MCQ Question For Class 12 Mathematics Chapter 9 Differential Equations

Check the below NCERT MCQ Class 12 Mathematics Chapter 9 Differential Equations with Answers available with PDF free download. MCQ Questions for Class 12 Mathematics with Answers were prepared based on the latest syllabus and examination pattern issued by CBSE, NCERT and KVS. Our teachers have provided below Differential Equations Mathematics Class 12 Mathematics MCQs Questions with answers which will help students to revise and get more marks in exams

Differential Equations Class 12 Mathematics MCQ Questions with Answers

Refer below for MCQ Class 12 Mathematics Chapter 9 Differential Equations with solutions. Solve questions and compare with the answers provided below

Question. The degree of the differential equation (d³y/dx³)⁴ + (d²y/dx²)⁵ + dy/dx + y = 0 is
(a) 2
(b) 4
(c) 6
(d) 8

Question. Differential equation of all straight lines which are at a constant distance from the origin is
(a) (y + xy₁)² = p²(1+y₁₂)
(b) (y – xy₁)² = p²(1-y₁₂)
(c) (y – xy₁)² = p²(1+y₁₂)
(d) None of these

Question. The orthogonal trajectories of the family of curve a^n-1 y=xⁿ are given by (a is the arbitrary constant)
(a) xⁿ +n²_y = constant
(b) ny² + x²  = constanat
(c) n²x+yⁿ = constant
(d) n²x- yⁿ  = constant

Question. The degree of the differential equation satisfying the relation

is
(a) 1
(b) 2
(c) 3
(d) None of these

Question. A tangent to the curve y=f (x) cuts the line y = x at a point which is at a distance of 1 unit from y-axis. The equation of the curve is
(a) x-1/y-1=C
(b) x/y= C
(c) xy = C
(d) None of these

Question. A curve passes through the point (0,1) and the gradient at(x ,y )on it is y( xy-1). The equation of the curve is
(a) y(x-1)=1
(b) y(x+1)=1
(c)x(y+1)=1
(d) x(y-1)=1

Question. The integrating factor of the differentiable equation (xy-1)dy/dx+y2=0 is
(a) 1/x
(b) 1/y
(c) 1/xy
(d) xy

Question. The equation of a curve passing through (2,7/2)  and having gradient 1-1/x² at (x,y) is
(a) y =x²+x+1
(b) xy =x²+x+1
(c) xy=x+1
(d) None of these

Question. A function y =f (x) has a second order derivative f ‘(x)=6(x-1). If is its graph passes through the point (2,1) and at the point the tangent to the graph is y= 3x-5, then the function
(a) (x+1)³
(b) (x-1)³
(c) (x+1)³
(d) (x-1)²

Question. An inverted conical tank of 2 m radius and 4 m height is initially full of water has an outlet at bottom. The outlet is opened at some instant. The rate of flow through the outlet at any time t is 6 h ^3/2,where h is height of water level above the outlet at time t.Then, the time it takes to empty the tank, is
(a) 2π/9unit
(b) π/9 unit
(c) 2π/8unit
(d) None of these

Question. The solution of the differential equation

Question. The solution of the differential equation

Question. The differential equation of the family of curves for which the length of the normal is equal to a constant k, is given by

Question. The equation of the curve for which the square of the ordinate is twice the rectangle contained by the abscissa and the intercept of the normal on x-axis and passing through (2,1) is
(a) x²+y²-x=0 
(b) 4x²+ 2y²-9y=0
(c) 2x²+4y²-9x=0
(d) 4x²+2y²-9x=0

Question. A curve y = f(x) passes through the point P(1,1)  The normal to the curve at point P is a (y-1) + (x-1)=0. If the slope of the tangent at any point on the curve is proportional to the ordinate at that point, then the equation of the curve is
(a)y=e^ax-1
 (b) y=e^ax+1
(c)y=e^ax+a
(d) y=e^a(x-1)

Question. The differential equation representing all lines at a distance p from the origin is

