Differential Equations Test

Differential Equations

This is Differential Equations Test-03 for CBSE class 12 Maths.. There are 15 questions in this test with each question having around four answer choices.

Questions & Answers

1
Find a particular solution of\(\cos \left( {\frac{{dy}}{{dx}}\;} \right) = a\;(a\; \in \;R);\;y\; = \;1 \) when \(\;x\; = 0\)
  • A
    \(cos\frac{{y - 3}}{x} = a\)
  • B
    \(cos\frac{{y - 10}}{x} = a\)
  • C
    \(cos\frac{{y - 4}}{x} = a\)
  • D
    \(cos\frac{{y - 1}}{x} = a\;\;\)
    Correct
2
Find a particular solution of\(\frac{{dy}}{{dx}} = y\tan x\;;y = 1\;when\;x = 0\)
  • A
    y = sin x
  • B
    y = tan x
  • C
    y = cos x
  • D
    y = sec x
    Correct
3
For the differential equation\(xy\frac{{dy}}{{dx}} = \left( {x + 2\;} \right)(\;y + 2)\) find the solution curve passing through the point (1, –1).
  • A
    \(y{\text{ }}--{\text{ }}x{\text{ }} - {\text{ }}2{\text{ }} = {\text{ }}log{\text{ }}\left( {{x^2}{{\left( {y{\text{ }} - {\text{ }}2} \right)}^2}} \right)\)
  • B
    \(\begin{array}{*{20}{l}} {y{\text{ }}--{\text{ }}x{\text{ }} + {\text{ }}2{\text{ }} = {\text{ }}log{\text{ }}\left( {{x^2}{{\left( {y{\text{ }} + {\text{ }}2} \right)}^2}} \right)} \end{array}\)
    Correct
  • C
    \(\begin{array}{*{20}{l}} {y{\text{ }} + {\text{ }}x{\text{ }} + {\text{ }}2{\text{ }} = {\text{ }}log{\text{ }}\left( {{x^2}{{\left( {y{\text{ }} + {\text{ }}2} \right)}^2}} \right)} \end{array}\)
  • D
    \(\begin{array}{*{20}{l}} {y{\text{ }}--{\text{ }}x{\text{ }} - {\text{ }}2{\text{ }} = {\text{ }}log{\text{ }}\left( {{x^2}{{\left( {y{\text{ }} + {\text{ }}2} \right)}^2}} \right)} \end{array}\)
4
Find the particular solution of the differential equation \(\log \left( {\frac{{dy}}{{dx}}} \right) = 3x + 4y\), given that y = 0 and x = 0.
  • A
    \(4{e^{3x}} + 3{e^{ - 4y}} - 7 = 1\)
  • B
    \(4{e^{3x}} + 3{e^{ - 4y}} - 7 = 0\)
    Correct
  • C
    \(4{e^{3x}} + 3{e^{ - 4y}} + 7 = 0\)
  • D
    \(4{e^{3x}} - 3{e^{ - 4y}} - 7 = 0\)
5
In a bank, principal increases continuously at the rate of r% per year. Find the value of r if Rs 100 double itself in 10 years (loge2 = 0.6931).
  • A
    8.93%
  • B
    9.93%
  • C
    6.93%
    Correct
  • D
    7.93%
6
In a bank, principal increases continuously at the rate of 5% per year. An amount of Rs1000 is deposited with this bank, how much will it worth after 10 years \(\left( {{e^{0.5}} = {\text{ }}1.648} \right).\)
  • A
    Rs 1848
  • B
    Rs 1648
    Correct
  • C
    Rs 1948
  • D
    Rs 1748
7
Solution of \(\frac{{dy}}{{dx}} = 1 + x + y + xy\) is
  • A
    \(\log \left| {1 + y} \right| = 2x + \frac{{{x^2}}}{2} + C\)
  • B
    None of these
  • C
    \(\log \left| {1 + y} \right| = x + \frac{{{x^2}}}{2} + C\)
    Correct
  • D
    \(\log \left| {1 + y} \right| = x + \frac{{{x^2}}}{2} + Cy\)
8
Find the particular solution for (x + y)dy + (x –y) dx =0; y=1 when x =1
  • A
    \(\log \left( {{x^2} - \;{y^2}} \right) + \;2{\tan ^{ - 1}}\frac{y}{x}\; \\ = \;\frac{\pi }{2} + \log 2\)
  • B
    \(\log \left( {{x^2} + \;{y^2}} \right) - \;2{\tan ^{ - 1}}\frac{y}{x}\; \\ = \;\frac{\pi }{2} + \log 2\)
  • C
