Differential Equations · JEE Main previous year questions

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2024 22 questions

Q162024 · 9 Apr 2024 · Shift 1
The solution curve, of the differential equation $2 y \frac{d y}{d x} + 3 = 5 \frac{d y}{d x}$ , passing through the point $(0, 1)$ is a conic, whose vertex lies on the line :
1. A. $2 x + 3 y = - 9$
2. B. $2 x + 3 y = - 6$
3. C. $2 x + 3 y = 9$
4. D. $2 x + 3 y = 6$
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Answer: C
Q62024 · 8 Apr 2024 · Shift 1
Let $f (x)$ be a positive function such that the area bounded by $y = f (x), y = 0$ from $x = 0$ to $x = a > 0$ is $e^{- a} + 4 a^{2} + a - 1$ . Then the differential equation, whose general solution is $y = c_{1} f (x) + c_{2}$ , where $c_{1}$ and $c_{2}$ are arbitrary constants, is
1. A. $(8 e^{x} + 1) \frac{d ^{2} y}{d x ^{2}} - \frac{d y}{d x} = 0$
2. B. $(8 e^{x} + 1) \frac{d ^{2} y}{d x ^{2}} + \frac{d y}{d x} = 0$
3. C. $(8 e^{x} - 1) \frac{d ^{2} y}{d x ^{2}} - \frac{d y}{d x} = 0$
4. D. $(8 e^{x} - 1) \frac{d ^{2} y}{d x ^{2}} + \frac{d y}{d x} = 0$
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Answer: B
Q92024 · 8 Apr 2024 · Shift 1
Let $y = y (x)$ be the solution of the differential equation $(1 + y^{2}) e^{t a n x} d x + cos^{2} x (1 + e^{2 t a n x}) d y = 0, y (0) = 1$ . Then $y (\frac{π}{4})$ is equal to
1. A. $\frac{1}{e ^{2}}$
2. B. $\frac{2}{e ^{2}}$
3. C. $\frac{2}{e}$
4. D. $\frac{1}{e}$
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Answer: D
Q12024 · 8 Apr 2024 · Shift 2
Let $y = y (x)$ be the solution curve of the differential equation $sec y \frac{d y}{d x} + 2 x sin y = x^{3} cos y, y (1) = 0$ . Then $y (3)$ is equal to:
1. A. $\frac{π}{6}$
2. B. $\frac{π}{12}$
3. C. $\frac{π}{3}$
4. D. $\frac{π}{4}$
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Answer: D
Q102024 · 5 Apr 2024 · Shift 2
If $y (θ) = \frac{2 c o s θ + c o s 2 θ}{c o s 3 θ + 4 c o s 2 θ + 5 c o s θ + 2}$ , then at $θ = \frac{π}{2}, y^{''} + y^{'} + y$ is equal to :
1. A. $\frac{1}{2}$
2. B.1
3. C. $\frac{3}{2}$
4. D.2
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Answer: D
Q212024 · 5 Apr 2024 · Shift 2· Numerical answer
Let $y = y (x)$ be the solution of the differential equation $\frac{d y}{d x} + \frac{2 x}{( 1 + x ^{2} ) ^{2}} y = x e^{\frac{1}{( 1 + x ^{2} )}}; y (0) = 0$ . Then the area enclosed by the curve $f (x) = y (x) e^{- \frac{1}{( 1 + x ^{2} )}}$ and the line $y - x = 4$ is _________.
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Answer: 18
Q132024 · 4 Apr 2024 · Shift 1
If the solution $y = y (x)$ of the differential equation $(x^{4} + 2 x^{3} + 3 x^{2} + 2 x + 2) d y - (2 x^{2} + 2 x + 3) d x = 0$ satisfies $y (- 1) = - \frac{π}{4}$ , then $y (0)$ is equal to :
1. A. $- \frac{π}{12}$
2. B. $\frac{π}{2}$
3. C.0
4. D. $\frac{π}{4}$
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Answer: D
Q212024 · 4 Apr 2024 · Shift 1· Numerical answer
Let the solution $y = y (x)$ of the differential equation $\frac{d y}{d x} - y = 1 + 4 sin x$ satisfy $y (π) = 1$ . Then $y (\frac{π}{2}) + 10$ is equal to ____.
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Answer: 7
Q112024 · 4 Apr 2024 · Shift 2
Let $y = y (x)$ be the solution of the differential equation $(x^{2} + 4)^{2} d y + (2 x^{3} y + 8 x y - 2) d x = 0$ . If $y (0) = 0$ , then $y (2)$ is equal to
1. A. $2 π$
2. B. $\frac{π}{8}$
3. C. $\frac{π}{16}$
4. D. $\frac{π}{32}$
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Answer: D
Q272024 · 4 Apr 2024 · Shift 2· Numerical answer
Let $y = y (x)$ be the solution of the differential equation $(x + y + 2)^{2} d x = d y, y (0) = - 2$ . Let the maximum and minimum values of the function $y = y (x)$ in $[0, \frac{π}{3}]$ be $α$ and $β$ , respectively. If $(3 α + π)^{2} + β^{2} = γ + δ 3, γ, δ \in Z$ , then $γ + δ$ equals _________.
