JEE Mains · Maths · STD 12 - 1. relation and function
Let \(f (x) = a^x (a > 0)\) be written as \(f( x) = f_1( x) + f_2( x)\) , where \(f_1( x)\) is an even function and \(f_2( x)\) is an odd function. Then \(f_1( x + y) + f_1( x - y )\) equals
- A \(2{f_1}\left( x \right){f_2}\left( y \right)\)
- B \(2{f_1}\left( x \right){f_1}\left( y \right)\)
- C \(2{f_1}\left( {x + y} \right){f_2}\left( {x - y} \right)\)
- D \(2{f_1}\left( {x + y} \right){f_1}\left( {x - y} \right)\)
Answer & Solution
Correct Answer
(B) \(2{f_1}\left( x \right){f_1}\left( y \right)\)
Step-by-step Solution
Detailed explanation
\({f_1}\left( x \right) = \frac{{{a^x} + {a^{ - x}}}}{2}\) and \({f_2}\left( x \right) = \frac{{{a^x} - {a^{ - x}}}}{2}\) \({f_1}\left( {x + y} \right) + {f_1}\left( {x - y} \right)\) \( = \frac{1}{2}\left( {{a^{x + y}} + {a^{ - x - y}} + {a^{x - y}} + {a^{ - x + y}}} \right)\)…
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 system of linear equations \(2 x+y-z=3\) \(x-y-z=\alpha\) \(3 x+3 y+\beta z=3\) has infinitely many solution, then \(\alpha+\beta-\alpha \beta\) is equal to .... .JEE Mains 2021 Medium
- 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
- If the sum of the first 10 terms of the series \(\frac{4.1}{1+4.1^4}+\frac{4.2}{1+4.2^4}+\frac{4.3}{1+4.3^4}+\ldots\) is \(\frac{m}{n}\), where \(\operatorname{gcd}(m, n)=1\), then \(m+n\) is equal to______JEE Mains 2025 Medium
- If \(\hat x,\,\hat y\) and \(\hat z\) are three unit vectors in three dimensional space , then the minimum value of \({\left| {\hat x + \hat y} \right|^2}\, + \,{\left| {\hat y + \hat z} \right|^2}\, + \,{\left| {\hat z + \hat x} \right|^2}\)JEE Mains 2014 Hard
- The distance between the two points \(A\) and \(A ^{\prime}\) which lie on \(y =2\) such that both the line segments \(AB\) and \(A ^{\prime} B\) (where \(B\) is the point \((2,3)\) ) subtend angle \(\frac{\pi}{4}\) at the origin, is equal toJEE Mains 2022 Medium
- Let the points of intersections of the lines \(x-y+1=0\), \(x-2 y+3=0\) and \(2 x-5 y+11=0\) are the mid points of the sides of a triangle \(A B C\). Then the area of the triangle \(\mathrm{ABC}\) is .... .JEE Mains 2021 Hard
More PYQs from JEE Mains
- Let \( \vec{a}=2\hat{i}-5\hat{j}+5\hat{k} \) and \( \vec{b}=\hat{i}-\hat{j}+3\hat{k}. \) If \( \vec{c} \) is a vector such that
\( 2(\vec{a}\times\vec{c})+3(\vec{b}\times\vec{c})=\vec{0} \) and \( (\vec{a}-\vec{b})\cdot\vec{c}=-97, \) then \( |\vec{c}\times \hat{k}|^{2} \) is equal toJEE Mains 2026 Easy - If \(\alpha=\lim _{x \rightarrow 0^{+}}\left(\frac{e^{\sqrt{\tan x}}-e^{\sqrt{x}}}{\sqrt{\tan x}-\sqrt{x}}\right)\) and \(\beta=\lim _{x \rightarrow 0}(1+\sin x)^{\frac{1}{2} \cot x}\) are the roots of the quadratic equation \(a x^2+b x-\sqrt{e}=0\), then 12 \(\log _e(a+b)\) is equal to .............JEE Mains 2024 Hard
- The sum of the infinite series \(1+\frac{5}{6}+\frac{12}{6^{2}}+\frac{22}{6^{3}}+\frac{35}{6^{4}}+\frac{51}{6^{5}}+\frac{70}{6^{6}}+\ldots .\) is equal toJEE Mains 2022 Hard
- Let the sum of the first \(n\) terms of a non-constant \(A.P., a_1, a_2, a_3, ……\) be \(50\,n\, + \,\frac{{n\,(n\, - 7)}}{2}A,\) where \(A\) is a constant. If \(d\) is the common difference of this \(A.P.,\) then the ordered pair \((d,a_{50})\) is equal toJEE Mains 2019 Hard
- Let \(a_1, a_2, a_3, ……\) be and \(A.P\) with \(a_6 = 2.\) Then the common difference of this \(A.P.,\) which maximizes the product \(a_1a_4a_5\) isJEE Mains 2019 Hard
- If tangents are drawn to the ellipse \(x^2 + 2y^2 = 2\) at all points on the ellipse other than its four vertices than the mid points of the tangents intercepted between the coordinate axes lie on the curveJEE Mains 2019 Hard