CBSE Class 12 Mathematics Differentials Equation MCQs Set 10

Question. The differential equation \( \frac{dy}{dx} = \frac{\sqrt{1 - y^2}}{y} \) determines a family of circle with
(a) variable radii and fixed centre (0, 1)
(b) variable radii and fixed centre (0, -1)
(c) fixed radius 1 and variable centre on x-axis
(d) fixed radius 1 and variable centre on y-axis
Answer: (c) fixed radius 1 and variable centre on x-axis

Question. If \( y' = y + 1 \), \( y(0) = 1 \), then \( y(\ln 2) = \)
(a) 1
(b) 2
(c) 3
(d) 4
Answer: (c) 3

Question. The general solution of the differential equation \( x(1 + y^2)dx + y(1 + x^2)dy = 0 \) is
(a) \( (1 + x^2)(1 + y^2) = 0 \)
(b) \( (1 + x^2)(1 + y^2) = C \)
(c) \( (1 + x^2) = C(1 + y^2) \)
(d) \( (1 + y^4) = C(1 + x^2) \)
Answer: (b) \( (1 + x^2)(1 + y^2) = C \)

Question. If \( \frac{dy}{dx} = y \sin 2x \), \( y(0) = 1 \), then solution is
(a) \( y = e^{\sin^2 x} \)
(b) \( y = \sin^2 x \)
(c) \( y = \cos^2 x \)
(d) \( y = e^{\cos^2 x} \)
Answer: (a) \( y = e^{\sin^2 x} \)

Question. The general solution of the differential equation \( \frac{dy}{dx} = \frac{x^2}{y^2} \) is
(a) \( x^3 - y^3 = c \)
(b) \( x^3 + y^3 = c \)
(c) \( x^2 + y^2 = c \)
(d) \( x^2 - y^2 = c \)
Answer: (a) \( x^3 - y^3 = c \)

Question. The general solution of the differential equation \( \frac{dy}{dx} = \frac{y}{x} \) is
(a) \( y = \frac{k}{x} \)
(b) \( y = k \log x \)
(c) \( y = k x \)
(d) \( \log y = k x \)
Answer: (c) \( y = k x \)

Question. If \( \frac{dy}{dx} = \frac{x+y}{x} \), \( y(1) = 1 \), then \( y = \)
(a) \( x + \ln x \)
(b) \( x^2 + x \ln x \)
(c) \( x e^{x-1} \)
(d) \( x + x \ln x \)
Answer: (d) \( x + x \ln x \)

Question. The order and degree respectively of the differential equation \( \frac{d^2y}{dx^2} + \left(\frac{dy}{dx}\right)^{1/4} + x^{1/5} = 0 \), are
(a) 2 and not defined
(b) 2 and 2
(c) 2 and 3
(d) 3 and 3
Answer: (a) 2 and not defined

Question. Integrating factor of \( x\frac{dy}{dx} - y = x^4 - 3x \) is
(a) \( x \)
(b) \( \log x \)
(c) \( \frac{1}{x} \)
(d) \( -x \)
Answer: (c) \( \frac{1}{x} \)

Question. The number of solutions of \( \frac{dy}{dx} = \frac{y+1}{x-1} \), when \( y(1) = 2 \) is
(a) None of the options
(b) one
(c) two
(d) infinite
Answer: (b) one

Question. Which of the following is a second order differential equation?
(a) \( (y')^2 + x = y^2 \)
(b) \( y' y'' + y = \sin x \)
(c) \( y''' + (y'')^2 + y = 0 \)
(d) \( y' = y^2 \)
Answer: (b) \( y' y'' + y = \sin x \)

Question. Integrating factor of the differential equation \( \frac{dy}{dx} + y\tan x - \sec x = 0 \) is
(a) \( \cos x \)
(b) \( \sec x \)
(c) \( e^{\cos x} \)
(d) \( e^{\sec x} \)
Answer: (b) \( \sec x \)

Question. The solution of the differential equation \( \frac{dy}{dx} = \frac{1 + y^2}{1 + x^2} \) is
(a) \( y = \tan^{-1} x \)
(b) \( y - x = k(1 + xy) \)
(c) \( x = \tan^{-1} y \)
(d) \( \tan (xy) = k \)
Answer: (b) \( y - x = k(1 + xy) \)

Question. The integrating factor of the differential equation \( \frac{dy}{dx} + y = \frac{1+y}{x} \) is
(a) \( \frac{x}{e^x} \)
(b) \( \frac{e^x}{x} \)
(c) \( x e^x \)
(d) \( e^x \)
Answer: (b) \( \frac{e^x}{x} \)

Question. The solution of \( x\frac{dy}{dx} + y = e^x \) is
(a) \( y = \frac{e^x}{x} + \frac{k}{x} \)
(b) \( y = x e^x + c x \)
(c) \( y = x e^x + k \)
(d) \( x = \frac{e^y}{y} + \frac{k}{y} \)
Answer: (a) \( y = \frac{e^x}{x} + \frac{k}{x} \)

Question. Differential equation having solution \( y = Ax + B^3 \) is of order
(a) 3
(b) 2
(c) 1
(d) not defined
Answer: (b) 2

Question. Integrating factor of the differential equation \( \cos x \frac{dy}{dx} + y \sin x = 1 \) is
(a) \( \cos x \)
(b) \( \tan x \)
(c) \( \sec x \)
(d) \( \sin x \)
Answer: (c) \( \sec x \)

Question. Solution of \( \frac{dy}{dx} - y = 1 \), \( y(0) = 1 \) is given by
(a) \( xy = -e^x \)
(b) \( xy = -e^{-x} \)
(c) \( xy = -1 \)
(d) \( y = 2 e^x - 1 \)
Answer: (d) \( y = 2 e^x - 1 \)

Question. Integrating factor of the differential equation \( (1-x^2)\frac{dy}{dx} - xy = 1 \) is
(a) \( -x \)
(b) \( \frac{x}{1+x^2} \)
(c) \( \sqrt{1-x^2} \)
(d) \( \frac{1}{2}\log(1-x^2) \)
Answer: (c) \( \sqrt{1-x^2} \)

Case Based MCQs

Case I : Read the following passage and answer the questions.
In a college hostel accommodating 1000 students, one of the hostellers came in carrying Corona virus, and the hostel was isolated. The rate at which the virus spreads is assumed to be proportional to the product of the number of infected students and remaining students. There are 50 infected students after 4 days.

