CBSE Class 12 Mathematics Differentials Equation MCQs Set 02

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

Answer : B

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 the options

Answer : C

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

Answer : B

Question. The degree of the differential equation satisfying the relation

(a) 1
(b) 2
(c) 3
(d) None of the options

Answer : A

Question.

Answer : A, B

Question.

Answer : A, C

Question. The solution of the differential equation

Answer : A, B

Question.

Answer : B, C

Question. The value of

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

Answer : B

Question. The differential equation obtained by eliminating arbitrary constants from y = aebx is

Answer : C

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

Answer : B

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

Answer : C

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

Answer : D

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

Answer : C

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

Answer : B

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

Answer : B

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

Answer : B

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

Answer : D

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

Answer : A

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

Answer : B

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

Answer : A

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

Answer : C

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

Answer : A

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

Answer : A

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

Answer : C

Question. Solution of the differential equation

Answer : A

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

Answer : C

Question.

Answer : A, B

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

Answer : B

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

Answer : C

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

Answer : B

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

Answer : C

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

Answer : A

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 the options

Answer : A

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

Answer : D

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

Answer : B

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

Answer : B
 

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.

Answer : A

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.

Answer : D

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.

Answer : C

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.

Answer : B

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 equationAnswer

Answer : A