MCQOPTIONS
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MCQOPTIONS
This section includes 88 Mcqs, each offering curated multiple-choice questions to sharpen your BITSAT knowledge and support exam preparation. Choose a topic below to get started.
| 51. |
If \({\rm{u}} = {\sin ^{ - 1}}(\frac{{{\rm{x + y}}}}{{\sqrt {\rm{x}} {\rm{ + }}\sqrt {\rm{y}} }})\) then by Euler's theorem, \({\rm{x}}\frac{{\partial {\rm{u}}}}{{\partial {\rm{x}}}} + {\rm{y}}\frac{{\partial {\rm{u}}}}{{\partial {\rm{y}}}}\) equals to |
| A. | \(\frac{1}{2}\sin {\rm{u}}\) |
| B. | \(\frac{1}{2}\tan {\rm{u}}\) |
| C. | x + y |
| D. | sin x + sin y |
| Answer» C. x + y | |
| 52. |
If all the terms of a PDF contain the dependent variable or its partial derivatives then such a PDF is called |
| A. | Quasi – linear equation |
| B. | Non-homogeneous partial differential equation |
| C. | Homogeneous partial differential equation |
| D. | Linear equation |
| Answer» C. Homogeneous partial differential equation | |
| 53. |
General solution of the Cauchy-Euler equation \(x^2\frac{d^2y}{dx^2}-7x\dfrac{dy}{dx}+16y=0\) is |
| A. | y = c1x2 + c2x4 |
| B. | y = c1x2 + c2x-4 |
| C. | y = (c1 + c2 In x) x4 |
| D. | y = c1x4 + c2x-4 In x |
| Answer» D. y = c1x4 + c2x-4 In x | |
| 54. |
One dimensional wave equation is |
| A. | \(\frac{{{\partial ^2}y}}{{\partial {t^2}}} = {\alpha ^2}\frac{{{\partial ^2}y}}{{\partial {x^2}}}\) |
| B. | \(\frac{{\partial y}}{{\partial t}} = {\alpha ^2}\frac{{{\partial ^2}y}}{{\partial {x^2}}}\) |
| C. | \(\frac{{{\partial ^2}y}}{{\partial {t^2}}} = -\frac{{{\partial ^2}y}}{{\partial {x^2}}}\) |
| D. | None of these |
| Answer» B. \(\frac{{\partial y}}{{\partial t}} = {\alpha ^2}\frac{{{\partial ^2}y}}{{\partial {x^2}}}\) | |
| 55. |
If xdy = y(dx + ydy) ; y(1) = 1 and y(x) > 0, then what is y(-3) equal to? |
| A. | 3 |
| B. | 2 |
| C. | 1 |
| D. | 0 |
| Answer» B. 2 | |
| 56. |
If y(x) satisfies the differential equation\((\sin x) \frac{dy}{dx}+y\cos x = 1\)subject to the condition y(π/2) = π/2, then y(π/6) is |
| A. | 0 |
| B. | π/6 |
| C. | π/3 |
| D. | π/2 |
| Answer» D. π/2 | |
| 57. |
Particular integral of\(\frac{{{d^2}y}}{{d{x^2}}} + 3\frac{{dy}}{{dx}} + 2y = 5\) is |
| A. | 2 / 5 |
| B. | 1 / 5 |
| C. | 5 / 2 |
| D. | 3 / 2 |
| Answer» D. 3 / 2 | |
| 58. |
General solution of \(x^2\frac{d^2y}{dx^2}+x\frac{dy}{dx}-y=0\) is |
| A. | \(y=\frac{C_1}{x}+\frac{C_2}{x^3}\) |
| B. | \(y={C_1}{x^2}+\frac{C_2}{x}\) |
| C. | \(y={C_1}{x}+\frac{C_2}{x}\) |
| D. | \(y={C_1}{x}+\frac{C_2}{x^3}\) |
| Answer» D. \(y={C_1}{x}+\frac{C_2}{x^3}\) | |
| 59. |
Differential equation are equations that involve ____________ variables and their ___________ with respect to the ____________ variables. |
| A. | dependent, derivatives, independent |
| B. | independent, derivatives, independent |
