CBSE Class 12 Mathematics Differentials Equation MCQs Set 04

Question: If y(t) is a solution of(1+t)dy/dt – ty = 1 and y(0) = –1, then the value of y (1) is
(a) 1/2
(b) -1/2
(c) 2
(d) 1
Answer: b

Question: The equation of the curve through the point (1, 2) and whose slope is y-1/x²+x , is
(a) (y – 1)(x +1) – 2x = 0
(b) 2x(y – 1) + x +1 = 0
(c) x(y -1)(x +1) + 2 = 0
(d) None of these
Answer: a

Question: The differential equation (1+ y² )x dx – (1+ x² )ydy = 0 represents a family of :
(a) ellipses of constant eccentricity
(b) ellipses of variable eccentricity
(c) hyperbolas of constant eccentricity
(d) hyperbolas of variable eccentricity
Answer: d

Question: The solution of ydx- xdy =xydx – = is
(a) y= Cxe ^-x
(b) 2y= Cxe ^-x
(c) y= 3Cxe ^-x
(d) y2 =Cxe ^-x
Answer: a

Question: The differential equation of all non-horizontal lines in a plane is
(a) d²y/dx²
(b) d²x/dy² = 0
(c) dy/dx = 0
(d) dx/dy = 0
Answer: b

Question: The differential equation y dy/dx+x=c represents family of
(a) hyperbolas
(b) parabolas
(c) ellipses
(d) circles
Answer: d

Question: The differential equation representing the family of curves y² = 2c ( x+√c) , where c > 0, is a parameter, is of order and degree as follows :
(a) order 1, degree 2
(b) order 1, degree 1
(c) order 1, degree 3
(d) order 2, degree 2
Answer: c

Question: The order of the differential equation of a family of curves represented by an equation containing four arbitrary constants, will be
(a) 2
(b) 4
(c) 6
(d) None of these
Answer: b

Question: Consider the following statements
I. The order of the differential equation dy/dx = e^x is 1.
II. The order of the differential equation d²y/dx² + y = 0 is 2.
III. The order of the differential equation (d³y/dx³) + x2 (d²y/dx²)3 = 0 is 3.
Choose correct option.
(a) I and II are true
(b) II and III are true
(c) I and III are true
(d) All are true
Answer: d

Question: The solution of the equation dy/dx=x(2log x+1)/sin y +y cos y is
Answer: a

Question: The solution of dy/dx + = xy= xy² is
Answer: b

Question: For the function y = Bx² to be the solution of differential equation (dy/dx)³ – 15x² dy/dx – 2xy = 0, the value of B is __________, given that B ≠ 0.
(a) 2
(b) 4
(c) 6
(d) 8
Answer: a

Question: The expression satisfying the differential equation (x²-1)dy/dx + 2xy = 1 is
(a) x²y – xy² = c
(b) (y² -1)x = y + c
(c) (x² -1) y = x + c
(d) None of these
Answer: c

 Question: General solution of the differential equation dy/dx + y g' (x) = g(x). g' (x), where g(x) is a function of x is
(a) g(x) - log[1- y - g(x)] = C
(b) g(x) - log[1+ y - g(x)] = C
(c) g(x) +[1+ y - logg(x)] = C
(d) g(x) + log[1+ y - g(x)] = C
Answer: d

Question: The particular solution of log dy/dx = 3x + 4y, y(0) = 0 is
(a) e^3x + 3e^–4y = 4
(b) 4e^3x – 3^–4y = 3
(c) 3e^3x + 4e^4y = 7
(d) 4e^3x + 3e^–4y = 7 
Answer: d

Question: The equation of the curve passing through the point (1, 1) whose differential equation is x dy = (2x² + 1) dx (x ≠ 0) is
(a) x² = y + log |x|
(b) y = x² + log |x|
(c) y² = x + log |x|
(d) y = x + log |x|
Answer: b

Question: The differential equation representing the family of parabolas having vertex at origin and axis along positive direction of x-axis is
(a) y²y'' – 2xy' = 0
(b) y² – 2xyy'' = 0
(c) y² – 2xyy' = 0
(d) None of these
Answer: c

Question: The order and degree of the differential equation y = x dy/dx + √a²(dy/dx)² + b² is
(a) order = 1, degree = 2
(b) order = 2, degree = 1
(c) order = 2, degree = 2
(d) None of these
Answer: a

Question: If dx +dy = (x + y) (dx – dy), then log (x + y) is equal to
(a) x + y + C
(b) x + 2y + C
(c) x – y + C
(d) 2x + y + C
Answer: c

Question: For the function y = Bx² to be the solution of differential equation (dy/dx)³ - 15x² dy/dx - 2xy = 0, the value of B is __________, given that B ≠ 0.
(a) 2
(b) 4
(c) 6
(d) 8
Answer: a

Question: A homogeneous differential equation of the dx/dy = h (x/y) can be solved by making the substitution
(a) y = vx
(b) v = yx
(c) x = vy
(d) x = v
Answer: c

