JEE Mains · Maths · STD 12 - 1. relation and function
Let \(\mathrm{f}: \mathrm{R} \rightarrow \mathrm{R}\) be a function which satisfies \(\mathrm{f}(\mathrm{x}+\mathrm{y})=\mathrm{f}(\mathrm{x})+\mathrm{f}(\mathrm{y}) \forall \mathrm{x}, \mathrm{y} \in \mathrm{R} .\) If \(\mathrm{f}(1)=2\) and \(g(n)=\sum \limits_{k=1}^{(n-1)} f(k), n \in N\) then the value of \(n,\) for which \(\mathrm{g}(\mathrm{n})=20,\) is
- A \(5\)
- B \(9\)
- C \(20\)
- D \(4\)
Answer & Solution
Correct Answer
(A) \(5\)
Step-by-step Solution
Detailed explanation
\(f(x+y)=f(x)+f(y)\) \(\Rightarrow f(n)=n f(1)\) \(f(n)=2 n\) \(g(n)=\sum_{k=1}^{n-1} 2 n=2\left(\frac{(n-1) n}{2}\right)=n(n-1)\) \(g(n)=20 \Rightarrow n(n-1)=20\) \(n=5\)
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 domain of the function \(\sin ^{-1}\left(\frac{3 x-22}{2 x-19}\right)+\log _e\left(\frac{3 x^2-8 x+5}{x^2-3 x-10}\right)\) is \((\alpha, \beta]\), then \(3 \alpha+10 \beta\) is equal to :JEE Mains 2024 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
- Let \(A\) be a square matrix of order 3 such that \(\operatorname{det}(A)=-2\) and \(\operatorname{det}(3 \operatorname{adj}(-6 \operatorname{adj}(3 A)))=2^{\mathrm{m}+\mathrm{n}} \cdot 3^{\mathrm{mn}}, \mathrm{m}\gt\mathrm{n}\). Then \(4 \mathrm{~m}+2 \mathrm{n}\) is equal to _______JEE Mains 2025 Medium
- Let \(a\) be an integer such that all the real roots of the polynomial \(2 x^{5}+5 x^{4}+10 x^{3}+10 x^{2}+10 x+10\) lie in the interval \((a, a+1) .\) Then, \(| a |\) is equal to ...... .JEE Mains 2021 Hard
- A letter is known to have arrived by post either from KANPUR or from ANANTPUR. On the envelope just two consecutive letters AN are visible. The probability, that the letter came from ANANTPUR, is:JEE Mains 2026 Hard
- If \(\lim\limits _{x \rightarrow 1} \frac{\sin \left(3 x^{2}-4 x+1\right)-x^{2}+1}{2 x^{3}-7 x^{2}+a x+b}=-2\), then the value of \((a-b)\) is equal toJEE Mains 2022 Hard
More PYQs from JEE Mains
- The foot of perpendicular from the origin \(O\) to a plane \(P\) which meets the co-ordinate axes at the points \(A, B, C\) is \((2, a, 4), a \in N\). If the volume of the tetrahedron \(OABC\) is \(144\) unit \(^3\), then which of the following points is \(NOT\) on \(P\) ?JEE Mains 2023 Hard
- From the top \(A\) of a vertical wall \(AB\) of height 30 \(m\), the angles of depression of the top \(P\) and bottom \(Q\) of a vertical tower \(P Q\) are \(15^{\circ}\) and \(60^{\circ}\) respectively. \(B\) and \(Q\) are on the same horizontal level. If \(C\) is a point on \(A B\) such that \(C B=P Q\), then the area (in \(m ^2\) ) of the quadrilateral \(BCPQ\) is equal toJEE Mains 2023 Hard
- Let \(C\) be the circle with centre at \((1, 1)\) and radius \(= 1\). If \(T\) is the circle centred at \((0, y),\) passing through origin and touching the circle \(C\) externally, then the radius of \(T\) is equalJEE Mains 2014 Hard
- If \(a+b+c=1, a b+b c+c a=2\) and \(a b c=3\), then the value of \(a^{4}+b^{4}+c^{4}\) is equal to \(....\)JEE Mains 2021 Hard
- Let the equations of two adjacent sides of a parallelogram \(A B C D\) be \(2 x-3 y=-23\) and \(5 x+4 y\) \(=23\). If the equation of its one diagonal \(AC\) is \(3 x +\) \(7 y=23\) and the distance of A from the other diagonal is \(d\), then \(50 d ^2\) is equal to \(........\).JEE Mains 2023 Hard
- Let the shortest distance between the lines \(L : \frac{ x -5}{-2}=\frac{ y -\lambda}{0}=\frac{ z +\lambda}{1}, \lambda \geq 0\) and \(L _1: x +1= y -\) \(1=4-z\) be \(2 \sqrt{6}\). If \((\alpha, \beta, \gamma)\) lies on \(L\), then which of the following is NOT possible?JEE Mains 2023 Hard