    \(\log \left( {{x^2} - \;{y^3}} \right) - \;2{\tan ^{ - 1}}\frac{y}{x}\; \\ = \;\frac{\pi }{2} - \log 2\)
  • D
    \(\log \left( {{x^2} + \;{y^2}} \right) + \;2{\tan ^{ - 1}}\frac{y}{x}\; \\ = \;\frac{\pi }{2} + \log 2\)
    Correct
9
General solution of\(x\frac{{dy}}{{dx}} + y - x + xycot\;x = 0\;(x\; \ne 0)\) is
  • A
    \(y = \frac{1}{x} - \cot x - \frac{C}{{x\sin x}}\)
  • B
    \(y = \frac{1}{x} + \cot x + \frac{C}{{x\sin x}}\)
  • C
    \(y = \frac{1}{x} + \cot x - \frac{C}{{x\sin x}}\)
  • D
    \(y = \frac{1}{x} - \cot x + \frac{C}{{x\sin x}}\)
    Correct
10
To form a differential equation from a given function
  • A
    Differentiate thefunction once and eliminate the arbitrary constants
  • B
    Differentiate thefunction once and add values to arbitrary constants
  • C
    Differentiate thefunction twice and eliminate the arbitrary constants
  • D
    Differentiate thefunction successively as many times as the number of arbitrary constants inthe given function and eliminate the arbitrary constants.
    Correct
11
Forming a differential equation representing the given family of curves by eliminating arbitrary constants a and b from\(\frac{x}{a} + \;\frac{y}{b} = 1\)yields the differential equation
  • A
    y″ = 2y
  • B
    \(y''={{y}^{3}}\)
  • C
    y″ = 0
    Correct
  • D
    y″ = y
12
Forming a differential equation representing the given family of curves by eliminating arbitrary constants a and b from \({y^2} = {\text{ }}a\left( {{b^2}--{\text{ }}{x^2}} \right)\;\) yields the differential equation
  • A
    \(x\text{ }y\prime\prime \text{ }+\text{ }x\text{ }\left( y\prime \right){}^\text{2}\text{ }\text{ }y\text{ }y\prime \text{ }=\text{ }0\)
  • B
    \(xy\text{ }y\prime\prime \text{ }+\text{ }2x\text{ }\left( y\prime \right){}^\text{2}\text{ }\text{ }y\text{ }y\prime \text{ }=\text{ }\)
  • C
    \(\begin{array}{*{35}{l}} y\text{ }y\prime\prime \text{ }+\text{ }x\text{ }\left( y\prime \right){}^\text{2}\text{ }\text{ }y\text{ }y\prime \text{ }=\text{ }0 \\ \end{array}\)
  • D
    \(xy\text{ }y\prime\prime \text{ }+\text{ }x\text{ }\left( y\prime \right){}^\text{2}\text{ }\text{ }y\text{ }y\prime \text{ }=\text{ }0 \)
    Correct
13
Forming a differential equation representing the given family of curves by eliminating arbitrary constants a and b from \(y{\text{ }} = {\text{ }}a{\text{ }}{e^{3x}} + {\text{ }}b{e^{ - 2x}}\) yields the differential equation
  • A
    y″ + y′– 6y = 0
  • B
    y″ + y′+ 6y = 0
  • C
    y″ – y′– 6y = 0
    Correct
  • D
    y″ – y′+ 6y = 0
14
Forming a differential equation representing the given family of curves by eliminating arbitrary constants a and b from \(y{\text{ }} = {\text{ }}{e^{2x}}\left( {a{\text{ }} + {\text{ }}bx} \right)\) yields the differential equation
  • A
    y″ + 4y′ - 4y = 0
  • B
    y″ – 4y′ - 4y = 0
  • C
    y″ – 4y′ + 4y = 0
    Correct
  • D
    y″ + 4y′ + 4y = 0
15
Forming a differential equation representing the given family of curves by eliminating arbitrary constants a and b from y = ex (a cosx + b sinx) yields the differential equation
  • A
    y″ – 2y′ - 2y = 0
  • B
    y″ +2y′ + 2y = 0
  • C
    y″ + 2y′ - 2y = 0
  • D
    y″ – 2y′ + 2y = 0
    Correct