CBSE Class 12 Mathematics Differentials Equation MCQs Set 09

Question. The order of the differential equation whose solution is \( y = a\cos x + b\sin x + c e^{-x} \) is
(a) 3
(b) 2
(c) 1
(d) None of the options
Answer: (a) 3

Question. The order of the differential equation whose general solution is given by \( y = (C_1 + C_2)\cos(x + C_3) - C_4 e^{x+C_5} \), where \( C_1, C_2, C_3, C_4, C_5 \) are arbitrary constants, is
(a) 5
(b) 4
(c) 3
(d) 2
Answer: (c) 3

Question. The differential equation of all circles of radius \( a \) is of order
(a) 2
(b) 3
(c) 4
(d) None of the options
Answer: (a) 2

Question. The differential equation of all parabolas whose axis of symmetry is along x-axis is of order
(a) 3
(b) 1
(c) 2
(d) None of the options
Answer: (c) 2

Question. The degree of differential equation of all curves having normal of constant length \( c \), is
(a) 1
(b) 3
(c) 4
(d) None of the options
Answer: (d) None of the options

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

Question. The differential equation whose solution is \( y = A e^{3x} + B e^{-3x} \) is given by
(a) \( y_2 - 3y_1 + 3y = 0 \)
(b) \( xy_2 + 3y_1 - xy + x^2 + 3 = 0 \)
(c) \( y_2 - 9y = 0 \)
(d) \( (y_1)^3 - 3y(xy_2 - 3y) = 0 \)
Answer: (c) \( y_2 - 9y = 0 \)

Question. The differential equation satisfied by \( y = \frac{A}{x} + B \) is (\( A, B \) are parameters)
(a) \( x^2 y_1 = y \)
(b) \( xy_1 + 2y_2 = 0 \)
(c) \( xy_2 + 2y_1 = 0 \)
(d) None of the options
Answer: (c) \( xy_2 + 2y_1 = 0 \)

Question. The differential equation whose solution represents the family \( xy = A e^{ax} + B e^{-ax} \) is
(a) \( x\left(\frac{d^2y}{dx^2}\right)^2 + 2\frac{dy}{dx} = xy \)
(b) \( x\left(\frac{d^2y}{dx^2}\right)^2 + 2\frac{dy}{dx} = a^2 xy \)
(c) \( x\frac{d^2y}{dx^2} + 2\frac{dy}{dx} = a^2 xy \)
(d) None of the options
Answer: (c) \( x\frac{d^2y}{dx^2} + 2\frac{dy}{dx} = a^2 xy \)

Question. The differential equation which represent the family of curves \( y = a e^{bx} \), where \( a \) and \( b \) are arbitrary constants, is
(a) \( y' = y^2 \)
(b) \( y'' = y y' \)
(c) \( y y'' = y' \)
(d) \( y y'' = (y')^2 \)
Answer: (d) \( y y'' = (y')^2 \)

Question. The differential equation having solution as \( y = 17e^x + a e^{-x} \) is
(a) \( y'' - x = 0 \)
(b) \( y'' - y = 0 \)
(c) \( y' - y = 0 \)
(d) \( y' - x = 0 \)
Answer: (b) \( y'' - y = 0 \)

Question. The order of the differential equation of all tangent lines to the parabola \( y = x^2 \), is
(a) 1
(b) 2
(c) 3
(d) 4
Answer: (a) 1

Question. The order of the differential equation whose general solution is given by \( y = (A + B)\cos(x + C) + D e^x \) is
(a) 4
(b) 3
(c) 2
(d) 1
Answer: (b) 3

Question. The differential equation of all circles in the first quadrant which touch the coordinate axes is of order
(a) 1
(b) 2
(c) 3
(d) None of the options
Answer: (a) 1

Question. The degree of the differential equation \( \left(\frac{d^2y}{dx^2}\right)^{2/3} + 4 - 3\frac{dy}{dx} = 0 \) is
(a) 2
(b) 1
(c) 3
(d) None of the options
Answer: (a) 2

Question. The solution of the differential equation \( \frac{dy}{dx} = \frac{x^2 + y^2 + 1}{2xy} \) satisfying \( y(1) = 1 \), is
(a) a hyperbola
(b) a circle
(c) \( y^2 = x(1 + x) - 10 \)
(d) \( (x - 2)^2 + (y - 3)^2 = 5xy \)
Answer: (a) a hyperbola

Question. Solution of the differential equation \( \frac{x + \frac{x^3}{3!} + \frac{x^5}{5!} + \dots}{1 + \frac{x^2}{2!} + \frac{x^4}{4!} + \dots} = \frac{dx - dy}{dx + dy} \), is
(a) \( 2ye^{2x} = C \cdot e^{2x} + 1 \)
(b) \( 2ye^{2x} = C \cdot e^{2x} - 1 \)
(c) \( ye^{2x} = C \cdot e^{2x} + 2 \)
(d) None of the options
Answer: (b) \( 2ye^{2x} = C \cdot e^{2x} - 1 \)