Question. The equation of the curve in which the portion of y-axis cut off between the origin and the tangent varies as the cube of the abscissa of the point of contact is

Question. A tangent and a normal to curve at any point P meet the x and y axes at A, B and C, D, respectively. If the centre of circle through O,C, P  and B lies on the line y =x (O is the origin), then the differential equation of all such curves is
(a) dy/dx = y-x/y+x
(b) dy/dx=y²-x²/y²+x²
(c) dy/dx=x-y/xy
(d) dy/dx = xy/y+x

Question. Let y f x = (x) be a curve passing through (e ,e^e), which satisfy the differential equation

(a) e
(b) 1
(c) 0
(d) 2

Question. Solution of the differential equation

Question. A particle of mass m is moving in straight line is acted on by an attrctive force mk² a²/x² for x ≥ a and 2mk²x/a for x<a. If the particle starts from rest at the point x= 2a then it will reach the point x = 0 with a speed
(a) k√a
(b)k√2a
(c) k√a3
(d) 1/k√a

Question.

Question. A curve is such that the portion of the x-axis cut off between the origin and tangent at a point is twice the abscissa and which passes through the point (1,2) The equation of the curve is
(a) xy = 1
(b) xy = 2
(c) xy = 3
(d) xy = 0

Question.

Question.

Question. The solution of the differential equation

Question.

Directions (Q. Nos. 26-29) Let y f x = ( ) and y g x = ( ) be the pair of curves such that(i) the tangents at point with equal abscissae intersect on y-axis.
(ii) the normal drawn at points with equal abscissae intersect on x-axis and
(iii) one curve passes through (1, 1) and the other one passes through (2, 3), then

Question. The curve g (x) is given by
(a) x-1/x
(b) x+2/x
(c) x²-1/x²
(d) x²+1/x²

Question. The number of positive integral solutions for f (x) =g(x), are
(a) 4
(b) 5
(c) 6
(d) None of these

Question. The curve f x) is given by
(a) 2/x-x
(b) 2x²-1/x
(c) 2/2×2-x
(d) 2/x-x2

Question. The value of

(a) 2
(b) 3
(c) 4
(d) 5

Question. Solution of differential equation (x² – 2x + 2y²) dx + 2xy dy = 0 is
(a) y² = 2x – 1/4 x² + c/x²
(b) y² = 2/3 x – x² + c/x²
(c) y² = 2/3 x – x^2/4 + c/x²
(d) None of these

Question. The solution of x³ dy/dx + 4x² tan y = e^x sec y satisfying y (1) = 0 is :
(a) tan y = (x – 2) e^x log x
(b) sin y = e^x (x -1)x^-4
(c) tan y=(x -1)e^xx^-3
(d) sin y = e^x (x -1) x^-3

Question. The order and degree of the differential equation d²y/dx² + (dy/dx)^1/4 + x^1/5 = 0 , respectively, are
(a) 2 and not defined
(b) 2 and 2
(c) 2 and 3
(d) 3 and 3

Question. tan–1 x + tan–1 y = c is the general solution of the differential equation
(a) dy/dx = 1+y²/1+x²
(b) dy/dx = 1 + x²/1+y²
(c) (1 + x²) dy + (1 + y²) dx = 0
(d) (1 + x²) dx + (1 + y²) dy = 0

Question. The differential equation obtained by eliminating the arbitrary constants a and b from xy = ae^x + be^–x is

Question. To solve first order linear differential equation, we use following steps
I. Write the solution of the given differential equation as y (IF) = ∫(Qx IF)dx + C
II. Write the given differential equation in the from dy/dx + Py = Q , where P and Q are constants or functions of x only.
III. Find the integrating factor (IF) _e∫ P dx The correct order of the above steps is
(a) II, III, I
(b) II, I, III
(c) III, I, II
(d) I, III, II

Question. The order of the differential equation whose solution is :
y = a cos x + b sin x + ce^–x is :
(a) 3
(b) 2
(c) 1
(d) None of these

