LFM Pure

jee-main 2014 Q86 Applied Modeling with Differential Equations View

Let the population of rabbits surviving at a time $t$ be governed by the differential equation $\frac { d p ( t ) } { d t } = \frac { 1 } { 2 } \{ p ( t ) - 400 \}$. If $p ( 0 ) = 100$, then $p ( t )$ equals
(1) $600 - 500 e ^ { \frac { t } { 2 } }$
(2) $400 - 300 e ^ { \frac { - t } { 2 } }$
(3) $400 - 300 e ^ { t / 2 }$
(4) $300 - 200 e ^ { \frac { - t } { 2 } }$

jee-main 2014 Q86 Higher-Order and Special DEs (Proof/Theory) View

If the differential equation representing the family of all circles touching $x$-axis at the origin is $\left( x ^ { 2 } - y ^ { 2 } \right) \frac { d y } { d x } = g ( x ) y$, then $g ( x )$ equals
(1) $\frac { 1 } { 2 } x ^ { 2 }$
(2) $2 x$
(3) $\frac { 1 } { 2 } x$
(4) $2 x ^ { 2 }$

jee-main 2016 Q84 Integral Equations Reducible to DEs View

For $x \in R , x \neq 0$, if $y ( x )$ is a differentiable function such that $x \int _ { 1 } ^ { x } y ( t ) d t = ( x + 1 ) \int _ { 1 } ^ { x } t y ( t ) d t$, then $y ( x )$ equals (where $C$ is a constant)
(1) $C x ^ { 3 } e ^ { \frac { 1 } { x } }$
(2) $\frac { C } { x ^ { 2 } } e ^ { - \frac { 1 } { x } }$
(3) $\frac { C } { x } e ^ { - \frac { 1 } { x } }$
(4) $\frac { C } { x ^ { 3 } } e ^ { - \frac { 1 } { x } }$

jee-main 2017 Q85 Solving Separable DEs with Initial Conditions View

If $(2 + \sin x)\dfrac{dy}{dx} + (y + 1)\cos x = 0$ and $y(0) = 1$, then $y\left(\dfrac{\pi}{2}\right)$ is equal to
(1) $\dfrac{1}{3}$
(2) $-\dfrac{2}{3}$
(3) $-\dfrac{1}{3}$
(4) $\dfrac{4}{3}$

jee-main 2017 Q85 First-Order Linear DE: General Solution View

The curve satisfying the differential equation, $y d x - \left( x + 3 y ^ { 2 } \right) d y = 0$ and passing through the point $( 1,1 )$ also passes through the point
(1) $\left( \frac { 1 } { 4 } , - \frac { 1 } { 2 } \right)$
(2) $\left( - \frac { 1 } { 3 } , \frac { 1 } { 3 } \right)$
(3) $\left( \frac { 1 } { 4 } , \frac { 1 } { 2 } \right)$
(4) $\left( \frac { 1 } { 3 } , - \frac { 1 } { 3 } \right)$

jee-main 2018 Q80 Qualitative Analysis of DE Solutions View

Let $S = \left\{ ( \lambda , \mu ) \in R \times R : f ( t ) = \left( | \lambda | e ^ { | t | } - \mu \right) \sin ( 2 | t | ) , t \in R \right.$ is a differentiable function $\}$. Then, $S$ is a subset of :
(1) $( - \infty , 0 ) \times R$
(2) $R \times [ 0 , \infty )$
(3) $[ 0 , \infty ) \times R$
(4) $R \times ( - \infty , 0 )$

jee-main 2018 Q80 Qualitative Analysis of DE Solutions View

Let $S = \left\{ ( \lambda , \mu ) \in R \times R : f ( t ) = \left( | \lambda | e ^ { t } - \mu \right) \cdot \sin ( 2 | t | ) , t \in R \right.$, is a differentiable function $\}$. Then $S$ is a subest of?
(1) $R \times [ 0 , \infty )$
(2) $( - \infty , 0 ) \times R$
(3) $[ 0 , \infty ) \times R$
(4) $R \times ( - \infty , 0 )$

jee-main 2019 Q1 Solving Separable DEs with Initial Conditions View

A particle is moving with a velocity $\vec { v } = K y \hat { i } + x \hat { j }$, where $K$ is a constant. The general equation for its path is:
(1) $y ^ { 2 } = x +$ constant
(2) $x y =$ constant
(3) $y = x ^ { 2 } +$ constant
(4) $y ^ { 2 } = x ^ { 2 } +$ constant

jee-main 2019 Q80 Integral Equations Reducible to DEs View

Let $f:[0,1] \rightarrow R$ be such that $f(xy) = f(x) \cdot f(y)$, for all $x,y \in [0,1]$, and $f(0) \neq 0$. If $y = y(x)$ satisfies the differential equation, $\frac{dy}{dx} = f(x)$ with $y(0) = 1$ then $y\left(\frac{1}{4}\right) + y\left(\frac{3}{4}\right)$ is equal to:
(1) 5
(2) 2
(3) 3
(4) 4

