CUET Mathematics Calculus > Differential Equations PYQs

$2x^2 - y = C$; C is an arbitrary constant

$2x^2 - \frac{1}{y} = C$; C is an arbitrary constant

$2x^2 + \frac{1}{y^2} = C$; C is an arbitrary constant

$2x^2 + \frac{1}{y} = C$; C is an arbitrary constant

(A) - (II), (B) - (IV), (C) - (III), (D) - (I)

(A) - (III), (B) - (IV), (C) - (I), (D) - (II)

(A) - (III), (B) - (IV), (C) - (II), (D) - (I)

(A) - (II), (B) - (III), (C) - (IV), (D) - (I)

(A) and (C) only

(B) and (D) only

(B) and (C) only

(A) and (D) only

0

1

3

4

3

4

5

2

$y = x + \log|x|$

$y = 2 + \log|x|$

$y = x^2 + \log|x|$

$y = x^2 + \log|x| + 1$

2

3

4

5

(A), (B) and (C) only

(B), (C) and (D) only

(B) and (D) only

(A), (C) and (D) only

1

9

5

7

(A) - (II), (B) - (I), (C) - (IV), (D) - (III)

(A) - (II), (B) - (I), (C) - (III), (D) - (IV)

(A) - (I), (B) - (II), (C) - (III), (D) - (IV)

(A) - (III), (B) - (IV), (C) - (II), (D) - (I)

$|y| = |c\log_e(xy)|$

$|x| = |cy|$

$|x| = |c\log_e y|$

$|y| = |c\log_e x|$

$3xy + x^3 = C$, C is an arbitrary constant

$3xy + y^3 = C$, C is an arbitrary constant

$3xy - y^3 = C$, C is arbitrary constant

$3xy - x^3 = C$, C is arbitary constant

(A) - (I), (B) - (IV), (C) - (II), (D) - (III)

(A) - (II), (B) - (I), (C) - (III), (D) - (IV)

(A) - (III), (B) - (IV), (C) - (II), (D) - (I)

(A) - (IV), (B) - (I), (C) - (II), (D) - (III)

$y = \frac{x}{2}\log_e(x^2 + y^2) + C$ : C is an arbitary constant

$y = \sqrt{x^2 + y^2} + C$ : C is an arbitrary constant

$y = \frac{1}{2}\log_e(x^2 + y^2) + c$ : C is an arbitrary constant

$\tan^{-1}\frac{y}{x} = \frac{1}{2}\log_e(x^2 + y^2) + C$: C is an arbitrary constant

$e^{-x/2}$

$e^{-x^2/2}$

$e^{x^2/2}$

$e^{x/2}$

$\sqrt{x} + \sqrt{y} = C$; C is an arbitrary constant

$\sqrt{x} - \sqrt{y} = C$; C is an arbitrary constant

$\sqrt{\frac{y}{x}} + C = 0$; C is arbitrary constant

$\sqrt{\frac{x}{y}} + C = 0$; C is an arbitrary constant

1

2

3

4

$\tan\left(\frac{y}{x}\right) = cx$, where c is an arbitary constant

$\sin\left(\frac{y}{x}\right) = cx$, where c is an arbitary constant

$\sin\left(\frac{x}{y}\right) = cx$, where c is an arbitary constant

$\sin\left(\frac{y}{x}\right) = cx^2$, where c is an arbitary constant

1

2

3

4

(A) and (B) only

(A), (B) and (C) only

(A), (B) and (D) only

(A), (C) and (D) only

$5e^{5x} + 2e^{-2y} + C = 0$ : C is an arbitrary constant

$2e^{5x} + 5e^{-2y} + C = 0$ : C is an arbitrary constant

$2e^{-5x} + 5e^{2y} + C = 0$ : C is an arbitrary constant

$5e^{-5x} + 2e^{2y} + C = 0$ : C is an arbitrary constant

(A) and (C) only

(A), (B) and (C) only

(B), (C) and (D) only

(C) and (D) only

3

6

2

9

$x^2\frac{d^2y}{dx^2} + x\frac{dy}{dx} + y = 0$

$Bx^2\frac{d^2y}{dx^2} - x\frac{dy}{dx} - y = 0$

$x^2\frac{d^2y}{dx^2} + x\frac{dy}{dx} + y = 0$

$x^2\frac{d^2y}{dx^2} + x\frac{dy}{dx} - y = 0$

(A) - (III), (B) - (II), (C) - (IV), (D) - (I)

(A) - (I), (B) - (II), (C) - (IV), (D) - (III)

(A) - (III), (B) - (IV), (C) - (II), (D) - (I)

(A) - (I), (B) - (III), (C) - (IV), (D) - (II)