Question: The solution of the differential equation dy/dx = e^x–y + x²e^–y is
(a) e^x = y³/3 = e^y + c
(b) e^y = x²/3 + e^x + c
(c) e^y = x³/3 + e^x + c
(d) None of these
Answer: c

Question: In the particular solution of differential equation dy/dx = 1/x(3y² - 1) , the value of constant term is ________, given that y = 2 when x = 1.
(a) 2
(b) 4
(c) 6
(d) 8
Answer: c

Question: The differential equation representing the family of curves y = A cos (x + B), where A, B are parameters, is
(a) d²y/dx² + y = 0
(b) d²y/dx² - y = 0
(c) d²y/dx² = dy/dx + y
(d) dy/dx + y = 0
Answer: a

 Question: Solution of the differential equation xdy – ydx = √x²+y² dx is
(a) y = cx²
(b) y = cx² + √x² + y²
(c) y + √x² + y² = cx²
(d) y - √x² - y² = c
Answer: c

 Question: If the I.F. of the differential equation dy/dx + 5y = cos x is ∫ e^Adx , then A =
(a) 0
(b) 1
(c) 3
(d) 5
Answer: d

 Question: If y = ex (sin x + cos x), then the value of d²y/dx² - 2 dy/dx + 2y , is
(a) 0
(b) 1
(c) 2
(d) 3
Answer: a

Question: The order of the differential equation of all tangent lines to the parabola y = x², is
(a) 1
(b) 2
(c) 3
(d) 4
Answer: a

Question: The solution of the equation dy/dx = 3x-4y- 2/3x-4y- 3 is
(a) (x – y²) + c = log (3x – 4y + 1)
(b) x – y + c = log (3x – 4y + 4)
(c) (x – y + c) = log (3x – 4y – 3)
(d) x – y + c = log (3x – 4y + 1)
Answer: d

Question: A differential equation of the form dy/dx = F(x, y) is said to be homogeneous, if F(x, y) is a homogeneous function of degree,
(a) 0
(b) 1
(c) 2
(d) 3
Answer: a

 Question: The order of the differential equation whose general solution is given by
y = (C₁ + C₂) cos (x +C₃) - C₄e^x+C₅ where C₁, C₂, C₃, C₄, C₅ are arbitrary constant, is
(a) 5
(b) 4
(c) 3
(d) 2
Answer: c

Question: Solution of differential equation xdy – ydx = 0 represents:
(a) rectangular hyperbola.
(b) parabola whose vertex is at origin.
(c) circle whose centre is at origin.
(d) straight line passing through origin.
Answer: d

Question: The differential equation dy/dx = √1-y²/y determines a family of circle with
(a) variable radii and fixed centre (0, 1)
(b) variable radii and fixed centre (0, –1)
(c) fixed radius 1 and variable centre on x-axis
(d) fixed radius 1 and variable centre on y-axis
Answer: c

Question: Family y = Ax + A₃ of curves will correspond to a differential equation of order
(a) 3
(b) 2
(c) 1
(d) not infinite
Answer: b

Question: The order and degree of the differential equation d²y/dx² + (dy/dx)^1/3 + x^1/4 - 0 is
(a) order = 3, degree = 2
(b) order = 2, degree = 3
(c) order = 2, degree = 2
(d) order = 3, degree = 3
Answer: b

 Question: The order and degree of the differential equation d4y/dx4 + sin (y'') = 0 are respectively
(a) 4 and 1
(b) 1 and 2
(c) 4 and 4
(d) 4 and not defined
Answer: d

Question: General solution of dy/dx + 2xy/1+x² = 1/(1+x²)² is
(a) y(1 + x²) = c + tan^–1 x
(b) y/1+x² + c tan^-1 x
(c) y log (1 + x²) = c + tan^–1 x
(d) y (1 + x²) = c + sin^–1 x
Answer: a

 Question: The differential equation which represent the family of curves y = aebx, where a and b are arbitrary constants.
(a) y' = y²
(b) y'' = y y'
(c) y y'' = y'
(d) y y'' = (y')²
Answer: d

 Question: If (1+e^x/y) dx + (1-x/y)e^x/y dy = 0 , then
(a) x – ye^x/y = c
(b) y – xe^x/y = c
(c) x + ye^x/y = c
(d) y + xex/y = c
Answer: c

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: The degree of the differential equation d²y/dx² + 3(dy/dx)² = x² log (d²y/dx²) is not defined.
Reason : If the differential equation is a polynomial in terms of its derivatives, then its degree is defined.
Answer: a

Question: Assertion : The number of arbitrary constants in the solution of differential equation d²y/dx2 = 0 are 2.
Reason: The solution of a differential equation contains as many arbitrary constants as is the order of differential equation.
Answer: a

Question: Assertion : The degree of the differential equation d³y/dx³ + 2(d²y/dx²)^3/2 + 2y = 0 is zero
Reason: The degree of a differential equation is not defined if it is not a polynomial eq in its derivatives.
Answer: d