jee-main 2019 Q83 Solving Separable DEs with Initial Conditions View

Given that the slope of the tangent to a curve $y = y ( x )$ at any point $( x , y )$ is $\frac { 2 y } { x ^ { 2 } }$. If the curve passes through the centre of the circle $x ^ { 2 } + y ^ { 2 } - 2 x - 2 y = 0$, then its equation is
(1) $x ^ { 2 } \log _ { e } | y | = - 2 ( x - 1 )$
(2) $x \log _ { e } | y | = 2 ( x - 1 )$
(3) $x \log _ { e } | y | = - 2 ( x - 1 )$
(4) $x \log _ { e } | y | = x - 1$

jee-main 2019 Q85 Solving Separable DEs with Initial Conditions View

If a curve passes through the point $( 1 , - 2 )$ and has slope of the tangent at any point $( x , y )$ on it as $\frac { x ^ { 2 } - 2 y } { x }$, then the curve also passes through the point
(1) $( \sqrt { 3 } , 0 )$
(2) $( - 1,2 )$
(3) $( - \sqrt { 2 } , 1 )$
(4) $( 3,0 )$

jee-main 2019 Q86 First-Order Linear DE: General Solution View

The solution of the differential equation $x \frac { d y } { d x } + 2 y = x ^ { 2 } , ( x \neq 0 )$ with $y ( 1 ) = 1$, is
(1) $y = \frac { x ^ { 3 } } { 5 } + \frac { 1 } { 5 x ^ { 2 } }$
(2) $y = \frac { 3 } { 4 } x ^ { 2 } + \frac { 1 } { 4 x ^ { 2 } }$
(3) $y = \frac { x ^ { 2 } } { 4 } + \frac { 3 } { 4 x ^ { 2 } }$
(4) $y = \frac { 4 } { 5 } x ^ { 3 } + \frac { 1 } { 5 x ^ { 2 } }$

jee-main 2019 Q86 First-Order Linear DE: General Solution View

If $\cos x \frac { d y } { d x } - y \sin x = 6 x , \left( 0 < x < \frac { \pi } { 2 } \right)$ and $y \left( \frac { \pi } { 3 } \right) = 0$, then $y \left( \frac { \pi } { 6 } \right)$ is equal to
(1) $- \frac { \pi ^ { 2 } } { 4 \sqrt { 3 } }$
(2) $\frac { \pi ^ { 2 } } { 2 \sqrt { 3 } }$
(3) $- \frac { \pi ^ { 2 } } { 2 }$
(4) $- \frac { \pi ^ { 2 } } { 2 \sqrt { 3 } }$

jee-main 2020 Q67 Solving Separable DEs with Initial Conditions View

If $y = y(x)$ is the solution of the differential equation, $e ^ { y } \left( \frac { d y } { d x } - 1 \right) = e ^ { x }$ such that $y(0) = 0$, then $y(1)$ is equal to
(1) $1 + \log _ { e } 2$
(2) $2 + \log _ { e } 2$
(3) $2e$
(4) $\log _ { e } 2$

jee-main 2020 Q67 Solving Separable DEs with Initial Conditions View

The solution curve of the differential equation, $\left( 1 + e ^ { - x } \right) \left( 1 + y ^ { 2 } \right) \frac { d y } { d x } = y ^ { 2 }$ which passes through the point $( 0,1 )$, is
(1) $y ^ { 2 } + 1 = y \left( \log _ { e } \left( \frac { 1 + e ^ { - x } } { 2 } \right) + 2 \right)$
(2) $y ^ { 2 } + 1 = y \left( \log _ { e } \left( \frac { 1 + e ^ { x } } { 2 } \right) + 2 \right)$
(3) $y ^ { 2 } = 1 + y \log _ { e } \left( \frac { 1 + e ^ { x } } { 2 } \right)$
(4) $y ^ { 2 } = 1 + y \log _ { e } \left( \frac { 1 + e ^ { - x } } { 2 } \right)$

jee-main 2020 Q68 Solving Separable DEs with Initial Conditions View

If $\frac { d y } { d x } = \frac { x y } { x ^ { 2 } + y ^ { 2 } } ; y ( 1 ) = 1$; then a value of $x$ satisfying $y(x) = e$ is:

jee-main 2020 Q68 Solving Separable DEs with Initial Conditions View

Let $y = y(x)$ be the solution of the differential equation, $\frac{2 + \sin x}{y + 1} \cdot \frac{dy}{dx} = -\cos x, y > 0, y(0) = 1$. If $y(\pi) = a$ and $\frac{dy}{dx}$ at $x = \pi$ is $b$, then the ordered pair $(a, b)$ is equal to
(1) $\left(2, \frac{3}{2}\right)$
(2) $(1, -1)$
(3) $(1, 1)$
(4) $(2, 1)$