| C. | dependent, derivatives, dependent |
| D. | none of these |
| Answer» B. independent, derivatives, independent | |
| 60. |
Consider the following statements about the linear dependence of the real valued functions y1 = 1, y2 = x and y3 = x2, over the field of real numbers.I. y1, y2 and y3 are linearly independent on – 1 ≤ x ≤ 0II. y1, y2 and y3 are linearly dependent on 0 ≤ x ≤ 1III. y1, y2 and y3 are linearly independent on 0 ≤ x ≤ 1IV. y1, y2 and y3 are linearly dependent on – 1 ≤ x ≤ 0Which one among the following is correct? |
| A. | Both I and II are true |
| B. | Both I and III are true |
| C. | Both II and IV are true |
| D. | Both III and IV are true |
| Answer» C. Both II and IV are true | |
| 61. |
Consider the differential equation \(\frac{{dx}}{{dt}} = \sin \left( x \right)\), with the initial condition x(0) = 0. The solution to this ordinary differential equation is |
| A. | x(t) = 0 |
| B. | x(t) = sin (t) |
| C. | x(t) = cos (t) |
| D. | x(t) = sin (t) – cos (t) |
| Answer» B. x(t) = sin (t) | |
| 62. |
Bernoulli's differential equation is given by |
| A. | \(\rm \dfrac{dy}{dx}+P(x) y=Q(x)y^n\) |
| B. | \(\rm \dfrac{dy}{dx}+y=Q(x)y^2+R(x)\) |
| C. | \(\rm \dfrac{dy}{dx}+P(x)y=Q(x)y^2\) |
| D. | \(\rm \dfrac{dy}{dx}+P(x)y=Q(x)\) |
| Answer» B. \(\rm \dfrac{dy}{dx}+y=Q(x)y^2+R(x)\) | |
| 63. |
A complete solution of partial differential equation \(\frac {\partial z}{\partial x} - 3x^2 = \left(\frac {\partial z}{\partial y}\right)^2 - y\) will be ________, where a and b are arbitrary constants. |
| A. | z2 = ax2 + by2 + 1 |
| B. | \(z = ax + x^3 + \left(\frac 2 3\right) (a + y)^{\frac 3 2} + b\) |
| C. | z1 / 2 = (x + a)1 / 2 + (y + b)1 / 2 |
| D. | z = (ax2 + by2)3 / 2 + 2 |
| Answer» C. z1 / 2 = (x + a)1 / 2 + (y + b)1 / 2 | |
| 64. |
Particular integral of \(\frac{{{d^2}y}}{{d{x^2}}} + 3\frac{{dy}}{{dx}} + 2y = {e^{ - 2x}} + \sin x\) is |
| A. | \(x{e^{ - 2x}} + \frac{1}{{10}}\left( {\sin x + 3\cos x} \right)\) |
| B. | \( - x{e^{ - 2x}} + \frac{1}{{10}}\left( {\sin x - 3\cos x} \right)\) |
| C. | \( - x{e^{ - 2x}} + \frac{1}{{10}}\left( {\cos x - 3\sin x} \right)\) |
| D. | \( - x{e^{ - 2x}} + \frac{1}{{10}}\left( {\sin x + 3\cos x} \right)\) |
| Answer» C. \( - x{e^{ - 2x}} + \frac{1}{{10}}\left( {\cos x - 3\sin x} \right)\) | |
| 65. |
Match the following:a. \(\frac{{{\partial ^2}u}}{{\partial {t^2}}} = C\frac{{{\partial ^2}u}}{{\partial {x^2}}}\)I. Two-dimensional Poisson equationb. \(\frac{{\partial u}}{{\partial t}} = C\frac{{{\partial ^2}u}}{{\partial {x^2}}}\)II. One-dimensional wave equationc. \(\frac{{{\partial ^2}u}}{{\partial {x^2}}} + \frac{{{\partial ^2}u}}{{\partial {y^2}}} = 0\)III. One-dimensional heat equationd. \(\frac{{{\partial ^2}u}}{{\partial {x^2}}} + \frac{{{\partial ^2}a}}{{\partial y}} = f\left( {x,y} \right)\)IV. Two-dimensional Laplace equation |
