CBSE Class 12 Mathematics Differentials Equation MCQs Set 07

Question. The degree of the differential equation \( \left(1 + \frac{dy}{dx}\right)^3 = \left(\frac{d^2y}{dx^2}\right)^2 \) is ]
(a) 1
(b) 2
(c) 3
(d) 4
Answer: (b) 2

Question. The degree of the differential equation \( \frac{d^2y}{dx^2} + 3 \left(\frac{dy}{dx}\right)^2 = x^2 \log \left(\frac{d^2y}{dx^2}\right) \) is 
(a) 1
(b) 2
(c) 3
(d) Not defined
Answer: (d) Not defined

Question. The order and degree of differential equation \( \left[1 + \left(\frac{dy}{dx}\right)^2\right]^2 = \frac{d^2y}{dx^2} \) respectively, are
(a) 1, 2
(b) 2, 2
(c) 2, 1
(d) 4, 2
Answer: (c) 2, 1

Question. The order of the differential equation of all circles of given radius \( a \) is
(a) 1
(b) 2
(c) 3
(d) 4
Answer: (b) 2

Question. The solution of the differential equation \( 2x . \frac{dy}{dx} - y = 3 \) represents a family of
(a) straight lines
(b) circles
(c) parabolas
(d) ellipses
Answer: (c) parabolas

Question. The integrating factor of the differential equation \( \frac{dy}{dx} (x \log x) + y = 2 \log x \) is 
(a) \( e^x \)
(b) \( \log x \)
(c) \( \log(\log x) \)
(d) \( x \)
Answer: (b) \( \log x \)

Question. A solution of the differential equation \( \left(\frac{dy}{dx}\right)^2 - x \frac{dy}{dx} + y = 0 \) is 
(a) \( y = 2 \)
(b) \( y = 2x \)
(c) \( y = 2x - 4 \)
(d) \( y = 2x^2 - 4 \)
Answer: (c) \( y = 2x - 4 \)

Question. Which of the following is not a homogeneous function of \( x \) and \( y \)?
(a) \( x^2 + 2xy \)
(b) \( 2x - y \)
(c) \( \cos^2\left(\frac{y}{x}\right) + \frac{y}{x} \)
(d) \( \sin x - \cos y \)
Answer: (d) \( \sin x - \cos y \)

Question. Solution of the differential equation \( \frac{dx}{x} + \frac{dy}{y} = 0 \) is
(a) \( \frac{1}{x} + \frac{1}{y} = c \)
(b) \( \log x . \log y = c \)
(c) \( xy = c \)
(d) \( x + y = c \)
Answer: (c) \( xy = c \)

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

Question. The degree of the differential equation \( \left(\frac{d^2y}{dx^2}\right)^2 + \left(\frac{dy}{dx}\right)^2 = x \sin \left(\frac{dy}{dx}\right) \) is
(a) 1
(b) 2
(c) 3
(d) Not defined
Answer: (d) Not defined

Question. The degree of the differential equation \( \left[1 + \left(\frac{dy}{dx}\right)^2\right]^{3/2} = \frac{d^2y}{dx^2} \) is
(a) 4
(b) \( \frac{3}{2} \)
(c) Not defined
(d) 2
Answer: (d) 2

Question. The order and degree of a differential equation \( \frac{d^2y}{dx^2} + \left(\frac{dy}{dx}\right)^{\frac{1}{4}} + x^{\frac{1}{5}} = 0 \), respectively, 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. If \( y = e^{-x}(A \cos x + B \sin x) \), then it is a solution of
(a) \( \frac{d^2y}{dx^2} + 2 \frac{dy}{dx} = 0 \)
(b) \( \frac{d^2y}{dx^2} - 2 \frac{d^2y}{dx^2} + 2y = 0 \)
(c) \( \frac{d^2y}{dx^2} + 2 \frac{dy}{dx} + 2y = 0 \)
(d) \( \frac{d^2y}{dx^2} + 2y = 0 \)
Answer: (c) \( \frac{d^2y}{dx^2} + 2 \frac{dy}{dx} + 2y = 0 \)

Question. Differential equation which as solution of the form \( y = A \cos \alpha x + B \sin \alpha x \), where \( A \) and \( B \) are arbitrary constants is
(a) \( \frac{d^2y}{dx^2} - \alpha^2 y = 0 \)
(b) \( \frac{d^2y}{dx^2} + \alpha^2 y = 0 \)
(c) \( \frac{d^2y}{dx^2} + \alpha y = 0 \)
(d) \( \frac{d^2y}{dx^2} - \alpha y = 0 \)
Answer: (b) \( \frac{d^2y}{dx^2} + \alpha^2 y = 0 \)

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. 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) \( y = 2e^x - 1 \)
Answer: (d) \( y = 2e^x - 1 \)

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

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

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) \( \sqrt{1 - x^2} \)

Assertion-Reason Questions

The following questions consist of two statements—Assertion(A) and Reason(R). Answer these questions selecting the appropriate option given below:
(a) Both A and R are true and R is the correct explanation for A.
(b) Both A and R are true and R is not the correct explanation for A.
(c) A is true but R is false.
(d) A is false but R is true.

