JEE Mains · Maths · STD 12 - 5. continuity and differentiation
Let \(\mathrm{f}: \mathbb{R}-\{0\} \rightarrow \mathbb{R}\) be a function satisfying \(f\left(\frac{x}{y}\right)=\frac{f(x)}{f(y)}\) for all \(x, y, f(y) \neq 0\). If \(f^{\prime}(1)=2024\) then
- A \(\mathrm{xf}^{\prime}(\mathrm{x})-2024 \mathrm{f}(\mathrm{x})=0\)
- B \(x f^{\prime}(x)-2024 f(x)=0\)
- C \(\mathrm{xf}^{\prime}(\mathrm{x})+\mathrm{f}(\mathrm{x})=2024\)
- D \(x f^{\prime}(x)-2023 f(x)=0\)
Answer & Solution
Correct Answer
(A) \(\mathrm{xf}^{\prime}(\mathrm{x})-2024 \mathrm{f}(\mathrm{x})=0\)
Step-by-step Solution
Detailed explanation
\(f\left(\frac{x}{y}\right)=\frac{f(x)}{f(y)}\) \(\mathrm{f}^{\prime}(1)=2024\) \({f}(1)=1\) Partially differentiating w. r. t. \(x\) \(f^{\prime}\left(\frac{x}{y}\right) \cdot \frac{1}{y}=\frac{1}{f(y)} f^{\prime}(x)\) \( y \rightarrow x \)…
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
- The equations of the sides \(AB\) and \(AC\) of a triangle \(ABC\) are \((\lambda+1) x +\lambda y =4 \text { and } \lambda x +(1-\lambda) y +\lambda=0\) respectively. Its vertex \(A\) is on the \(y\)-axis and its orthocentre is \((1,2)\). The length of the tangent from the point \(C\) to the part of the parabola \(y^2=6 x\) in the first quadrant isJEE Mains 2023 Hard
- If \(f(x)\, = \,2\,{\tan ^{ - 1}}\,x\, + \,{\sin ^{ - 1}}\,\left( {\frac{{2x}}{{1 + {x^2}}}} \right),x > 1\,\) then \(f\,(5)\) is equal toJEE Mains 2015 Hard
- A relation on the set \(A\, = \,\{ x\,:\,\left| x \right|\, < \,3,\,x\, \in Z\} ,\) where \(Z\) is the set of integers is defined by \(R= \{(x, y) : y = \left| x \right|, x \ne - 1\}\). Then the number of elements in the power set of \(R\) isJEE Mains 2014 Hard
- The distance of the line \(\frac{x-2}{2}=\frac{y-6}{3}=\frac{z-3}{4}\) from the point \((1,4,0)\) along the line \(\frac{x}{1}=\frac{y-2}{2}=\frac{z+3}{3}\) is :JEE Mains 2025 Hard
- The locus of the mid points of the chords of the circle \(C_1:(x-4)^2+(y-5)^2=4\) which subtend an angle \(\theta_i\) at the centre of the circle \(C_1\), is a circle of radius \(r_i\). If \(\theta_1=\frac{\pi}{3}, \theta_3=\frac{2 \pi}{3}\) and \(r_1^2=r_2^2+r_3^2\), then \(\theta_2\) is equal toJEE Mains 2023 Hard
- \(\lim _{x \rightarrow 0}\left(\left(\frac{1-\cos ^2(3 x)}{\cos ^3(4 x)}\right)\left(\frac{\sin ^3(4 x)}{\left.\left(\log _e(2 x+1)\right)^5\right)}\right)\right)\) is equal to \(.........\).JEE Mains 2023 Hard
More PYQs from JEE Mains
- A spherical iron ball of \(10 \;\mathrm{cm}\) radius is coated with a layer of ice of uniform thickness the melts at a rate of \(50\; \mathrm{cm}^{3} / \mathrm{min}\). When the thickness of ice is \(5 \;\mathrm{cm},\) then the rate (in \(\mathrm{cm} / \mathrm{min.}\) ) at which of the thickness of ice decreases, isJEE Mains 2020 Medium
- If \(\dfrac{\pi}{4} + \displaystyle\sum_{p=1}^{11} \tan^{-1}\left(\dfrac{2^{p-1}}{1 + 2^{2p-1}}\right) = \alpha\), then \(\tan\alpha\) is equal to __________.JEE Mains 2026 Hard
- If \(S _{ n }=4+11+21+34+50+\ldots\) to \(n\) terms, then \(\frac{1}{60}\left( S _{29}- S _9\right)\) is equal to \(.......\).JEE Mains 2023 Hard
- Let \(\mathrm{y}=\mathrm{y}(\mathrm{x})\) be a solution curve of the differential equation \((y+1) \tan ^{2} x d x+\tan x d y+y d x=0\) \(x \in\left(0, \frac{\pi}{2}\right)\). If \(\lim _{x \rightarrow 0+} x y(x)=1\), then the value of \(\mathrm{y}\left(\frac{\pi}{4}\right)\) is :JEE Mains 2021 Hard
- Let the function \(f(x)=2 x^{2}-\log _{e} x, x>0\), be decreasing in \((0, a)\) and increasing in \((a, 4)\). A tangent to the parabola \(y ^{2}=4 ax\) at a point \(P\) on it passes through the point \((8 a, 8 a-1)\) but does not pass through the point \(\left(-\frac{1}{a}, 0\right)\). If the equation of the normal at \(P\) is \(\frac{ x }{\alpha}+\frac{ y }{\beta}=1\), then \(\alpha+\beta\) is equal to-JEE Mains 2022 Hard
- If the probability of hitting a target by a shooter, in any shot, is \(\frac{1}{3}\), then the minimum number of independent shots at the target required by him so that the probability of hitting the target at least once is greater than \(\frac{5}{6}\), isJEE Mains 2019 Hard