| A. | a-II, b-III, c-I, d-IV |
| B. | a-II, b-III, c-IV, d-I |
| C. | a-IV, b-I, c-III, d-II |
| D. | a-IV, b-III, c-II, d-I |
| Answer» C. a-IV, b-I, c-III, d-II | |
| 66. |
A differential equation is given as:\({x^2}\frac{{{d^2}y}}{{d{x^2}}} - 2x\frac{{dy}}{{dx}} + 2y = 4\)The solution of the differential equation in terms of arbitrary constants C1 and C2 is |
| A. | y = C1x2 + C2x + 2 |
| B. | \(y = \frac{{{C_1}}}{{{x^2}}} + {C_2}x + 2\) |
| C. | y = C1x2 + C2x + 4 |
| D. | \(y = \frac{{{C_1}}}{{{x^2}}} + {C_2}x + 4\) |
| Answer» B. \(y = \frac{{{C_1}}}{{{x^2}}} + {C_2}x + 2\) | |
| 67. |
An ordinary differential equation is given below:\(\left( {\frac{{dy}}{{dx}}} \right)\left( {xlnx} \right) = y\)The solution for the above equation is (Note: K denotes a constant in the options) |
| A. | y = Klnx |
| B. | y = Kxlnx |
| C. | y = Kxex |
| D. | y = Kxe-x |
| Answer» B. y = Kxlnx | |
| 68. |
If f(Z) is an analytical function and (r, θ) denotes the polar co-ordinates, then: |
| A. | \(\frac {\partial u}{\partial r} = \frac 1 r \frac {\partial v}{\partial \theta}\) and \(\frac {\partial v}{\partial r} = \frac {-1}{r} \frac {\partial u}{\partial \theta}\) |
| B. | \(\frac {\partial u}{\partial r} = \frac {-1} r \frac {\partial v}{\partial \theta}\) and \(\frac {\partial u}{\partial r} = \frac {1} r \frac {\partial v}{\partial \theta}\) |
| C. | \(\frac {\partial u}{\partial r} = -r \frac {\partial v}{\partial \theta}\) and \(\frac {\partial v}{\partial r} = r \frac {\partial u}{\partial \theta}\) |
| D. | \(\frac {\partial u}{\partial r} = r \frac {\partial v}{\partial \theta}\) and \(\frac {\partial v}{\partial r} = -r \frac {\partial u}{\partial \theta}\) |
| Answer» B. \(\frac {\partial u}{\partial r} = \frac {-1} r \frac {\partial v}{\partial \theta}\) and \(\frac {\partial u}{\partial r} = \frac {1} r \frac {\partial v}{\partial \theta}\) | |
| 69. |
A partial differential equation derived from the equation z = aeby sinbx will be: |
| A. | \(\frac{{\partial z}}{{\partial y}} = 2y{\left( {\frac{{\partial z}}{{\partial x}}} \right)^2}\) |
| B. | \(\left( {1 + \frac{{\partial z}}{{\partial y}}} \right)\frac{{\partial z}}{{\partial x}} = z\) |
| C. | \(\frac{{{\partial ^2}z}}{{\partial {x^2}}} + \frac{{{\partial ^2}z}}{{\partial {y^2}}} = 0\) |
| D. | \(2z = x\frac{{\partial z}}{{\partial x}} + y\frac{{\partial z}}{{\partial y}}\) |
| Answer» D. \(2z = x\frac{{\partial z}}{{\partial x}} + y\frac{{\partial z}}{{\partial y}}\) | |
| 70. |
Complimentary function of the differential equation \({x^2}\frac{{{d^2}y}}{{d{x^2}}} + 4x\frac{{dy}}{{dx}} + 2y = {e^{{x^2}}}\) |
| A. | c1x-1 + c2x-2 |
| B. | c1x + c2x2 |
| C. | c1x-1 + c2x2 |
| D. | c1x + c2x-2 |
| Answer» B. c1x + c2x2 | |
| 71. |
Let f(x) be a polynomial satisfying f(0) = 2, f'(0) = 3 and f''(x) = f(x). Then f(4) is equal to |