Question. Assertion (A) : The degree of the differential equation \( \frac{d^2y}{dx^2} = 1 + \sqrt{\frac{dy}{dx}} \) is 2.
Reason (R) : The degree of a differential equation is the degree of the highest order derivative occurring in the equation, when differential co-efficients are made free from radicals, fractions and it is written as a polynomial in differential coefficient.
Answer: (a) Both A and R are true and R is the correct explanation for A.

Question. Assertion (A) : \( y = a \sin x + b \cos x \) is general solution of \( y'' + y = 0 \).
Reason (R) : \( y = a \sin x + b \cos x \) is a trigonometric function.
Answer: (b) Both A and R are true and R is not the correct explanation for A.

Question. Assertion (A) : Solution of the differential equation \( (1 + x^2) \frac{dy}{dx} + y = \tan^{-1} x \) is \( y e^{\tan^{-1} x} = (\tan^{-1} x - 1)e^{\tan^{-1} x} + C \)
Reason (R) : The differential equation of the form \( \frac{dy}{dx} + Py = Q \), where \( P, Q \) be the functions of \( x \) or constant, is a linear type differential equation.
Answer: (b) Both A and R are true and R is not the correct explanation for A.

Question. Assertion (A) : Differential equation corresponding to all lines \( ax + by + c = 0 \) has the order 3.
Reason (R) : General solution of a differential equation of nth order contains \( n \) independent arbitrary constants.
Answer: (d) A is false but R is true.

Question. Assertion (A) : The integrating factor of differential equation \( \frac{dx}{dy} + (\tan y) . x = \sec^2 y \) is \( \sec y \).
Reason (R) : Linear differential equation of the form \( \frac{dx}{dy} + Px = Q \), where \( P, Q = f(y) \) or constant has integrating factor, IF = \( e^{\int P dy} \)
Answer: (a) Both A and R are true and R is the correct explanation for A.

Solutions of Assertion-Reason Questions

Question.
Answer: We have,
\( \frac{d^2y}{dx^2} = 1 + \sqrt{\frac{dy}{dx}} \)

\( \implies \) \( \left( \frac{d^2y}{dx^2} - 1 \right)^2 = \left( \sqrt{\frac{dy}{dx}} \right)^2 \) [Squaring both sides]

\( \implies \) \( \left( \frac{d^2y}{dx^2} \right)^2 - 2 \frac{d^2y}{dx^2} + 1 = \frac{dy}{dx} \)

\( \implies \) \( \left( \frac{d^2y}{dx^2} \right)^2 - 2 \frac{d^2y}{dx^2} - \frac{dy}{dx} + 1 = 0 \)
\( \therefore \) Degree = 2
Clearly, both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).
Hence, (a) is the correct option.

Question.
Answer: \( y = a \sin x + b \cos x \)
\( y' = a \cos x - b \sin x \) [Differentiate w.r.t. \( x \)]
\( y'' = -a \sin x - b \cos x = -(a \sin x + b \cos x) = -y \) [Differentiate w.r.t. \( x \)]
\( \implies \) \( y'' + y = 0 \)
Clearly, both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of Assertion (A).
Hence, (b) is the correct option.

Question.
Answer: \( (1 + x^2) \frac{dx}{dy} + y = \tan^{-1} x \)
\( \implies \) \( \frac{dy}{dx} + \frac{1}{1 + x^2} . y = \frac{\tan^{-1} x}{1 + x^2} \)
\( \therefore \) IF = \( e^{\int \frac{1}{1+x^2} dx} = e^{\tan^{-1} x} \)
Solution will be \( y \times e^{\tan^{-1} x} = \int \frac{\tan^{-1} x}{1 + x^2} \times e^{\tan^{-1} x} dx \dots(i) \)
Let \( e^{\tan^{-1} x} = t \)

\( \implies \) \( \frac{e^{\tan^{-1} x}}{1 + x^2} dx = dt \) and \( \log (e^{\tan^{-1} x}) = \log t \)

\( \implies \) \( \tan^{-1} x = \log t \)
From equation (i), \( \int \log t . dt = t \log t - t + C \)
\( y e^{\tan^{-1} x} = e^{\tan^{-1} x} (\tan^{-1} x - 1) + C \)
Clearly, both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).
Hence, (b) is the correct option.

Question.
Answer: Clearly, Assertion (A) is false and Reason (R) is true.
Hence, (d) is the correct option

Question.
Answer: \( \frac{dx}{dy} + (\tan y) . x = \sec^2 y \)
Here, IF = \( e^{\int \tan y dy} = e^{\log \sec y} = \sec y \)
Clearly, both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).
Hence, (a) is the correct option.