
- Let \(X=\{\mathrm{x} \in \mathrm{N}: 1 \leq \mathrm{x} \leq 17\}\) and \(\mathrm{Y}=\{\mathrm{ax}+\mathrm{b}: \mathrm{x} \in \mathrm{X}\) and \(\mathrm{a}, \mathrm{b} \in \mathrm{R}, \mathrm{a}>0\} .\) If mean and variance of elements of \(Y\) are \(17\) and \(216\) respectively then \(a + b\) is equal toJEE Mains 2020 Hard
- If the point of intersections of the ellipse \(\frac{ x ^{2}}{16}+\frac{ y ^{2}}{ b ^{2}}=1\) and the circle \(x ^{2}+ y ^{2}=4 b , b > 4\) lie on the curve \(y^{2}=3 x^{2},\) then \(b\) is equal to:JEE Mains 2021 Hard
- One vertex of a rectangular parallelopiped is at the origin \(O\) and the lengths of its edges along \(x , y\) and \(Z\) axes are \(3,4\) and \(5\) units respectively. Let \(P\) be the vertex \((3,4,5)\). Then the shortest distance between the diagonal \(OP\) and an edge parallel to \(Z\) axis, not passing through \(O\) or \(P\) is:JEE Mains 2023 Hard
- Let \(S_{1}=\left\{z_{1} \in C:\left|z_{1}-3\right|=\frac{1}{2}\right\}\) and \(S_{2}=\left\{z_{2} \in C:\left|z_{2}-\right| z_{2}+1||=\left|z_{2}+\right| z_{2}-1||\right\} . \quad\) Then, for \(z_{1} \in S_{1}\) and \(z_{2} \in S_{2}\), the least value of \(\left|z_{2}-z_{1}\right|\) is.JEE Mains 2022 Hard
More PYQs from JEE Mains
- Let \( f, \mathrm{~g}: \mathrm{R} \rightarrow \mathrm{R}\) be defined as : \(f(\mathrm{x})=|\mathrm{x}-1|\) and \(g(x)=\left\{\begin{array}{cc}\mathrm{e}^{\mathrm{x}}, & \mathrm{x} \geq 0 \\ \mathrm{x}+1, & \mathrm{x} \leq 0\end{array}\right.\). Then the function \(f(\mathrm{~g}(\mathrm{x}))\) isJEE Mains 2024 Hard
- \(\mathop {\lim }\limits_{n \to \infty } \left( {\frac{{{{\left( {n + 1} \right)}^{1/3}}}}{{{n^{4/3}}}} + \frac{{{{\left( {n + 2} \right)}^{1/3}}}}{{{n^{4/3}}}} + .... + \frac{{{{\left( {2n} \right)}^{1/3}}}}{{{n^{4/3}}}}} \right)\) is equal toJEE Mains 2019 Hard
- Five numbers \(x _{1}, x _{2}, x _{3}, x _{4}, x _{5}\) are randomly selected from the numbers \(1,2,3, \ldots \ldots, 18\) and are arranged in the increasing order \(\left( x _{1}< x _{2}< x _{3}< x _{4}< x _{5}\right)\). The probability that \(x_{2}=7\) and \(x_{4}=11\) isJEE Mains 2022 Hard
- If \(A = \dfrac{\sin 3^\circ}{\cos 9^\circ} + \dfrac{\sin 9^\circ}{\cos 27^\circ} + \dfrac{\sin 27^\circ}{\cos 81^\circ}\) and \(B = \tan 81^\circ - \tan 3^\circ\), then \(\dfrac{B}{A}\) is equal to _____.JEE Mains 2026 Medium
- The sum of the coefficients of \(x^{499}\) and \(x^{500}\) in \((1+x)^{1000}+x(1+x)^{999}+x^{2}(1+x)^{998}+.......+x^{1000}\) isJEE Mains 2026 Easy
- Let \(\alpha, \beta\) and \(\gamma\) be three positive real numbers. Let \(f ( x )=\alpha x ^{5}+\beta x ^{3}+\gamma x , x \in R \quad\) and \(\quad g : R \rightarrow R\) be such that \(g(f(x))=x\) for all \(x \in R\). If \(a_{1}, a_{2}, a_{3}, \ldots, a_{n}\) be in arithmetic progression with mean zero, then the value of \(f\left(g\left(\frac{1}{n} \sum_{i=1}^{n} f\left(a_{i}\right)\right)\right)\) is equal to.JEE Mains 2022 Hard