Question. If n(t) denote the number of students infected by Corona virus at any time t, then maximum value of n(t) is
(a) 50
(b) 100
(c) 500
(d) 1000
Answer: (d) 1000

Question. \( \frac{dn}{dt} \) is proportional to
(a) \( n(1000 - n) \)
(b) \( n(1000 + n) \)
(c) \( n(100 - n) \)
(d) \( n(100 + n) \)
Answer: (a) \( n(1000 - n) \)

Question. The value of n(4) is
(a) 1
(b) 50
(c) 100
(d) 1000
Answer: (b) 50

Question. The most general solution of differential equation formed in given situation is
(a) \( \frac{1}{1000}\log\left(\frac{1000-n}{n}\right) = \lambda t + c \)
(b) \( \log\left(\frac{n}{100-n}\right) = \lambda t + c \)
(c) \( \frac{1}{1000}\log\left(\frac{n}{1000-n}\right) = \lambda t + c \)
(d) None of the options
Answer: (c) \( \frac{1}{1000}\log\left(\frac{n}{1000-n}\right) = \lambda t + c \)

Question. The value of n at any time is given by
(a) \( n(t) = \frac{1000}{1 + 999e^{-0.9906t}} \)
(b) \( n(t) = \frac{1000}{1 - 999e^{-0.9906t}} \)
(c) \( n(t) = \frac{100}{1 - 999e^{-0.996t}} \)
(d) \( n(t) = \frac{100}{999 + e^{1000t}} \)
Answer: (a) \( n(t) = \frac{1000}{1 + 999e^{-0.9906t}} \)

Assertion & Reasoning Based MCQs

Directions: In these questions, a statement of Assertion is followed by a statement of Reason is given. Choose the correct answer out of the following choices:
(a) Assertion and Reason both are correct statements and Reason is the correct explanation of Assertion.
(b) Assertion and Reason both are correct statements but Reason is not the correct explanation of Assertion.
(c) Assertion is correct statement but Reason is wrong statement.
(d) Assertion is wrong statement but Reason is correct statement.

Question. Assertion: The elimination of four arbitrary constants in \( y = (c_1 + c_2 + c_3 e^{c_4})x \) results into a differential equation of the first order \( x \frac{dy}{dx} = y \).
Reason: Elimination of n arbitrary constants requires in general, a differential equation of the \( n^{\text{th}} \) order.
(a) Assertion and Reason both are correct statements and Reason is the correct explanation of Assertion.
(b) Assertion and Reason both are correct statements but Reason is not the correct explanation of Assertion.
(c) Assertion is correct statement but Reason is wrong statement.
(d) Assertion is wrong statement but Reason is correct statement.
Answer: (a) Assertion and Reason both are correct statements and Reason is the correct explanation of Assertion.

Question. Assertion: 'x' is not an integrating factor for the differential equation \( x \frac{dy}{dx} + 2y = e^x \).
Reason: \( x \left( x \frac{dy}{dx} + 2y \right) = \frac{d}{dx} (x^2 y) \).
(a) Assertion and Reason both are correct statements and Reason is the correct explanation of Assertion.
(b) Assertion and Reason both are correct statements but Reason is not the correct explanation of Assertion.
(c) Assertion is correct statement but Reason is wrong statement.
(d) Assertion is wrong statement but Reason is correct statement.
Answer: (b) Assertion and Reason both are correct statements but Reason is not the correct explanation of Assertion.

Question. Assertion: \( x \sin x \frac{dy}{dx} + (x + x \cos x + \sin x)y = \sin x \), \( y\left(\frac{\pi}{2}\right) = 1 - \frac{2}{\pi} \Rightarrow \lim_{x \to 0} y(x) = \frac{1}{3} \).
Reason: The differential equation is linear with integrating factor \( x(1 - \cos x) \).
(a) Assertion and Reason both are correct statements and Reason is the correct explanation of Assertion.
(b) Assertion and Reason both are correct statements but Reason is not the correct explanation of Assertion.
(c) Assertion is correct statement but Reason is wrong statement.
(d) Assertion is wrong statement but Reason is correct statement.
Answer: (a) Assertion and Reason both are correct statements and Reason is the correct explanation of Assertion.

Question. Assertion: If \( \frac{dy}{dx} + xy = x^3 y^3 \), \( x > 0 \), \( y \ge 0 \) and \( y(0) = 1 \), then \( y(1) = \frac{1}{\sqrt{2}} \).
Reason: The differential equation is linear with integrating factor \( e^x \).
(a) Assertion and Reason both are correct statements and Reason is the correct explanation of Assertion.
(b) Assertion and Reason both are correct statements but Reason is not the correct explanation of Assertion.
(c) Assertion is correct statement but Reason is wrong statement.
(d) Assertion is wrong statement but Reason is correct statement.
Answer: (c) Assertion is correct statement but Reason is wrong statement.