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Answer: 31
Q142024 · 1 Feb 2024 · Shift 1
Let $y = y (x)$ be the solution of the differential equation $\frac{d y}{d x} = 2 x (x + y)^{3} - x (x + y) - 1, y (0) = 1$ . Then, $(\frac{1}{2} + y (\frac{1}{2}))^{2}$ equals :
1. A. $\frac{4}{4 + e}$
2. B. $\frac{3}{3 - e}$
3. C. $\frac{2}{1 + e}$
4. D. $\frac{1}{2 - e}$
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Answer: D
Q72024 · 31 Jan 2024 · Shift 1
Let $y = y (x)$ be the solution of the differential equation $\frac{d y}{d x} = \frac{( t a n x ) + y}{s i n x ( s e c x - s i n x t a n x )}, x \in (0, \frac{π}{2})$ satisfying the condition $y (\frac{π}{4}) = 2$ . Then, $y (\frac{π}{3})$ is
1. A. $3 (2 + lo g_{e} 3)$
2. B. $3 (1 + 2 lo g_{e} 3)$
3. C. $3 (2 + lo g_{e} 3)$
4. D. $\frac{3}{2} (2 + lo g_{e} 3)$
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Answer: C
Q162024 · 31 Jan 2024 · Shift 1
The solution curve of the differential equation $y \frac{d x}{d y} = x (lo g_{e} x - lo g_{e} y + 1), x > 0, y > 0$ passing through the point $(e, 1)$ is
1. A. $lo g_{e} \frac{y}{x} = y^{2}$
2. B. $lo g_{e} \frac{y}{x} = x$
3. C. $lo g_{e} \frac{x}{y} = y$
4. D. $2 lo g_{e} \frac{x}{y} = y + 1$
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Answer: C
Q222024 · 31 Jan 2024 · Shift 2· Numerical answer
Let $y = y (x)$ be the solution of the differential equation $sec^{2} x d x + (e^{2 y} tan^{2} x + tan x) d y = 0, 0 < x < \frac{π}{2}, y (π /4) = 0.$ If $y (π /6) = α$ , then $e^{8 α}$ is equal to ____.
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Answer: 9
Q152024 · 30 Jan 2024 · Shift 1
Let $y = y (x)$ be the solution of the differential equation $sec x d y + {2 (1 - x) tan x + x (2 - x)} d x = 0$ such that $y (0) = 2$ . Then $y (2)$ is equal to:
1. A. $2 {sin (2) + 1}$
2. B.2
3. C.1
4. D. $2 {1 - sin (2)}$
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Answer: B
Q222024 · 30 Jan 2024 · Shift 1· Numerical answer
Let $y = y (x)$ be the solution of the differential equation $(1 - x^{2}) d y = [x y + (x^{3} + 2) 3 (1 - x^{2})] d x, - 1 < x < 1, y (0) = 0$ . If $y (\frac{1}{2}) = \frac{m}{n}$ , m and n are co-prime numbers, then $m + n$ is equal to _________.
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Answer: 97
Q302024 · 29 Jan 2024 · Shift 1· Numerical answer
If the solution curve $y = y (x)$ of the differential equation $(1 + y^{2}) (1 + lo g_{e} x) d x + x d y = 0$ , $x > 0$ passes through the point $(1, 1)$ and $y (e) = \frac{α - t a n ( \frac{3}{2} )}{β + t a n ( \frac{3}{2} )}$ , then $α + 2 β$ is _______.
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Answer: 3
Q122024 · 29 Jan 2024 · Shift 2
If $sin (\frac{y}{x}) = lo g_{e} ∣ x ∣ + \frac{α}{2}$ is the solution of the differential equation $x cos (\frac{y}{x}) \frac{d y}{d x} = y cos (\frac{y}{x}) + x$ and $y (1) = \frac{π}{3}$ , then $α^{2}$ is equal to
1. A.12
2. B.9
3. C.4
4. D.3
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Answer: D
Q62024 · 27 Jan 2024 · Shift 1
Let $x = x (t)$ and $y = y (t)$ be solutions of the differential equations $\frac{d x}{d t} + a x = 0$ and $\frac{d y}{d t} + b y = 0$ respectively, $a, b \in R$ . Given that $x (0) = 2$ ; $y (0) = 1$ and $3 y (1) = 2 x (1)$ , the value of $t$ , for which $x (t) = y (t)$ , is :
1. A. $lo g_{\frac{2}{3}} 2$
2. B. $lo g_{\frac{4}{3}} 2$
3. C. $lo g_{4} 3$
4. D. $lo g_{3} 4$
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Answer: B
Q222024 · 27 Jan 2024 · Shift 1· Numerical answer
If the solution of the differential equation $(2 x + 3 y - 2) d x + (4 x + 6 y - 7) d y = 0$ , $y (0) = 3$ , is $α x + β y + 3 lo g_{e} ∣2 x + 3 y - γ ∣ = 6$ , then $α + 2 β + 3 γ$ is equal to ____.
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Answer: 29
Q122024 · 27 Jan 2024 · Shift 2
If $y = y (x)$ is the solution curve of the differential equation $(x^{2} - 4) d y - (y^{2} - 3 y) d x = 0$ , $x > 2, y (4) = \frac{3}{2}$ and the slope of the curve is never zero, then the value of $y (10)$ equals :
1. A. $\frac{3}{1 + ( 8 ) ^{1/4}}$
2. B. $\frac{3}{1 - ( 8 ) ^{1/4}}$
3. C. $\frac{3}{1 - 2 2}$
4. D. $\frac{3}{1 + 2 2}$
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Answer: A
Q302024 · 27 Jan 2024 · Shift 2· Numerical answer
If the solution curve, of the differential equation $\frac{d y}{d x} = \frac{x + y - 2}{x - y}$ passing through the point $(2, 1)$ is $tan^{- 1} (\frac{y - 1}{x - 1}) - \frac{1}{β} lo g_{e} (α + (\frac{y - 1}{x - 1})^{2}) = lo g_{e} ∣ x - 1∣$ , then $5 β + α$ is equal to _______.
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Answer: 11