Previous Year Papers

Download Solved Question Papers Free for Offline Practice and view Solutions Online.

Test Series

Take Zigya Full and Sectional Test Series. Time it out for real assessment and get your results instantly.

Test Yourself

Practice and master your preparation for a specific topic or chapter. Check you scores at the end of the test.

Differential Equations

Multiple Choice Questions

91.

The solution of the differential equation $\frac{d y}{d x} = \sin (x + y) \tan (x + y) - 1$ is

$\csc (x + y) + \tan (x + y) = x + c$
$x + \csc (x + y) = c$
$x + \tan (x + y) = c$
$x + \sec (x + y) = c$

92.

The differential equation of all straight lines touching the circle x² + y² = a² is

${(y - \frac{dy}{dx})}^{2} = a^{2} [1 + {(\frac{dy}{dx})}^{2}]$
${(y - x \frac{dy}{dx})}^{2} = a^{2} [1 + {(\frac{dy}{dx})}^{2}]$
$(y - x \frac{dy}{dx}) = a^{2} [1 + \frac{dy}{dx}]$
$(y - \frac{dy}{dx}) = a^{2} [1 - \frac{dy}{dx}]$

93.

The differential equation $|\frac{dy}{dx}| + |y| + 3 = 0$ admits

infinite number of solutions
no solutions
a unique solution
many solutions

94.

Solution of the differential equation $xdy - ydx - \sqrt{x^{2} + y^{2}} dx = 0$

$y - \sqrt{x^{2} + y^{2}} = {cx}^{2}$
$y + \sqrt{x^{2} + y^{2}} = {cx}^{2}$
$y + \sqrt{x^{2} + y^{2}} = {cy}^{2}$
$x - \sqrt{x^{2} + y^{2}} = {cy}^{2}$

95.

If $xsin (\frac{x}{y}) dy = [ysin (\frac{y}{x}) - x] dx$ and $y (1) = \frac{π}{2}$ , then the value of $\cos (\frac{y}{x})$ is equal to :

x
$\frac{1}{x}$
$\log (x)$
e^x

96.
The differential equation of the system of all circles of radius r in the xy plane is :

${[1 + {(\frac{dy}{dx})}^{3}]}^{2} = r^{2} {(\frac{d^{2} y}{{dx}^{2}})}^{2}$

${[1 + {(\frac{dy}{dx})}^{3}]}^{2} = r^{2} {(\frac{d^{2} y}{{dx}^{2}})}^{3}$

${[1 + {(\frac{dy}{dx})}^{2}]}^{3} = r^{2} {(\frac{d^{2} y}{{dx}^{2}})}^{2}$

${[1 + {(\frac{dy}{dx})}^{2}]}^{3} = r^{2} {(\frac{d^{2} y}{{dx}^{2}})}^{3}$

${[1 + {(\frac{dy}{dx})}^{2}]}^{3} = r^{2} {(\frac{d^{2} y}{{dx}^{2}})}^{2}$

The equation of the family of circles of radius r is

(x - a)² + (y - b)² = r² ...(i)

where a and b are arbitrary constants.

On differentiating Eq. (i) w.r.t. x, we get

2(x - a) + 2(y - b) $\frac{dy}{dx}$ = 0

$\Rightarrow (x - a) + (y - b) \frac{dy}{dx} = 0$ ...(ii)

On differentiating Eq. (ii) w.r.t. x, we get

$1 + (y - b) \frac{d^{2} y}{d x^{2}} + {(\frac{d y}{d x})}^{2} = 0 \Rightarrow (y - b) = - \frac{1 + {(\frac{dy}{dx})}^{2}}{\frac{d^{2} y}{{dx}^{2}}} . . . (iii)$

On putting the value of (y - b) in Eq. (ii), we get

$(x - a) = \frac{[1 + {(\frac{dy}{dx})}^{2}] \frac{dy}{dx}}{\frac{d^{2} y}{{dx}^{2}}} . . . (iv)$

On putting the value of (y - b) and (x - a), in Eq. (i), we get

$\frac{{[1 + {(\frac{dy}{dx})}^{2}]}^{2} {(\frac{dy}{dx})}^{2}}{{(\frac{d^{2} y}{{dx}^{2}})}^{2}} + \frac{{[1 + {(\frac{dy}{dx})}^{2}]}^{2}}{{(\frac{d^{2} y}{{dx}^{2}})}^{2}} = r^{2} \Rightarrow {[1 + {(\frac{dy}{dx})}^{2}]}^{3} = r^{2} {[\frac{d^{2} y}{{dx}^{2}}]}^{2} This is the required differential equation .$

97.

The general solution of the differential equation

$\frac{d^{2} y}{{dx}^{2}} + 2 \frac{d y}{d x} + y = 2 e^{3 x}$ is given by

$y = (c_{1} + c_{2} x) e^{x} + \frac{e^{3 x}}{8}$
$y = (c_{1} + c_{2} x) e^{- x} + \frac{e^{- 3 x}}{8}$
$y = (c_{1} + c_{2} x) e^{- x} + \frac{e^{3 x}}{8}$
$y = (c_{1} + c_{2} x) e^{x} + \frac{e^{- 3 x}}{8}$

98.

The solution of the differential ydx + (x - y³)dy = 0 is:

xy = $\frac{1}{3} y^{3} + c$
xy = y⁴ + c
y⁴ = 4xy + c
4y = y³ + c

99.

If the distances covered by a particle in time t is proportional to the cube root of its velocity, then the acceleration is

a constant
$\propto s^{3}$
$\propto \frac{1}{s^{3}}$
$\propto s^{5}$

100.

The solution of the differential equation $\frac{dy}{dx} = \frac{y}{x} + \frac{ϕ (\frac{y}{x})}{ϕ' (\frac{y}{x})}$ is

$xϕ (\frac{y}{x}) = k$
$ϕ (\frac{y}{x}) = kx$
$yϕ (\frac{y}{x}) = k$
$ϕ (\frac{y}{x}) = ky$

Previous Year Papers

Test Series

Test Yourself

Multiple Choice Questions

96. The differential equation of the system of all circles of radius r in the xy plane is : 1 + dydx32 = r2d2ydx22 1 + dydx32 = r2d2ydx23 1 + dydx23 = r2d2ydx22 1 + dydx23 = r2d2ydx23