Differential Equations Test

Differential Equations

This is Differential Equations Test-02 for CBSE class 12 Maths.. There are 15 questions in this test with each question having around four answer choices.

Questions & Answers

1
General solution of a given differential equation
  • A
    does not contain arbitrary constants
  • B
    contains exactly one arbitrary constant
  • C
    contains arbitrary constants depending on the order of the differential equation
    Correct
  • D
    contains exactly two arbitrary constants
2
Particular solution of a given differential equation
  • A
    does not contain arbitrary constants
    Correct
  • B
    can contain exactly one arbitrary constant
  • C
    can contain exactly two arbitrary constants
  • D
    can contain arbitrary constants
3
The number of arbitrary constants in the general solution of a differential equation of fourth order are:
  • A
    1
  • B
    2
  • C
    3
  • D
    4
    Correct
4
The number of arbitrary constants in the particular solution of a differential equation of third order are:
  • A
    0
    Correct
  • B
    1
  • C
    3
  • D
    2
5
General solution of \(\frac{{dy}}{{dx}} = \;\frac{{1 - cosx}}{{1 + cosx}}\) is
  • A
    \(y{\text{ }} = {\text{ }}2tan\left( {\frac{x}{2}} \right){\text{ }} + {\text{ }}x\; + {\text{ }}c\)
  • B
    \(y{\text{ }} = {\text{ }}2ta{n^2}\left( {\frac{x}{2}} \right){\text{ }}--{\text{ }}x\; + {\text{ }}c\)
  • C
    \(\begin{array}{*{20}{l}} {y{\text{ }} = {\text{ }}2tan\left( {\frac{x}{2}} \right){\text{ }}--{\text{ }}x\; + {\text{ }}c} \end{array}\)
    Correct
  • D
    y = 2tan (x) – x + c
6
General solution of \(\frac{{dy}}{{dx}} = \sqrt[2]{{4 - \;{y^2}}}\)( -2 < y < 2) is
  • A
    y = 2sin(x + c)
    Correct
  • B
    y = 2cosx + c
  • C
    \(\begin{array}{*{20}{l}} {y{\text{ }} = {\text{ }}2sin{x^2} + {\text{ }}c} \end{array}\)
  • D
    y = 2sin(2x) + c
7
General solution of \(\;\frac{{dy}}{{dx}} + y = 1\;(y\; \ne 1)\) is
  • A
    \(y{\text{ }} = {\text{ }}1{\text{ }} + {\text{ }}A{e^x}\)
  • B
    \(\begin{array}{*{20}{l}} {y{\text{ }} = {\text{ }}B{\text{ }} + {\text{ }}A{e^{ - x}}} \end{array}\)
  • C
    \(y{\text{ }} = {\text{ }}1{\text{ }} + {\text{ }}A{e^{ - x}}\)
    Correct
  • D
    \(y{\text{ }} = {\text{ }}1{\text{ }} + {\text{ }}A{e^{ - 3x}}\)
8
General solution of \(se{c^2}xtany{\text{ }}dx{\text{ }} + {\text{ }}se{c^2}y{\text{ }}tanxdy{\text{ }} = {\text{ }}0\) is
  • A
    y = tanx tan2y + C
  • B
    tanxtany = C
    Correct
  • C
    y = tan2x tany + C
  • D
    \(\begin{array}{*{20}{l}} {y{\text{ }} = {\text{ }}tan{x^2}tany{\text{ }} + {\text{ }}C} \end{array}\)
9
General solution of \(\left( {{e^x} + {\text{ }}{e^{ - x}}} \right){\text{ }}dy{\text{ }} - {\text{ }}\left( {{e^x} - {\text{ }}{e^{ - x}}} \right){\text{ }}dx{\text{ }} = {\text{ }}0\)
