Let f : ℝ → ℝ and g : ℝ → ℝ be functions satisfying f x + y = f x + f y + f x f y and f x = x g x for all x , y ∈ ℝ . If lim x → 0 g x = 1 , then which of the following statements is/are TRUE?

The derivative f ' 0 is equal to 1

The derivative f ' 1 is equal to 1

A solution curve of differential equation x 2 + x y + 4 x + 2 y + 4 d y d x - y 2 = 0 , x > 0 passes through the point ( 1 , 3 ) . Then the solution curve -

intersects y = x + 2 exactly at two points

For any real numbers α and β , let y α , β x , x ∈ R , be the solution of the differential equation d y d x + α y = x e β x , y 1 = 1 Let S = y α , β x : α , β ∈ R . Then which of the following functions belong(s) to the set S ?

f x = e x 2 x - 1 2 + e - e 2 4 e - x

f x = - x 2 2 e - x + e + 1 2 e - x

f x = e x 2 1 2 - x + e + e 2 4 e - x

Let f : ℝ → ℝ and g : ℝ → ℝ be functions satisfying f x + y = f x + f y + f x f y and f x = x g x for all x , y ∈ ℝ . If lim x → 0 g x = 1 , then which of the following statements is/are TRUE?

The derivative f ' 0 is equal to 1

The derivative f ' 1 is equal to 1

Let f : 0 , ∞ → R be a differentiable function such that f ' x = 2 - f x x ∀ x ∈ ( 0 , ∞ ) and f 1 ≠ 1 . Then,

lim x → 0 + ⁡ f ′ 1 x = 1

lim x → 0 + ⁡ x 2 f ′ x = 0

HARD

JEE Advanced

IMPORTANT

Earn 100

Let $b$ be a nonzero real number. Suppose $f : ℝ \to ℝ$ is a differentiable function such that $f (0) = 1$ . If the derivative $f^{'}$ of $f$ satisfies the equation $f^{'} (x) = \frac{f (x)}{b^{2} + x^{2}}$ for all $x \in ℝ,$ then which of the following statements is/are TRUE?

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Important Questions on Differential Equations

HARD

JEE Advanced

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 28 - Differential Equations>JEE Advanced Paper 2 - 2020>Q 3

Let $f : ℝ \to ℝ$ and $g : ℝ \to ℝ$ be functions satisfying $f (x + y) = f (x) + f (y) + f (x) f (y) and f (x) = x g (x)$ for all $x, y \in ℝ .$ If $\lim_{x \to 0} g (x) = 1,$ then which of the following statements is/are TRUE?

HARD

JEE Advanced

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 28 - Differential Equations>JEE Advanced Paper 2 - 2018>Q 4

Let

f : (0, π) \to R

be a twice differentiable function such that

\lim_{t \to x} \{\frac{f (x) \sin t - f (t) \sin x}{t - x}\} = \sin^{2} x

for all

x \in (0, π)

. If

f (\frac{π}{6}) = - \frac{π}{12}

, then which of the following statement(s) is (are) TRUE ?

MEDIUM

JEE Advanced

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 28 - Differential Equations>JEE Advanced Paper 2 - 2018>Q 5

Let

f : R \to R

be a differentiable function with

f (0) = 0

. If

y = f (x)

satisfies the differential equation

\frac{d y}{d x} = (2 + 5 y) (5 y - 2) .

If the value of

\lim_{x \to - \infty} f (x) = λ

, then

10 λ

is equal to

HARD

JEE Advanced

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 28 - Differential Equations>JEE Advanced Paper 2 - 2018>Q 6

Let

f : R \to R

be a differentiable function with

f (0) = 1

and satisfying the equation

f (x + y) = f (x) f^{'} (y) + f^{'} (x) f (y)

for all

x, y \in R

. Then, the value of

\log_{e} (f (4))

HARD

JEE Advanced

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 28 - Differential Equations>JEE Advanced Paper 2 - 2017>Q 7

y = y (x)

satisfies the differential equation

8 \sqrt{x} (\sqrt{9 + \sqrt{x}}) d y = {(\sqrt{4 + \sqrt{9 + \sqrt{x}}})}^{- 1} d x

and

x > 0, y (0) = \sqrt{7},

then

y (256) =

HARD

JEE Advanced

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 28 - Differential Equations>JEE Advanced Paper 2 - 2014>Q 8

The function

y = f (x)

is the solution of the differential equation

\frac{d y}{d x} + \frac{x y}{x^{2} - 1} = \frac{x^{4} + 2 x}{\sqrt{1 - x^{2}}}

(- 1, 1)

satisfying

f (0) = 0 .

Then

\int_{- \frac{\sqrt{3}}{2}}^{\frac{\sqrt{3}}{2}} f (x) d x

HARD

JEE Advanced

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 28 - Differential Equations>JEE Advanced Paper 1 - 2018>Q 9

Let

f : R \to R a n d g : R \to R

be two non-constant differentiable functions. If

f^{'} (x) = (e^{(f (x)) - g (x)}) g^{'} (x) f o r a l l x \in R

, and

f (1) = g (2) = 1

then which of the following statement(s) is (are) TRUE?

HARD

JEE Advanced

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 28 - Differential Equations>JEE Advanced Paper 1 - 2016>Q 10

A solution curve of differential equation

(x^{2} + x y + 4 x + 2 y + 4) \frac{d y}{d x} - y^{2} = 0, x > 0

passes through the point

(1, 3)

. Then the solution curve -

Let b be a nonzero real number. Suppose f:ℝ→ℝ is a differentiable function such that f0=1. If the derivative f' of f satisfies the equation f'x=fxb2+x2 for all x∈ℝ, then which of the following statements is/are TRUE?

Important Questions on Differential Equations

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Let $b$ be a nonzero real number. Suppose $f : ℝ \to ℝ$ is a differentiable function such that $f (0) = 1$ . If the derivative $f^{'}$ of $f$ satisfies the equation $f^{'} (x) = \frac{f (x)}{b^{2} + x^{2}}$ for all $x \in ℝ,$ then which of the following statements is/are TRUE?