$y = e^{x^2}$

$y = e^{4x^2}$

$y = e^{8x^2}$

$y = e^x$

(A), (C) and (D) only

(A) and (B) only

(A) and (D) only

(A), (B) and (D) only

0

4

2

1

$y = \frac{1}{x^3}$

$y = \frac{1}{x^2}$

$x = \frac{1}{y}$

$x = \frac{1}{y^2}$

$(4x + 5)dy + (3y - 4)dx = 0$

$x^2y dx - (x^3 + y^3)dy = 0$

$(x^3 + 2y^2x)dy + 2xydx = 0$

$x^2 dy - y^2 dx = \sqrt{x^2 + y^2} dx$

$y + \log_e\left|\frac{y-2}{x}\right| = c$, ($c$ is an arbitrary constant)

$\frac{y}{x} + \log_e\left|\frac{(y-x)^2}{x}\right| = c$, ($c$ is an arbitrary constant)

$y - \log_e\left|\frac{y-x}{x}\right| = c$, ($c$ is an arbitrary constant)

$y^2 + \log_e\left|\frac{(y-x)^2}{x}\right| = c$, ($c$ is an arbitrary constant)

(A) - (IV), (B) - (III), (C) - (II), (D) - (I)

(A) - (III), (B) - (I), (C) - (II), (D) - (IV)

(A) - (I), (B) - (II), (C) - (III), (D) - (IV)

(A) - (III), (B) - (IV), (C) - (I), (D) - (II)

order 2, degree 2

order 2, degree not defined

order not defined, degree 2

order 1, degree 2

straight line passing through origin

a circle whose centre is at origin

a rectangular hyperbola

parabola whose vertex is at origin

$y = e^{-x^2/2} + 1$

$y = (x+1)e^{x^2/2}$

$y = xe^{x^2/2} + 1$

$y = (x-1)e^{x^2/2}$

(A) - (IV), (B) - (III), (C) - (II), (D) - (I)

(A) - (II), (B) - (IV), (C) - (III), (D) - (I)

(A) - (IV), (B) - (II), (C) - (I), (D) - (III)

(A) - (II), (B) - (III), (C) - (IV), (D) - (I)

20°F

40°F

30° F

50°F

(A) - (III), (B) - (I), (C) - (IV), (D) - (II)

(A) - (II), (B) - (I), (C) - (IV), (D) - (III)

(A) - (III), (B) - (I), (C) - (II), (D) - (IV)

(A) - (II), (B) - (I), (C) - (III), (D) - (IV)

$\tan^{-1} y + x + \frac{x^3}{3} = C$

$\tan^{-1} y - x - \frac{x^3}{3} = C$

$\tan^{-1} x - y - \frac{y^3}{3} = C$

$\tan^{-1} x + y + \frac{y^3}{3} = C$

$x = e^{ \tan\frac{y}{x}+1}$

$x = e^{\tan\frac{y}{x}-1}$

$x = e^{\cot\frac{y}{x}-1}$

$x = e^{\cot\frac{y}{x}+1}$

(A) - (IV), (B) - (II), (C) - (III), (D) - (I)

(A) - (IV), (B) - (II), (C) - (I), (D) - (III)

(A) - (II), (B) - (IV), (C) - (I), (D) - (III)

(A) - (II), (B) - (IV), (C) - (III), (D) - (I)

(A) - (I), (B) - (II), (C) - (III), (D) - (IV)

(A) - (II), (B) - (III), (C) - (IV), (D) - (I)

(A) - (IV), (B) - (III), (C) - (II), (D) - (I)

(A) - (IV), (B) - (III), (C) - (II), (D) - (IV)

$r = r_0 e^{t^2/2}$

$r = -r_0 e^{t^2/2}$

$r = r_0 e^{-t^2/2}$

$r = -r_0 e^{-t^2/2}$

$y = x\log_e x + C$ ; C is an arbitrary constant

$y = x(\log_e x + C)$ ; C is an arbitrary constant

$y = x((\log_e x)^2 + C)$ ; C is an arbitrary constant

$y = x(2(\log_e x)^2 + C)$ ; C is an arbitrary constant

(A) only

(B) and (C) only

(B) and (D) only

(A), (B), (C) and (D)

$\frac{d^2y}{dx^2} + 9y = 0$

$\frac{d^2y}{dx^2} + 3y = 0$

$\frac{d^2y}{dx^2} = 9y$

$\frac{d^2y}{dx^2} = 3y$

(A) - (IV), (B) - (I), (C) - (III), (D) - (II)

(A) - (I), (B) - (IV), (C) - (II), (D) - (III)

(A) - (II), (B) - (III), (C) - (IV), (D) - (I)

(A) - (III), (B) - (IV), (C) - (II), (D) - (I)

$|(x - 1)(e^y - 2)| = 1$

$|(x + 1)(e^y - 2)| = 1$

$|(x + 1)(e^y + 2)| = 1$

$|(x - 2)(e^y - 2)| = 1$

(A) - (IV), (B) - (I), (C) - (II), (D) - (III)

(A) - (IV), (B) - (III), (C) - (II), (D) - (I)

(A) - (I), (B) - (II), (C) - (IV), (D) - (III)

(A) - (III), (B) - (IV), (C) - (II), (D) - (I)

$a = -b$

$a = b$

$a = -2b$

$a = 3b$

(A) - (IV), (B) - (I), (C) - (II), (D) - (III)

