Bihar Board 12th Maths Objective Answers Chapter 9 Differential Equations

Bihar Board 12th Maths Objective Questions and Answers

Bihar Board 12th Maths Objective Answers Chapter 9 Differential Equations

Question 1.
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) 1

Question 2.
The differential equation of all parabolas whose axis of symmetry is along the axis of the x-axis is of order
(a) 3
(b) 1
(c) 2
(d) none of these
Answer:
(c) 2

Question 3.
The degree of the equation satisfying the relation \(\sqrt{1+x^{2}}+\sqrt{1+y^{2}}=\lambda(\sqrt{1+y^{2}}-y \sqrt{1+x^{2}})\) is
(a) 1
(b) 2
(c) 3
(d) none of these
Answer:
(a) 1

Question 4.
The degree of the differential equation \(\left(\frac{d^{2} y}{d x^{2}}\right)^{2 / 3}+4-\frac{3 d y}{d x}=0\) is
(a) 2
(b) 1
(c) 3
(d) none of these
Answer:
(a) 2

Question 5.
The differential equation whose solution is (x – h)² + (y – k)² = a² is (a is a constant)

Answer:
(b) \(\left[1+\left(\frac{d y}{d x}\right)^{2}\right]^{3}=a^{2}\left(\frac{d^{2} y}{d x^{2}}\right)^{2}\)

Question 6.
The differential equation satisfied by y = \(\frac{A}{x}\) + B is (A, B are parameters)
(a) x² y₁ = y
(b) xy₁ + 2y₂ = 0
(c) xy₂ + 2y₁ = 0
(d) none of these
Answer:
(c) xy₂ + 2y₁ = 0

Question 7.
The differential equation whose solution represents the family xy = Ae^ax + Be^-ax

Answer:
(c) \(x \frac{d^{2} y}{d x^{2}}+2 \frac{d y}{d x}=a^{2} x y\)

Question 8.
The differential equation having solution is y = 17e^x + ae^-x is
(a) y” – x = 0
(b) y” – y = 0
(c) y’ – y = 0
(d) y’ – x = 0
Answer:
(b) y” – y = 0

Question 9.
The solution of a differential equation is y = c₁e^4x + c₂e^3x, the differential equation is given by

Answer:
(c) \(\frac{d^{2} y}{d x^{2}}-7 \frac{d y}{d x}+12 y=0\)

Question 10.
The differential equation satisfied by

Answer:
(b) \(\frac{d y}{d x}=\frac{1+y^{2}}{1+x^{2}}\)

Question 11.
The differential equation of all ‘Simple Harmonic Motions’ of given period \(\frac{2 \pi}{n}\) is

Answer:
(b) \(\frac{d^{2} x}{d t^{2}}+n^{2} x=0\)

Question 12.
The differential equation of all parabolas whose axes are parallel to y-axis is

Answer:
(a) \(\frac{d y}{d x}=-\frac{c^{2}}{x^{2}}\)

Question 13.
The differential equation of all non-horizontal lines in a plane is

Answer:
(b) \(\frac{d^{2} x}{d y^{2}}=0\)

Question 14.
The differential equation of all circles which pass through the origin and whose centre lies on y-axis is

Answer:
(a) \(\left(x^{2}-y^{2}\right) \frac{d y}{d x}-2 x y=0\)

Question 15.
The differential equation of the family of circles touching the x-axis at origin is given by

Answer:
(b) \(y^{\prime}=\frac{2 x y}{x^{2}-y^{2}}\)

Question 16.
The differential equation representing the family of ellipses with centre at origin and foci on x-axis is given as
(a) xy’ + y = 0
(b) x²y²(y”)² + yy’= 0
(c) xyy” + x(y’)² – yy’ = 0
(d) None of these
Answer:
(b) x²y²(y”)² + yy’= 0

Question 17.
The differential equation of all parabolas whose axes are along x-axis is
(a) \(y_{2}^{2}+y_{1}=0\)
(b) \(y_{1}^{2}+y_{2}=0\)
(c) \(y_{1}^{2}+y_{1} y_{2}=0\)
(d) \(y_{1}^{2}+y y_{2}=0\)
Answer:
(d) \(y_{1}^{2}+y y_{2}=0\)

Question 18.
The equation of family of curves for which the length of the normal is equal to the radius vector is
(a) \(y^{2} \mp x^{2}=k^{2}\)
(b) \(y \pm x=k\)
(c) y² = kx
(d) none of these
Answer:
(a) \(y^{2} \mp x^{2}=k^{2}\)

