JEE Mains · Maths · STD 11 - 4.2 Quadratic equations and inequations
Let \(\alpha, \alpha + 2, \alpha \in \mathbb{Z}\), be the roots of the quadratic equation \(x(x+2) + (x+1)(x+3) + (x+2)(x+4) + \ldots + (x+n-1)(x+n+1) = 4n\) for some \(n \in \mathbb{N}\). Then \(n + \alpha\) is equal to :
- A \(0\)
- B \(1\)
- C \(2\)
- D \(3\)
Answer & Solution
Correct Answer
(C) \(2\)
Step-by-step Solution
Detailed explanation
The given equation is: \(\sum_{k=0}^{n-1} (x+k)(x+k+2) = 4n\) Expanding the terms inside the summation: \(\sum_{k=0}^{n-1} (x^2 + 2(k+1)x + k(k+2)) = 4n\) Summing each term separately: \(n x^2 + 2x \sum_{k=0}^{n-1} (k+1) + \sum_{k=0}^{n-1} (k^2 + 2k) = 4n\) Using the standard…
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
- \(16 \sin \left(20^{\circ}\right) \sin \left(40^{\circ}\right) \sin \left(80^{\circ}\right)\) is equal toJEE Mains 2022 Medium
- The number of values of \(\theta \in (0,\pi)\) for which the system of linear equations
\(x + 3y + 7z = 0\)
\(-x + 4y + 7z = 0\)
\((sin\,3\theta )x + (cos\,2\theta )y + 2z = 0\) has a non-trivial solution, isJEE Mains 2019 Hard - The value of \(\frac{8}{\pi} \int \limits_0^{\frac{\pi}{2}} \frac{(\cos x)^{2023}}{(\sin x)^{2023}+(\cos x)^{2023}} d x\) is \(.............\).JEE Mains 2023 Easy
- Let \(A=\left(\begin{array}{cc}1+ i & 1 \\ - i & 0\end{array}\right)\) where \(i =\sqrt{-1}\) Then, the number of elements in the set \(\left\{ n \in\{1,2, \ldots, 100\}: A ^{ n }= A \right\}\) isJEE Mains 2022 Hard
- Let \(\mathrm{A}=\{-3,-2,-1,0,1,2,3\}\) and R be a relation on \(A\) defined by \(x R y\) if and only if \(2 x-y \in\{0,1\}\). Let \(l\) be the number of elements in R. Let \(m\) and \(n\) be the minimum number of elements required to be added in R to make it reflexive and symmetric relations, respectively. Then \(l+\mathrm{m} \mathrm{n}\) is equal to :-JEE Mains 2025 Easy
- Let \(f:[-1,2] \rightarrow \mathrm{R}\) be given by \(f(x)=2 x^2+x+\left[x^2\right]-[x]\), where \([t]\) denotes the greatest integer less than or equal to \(t\). The number of points, where \(f\) is not continuous, is :JEE Mains 2024 Hard
More PYQs from JEE Mains
- If \(\left| {\begin{array}{*{20}{c}}
{{a^2}}&{{b^2}}&{{c^2}} \\
{{{(a + \lambda )}^2}}&{{{(b + \lambda )}^2}}&{{{(c + \lambda )}^2}} \\
{{{(a - \lambda )}^2}}&{{{(b - \lambda )}^2}}&{{{(c - \lambda )}^2}}
\end{array}} \right|\) \( = \,k\lambda \,\,\left| {{\mkern 1mu} {\mkern 1mu} \begin{array}{*{20}{c}}
{{a^2}}&{{b^2}}&{{c^2}} \\
a&b&c \\
1&1&1
\end{array}} \right|,\lambda \, \ne \,0\) then \(k\) is equal toJEE Mains 2014 Hard - The maximum area of a right angled triangle with hypotenuse \(h\) isJEE Mains 2013 Hard
- The length of the latus rectum of a parabola, whose vertex and focus are on the positive \(x\)-axis at a distance \(\mathrm{R}\) and \(\mathrm{S}(\,>\,\mathrm{R})\) respectively from the origin, is:JEE Mains 2021 Medium
- Let \(P_n=\alpha^n+\beta^n, n \in \mathbf{N}\). If \(P_{10}=123, P_9=76\), \(P_8=47\) and \(P_1=1\), then the quadratic equation having roots \(\frac{1}{\alpha}\) and \(\frac{1}{\beta}\) is :JEE Mains 2025 Medium
- Let \(\vec{a}=\hat{i}+\alpha \hat{j}+3 \hat{k}\) and \(\vec{b}=3 \hat{i}-\alpha \hat{j}+\hat{k} \cdot\) If the area of the parallelogram whose adjacent sides are represented by the vectors \(\vec{a}\) and \(\vec{b}\) is \(8 \sqrt{3}\) square units, then \(\overrightarrow{ a } \cdot \overrightarrow{ b }\) is equal to ....... .JEE Mains 2021 Medium
- A light ray emits from the origin making an angle \(30^{\circ}\) with the positive \(x\)-axis. After getting reflected by the line \(x + y =1\), if this ray intersects \(x\)-axis at \(Q\), then the abscissa of \(Q\) isJEE Mains 2023 Hard