JEE Mains · Maths · STD 11 - 4.2 Quadratic equations and inequations
If \(\alpha \) and \(\beta \) be the roots of the equation \(x^2 - 2x + 2 = 0\) , then the least value of \(n\) for \({\left( {\frac{\alpha }{\beta }} \right)^n} = 1\) is
- A \(4\)
- B \(2\)
- C \(5\)
- D \(3\)
Answer & Solution
Correct Answer
(A) \(4\)
Step-by-step Solution
Detailed explanation
\((x-1)^{2}+1=0 \Rightarrow x=1+i, 1-i\) \(\therefore\left(\frac{\alpha}{\beta}\right)^{n}=1 \Rightarrow(\pm i)^{n}=1\) \(n\) (least natural number) \(=4\)
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 \(\mathrm{x}=\frac{\mathrm{m}}{\mathrm{n}}\) ( \(\mathrm{m}, \mathrm{n}\) are co-prime natural numbers) be a solution of the equation \(\cos \left(2 \sin ^{-1} x\right)=\frac{1}{9}\) and let \(\alpha, \beta(\alpha>\beta)\) be the roots of the equation \(\mathrm{mx}^2-\mathrm{nx}-\) \(\mathrm{m}+\mathrm{n}=0\). Then the point \((\alpha, \beta)\) lies on the lineJEE Mains 2024 Medium
- Let \(f: \mathbb{R} \rightarrow \mathbb{R}\) be such that \(f(xy) = f(x)f(y)\), for all \(x, y \in \mathbb{R}\) and \(f(0) \neq 0\). Let \(g: [1, \infty) \rightarrow \mathbb{R}\) be a differentiable function such that
\(x^2 g(x) = \int\limits_1^x (t^2 f(t) - tg(t))\,dt\).
Then \(g(2)\) is equal to :JEE Mains 2026 Hard - Let \(\mu\) be the mean and \(\sigma\) be the standard deviation of the distribution
where \(\sum f_i=62\). if \([x]\) denotes the greatest integer \(\leq x\), then \(\left[\mu^2+\sigma^2\right]\) is equal \(.........\).\(X_i\) \(0\) \(1\) \(2\) \(3\) \(4\) \(5\) \(f_i\) \(k+2\) \(2k\) \(K^{2}-1\) \(K^{2}-1\) \(K^{2}-1\) \(k-3\) JEE Mains 2023 Hard - The value of \( \frac{^{100}C_{50}}{51} + \frac{^{100}C_{51}}{52} + \dots + \frac{^{100}C_{100}}{101} \) is:JEE Mains 2026 Medium
- Let \([t]\) denote the greatest integer less than or equal to \(\mathrm{t}\). Let \(\mathrm{f}(\mathrm{x})=\mathrm{x}-[\mathrm{x}], \mathrm{g}(\mathrm{x})=1-\mathrm{x}+[\mathrm{x}]\), and \(h(x)=\min \{f(x), g(x)\}, x \in[-2,2]\). Then \(h\) is :JEE Mains 2021 Hard
- The number of elements in the set \(\left\{n \in Z :\left|n^2-10 n+19\right| < 6\right\}\) is \(...........\)JEE Mains 2023 Hard
More PYQs from JEE Mains
- For \(k \in N\), let \(\frac{1}{\alpha(\alpha+1)(\alpha+2) \ldots(\alpha+20)}=\sum_{k=0}^{20} \frac{A_{k}}{a+k}\), where \(a\,>\,0\). Then the value of \(100\left(\frac{A_{14}+A_{15}}{A_{13}}\right)^{2}\) is equal to \(....\)JEE Mains 2021 Hard
- An are \(P Q\) of a circle subtends a right angle at its centre \(O\). The mid point of the arc \(P Q\) is \(R\). If \(\overline{O P}=\vec{u}, \overline{O R}=\vec{v}\) and \(\overrightarrow{O Q}=\alpha \vec{u}+\beta \vec{v}\), then \(\alpha, \beta^2\) are the roots of the equationJEE Mains 2023 Hard
- A rod of length eight units moves such that its ends \(A\) and \(B\) always lie on the lines \(x-y+2=0\) and \(y+2=0\), respectively. If the locus of the point \(P\), that divides the rod \(A B\) internally in the ratio \(2: 1\) is \(9\left(x^2+\alpha y^2+\beta x y+\gamma x+28 y\right)-76=0\), then \(\alpha-\beta-\gamma\) is equal to :JEE Mains 2025 Hard
- Let the maximum and minimum values of \(\left(\sqrt{8 x-x^2-12}-4\right)^2+(x-7)^2, x \in R\) be \(M\) and \(m\) respectively. Then \(\mathrm{M}^2-\mathrm{m}^2\) 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
- Let \(\alpha\) and \(\beta\) be the roots of the equation \(\mathrm{px}^2+\mathrm{qx}-\) \(r=0\), where \(p \neq 0\). If \(p, q\) and \(r\) be the consecutive terms of a non-constant G.P and \(\frac{1}{\alpha}+\frac{1}{\beta}=\frac{3}{4}\), then the value of \((\alpha-\beta)^2\) is :JEE Mains 2024 Medium