CBSE Class 12 Mathematics Differentials Equation MCQs Set 06

Question. The degree of the differential equation satisfying √1− x² + √1− y² = a(x − y) is:
(a) 1
(b) 2
(c) 3
(d) 4
Answer: a

Question. The differential equation found by the elimination of the arbitrary constant K from the equation y = (x+K)e−x is:
(a) dy/dx−y = e−x
(b) dy/dx−ye^x = 1
(c) dy/dx+ ye^x = 1
(d) dy/dx + y = e^−x
Answer: d

 Question. The general solution of the differential equation (x + y)dx + xdy = 0 is:
(a) x² + y² = c
(b) 2x² – y² = c
(c) x² + 2xy = c
(d) y² + 2xy = c
Answer: c

Question. The elimination of the arbitrary constants A, B and C from y = A + Bx Ce^−x leads to the differential equation:
(a) y′′′ − y′ = 0
(b) y′′′ − y′′ + y′ = 0
(c) y′′′ + y′′ = 0
(d) y′′ + y′′ − y′ = 0
Answer: c

Question. The solution of the equation sin−1(dy/dx) = x+y is:
(a) tan(x + y) + sec(x + y) = x + c
(b) tan(x + y) − sec(x + y) = x + c
(c) tan(x + y) + sec(x + y) + x + c = 0
(d) None of these
Answer: b

Question. The differential equation y dy/dx + x = a (a is any constant) represents:
(a) A set of circles having centre on the y-axis
(b) A set of circles centre on the x-axis
(c) A set of ellipses
(d) None of these
Answer: b

Question. The solution of the differential equation x d²y/dx² = 1 given that y = 1 , dy/dx when x =1, is:
(a) y = x log x + x + 2
(b) y = x log x − x + 2
(c) y = x log x + x
(d) y = x log x − x
Answer: b

Question. The solution of the differential equation dy/dx = sec x(sec x + tan x) is:
(a) y = sec x + tan x + c
(b) y = sec x + cot x + c
(c) y = sec x − tan x + c
(d) None of these
Answer: a

Question. The order of the differential equation, whose general solution is y= c₁e^x + c₂e^2x + c₃e^3x + c₄e^x+c3 , where c₁,c₂,c₃, c₄, c₅ are arbitrary constants is:
(a) 5
(b) 4
(c) 3
(d) 2
Answer: c

Question. The solution of ydx − xdy + xdy + 3x²y²e^x3 dx = 0 is:
(a) x/y + ex3 = c
(b) x/y − ex3 = c
(c) −x/y + e^x3 = c
(d) None of these
Answer: a

Question. The solution of the differential equation cos²x d²y/dx² = 1 is:
(a) y = log cos x + cx
(b) y = logsec x + c₁x + c₂
(c) y = logsec x − c₁x + c₂
(d) None of these
Answer: b,c

Question. The order and degree of the differential equation x(dy/dx)3 + 2(d2y/dx2)3 + 3y +x = 0 are respectively:
(a) 3, 2
(b) 2, 1
(c) 2, 2
(d) 2, 3
Answer: c

Question. If y(x) is a solution of (2+sin x /1+y)dy/dx=-cos x and y(0) =1,then the value of y(π/2) is
(a) 1/3
(b) 1/2
(c) 1
(d) 0
Answer: a

Question. Solution of (xy cos xy + sin xy)dx + x² cos xy dy = 0 is:
(a) xsin(xy) = k
(b) xy sin(xy) = k
(c) x/y sin(xy)
(d) x sin(xy)+ xy cos xy = k
Answer: a

Question. Equation of curve through point (1, 0)which satisfies the differential equation (1+ y²)dx − xydy = 0 , is:
(a) x² + y² = 0 =1
(b) x² − y² = 0 =1
(c) 2x² + y² = 0 = 2
(d) None of these
Answer: b

Question. The equation of the curve which passes through the point (1, 1) and whose slope is given by 2y/x , is:
(a) y = x²
(b) x² − y² = 0
(c) 2x² + 2y² = 3
(d) None of these
Answer: a

Question. The order of differential equations of all parabolas having directrix parallel to x-axis is:
(a) 3
(b) 1
(c) 4
(d) 2
Answer: a