Question. The elimination of constants A, B and C from y = A + Bx – Ce–x leads the differential equation:
(a) y” + y”‘ = 0
(b) y” – y”‘ = 0
(c) y’ + e^x = 0
(d) y” + e^x = 0

Question. Which of the following equation has y = c₁ ex + c₂ e^–x as the general solution?
(a) d²y/dx² + y = 0
(b) d²y/dx² – y = 0
(c) d²y/dx² + 1 = 0
(d) d²y/dx² – 1 = 0

Question. The order and degree of the differential equation whose solution is y = cx +c₂ – 3c₃/2 + 2 , where c is a parameter, is
(a) order = 4, degree = 4
(b) order = 4, degree = 1
(c) order = 1, degree = 4
(d) None of these

Question. A first order-first degree differential equation is of the form
(a) d²y/dx² = F (x, y)
(b) d³y/dx² = F (x, y)
(c) (dy/dx)² = F (x, y)
(d) dy/dx = F (x, y)

Question. A steam boat is moving at velocity V when steam is shut off.
Given that the retardation at any subsequent time is equal to the magnitude of the velocity at that time. The velocity v in time t after steam is shut off is
(a) v = Vt
(b) v = Vt – V
(c) v = Ve^t
(d) v = Ve^–t

Question. A family of curves is given by the equation x²/a² + y²/b² = 1 . The differential equation representing this family of curves is given by xy d2y/dx2 + Ax (dy/dx)2 – y dy/dx = 0 . 
The value of A is
(a) 0
(b) 1
(c) 3
(d) 5

Question. The solution of the differential equation y’= y/x + φ(y/x)/φ'(y/x) is
(a) x φ (y/x) = k
(b) φ (y/x) = kx
(c) y φ (y/x) = k
(d) φ (y/x) = ky

Question. The equation of a curve whose tangent at any point on it different from origin has slope y + y/x , is
(a) y = e^x
(b) y = kx. e^x
(c) y = kx
(d) y = k. ex²

Question. The solution of the differential equation (1+e^x/y) dx + ex/y (1+x/y) dy = 0
(a) ye^x + x = C
(b) xe^y + y = C
(c) ye^y/x + y = C
(d) ye^x/y + y = C

Question. The solution of the differential equation ydx – xdy + 3x² y² e^x3 dx = 0
(a) x/y + e^x3 = c
(b) x/y – e^x3 = c
(c) y/x + e^x3 = c
(d) y/x – e^x3 = c

Question. The differential equation of the form dy/dx + Py = Q
where, P and Q are constants or functions of x only, is known as a
(a) first order differential equation
(b) linear differential equation
(c) first order linear differential equation
(d) None of the above

Question. Which of the following is/are first order linear differential equation?
(a) dy/dx + y = sin x
(b) dy/dx + (1/x) y = e^x
(c) dy/dx + (y/x log x) = 1/x 
(d) All the above

Question. If p and q are the order and degree of the differential equation y dy/dx + x³ d²y/dx² + xy = cos x, then
(a) p < q
(b) p = q
(c) p > q
(d) None of these

Question. The differential equation obtained by eliminating arbitrary constants from y = ae^bx is

Question. The degree of the differential equation y₃^2/3 + 2 + 3y₂ + y₁ = 0 is :
(a) 1
(b) 2
(c) 3
(d) None of these

Question. For the differential equation d²y/dx² + y = 0 , if there is a function y = φ (x) that will satisfy it, then the function y = φ (x) is called
(a) solution curve only
(b) integral curve only
(c) solution curve or integral curve
(d) None of the above

Question.

