CBSE Class 12 Mathematics Differentials Equation MCQs Set 03

Question. The solution of the differential equation

Answer : A

Question. The solution of the differential equation

Answer : A

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

Answer : B

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

Answer : D

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’ + ex = 0
(d) y” + ex = 0

Answer : A

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

Answer : B

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 the options

Answer : C

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)

Answer : D

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

Answer : D

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

Answer : B

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

Answer : A

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

Answer : A

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

Answer : A

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

Answer : A

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

Answer : D

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 the options

Answer : C

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

Answer : A

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

Answer : C

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

Answer : C

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

Answer : B

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 the options

Answer : B

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

Answer : B

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 the options

Answer : A

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)

Answer : D

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 the options

Answer : A

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

Answer : B

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

Answer : B

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

Answer : B

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

Answer : B

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

Answer : A

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

Answer : C

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

Answer : D