(A) - (II), (B) - (IV), (C) - (III), (D) - (I)

(A) - (II), (B) - (IV), (C) - (I), (D) - (III)

(A) - (I), (B) - (II), (C) - (IV), (D) - (III)

1

2

3

4

x

-x

$\frac{1}{x}$

$-\frac{1}{x}$

$(1 + y^2)(1 + x^2) = C$; where C is an arbitrary constant

$(1 - y^2)(1 - x^2) = C$; where C is an arbitrary constant

$(1 - y^2)(1 + x^2) = C$; where C is an arbitrary constant

$(1 + y^2)(1 - x^2) = C$; where C is an arbitrary constant

$\frac{d^2y}{dx^2} + 4\frac{dy}{dx} + 4y = 0$

$\frac{d^2y}{dx^2} - 4\frac{dy}{dx} - 4y = 0$

$\frac{d^2y}{dx^2} - 4\frac{dy}{dx} + 4y = 0$

$\frac{d^2y}{dx^2} + 4\frac{dy}{dx} - 4y = 0$

(A) - (I), (B) - (II), (C) - (III), (D) - (IV)

(A) - (II), (B) - (III), (C) - (IV), (D) - (I)

(A) - (IV), (B) - (III), (C) - (I), (D) - (II)

(A) - (IV), (B) - (III), (C) - (II), (D) - (I)

$be^{-by} + ae^{ax} = C$

$-be^{-by} + ae^{ax} = C$

$ae^{-by} + be^{ax} = C$

$e^{ax} + e^{by} = C$

$\log \left| \frac{y + \sqrt{1 + y^2}}{x + \sqrt{1 + x^2}} \right| = C$

$\log \left| \frac{y + \sqrt{1 + y^2}}{\sqrt{x + \sqrt{1 + x^2}}} \right| = C$

$\log \left| \frac{y + \sqrt{1 + y^2}}{\sqrt{x + \sqrt{1 + x^2}}} \right| = \frac{x}{2}\sqrt{1 + x^2} + C$

$\log \left| \frac{y + \sqrt{1 + y^2}}{x + \sqrt{1 + x^2}} \right| = \frac{x}{2}\sqrt{1 + x^2} + C$

(A) - (IV), (B) - (III), (C) - (II), (D) - (I)

(A) - (IV), (B) - (II), (C) - (I), (D) - (III)

(A) - (III), (B) - (IV), (C) - (I), (D) - (II)

(A) - (III), (B) - (II), (C) - (I), (D) - (IV)

$y^2 = 3x^3$

$y^3 = 3x^2$

$y^4 = 3x^2$

$y = 3x^2$

6

1

3

2

$3e³ˣ + 4e⁻⁴ʸ + 7 = 0$

$3e³ˣ - 4e⁻⁴ʸ - 7 = 0$

$4e³ˣ + 3e⁻⁴ʸ - 7 = 0$

$4e³ˣ - 3e⁻⁴ʸ + 7 = 0$

(B) and (D) only

(A) only

(C) only

(C) and (D) only

$y\sec x - \tan x = c$, where c is an arbitary constant

$y\sec x + \tan x = c$, where c is an arbitary constant

$y \tan x - \sec x = c$, where c is an arbitary constant

$y \tan x + \sec x = c$, where c is an arbitary constant

$xy^2 = x^2 + 1$

$y^2 = 2x^2 + 1$

$y^2 = 2x^2 + xy + 1$

$x^2 = 3y^2 + 2xy + 1$

(A) - (II), (B) - (III), (C) - (IV), (D) - (I)

(A) - (IV), (B) - (I), (C) - (III), (D) - (II)

(A) - (IV), (B) - (III), (C) - (II), (D) - (I)

(A) - (IV), (B) - (I), (C) - (II), (D) - (III)

(A), (B) and (C) only

(A), (D) and (E) only

(A) and (D) only

(B) and (C) only

$y = \frac{x^3}{7} + cx^{-4}$; $c$ is arbitrary constant.

$y = \frac{x^3}{7} + cx^{-7}$; $c$ is arbitrary constant.

$y = \frac{x^7}{7} + cx^{-4}$; $c$ is arbitrary constant.

$y = \frac{x^7}{7} + cx^{-3}$; $c$ is arbitrary constant.

(C) and (D) only

(A) only

(B) only

(C) only

$\log_e|y| + \frac{1}{x} + \frac{1}{y} - x = c$, where $c$ is constant of integration

$\log_e|y| - \frac{1}{x} + \frac{1}{y} + x = c$, where $c$ is constant of integration

$\log_e|x| - \frac{1}{x} + \frac{1}{y} + x = c$, where $c$ is constant of integration

$\log_e|x| + \frac{1}{x} + \frac{1}{y} + x = c$, where $c$ is constant of integration

$y = C(1 + e^x)$, C is an arbitary constant

$\frac{C}{e^x} = \frac{y}{1 + e^x}$, C is an arbitary constant

$C = y(1 + e^x)$, C is an arbitary constant