JEE Mathematics Differential Equations MCQs Set 04

Question. Solution of the equation (e^x +1) ydy = ( y +1)e^x dx is
(a) c ( y +1)(e^x +1) = e^y
(b) c ( y +1)(e^x -1)- e^y = 0
(c)   c( y +1)(e^x -1)+ e^y = 0
(d) None of these

Answer: A

Question. Solution of (x + y -1)dx + (2x + 2y - 3)dy = 0 is
(a) 2 y + x + log(x + y - 2) = c
(b) 2y + 2x + log(x + y - 2) = c
(c) y + 2x + log(x + y - 2) = c
(d) y + x + log(x + y - 2) = c

Answer: A

Question. The degree of the differential equation y = Ax + A³ is :
(a) 3
(b) 4
(c) 2
(d) 1

Answer: A

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' + e^x = 0
(c) y" – y"' = 0
(d) y" + e^x = 0

Answer: A

Question. The differential equation representing the family of curves y² = 2c (x + √c ) , where c is a positive parameter, is of
(1) order 1    (2) order 2
(3) degree 3 (4) degree 4
(a) 1 and 3 are correct
(b) 1 and 2 are correct
(c) 1, 2 and 3 are correct
(d) 2 and 4 are correct

Answer: A

Question. Differential equation dy/dx= f (x).g(y) can be solved by separating variable as dy/g(y)= f (x) dx
The equation of the curve passing through the point (1, 0) and satisfies the differential equation (1 + y²) dx – xy dy = 0, is
(a) x² – y² = 1
(b) x² – y² = 2
(c) x² + y² = 1
(d) x² + y² = 2

Answer: A

Question. Statement-1 : The equation of the curve through the point (1, 0) which satisfies differential equation (1 + y²) dx – xy dy = 0 is x² – y² = 1.
Statement-2 : D.E. dy/dx = f (x) . g (y) can be solved by separating variables as dy/g(y)=f (x) dx
(a) Statement-1 is True, Statement-2 is True; Statement-2 is a correct explanation for Statement-1.
(b) Statement-1 is True, Statement-2 is True; Statement-2 is NOT a correct explanation for Statement-1.
(c) Statement -1 is False, Statement-2 is True.
(d) Statement -1 is True, Statement-2 is False.

Answer: A

Question. Statement-1 : The differential equation of all circles in a plane must be of order 3.
Statement-2 : There is only one circle passing through three noncollinear points.
(a) Statement-1 is True, Statement-2 is True; Statement-2 is NOT a correct explanation for Statement-1.
(b) Statement -1 is False, Statement-2 is True.
(c) Statement -1 is True, Statement-2 is False.
(d) Statement-1 is True, Statement-2 is True; Statement-2 is a correct explanation for Statement-1.

Answer: A

Question. The solution of the equation dy/dx= x/2y-x is
(a) (x - y)(x + 2y)² = c
(b) y = (2y - x) + c
(c) y = x + c
(d) None of these

Answer: A

Question. Solution of the differential equation, ydx - xdy + xy² dx = 0 is
(a) 2x + x² y = λy
(b) 2y + y² x = λy
(c) 2y - y²x = λy
(d) None of these

Answer: A

Question. y+x²=dy/dx has the solution
(a) y + x² + 2x + 2 = ce^x
(b) y + x + x² + 2 = ce^2x
(c) y + x + 2x² + 2 = ce^x
(d) y² + x + x² + 2 = ce^x

Answer: A

Question. The slope of a curve at any point is the reciprocal of twice the ordinate at the point and it passes through the point (4, 3), then equation of the curve is
(a) y² = x + 5
(b) x² = y + 5
(c) y² = x - 5
(d) x² = y -5

Answer: A

Question. Solution of the differential equation x²y–x³ dy/dx=y⁴ cos x, when y(0) = 1 is :
(a) x³ = 3 y³ sin x
(b) none of these
(c) x³ ≠ y³ sin x
(d) y³ = 3x³ sin x

Answer: A

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

Answer: A

Question. The rate of increase of bacteria in a certain culture is proportional to the number present. If it doubles in 5 hours then in 25 hours, its number would be
(a) 32 times the original
(b) 8 times the original
(c) 16 times the original
(d) 64 times the original

Answer: A

Question. Statement 1 : The differential equation y³ dy + (x + y²) dx = 0 becomes homogeneous if we put y² = t.
Statement 2 : All differential equation of first order first degree becomes homogeneous, if we put y = tx.
(a) Statement -1 is True, Statement-2 is False.
(b) Statement -1 is False, Statement-2 is True.
(c) Statement-1 is True, Statement-2 is True; Statement-2 is NOT a correct explanation for Statement-1.
(d) Statement-1 is True, Statement-2 is True; Statement-2 is a correct explanation for Statement-1.

Answer: A