  • A
    \(\begin{array}{*{20}{l}} {y{\text{ }} = {\text{ }}log\left( {{e^{ - 2x}} + {\text{ }}{e^{ - x}}} \right){\text{ }} + {\text{ }}C} \end{array}\)
  • B
    \(\begin{array}{*{20}{l}} {y{\text{ }} = {\text{ }}log\left( {{e^{2x}} + {\text{ }}{e^{ - x}}} \right){\text{ }} + {\text{ }}C} \end{array}\)
  • C
    \(\begin{array}{*{20}{l}} {y{\text{ }} = {\text{ }}log\left( {{e^x} + {\text{ }}{e^{ - x}}} \right){\text{ }} + {\text{ }}C} \end{array}\)
    Correct
  • D
    \(y{\text{ }} = {\text{ }}\left( {{e^x} + {\text{ }}{e^{ - x}}} \right){\text{ }} + {\text{ }}C\)
10
General solution of \(\frac{{dy}}{{dx}} = \left( {\;1 + {x^2}} \right)(1 + {y^2})\) is
  • A
    \(si{n^{ - 1}}y\) = x + \(\frac{{{x^3}}}{3} + \;C\)
  • B
    \(ta{n^{ - 1}}y\) = x + \(\frac{{{x^3}}}{3} + \;C\)
    Correct
  • C
    \(co{t^{ - 1}}y\) = x + \(\frac{{{x^3}}}{3} + \;C\)
  • D
    \(co{s^{ - 1}}y\) = x + \(\frac{{{x^3}}}{3} + \;C\)
11
General solution of y log y dx – x dy = 0
  • A
    \(y{\text{ }} = {\text{ }}{e^{cx}}\)
    Correct
  • B
    \(y{\text{ }} = {\text{ }}{e^{cx}} + {\text{ }}{e^{ - cx}}\)
  • C
    \(\begin{array}{*{20}{l}} {y{\text{ }} = {\text{ }}{e^{ - cx}}} \end{array}\)
  • D
    \({y^2} = {\text{ }}{e^{cx}}\)
12
General solution of \(ydx + (x - {y^3})dy = 0\) is
  • A
    None of these
  • B
    \(xy = \frac{{{y^4}}}{4} + C\)
    Correct
  • C
    xy = 3y + C
  • D
    \(xy = \frac{{{y^4}}}{3} + C\)
13
General solution of \(\frac{{dy}}{{dx}} = \;si{n^{ - 1}}x\) is
  • A
    y = \(xsi{n^{ - 1}}x\) + \(\sqrt[2]{{1 - {x^2}}} + \;C\)
    Correct
  • B
    y = \(xsi{n^{ - 1}}x\) + \(\sqrt[2]{{1 - {x^3}}} + \;C\)
  • C
    y = \(xsi{n^{ - 1}}x\) + \(\sqrt[3]{{1 - {x^2}}} + \;C\)
  • D
    y = \(xsi{n^{ - 1}}x\) + \(\sqrt[2]{{1 - {x^4}}} + \;C\)
14
General solution of \({e^x}tan{\text{ }}y{\text{ }}dx{\text{ }} + {\text{ }}\left( {1{\text{ }}--{\text{ }}{e^x}} \right){\text{ }}se{c^2}y{\text{ }}dy{\text{ }} = {\text{ }}0\)
  • A
    \(tan{y^2} = C\left( {\;1 - {e^x}} \right)\)
  • B
    \(tan{x^2} = C\left( {\;1 - {e^x}} \right)\)
  • C
    \(tan{y^3} = C\left( {\;1 - {e^x}} \right)\)
  • D
    \(tany = C\left( {\;1 - {e^x}} \right)\;\)
    Correct
15
Find a particular solution of \(x\left( {{x^2} - {\text{ }}1} \right)\frac{{dx}}{{dy}}\)= 1; y =0 when x =2
  • A
    \(y = \;\frac{1}{2}\log \left( {\frac{{{x^2} - 1}}{{{x^2}}}} \right) - \;\frac{1}{2}log\frac{3}{7}\)
  • B
    \(y = \;\frac{1}{2}\log \left( {\frac{{{x^2} - 1}}{{{x^3}}}} \right) + \;\frac{1}{2}log\frac{3}{4}\)
  • C
    \(y = \;\frac{1}{2}\log \left( {\frac{{{x^3} - 1}}{{{x^2}}}} \right) - \;\frac{1}{2}log\frac{3}{4}\)
  • D
    \(y = \;\frac{1}{2}\log \left( {\frac{{{x^2} - 1}}{{{x^2}}}} \right) - \;\frac{1}{2}log\frac{3}{4}\)
    Correct