Let f : R - 0 R be a function satisfying f((x/y)) = (f(x)/f(y)) for all x, y, f(y) neq 0. If f'(1) = 2024, then

Year	2026	2025	2024	Total
Questions	13	17	16	46

Year

2026

2025

2024

Total

Questions

Solution & Explanation

Related Formula

Chain rule for partial differentiation: (\partial)/(\partial x) f((x)/(y)) = f'((x)/(y)) \cdot (1)/(y)

Core Logic

Given functional equation: $f((x)/(y)) = (f(x))/(f(y))$ First, put $x=1, y=1$ $x=1, y=1$: $f(1) = (f(1))/(f(1)) \Rightarrow f(1) = 1$ (since $f(y) \neq 0$ $f(y) \neq 0$ implies $f(1) \neq 0$ $f(1) \neq 0$).

Step 1: Partial Differentiation

Partially differentiate the given equation with respect to $x$ $x$, keeping $y$ $y$ constant:

f'((x)/(y)) \cdot (1)/(y) = (1)/(f(y)) \cdot f'(x)

Let $x \to y$ $x \to y$ so that $(x)/(y) \to 1$ $\frac{x}{y} \to 1$. Wait, let's substitute $y = x$ $y = x$ directly:

f'(1) \cdot (1)/(x) = (1)/(f(x)) \cdot f'(x)

Step 2: Forming the Final Equation

We are given $f'(1) = 2024$ $f'(1) = 2024$. Substituting this value:

2024 \cdot (1)/(x) = (f'(x))/(f(x))

Cross-multiplying yields:

2024 f(x) = x f'(x)

\Rightarrow x f'(x) - 2024 f(x) = 0

Pattern Recognition

Functional equations of the form $f(x/y) = f(x)/f(y)$ $f(x/y) = f(x)/f(y)$ inherently describe power functions ( $f(x) = x^k$ $f(x) = x^k$). Partial differentiation with respect to one variable quickly resolves the derivative form without limits.

Chapter Mix

Class 11 Maths: Functions Class 12 Maths: Differential Equations

More Differential Equations Previous-Year Questions — Page 10

Q11 jee_main_2024_31_jan_morning Linear Differential Equations

Let

y = y(x)

be the solution of the differential equation

(dy)/(dx) = (( x) + y)/( x( x - x x)), x in (0, (π)/(2))

satisfying the condition

y((π)/(4)) = 2

. Then,

y((π)/(3))

A. $\sqrt(3)(2 + ₑ\sqrt(3))$ $$\sqrt{3}(2 + \log_e\sqrt{3})$$
B. $\sqrt(3)2(2 + ₑ 3)$ $$\frac{\sqrt{3}}{2}(2 + \log_e 3)$$
C. $\sqrt(3)(1 + 2 ₑ 3)$ $$\sqrt{3}(1 + 2\log_e 3)$$
D. $\sqrt(3)(2 + ₑ 3)$ $$\sqrt{3}(2 + \log_e 3)$$

Solution

Core Logic

(dy)/(dx) = (( x)/( x) + y)/( x ((1)/( x) - ( ² x)/( x))) = ( x + y x)/( x (1 - ² x))

(dy)/(dx) = ( x + y x)/( x ² x) = ² x + (2y)/( 2x)

(dy)/(dx) - 2 (2x)y = ² x

Step 1: Integrating Factor

This is an LDE of form $(dy)/(dx) + Py = Q$ $\frac{dy}{dx} + Py = Q$.

I.F. = e\int -2 (2x) dx

Let $2x = t 2dx = dt$ $2x = t \implies 2dx = dt$.

I.F. = e-\int t dt = e-ln| (t/2)| = e-ln| x| = (1)/(| x|)

Step 2: Solution of LDE

y(I.F.) = \int Q(I.F.) dx + C

y(1)/( x) = \int ² x (1)/( x) dx + C

Let $x = t ² x dx = dt$ $\tan x = t \implies \sec^2 x dx = dt$.

y(1)/( x) = \int (dt)/(t) + C = ln| x| + C

y = x(ln| x| + C)

Step 3: Boundary Value

Given $y(π/4) = 2$ $y(\pi/4) = 2$:

2 = 1(ln 1 + C) C = 2

Thus, $y = x (ln| x| + 2)$ $y = \tan x (\ln|\tan x| + 2)$. At $x = π/3$ $x = \pi/3$:

y(π/3) = \sqrt(3)(ln\sqrt(3) + 2)

Chapter Mix

Class 12 Maths: Differential Equations

Let $f : R - 0 \to R$ $f : \mathbb{R} - \{0\} \to \mathbb{R}$ be a function satisfying $f((x)/(y)) = (f(x))/(f(y))$ for all $x, y$ $x, y$, $f(y) \neq 0$ $f(y) \neq 0$. If $f'(1) = 2024$ $f'(1) = 2024$, then

Solution & Explanation

Related Formula

Core Logic

Step 1: Partial Differentiation

Step 2: Forming the Final Equation

Pattern Recognition

Chapter Mix

Reference Study Guides

More Differential Equations Previous-Year Questions — Page 10

Solution

Core Logic

Step 1: Integrating Factor

Step 2: Solution of LDE

Step 3: Boundary Value

Chapter Mix

More Differential Equations Questions — jee_main_2024_30_january_evening

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Let f : R - 0 → R$f : \mathbb{R} - \{0\} \to \mathbb{R}$ be a function satisfying f((x)/(y)) = (f(x))/(f(y))$f\left(\frac{x}{y}\right) = \frac{f(x)}{f(y)}$ for all x, y$x, y$, f(y) ≠ 0$f(y) \neq 0$. If f'(1) = 2024$f'(1) = 2024$, then

Solution & Explanation

Related Formula

Core Logic

Step 1: Partial Differentiation

Step 2: Forming the Final Equation

Pattern Recognition

Chapter Mix

Reference Study Guides

More Differential Equations Previous-Year Questions — Page 10

Solution

Core Logic

Step 1: Integrating Factor

Step 2: Solution of LDE

Step 3: Boundary Value

Chapter Mix

More Differential Equations Questions — jee_main_2024_30_january_evening

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Let $f : R - 0 \to R$ $f : \mathbb{R} - \{0\} \to \mathbb{R}$ be a function satisfying $f((x)/(y)) = (f(x))/(f(y))$ for all $x, y$ $x, y$, $f(y) \neq 0$ $f(y) \neq 0$. If $f'(1) = 2024$ $f'(1) = 2024$, then