Codes
A B C D
(a) 5 4 1 3
(b) 2 4 1 3
(c) 4 2 3 1
(d) 4 3 2 1

Question. The degree of the equation ex d²y/dx² + sin (dy/dx) = 3 is
(a) 2
(b) 0
(c) not defined
(d) 1

Question. The equation of the curve satisfying the differential equation y² (x² + 1) = 2 xy, passing through the point (0, 1) and having slope of tangent at x = 0 as 3, is
(a) y = x³ + 3x + 1
(b) y = x² + 3x + 1
(c) y = x³ + 3x
(d) y = x³ + 1

Question. The degree of the differential equation satisfied by the curve √1+ x – a √1+ y = 1, is
(a) 0
(b) 1
(c) 2
(d) 3

Question. The female-male ratio of a village decreases continuously at the rate proportional to their ratio at any time. If the ratio of female : male of the villages was 980 : 1000 in 2001 and 920 : 1000 in 2011. What will be the ratio in 2021 ?
(a) 864 : 1000
(b) 864 : 100
(c) 1000 : 864
(d) 100 : 864

Question. An equation which involves variables as well as derivative of the dependent variable with respect to independent variable, is known as
(a) differential equation
(b) integral equation
(c) linear equation
(d) quadratic equation

Question. If y = (x+√1+x²)ⁿ , then (1 + x²) d²y/dx² + x dy/dx is
(a) n²y
(b) –n²y
(c) 0 – y
(d) 2x²y

Question. For y = cos kx to be a solution of differential equation d²y/dx² + 4y = 0, the value of k is
(a) 2
(b) 4
(c) 6
(d) 8

Question. The integrating factor of the differential equation x dy/dx – y = 2x² is
(a) e^–x
(b) e^–y
(c) 1/x
(d) x

Question. In a bank, principal increases continuously at the rate of 5% per year. In how many years ₹ 1000 double itself?
(a) 2
(b) 20
(c) 20 log_e2
(d) 2 log_e20

Question. The equation of curve through the point (1, 0), if the slope of the tangent to the curve at any point (x, y) is y-1/x²+x , is
(a) (y + 1) (x – 1) + 2x = 0
(b) (y + 1) (x – 1) – 2x = 0
(c) (y – 1) (x – 1) + 2x = 0
(d) (y – 1) (x + 1) + 2x = 0

Question. The general solution of the homogeneous differential equation of the type.
dy/dx = F(x, y) = g (y/x) , when y = v : x is

Question. In order to solve the differential equation x cos x dy/dx + y(xsin x cos x) = 1 the integrating factor is:
(a) x cos x
(b) x sec x
(c) x sin x
(d) x cosec x

Question. The diffrerential equation dy/dx + 1/x sin 2y = x³ cos² y represents a family of curves given by the equation
(a) x6 + 6x² = Ctan y
(b) 6x² tan y = x⁶ + C
(c) sin 2y = x³ cos² y + C
(d) none of these

Question.

Codes
A B C D E F
(a) 2 2 4 3 1 1
(b) 1 1 1 1 2 3
(c) 3 4 1 1 2 3
(d) 1 1 1 3 4 2

Assertion and Reason type Questions :

(a) Assertion is correct, Reason is correct; Reason is a correct explanation for assertion.
(b) Assertion is correct, Reason is correct; Reason is not a correct explanation for Assertion
(c) Assertion is correct, Reason is incorrect
(d) Assertion is incorrect, Reason is correct.

Question. Assertion : dy/dx + x²y = 5 is a first order linear differential equation.
Reason: If P and Q are functions of x only or constant then differential equation of the form dy/dx + Py = Q is a first order linear differential equation.

Question. Assertion: Order of the differential equation whose solution is y = c₁e ^{x + c₂} + c₃e ^x + c₄ is 4.
Reason: Order of the differential equation is equal to the number of independent arbitrary constant mentioned in the solution of differential equation.

Question. Assertion: The differential equation y³ dy + (x + y²) dx = 0 becomes homogeneous if we put y² = t.
Reason: All differential equation of first order first degree becomes homogeneous if we put y = tx.

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 always passing through these points.

Question. Assertion : The differential equation dy/dx + x = cos y and dy/dx + -2x/y = y² e-y are first order linear differential equations.
Reason : The differential equation of the form dy/dx + P1x = Q₁ where, P₁ and Q₁ are constants or functions of y only, is called first order linear differential equation.