| A. | \(5 \frac {(e^8 - 1)}{2e^4}\) |
| B. | \(\frac {(5e^8 - 1)}{2e^4}\) |
| C. | \(\frac {2e^4}{5e^8 - 1}\) |
| D. | \(\frac {2e^4}{5(e^8 + 1)}\) |
| Answer» C. \(\frac {2e^4}{5e^8 - 1}\) | |
| 72. |
Consider the following differential equation: \(\frac{{dy}}{{dt}} = - 5y\); Initial condition: y = 2 at t = 0The value of y at t = 3 is |
| A. | – 5e-10 |
| B. | 2e-10 |
| C. | 2e-15 |
| D. | -15e2 |
| Answer» D. -15e2 | |
| 73. |
If y = f(x) is solution of \(\frac{{{d^2}y}}{{d{x^2}}} = 0\) with the boundary conditions y = 5 at x = 0, and \(\frac{{dy}}{{dx}} = 2\) at x = 10, f(15) = _____ |
| A. | 15 |
| B. | 25 |
| C. | 35 |
| D. | 5 |
| Answer» D. 5 | |
| 74. |
Eliminating a and b from the (x - a)2 + (y - b)2 + z2 = c2 the partial differential equation is |
| A. | z2(p - q + 1) = c2 |
| B. | z2(p2 + q2 + 1) = c2 |
| C. | z2(p2 + q2) = c2 |
| D. | z2(p - q) = c2 |
| Answer» C. z2(p2 + q2) = c2 | |
| 75. |
If k is constant, the general solution of \(\frac{dy}{dx} - \frac{y}{x}=1\) will be in the form of |
| A. | y = x ln (kx) |
| B. | y = k ln (kx) |
| C. | y = xk ln (k) |
| D. | y = x ln (x) |
| Answer» B. y = k ln (kx) | |
| 76. |
Given that y1(x) = x-1 is one solution of 2x2y"+ 3xy' - y = 0, x > 0. Then the second linearly independent solution is |
| A. | x-2 |
| B. | x |
| C. | x1/2 |
| D. | x2 |
| Answer» D. x2 | |
| 77. |
If y(x) is a solution of the differential equation \(\frac{{dy}}{{dx}} + 4xy = {x^3},y(0) = 0\) then \(\mathop {\lim }\limits_{x \to 0} y(x)\) is |
| A. | 0 |
| B. | -2 |
| C. | 1 |
| D. | not existing |
| Answer» B. -2 | |
| 78. |
If y = e3x + e-5x find the value of \({d^2y\over dx^2}\) at x = 0 |
| A. | 8 |
| B. | 34 |
| C. | 16 |
| D. | -16 |
| Answer» C. 16 | |
| 79. |
A partial differential equation formed from the relationz = (x2 + a)(y2 + b) will be |
| A. | \(\frac{\delta z}{\delta x} \frac{\delta z}{ \delta y} = 4xyz\) |
| B. | \(\frac{\delta z}{\delta x} \frac{\delta z}{ \delta y} = 4xy\) |
| C. | \(\frac{\delta z}{\delta x} +\frac{\delta z}{ \delta y} = 4xyz\) |
| D. | \(\frac{\delta z}{\delta x} -\frac{\delta z}{ \delta y} = 4xy\) |
| Answer» B. \(\frac{\delta z}{\delta x} \frac{\delta z}{ \delta y} = 4xy\) | |
| 80. |
A curve passes through the point (x = 1, y = 0) and satisfies the differential equation \(\frac{{{\rm{dy}}}}{{{\rm{dx}}}} = \frac{{{{\rm{x}}^2} + {{\rm{y}}^2}}}{{2{\rm{y}}}} + \frac{{\rm{y}}}{{\rm{x}}}.\) The equation that describes the curve is |
| A. | \({\rm{In}}\left( {1 + \frac{{{{\rm{y}}^2}}}{{{{\rm{x}}^2}}}} \right) = {\rm{x}} - 1\) |
| B. | \(\frac{1}{2}{\rm{In}}\left( {1 + \frac{{{{\rm{y}}^2}}}{{{{\rm{x}}^2}}}} \right) = {\rm{x}} - 1\) |
| C. | \({\rm{In}}\left( {1 + \frac{{\rm{y}}}{{\rm{x}}}} \right) = {\rm{x}} - 1\) |
