JEE Mains · Maths · STD 12 - 1. relation and function
If \(f(x)=\frac{2^x}{2^x+\sqrt{2}}, \mathrm{x} \in \mathbb{R}\), then \(\sum_{\mathrm{k}=1}^{81} f\left(\frac{\mathrm{k}}{82}\right)\) is equal to
- A \(81 \sqrt{2}\)
- B 41
- C 82
- D \(\frac{81}{2}\)
Answer & Solution
Correct Answer
(D) \(\frac{81}{2}\)
Step-by-step Solution
Detailed explanation
\(f(x)=\frac{2^x}{2^x+\sqrt{2}} \) \( f(x)+f(1-x)=\frac{2^x}{2^x+\sqrt{2}}+\frac{2^{1-x}}{2^{1-x}+\sqrt{2}} \) \( =\frac{2^x}{2^x+\sqrt{2}}+\frac{2}{2+\sqrt{2} 2^x}=\frac{2^x+\sqrt{2}}{2^x+\sqrt{2}}=1\) Now,…
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 \(\lambda_1, \lambda_2\) be the values of \(\lambda\) for which the points \(\left(\frac{5}{2}, 1, \lambda\right)\) and \((-2,0,1)\) are at equal distance from the plane \(2 x+3 y-6 z+7=0\). if \(\lambda_1 > \lambda_2\), then the distance of the point \(\left(\lambda_1-\lambda_2, \lambda_2, \lambda_1\right)\) from the line \(\frac{x-5}{1}=\frac{y-1}{2}=\frac{z+7}{2}\) is \(............\).JEE Mains 2023 Hard
- The area of the region \(\left\{(x, y): y^2 \leq 4 x, x<4, \frac{x y(x-1)(x-2)}{(x-3)(x-4)}>0, x \neq 3\right\}\) isJEE Mains 2024 Hard
- If \(\int \frac{\left(\sqrt{1+x^2}+x\right)^{10}}{\left(\sqrt{1+x^2}-x\right)^9} d x=\)
\(\frac{1}{m}\left(\left(\sqrt{1+x^2}+x\right)^n\left(n \sqrt{1+x^2}-x\right)\right)+C\)
where C is the constant of integration and \(m, n \in N\), then \(\mathrm{m}+\mathrm{n}\) is equal toJEE Mains 2025 Hard - Consider the data on x taking the values \(0,2,4,8, \ldots, 2^{n}\) with frequencies \({ }^{n} C_{0},{ }^{n} C_{1},{ }^{n} C_{2}, \ldots\) \({ }^{ n } C _{ n }\) respectively. If the mean of this data is \(\frac{728}{2^{ n }},\) then \(n\) is equal toJEE Mains 2020 Hard
- \(\lim _{x \rightarrow 0} \frac{e^{2 |\text { sin } x | \mid}-2|\sin x|-1}{x^2}\)JEE Mains 2024 Hard
- If \(\displaystyle\sum_{k=1}^{n} a_k = 6n^3\), then \(\displaystyle\sum_{k=1}^{6} \left(\dfrac{a_{k+1} - a_k}{36}\right)^2\) is equal to _______.JEE Mains 2026 Medium
More PYQs from JEE Mains
- The sum of all those terms, of the anithmetic progression \(3,8,13, \ldots \ldots .373\), which are not divisible by \(3\),is equal to \(.......\).JEE Mains 2023 Hard
- \(\sum_{\mathrm{k}=0}^{20}\left({ }^{20} \mathrm{C}_{\mathrm{k}}\right)^{2}\) is equal to :JEE Mains 2021 Hard
- Let \(\mathrm{A}, \mathrm{B}\) and \(\mathrm{C}\) be three points on the parabola \(y^2=6 x\) and let the line segment \(A B\) meet the line \(L\) through \(\mathrm{C}\) parallel to the \(\mathrm{x}\)-axis at the point \(\mathrm{D}\). Let \(\mathrm{M}\) and \(\mathrm{N}\) respectively be the feet of the perpendiculars from \(\mathrm{A}\) and \(\mathrm{B}\) on \(\mathrm{L}\). Then \(\left(\frac{\mathrm{AM} \cdot \mathrm{BN}}{\mathrm{CD}}\right)^2\) is equal to ...........JEE Mains 2024 Hard
- If P is a point on the circle \( x^{2}+y^{2}=4 \), Q is a point on the straight line \( 5x+y+2=0 \) and \( x-y+1=0 \) is the perpendicular bisector of PQ, then 13 times the sum of abscissa of all such point P is ........... .JEE Mains 2026 Hard
- If \(f:R \to R\) is a differentiable function and \(f\left( 2 \right) = 6\), then \(\mathop {\lim }\limits_{x \to 2} \int\limits_6^{f\left( x \right)} {\frac{{2\,tdt}}{{\left( {x - 2} \right)}}} \) isJEE Mains 2019 Hard
- Let \(A\) be any \(3 \times 3\) invertible matrix. Then which one of the following is not always true ?JEE Mains 2017 Hard