Question 19.
The solution of the differential equation \(\frac{d y}{d x}=\frac{x^{2}+y^{2}+1}{2 x y}\) satisfying (1) = 1, is
(a) a hyperbola
(b) a circle
(c) y² = x(1 + x) – 10
(d) (x – 2)² + (y – 3)² = 5xy
Answer:
(a) a hyperbola

Question 20.
Given the differential equation \(\frac{d y}{d x}=\frac{6 x^{2}}{2 y+\cos y}\); y(1) = π
Mark out the correct statement.
(a) solution is y² – sin y = -2x³ + C
(b) solution is y² + sin y = 2x³ + C
(c) C = π²+ 2√2
(d) C = π² + 2
Answer:
(b) solution is y² + sin y = 2x³ + C

Question 21.
For the differential equation \(x \frac{d y}{d x}+2 y=x y \frac{d y}{d x}\),
(a) order is 1 and degree is 1
(b) solutio is ln(yx²) = C – y
(c) order is 1 and degree is 2
(d) solution is ln(xy²) = C + y
Answer:
(a) order is 1 and degree is 1

Question 22.
The particular solution In(\(\frac{d y}{d x}\)) = 3x + 4y, y(0) = 0 is
(a) e^3x + 3e^-4y = 4
(b) 4e^3x – 3e^-4y = 3
(c) 3e^3x + 4e^4y = 7
(d) 4e^3x + 3e^-4y = 7
Answer:
(d) 4e^3x + 3e^-4y = 7

Question 23.
The solution of the differential equation

Answer:
(c) y = x tan(C – x)

Question 24.
The solution of the differential equation

Answer:
(d) None of these

Question 25.
The solution of the differential equation

Answer:
(c) \(y=x \tan \left(\frac{C-x^{2}-y^{2}}{2}\right)\)

Question 26.

Answer:
(b) \(c e^{y / 2}\)

Question 27.
If ydx – xdy + ln x dx = 0, y(1) = -1, then
(a) y + 1 + ln x = 0
(b) y + 1 + 2 ln x = 0
(c) 2(y + 1) + lnx = 0
(d) y + 1 – y ln x = 0
Answer:
(a) y + 1 + ln x = 0

Question 28.
The differential equation \(\frac{d y}{d x}=\sqrt{\frac{1-y^{2}}{y}}\) 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) fixed radius 1 and variable centre on x-axis

Question 29.
If y dx + y² dy = x dy, x ∈ R, y > 0 and y(1) = 1, then y(-3) =
(a) 3
(b) 2
(c) 1
(d) 5
Answer:
(a) 3

Question 30.
The solution of y dx + (x + x²y) dy = 0 is

Answer:
(b) \(-\frac{1}{x y}+\ln y=c\)

Question 31.
If (x + y)² \(\frac{d y}{d x}\) = a², y = 0 when x = 0, then y = a if \(\frac{x}{a}\) =
(a) 1
(b) tan 1
(c) tan 1 + 1
(d) tan 1 – 1
Answer:
(d) tan 1 – 1

Question 32.

Answer:
(a) e^x – 1

Question 33.
If sinx \(\frac{d y}{d x}\) + y cosx = x sinx, then (y – 1) sinx =
(a) c – x sinx
(b) c + xcosx
(c) c – x cos x
(d) c + x sin x
Answer:
(c) c – x cos x

Question 34.
The solution of the differential equation

Answer:
(c) \(e^{y}=\frac{x^{3}}{3}+e^{x}+c\)

Question 35.
The solution of the differential equation xdy + ydx = xydx when y(1) = 1 is

Answer:
(b) \(\frac{e^{x}}{e x}\)

Question 36.
The solution of differential equation (e^y + 1) cosx dx + e^y sinx dy = 0 is
(a) (e^y + 1) sinx = c
(b) e^x sinx = c
(c) (e^x + 1) cosx = c
(d) none of these
Answer:
(a) (e^y + 1) sinx = c

Question 37.
The solution of the differential equation \(\frac{d y}{d x}=\frac{x}{1+x^{2}}\) is

Answer:
(c) \(y=\log (\sqrt{1+x^{2}})+c\)

Question 38.

Answer:
(c) \(\frac{e^{6}+9}{2}\)

Question 39.

Answer:
(a) y = e sin²x

Question 40.
The general solution of the differential equation \(\frac{d y}{d x}=\frac{x^{2}}{y^{2}}\) is
(a) x³ – y³ = c
(b) x³ + y³ = c
(c) x² + y² = c
(d) x² – y² = c
Answer:
(a) x³ – y³ = c

Question 41.
The Solution of cos(x + y) dy = dx is

Answer:
(a) \(y=\tan \left(\frac{x+y}{2}\right)+C\)

Question 42.