jee-main 2020 Q68 Solving Separable DEs with Initial Conditions View

The solution of the differential equation $\frac { d y } { d x } - \frac { y + 3 x } { \log _ { e } ( y + 3 x ) } + 3 = 0$ is (where $C$ is a constant of integration)
(1) $x - \frac { 1 } { 2 } \left( \log _ { e } ( y + 3 x ) \right) ^ { 2 } = C$
(2) $x - \log _ { e } ( y + 3 x ) = C$
(3) $y + 3 x - \frac { 1 } { 2 } \left( \log _ { e } x \right) ^ { 2 } = C$
(4) $x - 2 \log _ { e } ( y + 3 x ) = C$

jee-main 2020 Q68 Solving Separable DEs with Initial Conditions View

If $y = y ( x )$ is the solution of the differential equation $\frac { 5 + e ^ { x } } { 2 + y } \cdot \frac { d y } { d x } + e ^ { x } = 0$ satisfying $y ( 0 ) = 1$ then value of $y \left( \log _ { e } 13 \right)$ is
(1) 1
(2) - 1
(3) 0
(4) 2

jee-main 2021 Q74 Integral Equations Reducible to DEs View

Let $f ( x ) = \int _ { 0 } ^ { x } e ^ { t } f ( t ) d t + e ^ { x }$ be a differentiable function for all $x \in R$. Then $f ( x )$ equals:
(1) $e ^ { \left( e ^ { x } - 1 \right) }$
(2) $e ^ { e ^ { x } } - 1$
(3) $2 e ^ { e ^ { x } } - 1$

jee-main 2021 Q76 Solving Separable DEs with Initial Conditions View

If a curve passes through the origin and the slope of the tangent to it at any point $( x , y )$ is $\frac { x ^ { 2 } - 4 x + y + 8 } { x - 2 }$, then this curve also passes through the point:
(1) $( 5,4 )$
(2) $( 4,4 )$
(3) $( 4,5 )$
(4) $( 5,5 )$

jee-main 2021 Q76 First-Order Linear DE: General Solution View

If the solution curve of the differential equation $\left( 2 x - 10 y ^ { 3 } \right) d y + y d x = 0$, passes through the points $( 0,1 )$ and $( 2 , \beta )$, then $\beta$ is a root of the equation? (1) $y ^ { 5 } - 2 y - 2 = 0$ (2) $y ^ { 5 } - y ^ { 2 } - 1 = 0$ (3) $2 y ^ { 5 } - y ^ { 2 } - 2 = 0$ (4) $2 y ^ { 5 } - 2 y - 1 = 0$

jee-main 2021 Q77 Solving Separable DEs with Initial Conditions View

Which of the following is true for $y ( x )$ that satisfies the differential equation $\frac { d y } { d x } = x y - 1 + x - y ; y ( 0 ) = 0$
(1) $y ( 1 ) = \mathrm { e } ^ { - \frac { 1 } { 2 } } - 1$
(2) $y ( 1 ) = e ^ { \frac { 1 } { 2 } } - e ^ { - \frac { 1 } { 2 } }$
(3) $y ( 1 ) = 1$
(4) $y ( 1 ) = e^{\frac{1}{2}} - 1$

jee-main 2021 Q77 Higher-Order and Special DEs (Proof/Theory) View

A differential equation representing the family of parabolas with axis parallel to $y$-axis and whose length of latus rectum is the distance of the point $( 2 , - 3 )$ from the line $3 x + 4 y = 5$, is given by: (1) $11 \frac { d ^ { 2 } x } { d y ^ { 2 } } = 10$ (2) $11 \frac { d ^ { 2 } y } { d x ^ { 2 } } = 10$ (3) $10 \frac { d ^ { 2 } y } { d x ^ { 2 } } = 11$ (4) $10 \frac { d ^ { 2 } x } { d y ^ { 2 } } = 11$

jee-main 2021 Q78 Solving Separable DEs with Initial Conditions View

If $\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 2 ^ { x + y } - 2 ^ { x } } { 2 ^ { y } } , y ( 0 ) = 1$, then $y ( 1 )$ is equal to :
(1) $\log _ { 2 } \left( 1 + \mathrm { e } ^ { 2 } \right)$
(2) $\log _ { 2 } ( 2 \mathrm { e } )$
(3) $\log _ { 2 } ( 2 + e )$
(4) $\log _ { 2 } ( 1 + e )$

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LFM Pure

Straight Lines & Coordinate Geometry 247

Circles 443

Simultaneous equations 79

Proof 711

Proof by induction 25

Sine and Cosine Rules 195

Radians, Arc Length and Sector Area 35

Trig Graphs & Exact Values 75

Standard trigonometric equations 106

Trig Proofs 53

Quadratic trigonometric equations 36

Trigonometric equations in context 18

Modulus function 49

Matrices 1553

Linear transformations 26

Invariant lines and eigenvalues and vectors 188

Reciprocal Trig & Identities 44

Addition & Double Angle Formulae 98

Harmonic Form 12

Small angle approximation 20

Chain Rule 113

Product & Quotient Rules 18

Differentiating Transcendental Functions 183

Connected Rates of Change 56

Standard Integrals and Reverse Chain Rule 43

Integration by Substitution 116

Integration by Parts 74

Implicit equations and differentiation 43

Differential equations 352

3x3 Matrices 144