Practice Test-I
Q:1. The degree and order of the differential equation of all the parabolas whose axis is x-axis are
(A) 2 , 1
(B) 1 , 2
(C) 3 , 2
(D) none of these
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Q:2. The solution of differential equation $ \frac{dy}{dx} = \frac{y}{x} + \frac{\phi(\frac{y}{x})}{\phi'(\frac{y}{x})}$ is
(A) $x \phi(\frac{y}{x}) = k $
(B) $ \phi(\frac{y}{x}) = k x $
(C) $ y \phi(\frac{y}{x}) = k $
(D) $ \phi(\frac{y}{x}) = k y $
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Q: 3. The differential equation for the family of curves y2 = a sinx + b cosx (a, b being parameters) is
(A) $ y \frac{d^2y}{dx^2} + (\frac{dy}{dx})^2 + y = 0 $
(B) $ 2y (\frac{d^2y}{dx^2}) + 2(\frac{dy}{dx})^2 + y = 0 $
(C) $ 2y (\frac{d^2y}{dx^2}) + 2(\frac{dy}{dx})^2 – y = 0 $
(D) none of these
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Q:4. If x intercept of any tangent is 3 times the x-coordinate of the point of tangency , then the equation of the curve, given that it passes through (1, 1) is
(A) $ y = \frac{1}{x} $
(B) $ y = \frac{1}{x^2} $
(C) $ y = \frac{1}{\sqrt{x}} $
(D) none of these
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Q:5. If (x – a)2 + (y – b)2 = c2 , then $ c \frac{d^2y}{dx^2}+ [1 + (\frac{dy}{dx})^2 ]^{3/2} $ is dependent on
(A) a
(B) b
(C) c
(D) none of these
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Q:6. The curve for which the normal at every point passes through a fixed point (h, k) is
(A) (x – h)2 + y = c
(B) x + y = c
(C) (x – h)2 + (y – k)2 = c
(D) none of these
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Q:7. The solution of $(x + 1)\frac{dy}{dx} + 1 = e^{(x-y)} $ is
(A) ey (x + 1) = c
(B) ey (x + 1) = ex + c
(C) ey (x + 1) = c ex
(D) None of these
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Q:8. Solution of y (xy + 1) dx + x (1 + xy + x2y2) dy = 0 is
(A) $logy = \frac{1}{x y} + c $
(B) $logy = \frac{1}{2 x^2 y^2} + c $
(C) $logy = \frac{1}{x y} + \frac{1}{2 x^2 y^2} + c $
(D) None of these.
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Q: 9. Solution of the differential equation, y dx – x dy + x y2 dx = 0 can be
(A) 2x + x2 y = λ y
(B) 2y + y2 x = λ y
(C) 2y – y2 x = λ y
(D) none of these
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Q:10. If y = y(x) and $ (\frac{2+ sinx}{y+1}) \frac{dy}{dx} = – cosx $ , y(0) = 1 , then y (π/2) equals,
(A) 1/3
(B) 2/3
(C) -1/3
(D) 1
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Q:11. The equation of the curve whose subnormal is equal to a constant a is:
(A) y = ax + b
(B) y2 = 2ax + 2b
(C) ay2 – x3 = a
(D) None of these
where b is an arbitrary constant.
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Q:12. Solution of the equation $\frac{dy}{dx} + \frac{1}{x} tany = \frac{1}{x^2} tany siny $ is
(A) x = siny (1 + 2cx2)
(B) 2x = siny (1 + cx2)
(C) 2x + siny (1 + cx2) = 0
(D) None of these
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Q: 13. If y = e4x + 2e-x satisfies the relation y3 + Ay’ + By = 0 , then the value of A and B are respectively
(A) 12, 13
(B) – 12 , 13
(C) – 13 , – 12
(D) – 12 , – 13
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Q:14. A solution of the differential equation $ (\frac{dy}{dx})^2 – x \frac{dy}{dx} + y = 0 $ is
(A) y = 2
(B) y = 2x
(C) y = 2x – 4
(D) y = 2x2 – 4
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Q:15. Equation to the curve such that the y-intercept cut off by the tangent at any arbitrary point is proportional to the square of the ordinate of the point of tangency is of the form
(A) $ \frac{a}{x} + \frac{b}{y^2} = 1 $
(B) $ \frac{a}{x^2} + \frac{b}{y^2} = 1 $
(C) $ \frac{a}{x} + \frac{b}{y} = 1 $
(D) none of these
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Q:16. The solution of the differential equation (1 + x2 y2)y dx + (x2 y2 – 1)x dy = 0 is
(A) $ xy = ln(\frac{x}{y}) + c $
(B) $ xy = 2ln\frac{y}{x} + c $
(C) $ x2 y2 = 2ln \frac{y}{x} + c $
(D) none of these
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Q:17. Tangent to a curve intersects the y-axis at a point P. A line perpendicular to this tangent through P passes through another point (1, 0). The differential equation of the curve is
(A) $ y \frac{dy}{dx} – x (\frac{dy}{dx})^2 = 0 $
(B) $ x \frac{d_2 y}{dx^2} + (\frac{dy}{dx})^2 = 1 $
(C) $ y \frac{dx}{dy} + x = 1 $
(D) none of these
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Q:18. Solution to the differential equation $\frac{x + \frac{x^3}{3!}+ \frac{x^5}{5!}+ …}{1 + \frac{x^2}{2!}+ \frac{x^4}{4!}+ …}$ is
(A) 2y e2x = C.e2x + 1
(B) 2y e2x = C.ex – 1
(C) y e2x = C.e2x + 2
(D) none of these
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Q:19. A curve is such that the mid point of the portion of the tangent intercepted between the point where the tangent is drawn and the point where the tangent meets y-axis, lies on the line y = x. If the curve passes through (1, 0), then the curve is
(A) 2y = x2 – x
(B) y = x2 – x
(C) y = x – x2
(D) y = 2(x – x2)
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Q:20. The solution of the differential equation dx (x2 – t) = dt , is
(A) t = (x – 1)2ex + c
(B) t = (x + 1)2ex + c
(C) t = (x2 + 2x – 1)ex + c
(D) none of these
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11. (B) 12. (B) 13. (C) 14. (C) 15. (C) 16. (C) 17. (A) 18. (B) 19. (C) 20. (A)