Answer:
(d) x + x ln x

Question 43.

Answer:
(c) √3e

Question 44.

Answer:
(d) \(\ln \frac{y}{x}=c x\)

Question 45.

Answer:
(d) \(\frac{\pi}{12}\)

Question 46.

Answer:
(d) \(\sec \frac{y}{x}=c x y\)

Question 47.

Answer:
(c) \(-2 \sqrt{\frac{x}{y}}=\ln c y\)

Question 48.

Answer:
(c) \(x+y e^{x / y}=c\)

Question 49.

Answer:
(a) \(x y=c e^{y / x}\)

Question 50.

Answer:
(c) Circle

Question 51.

Answer:
(c) \(\sqrt{x^{2}+y^{2}}+y=c x^{2}\)

Question 52.
The solution of the differential equation (x² + y²) dx – 2xy dy = 0 is

Answer:
(d) \(\frac{x^{2}-y^{2}}{x}=c\)

Question 53.
The solution of the differential equation x dy + (x + y) dx = 0 is

Answer:
(b) \(c=x y+\frac{x^{2}}{2}\)

Question 54.
The solution of differential equation \(\frac{d y}{d x}=\frac{x-y}{x+y}\) is
(a) x² – y² + 2xy + c = 0
(b) x² – y² – xy + c = 0
(c) x² – y² + xy + c = 0
(d) x² – y² – 2xy + c = 0
Answer:
(d) x² – y² – 2xy + c = 0

Question 55.

Answer:
(a) \(-\frac{1}{2}\)

Question 56.

Answer:
(c) \(\frac{e^{2}+1}{4}\)

Question 57.

Answer:
(d) 6

Question 58.

Answer:
(d) \(\frac{5}{2}\)

Question 59.

Answer:
(d) \(\frac{2}{e}\)

Question 60.

Answer:
(d) \(\frac{1}{4}\)

Question 61.

Answer:
(c) \(\frac{x}{y}-y^{2}=c\)

Question 62.


Answer:
(b) \(y=\frac{\sqrt{1+x^{2}}}{x}+\frac{c}{x}\)

Question 63.

Answer:
(c) \(y e^{-3 x}=-e^{-3 x} \frac{(2 \cos 2 x+3 \sin 2 x)}{13}+c\)

Question 64.

Answer:
(a) \(-\frac{1}{2}\)

Question 65.
The solution of the differential equation,

Answer:
(a) \(y=\sin \frac{1}{x}-\cos \frac{1}{x}\)

Question 66.
The degree of the differential equation

(a) 1
(b) 2
(c) 3
(d) not defined
Answer:
(d) not defined

Question 67.
The order and degree of the differential equation \(\frac{d^{2} y}{d x^{2}}+\left(\frac{d y}{d x}\right)^{\frac{1}{4}}+x^{\frac{1}{5}}=0\) respectively are
(a) 2 and not defined
(b) 2 and 2
(c) 2 and 3
(d) 3 and 3
Answer:
(a) 2 and not defined

Question 68.
The differential equation for y = A cos αx + B sin αx, where A and B are arbitrary constants is

Answer:
(b) \(\frac{d^{2} y}{d x^{2}}+\alpha^{2} y=0\)

Question 69.
Integrating factor of the differential equation

Answer:
(c) \(\sqrt{1-x^{2}}\)

Question 70.
Integrating factor of the differential equation \(\frac{d y}{d x}\) + y tanx – sec x = 0 is
(a) cos x
(b) sec x
(c) e^{cos x}
(d) e^{sec x}
Answer:
(b) sec x

Question 71.
The solution of the differential equation \(\frac{d y}{d x}=\frac{1+y^{2}}{1+x^{2}}\) is
(a) y = tan^-1 x
(b) y – x = k(1 + xy)
(c) x = tan^-1 y
(d) tan(xy) = k
Answer:
(b) y – x = k(1 + xy)

Question 72.
The solution of the differential equation cos x sin y dx + sin x cos y dy = 0 is
(a) \(\frac{\sin x}{\sin y}=c\)
(b) sin x sin y = c
(c) sin x + sin y = c
(d) cos x cos y = c
Answer:
(b) sin x sin y = c

Question 73.

Answer:
(c) \(e^{y}=e^{x^{2}}+c\)

Question 74.
Which of the following is the general solution of

Answer:
(a) y = (Ax + B) e^x

Question 75.

Answer:
(a) \(y\left(1+x^{2}\right)=c+\tan ^{-1} x\)