Question. Given the differential equation \( \frac{dy}{dx} = \frac{6x^2}{2y + \cos y} \); \( y(1) = \pi \). Which of the following option is correct?
(a) Solution is \( y^2 - \sin y = -2x^3 + C \)
(b) Solution is \( y^2 + \sin y = 2x^3 + C \)
(c) \( C = \pi + 2\sqrt{2} \)
(d) \( C = \pi^2 + 2 \)
Answer: (b) Solution is \( y^2 + \sin y = 2x^3 + C \)

Question. For the differential equation \( x \frac{dy}{dx} + 2y = xy \frac{dy}{dx} \),
(a) order is 1 and degree is 1
(b) solution is \( \ln(yx^2) = C - y \)
(c) order is 1 and degree is 2
(d) solution is \( \ln(xy^2) = C + y \)
Answer: (a) order is 1 and degree is 1

Case Based MCQs

Case I : Read the following passage and answer the questions.
A thermometer reading 80°F is taken outside. Five minutes later the thermometer reads 60°F. After another 5 minutes the thermometer reads 50°F. At any time t the thermometer reading be T°F and the outside temperature be S°F.

Question. If \( \lambda \) is positive constant of proportionality, then \( \frac{dT}{dt} \) is
(a) \( \lambda (T - S) \)
(b) \( \lambda (T + S) \)
(c) \( \lambda TS \)
(d) \( -\lambda (T - S) \)
Answer: (d) \( -\lambda (T - S) \)

Question. The value of T(5) is
(a) 30°F
(b) 40°F
(c) 50°F
(d) 60°F
Answer: (d) 60°F

Question. The value of T(10) is
(a) 50°F
(b) 60°F
(c) 80°F
(d) 90°F
Answer: (a) 50°F

Question. Find the general solution of differential equation formed in given situation.
(a) \( \log T = St + c \)
(b) \( \log (T - S) = -\lambda t + c \)
(c) \( \log S = tT + c \)
(d) \( \log (T + S) = \lambda t + c \)
Answer: (b) \( \log (T - S) = -\lambda t + c \)

Question. Find the value of constant of integration c in the solution of differential equation formed in given situation.
(a) \( \log (60 - S) \)
(b) \( \log (80 + S) \)
(c) \( \log (80 - S) \)
(d) \( \log (60 + S) \)
Answer: (c) \( \log (80 - S) \)

Case II : Read the following passage and answer the questions.
It is known that, if the interest is compounded continuously, the principal changes at the rate equal to the product of the rate of bank interest per annum and the principal. Let P denotes the principal at any time t and rate of interest be r% per annum.

Question. Find the value of \( \frac{dP}{dt} \).
(a) \( \frac{Pr}{1000} \)
(b) \( \frac{Pr}{100} \)
(c) \( \frac{Pr}{10} \)
(d) \( Pr \)
Answer: (b) \( \frac{Pr}{100} \)

Question. If \( P_0 \) be the initial principal, then find the solution of differential equation formed in given situation.
(a) \( \log\left(\frac{P}{P_0}\right) = \frac{rt}{100} \)
(b) \( \log\left(\frac{P}{P_0}\right) = \frac{rt}{10} \)
(c) \( \log\left(\frac{P}{P_0}\right) = rt \)
(d) \( \log\left(\frac{P}{P_0}\right) = 100rt \)
Answer: (a) \( \log\left(\frac{P}{P_0}\right) = \frac{rt}{100} \)

Question. If the interest is compounded continuously at 5% per annum, then in how many years will ₹ 100 double itself ? (\( \log_e 2 = 0.6931 \))
(a) 12.728 years
(b) 14.789 years
(c) 13.862 years
(d) 15.872 years
Answer: (c) 13.862 years

Question. At what interest rate will ₹ 100 double itself in 10 years? (\( \log_e 2 = 0.6931 \)).
(a) 9.66%
(b) 8.239%
(c) 7.341%
(d) 6.931%
Answer: (d) 6.931%

Question. How much will ₹ 1000 be worth at 5% interest after 10 years? (\( e^{0.5} = 1.648 \)).
(a) ₹ 1648
(b) ₹ 1500
(c) ₹ 1664
(d) ₹ 1572
Answer: (a) ₹ 1648

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 differential equation of all circles in a plane must be of order 3.
Reason: If three points are non-collinear, then only one circle passes through these points.
(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: Order of the differential equation whose solution is \( y = c_1 e^{x+c_2} + c_3 e^{x+c_4} \) is 4.
Reason: Order of the differential equation is equal to the number of independent arbitrary constants mentioned in the solution of the differential equation.
(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: (d) Assertion is wrong statement but Reason is correct statement.

Question. Assertion: \( y = a \sin x + b \cos x \) is a general solution of \( y'' + y = 0 \).
Reason: \( y = a \sin x + b \cos x \) is a trigonometric function.
(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.