JEE Mains · Maths · STD 11 - 7. binomial theoram
Suppose \(\sum \limits_{ r =0}^{2023} r ^{20023} C _{ r }=2023 \times \alpha \times 2^{2022}\). Then the value of \(\alpha\) is \(............\)
- A \(1011\)
- B \(1013\)
- C \(1012\)
- D \(1014\)
Answer & Solution
Correct Answer
(C) \(1012\)
Step-by-step Solution
Detailed explanation
using result \(\sum \limits_{r=0}^n r^{2 n} C_r=n(n+1) \cdot 2^{n-2}\) \(=2023 \times \alpha \times 2^{2022} \text { So, }\) \(\Rightarrow \alpha=1012\)
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 the coefficient of \(\mathrm{a}^{7} \mathrm{~b}^{8}\) in the expansion of \((a+2 b+4 a b)^{10}\) is \(K \cdot 2^{16}\), then \(K\) is equal .... .JEE Mains 2021 Hard
- Let \(A=\left[\begin{array}{lll}2 & 0 & 1 \\ 1 & 1 & 0 \\ 1 & 0 & 1\end{array}\right], B=\left[B_1, B_2, B_3\right]\), where \(B_1\), \(\mathrm{B}_2, \mathrm{~B}_3\) are column matrices, and \(\mathrm{AB}_1=\left[\begin{array}{l}1 \\ 0 \\ 0\end{array}\right]\), \(\mathrm{AB}_2=\left[\begin{array}{l}2 \\ 3 \\ 0\end{array}\right], \mathrm{AB}_3=\left[\begin{array}{l}3 \\ 2 \\ 1\end{array}\right]\) If \(\alpha=|B|\) and \(\beta\) is the sum of all the diagonal elements of \(B\), then \(\alpha^3+\beta^3\) is equal toJEE Mains 2024 Hard
- The line \(x =8\) is the directrix of the ellipse \(E: \frac{x^2}{a^2}+\frac{y^2}{b^2}=1\) with the corresponding focus \((2,0)\). If the tangent to \(E\) at the point \(P\) in the first quadrant passes through the point \((0,4 \sqrt{3})\) and intersects the \(x\)-axis at \(Q\), then \((3PQ)^2\) is equal to \(........\)JEE Mains 2023 Hard
- Let \(O\) be the origin, the point \(A\) be \(z_1=\sqrt{3}+2 \sqrt{2} i\), the point \(B\left(z_2\right)\) be such that \(\sqrt{3}\left|z_2\right|=\left|z_1\right|\) and \(\arg \left(z_2\right)=\arg \left(z_1\right)+\frac{\pi}{6}\). ThenJEE Mains 2025 Hard
- The length of the perpendicular from the origin, on the normal to the curve, \(x^{2}+2 x y-3 y^{2}=0\) at the point \((2,2)\) isJEE Mains 2020 Hard
- The line \(L_1\) is parallel to the vector \(\vec{a}=-3 \hat{i}+2 \hat{j}+4 \hat{k}\) and passes through the point \((7,6,2)\) and the line \(L_2\) is parallel to the vector \(\vec{b}=2 \hat{i}+\hat{j}+3 \hat{k}\) and passes through the point \((5,3,4)\). The shortest distance between the lines \(L_1\) and \(L_2\) is :JEE Mains 2025 Medium
More PYQs from JEE Mains
- The number of common tangents to the circles \({x^2} + {y^2} - 4x - 6y - 12 = 0\) and \({x^2} + {y^2} + 6x + 18y + 26 = 0\) isJEE Mains 2015 Hard
- Let \(\mathrm{f}: \mathrm{R}-\left\{\frac{-1}{2}\right\} \rightarrow \mathrm{R}\) and \(\mathrm{g}: \mathrm{R}-\left\{\frac{-5}{2}\right\} \rightarrow \mathrm{R}\) be defined as \(f(x)=\frac{2 x+3}{2 x+1}\) and \(g(x)=\frac{|x|+1}{2 x+5}\). Then the domain of the function \(fog\) is :JEE Mains 2024 Hard
- The equation of the curve passing through the origin and satisfying the differential equation \(\left( {1 + {x^2}} \right)\,\frac{{dy}}{{dx}} + 2xy = 4{x^2}\) isJEE Mains 2013 Hard
- Number of solutions of \( \sqrt{3}\cos 2\theta+8\cos \theta+3\sqrt{3}=0 \), \( \theta \in [-3\pi, 2\pi] \) is:JEE Mains 2026 Hard
- If \(S=\{z \in C:|z-i|=|z+i|=|z-1|\}\), then, \(n(S)\) is:JEE Mains 2024 Medium
- Let \( a_{1}, a_{2}, a_{3},..... \) be a G.P. of increasing positive terms such that \( a_{2} . a_{3} . a_{4}=64 \) and \( a_{1}+a_{3}+a_{5}=\frac{813}{7} \).
Then \( a_{3}+a_{5}+a_{7} \) is equal to:JEE Mains 2026 Hard