| D. | \(\frac{1}{2}{\rm{In}}\left( {1 + \frac{{\rm{y}}}{{\rm{x}}}} \right) = {\rm{x}} - 1\) |
| Answer» B. \(\frac{1}{2}{\rm{In}}\left( {1 + \frac{{{{\rm{y}}^2}}}{{{{\rm{x}}^2}}}} \right) = {\rm{x}} - 1\) | |
| 81. |
Consider the following second-order differential equation:y” – 4y’ + 3y = 2t – 3t2The particular solution of the differential equation is |
| A. | – 2 – 2t – t2 |
| B. | – 2t – t2 |
| C. | 2t – 3t2 |
| D. | – 2 – 2t – 3t2 |
| Answer» B. – 2t – t2 | |
| 82. |
Consider the differential equation \(\frac{{{\rm{dx}}}}{{{\rm{dt}}}} = 10{\rm{\;}}-{\rm{\;}}0.2{\rm{x}}\) with initial condition \({\rm{\;x}}\left( 0 \right){\rm{\;}} = {\rm{\;}}1\). The response x(t) for \({\rm{t}} > 0\) is |
| A. | \(2{\rm{\;}}-{\rm{\;}}{{\rm{e}}^{ - 0.2{\rm{t}}}}\) |
| B. | \(2{\rm{\;}}-{\rm{\;}}{{\rm{e}}^{0.2{\rm{t}}}}\) |
| C. | \(50{\rm{\;}}-{\rm{\;}}49{{\rm{e}}^{ - 0.2{\rm{t}}}}\) |
| D. | \(50{\rm{\;}}-{\rm{\;}}49{{\rm{e}}^{0.2{\rm{t}}}}\) |
| Answer» D. \(50{\rm{\;}}-{\rm{\;}}49{{\rm{e}}^{0.2{\rm{t}}}}\) | |
| 83. |
A partial differential equation requires |
| A. | exactly one independent variable |
| B. | two or more independent variables |
| C. | more than one dependent variable |
| D. | exactly one dependent variable |
| Answer» C. more than one dependent variable | |
| 84. |
Partial differential equation involves |
| A. | only one independent variable |
| B. | only two independent variables |
| C. | two or more than one independent variable |
| D. | no independent variable |
| Answer» D. no independent variable | |
| 85. |
A general solution of the differential equation \((x + y) \frac{dy}{dx} = x - y\) will be _____ where c is a constant. |
| A. | x2 + xy + y2 = c |
| B. | x2 - 2xy - y2 = c |
| C. | 2x2 + xy + y2 = c |
| D. | x2 + 2xy - y2 = c |
| Answer» C. 2x2 + xy + y2 = c | |
| 86. |
Cauchy’s linear differential equation \({x^n}\frac{{{d^n}y}}{{d{x^n}}} + {a_1}{x^{n - 1}}\frac{{{d^{n - 1}}}}{{d{x^{n - 1}}}} + \ldots + {a_n}y = f\left( x \right)\) can be reduced to a linear differential equation with constant coefficient by using substitution |
| A. | x = ez |
| B. | y = ez |
| C. | z = ex |
| D. | z = ey |
| Answer» B. y = ez | |
| 87. |
Fine the solution of \(\frac{{{d^2}y}}{{d{x^2}}} = y\) which passes through the origin and the point \(\left( {\ln2,\frac{3}{4}} \right)\) |
| A. | \(y = \frac{1}{2}{e^x} - {e^x}\) |
| B. | \(y = \frac{1}{2}\left( {{e^x} + {e^{ - x}}} \right)\) |
| C. | \(y = \frac{1}{2}\left( {{e^x} - {e^{ - x}}} \right)\) |
| D. | \(y = \frac{1}{2}{e^x} + {e^{ - x}}\) |
| Answer» D. \(y = \frac{1}{2}{e^x} + {e^{ - x}}\) | |
| 88. |
If sin \(u = \frac{x^3 + y^3}{\sqrt{x} +\sqrt{y}}\), then xux + yuy will be equal to: |
| A. | \(\frac{5}{2} tan \ u\) |
| B. | \(\frac{3}{2} tan \ u\) |
| C. | \(\frac{1}{2} cot \ u\) |
| D. | \(\frac{3}{2} cot \ u\) |
| Answer» B. \(\frac{3}{2} tan \ u\) | |