Question. Solution of differential equation x dy − y dx = 0 represents:
(a) Rectangular hyperbola
(b) Straight line passing through origin
(c) Parabola whose vertex is at origin
(d) Circle whose centre is at origin
Answer: b

Question. The degree of differential equation d2y/dx2 +(dy/dx)3 + 6y =0 is:
(a) 1
(b) 3
(c) 2
(d) 5
Answer: a

Question. The solution of the differential equation dy/dx = (4x + y +1)² is:
(a) 4x – y + 1 = 2 tan (2x – 2c)
(b) 4x – y – 1 = 2 tan (2x – 2c)
(c) 4x + y + 1 = 2 tan (2x + 2c)
(d) None of these
Answer: c

Question. The differential equation d²y/dx²=2 represents 
(a) a parabola whose axis is parallel to x-axis
(b) a parabola whose axis is parallel to y-axis
(c) a circle
(d) None of the above
Answer: b

Question. The solution of dy/dx=x2+y2+1/2xy, satisfying y(1)=1,
is given by a
(a) hyperbola
(b) circle
(c) ellipse
(d) parabola 
Answer: a

Question. If c is any arbitrary constant, then the general solution of the differential equation ydx − xdy = xy dx is given by:
(a) y = cx e^−x
(b) x = cye^−x
(c) y + e^x = cx
(d) ye^x = cx
Answer: d

Question. The general solution of the differential equation (2x − y +1) dx + (2y − x +1)dy = 0 is:
(a) x² + y² + xy − x + y = c
(b) x² + y² − xy + x + y = c
(c) x² + y² + 2xy − x + y = c
(d) x² + y² − 2xy + x − y = c
Answer: b

Question. A continuously differential function Φ (x) in (0,π) satisfying y’=1+y2,y(0)=0=y(π), is 
(a) tan x
(b) x (x-π)
(c) (x-π)(1-e^x)
(d) Not possible 
Answer: d

Question. The solution of the differential equation 3ex tan ydx + (1–e^x )sec²ydy = 0 is:
(a) tan y = c(1−e^x)³
(b) (1−e^x)³ tan y = c
(c) tan y = c(1−e^x)
(d) (1−e^x) tan y = c
Answer: c

Question. A particle starts at the origin and moves along the x–axis in such a way that its velocity at the point (x, 0) is given by the formula dy/dx = cos2πx . Then the particle never reaches the point on:
(a) x = 1/4
(b) x = 3/4
(c) x = 1/2
(d) x = 1
Answer: c

Question. If d²y/dx² = 0,then:
(a) y = ax + b
b) y² = ax + b
(c) y = log x
(d) y = e^x + c
Answer: a

Question. Family y = Ax + A³ of curve represented by the differential equation of degree:
(a) Three
(b) Two
(c) One
(d) None of these
Answer: a

Question. The second order differential equation is:
(a) y′² + x = y²
(b) y′y′′ + y = sin x
(c) y′′′ + y′′ + y = 0
(d) y′ = y
Answer: b

Question. The degree and order of the differential equation of the family of all parabolas whose axis is x–axis, are respectively:
(a) 2, 1
(b) 1, 2
(c) 3, 2
(d) 2, 3
Answer: b

Question. The differential equation for the line y = mx + c is: (where c is arbitrary constant)
(a) dy/dx = m
(b) dy/dx + m
(c) dy/dx = 0
(d) None of these
Answer: a

Question. Solution of y(2xy + e^x )dx = e^xdy is:
(a) yx² + e^x = cy
(b) xy² + e^x = cx
(c) xy² + e^−x = c
(d) None of these
Answer: a

 Question. The solution of the differential equation xcos ydy = (xe logx + ex)dx is:
(a) sin y = 1/x e^x + c
(b) sin y + e^x logx + c = 0
(c) sin y = e^x logx + c 
(d) None of these 
Answer: c

Question. The general solution of the differential equation (x + y) dx + xdy = 0 is:
(a) x² + y² = c
(b) 2x² − y² = c
(c) x² + 2xy = c
(d) y² + 2xy = c
Answer: c