MCQ 11 Mark
Directions: In these questions, a statement of Assertion is followed by a statement of Reason is given.Choose the correct answer out of the following choices:
Assertion: If $\frac{\text{dy}}{\text{dy}}+\text{xy}=\text{x}^3\text{y}^3,\text{x}>0,\text{y}\geq0$ and $\text{y}(0)=1,$ then $\text{y}(1)=\frac{1}{\sqrt{2}}$
Reason: The differential equation is linear with integrating factor $e^x$
- A
Assertion and Reason both are correct statements and Reason is the correct explanation of Assertion.
- B
Assertion and Reason both are correct statements but Reason is not the correct explanation of Assertion.
- ✓
Assertion is correct statement but Reason is wrong statement.
- D
Assertion is wrong statement but Reason is correct statement.
AnswerCorrect option: C. Assertion is correct statement but Reason is wrong statement.
$\frac{1}{\text{y}^3}\frac{\text{dy}}{\text{dx}}+\frac{\text{x}}{\text{y}^2}=\text{x}^3$
Put $\frac{1}{\text{y}^2}=\text{z}$
$\Rightarrow\frac{2}{\text{y}^3}\text{dy}=\text{dz}$
$\therefore\frac{\text{dz}}{\text{dx}}-2\text{xz}=-2\text{x}^3,$
which is a linear differential equation with $\text{I.F}=\text{e}^{\text{x}^2}$
$\therefore \text{ze}^{-\text{x}^2}=-\int\text{e}^{\text{x}^2}2\text{x}^3\text{dx} $
$\Rightarrow\text{ze}^{-\text{x}^2}=(\text{x}^2+1)\text{e}^{\text{-x}^2}+\text{C}$
$\Rightarrow\text{z}=\text{x}^2+1+\text{C}\text{e}^{\text{x}^2}$
$\therefore\frac{1}{\text{y}^2}=\text{x}^2+1+\text{C}\text{e}^{\text{x}^2}$
$\because\text{y}(0)=1\Rightarrow\text{c}=0$
$\therefore\text{y}^2=\frac{1}{\text{x}^2+1}$
$\Rightarrow\text{y}=\frac{1}{\sqrt{\text{x}^2+1}}$
$\Rightarrow\text{y}(1)=\frac{1}{\sqrt{2}}$
View full question & answer→MCQ 21 Mark
Directions: In these questions, a statement of Assertion is followed by a statement of Reason is given.Choose the correct answer out of the following choices:
Assertion: The elimination of four arbitrary constants in $\text{y}=(\text{c}_1+\text{c}_2+\text{c}_3\text{e}^\text{c}4)\text{x}$ results into a differential equation of the first order $\text{x}\frac{\text{dy}}{\text{dx}}=\text{y}$
Reason: Elimination of $n$ arbitrary constants requires in general, a differential equation of the $n^{th}$ order.
- ✓
Assertion and Reason both are correct statements and Reason is the correct explanation of Assertion.
- B
Assertion and Reason both are correct statements but Reason is not the correct explanation of Assertion.
- C
Assertion is correct statement but Reason is wrong statement.
- D
Assertion is wrong statement but Reason is correct statement.
AnswerCorrect option: A. Assertion and Reason both are correct statements and Reason is the correct explanation of Assertion.
Let $=(\text{c}_1+\text{c}_2+\text{c}_3\text{e}^\text{c}4)=\text{A}\text{(Constant)}$
Then, $\text{y} = \text{Ax}$
$\Rightarrow\frac{\text{dy}}{\text{dx}}=\text{A}\Rightarrow\frac{\text{dy}}{\text{dx}}=\frac{\text{y}}{\text{x}}$
$\Rightarrow\text{x}\frac{\text{dy}}{\text{dx}}=\text{y}$
View full question & answer→MCQ 31 Mark
Directions : In these questions, a statement of Assertion is followed by a statement of Reason is given.Choose the correct answer out of the following choices :
Assertion : Order of the differential equation whose solution is $\text{y}=\text{c}_1\text{e}^{\text{x}+\text{c}_2}+\text{c}_3\text{e}^{\text{x}+\text{c}_4}$ is $4.$
Reason : Order of the differential equation is equal to the number of independent arbitrary constants mentioned in the solution of the differential equation.
- A
Assertion and Reason both are correct statements and Reason is the correct explanation of Assertion.
- B
Assertion and Reason both are correct statements but Reason is not the correct explanation of Assertion.
- C
Assertion is correct statement but Reason is wrong statement.
- ✓
Assertion is wrong statement but Reason is correct statement.
AnswerCorrect option: D. Assertion is wrong statement but Reason is correct statement.
$\because \text{y}=(\text{c}_1\text{e}^{\text{c}2}+\text{c}_3\text{e}^{\text{c}4})\text{e}^\text{x}=\text{ce}^\text{x}$
$\therefore\frac{\text{dy}}{\text{dx}}=\text{ce}^\text{x}$
$\Rightarrow\frac{\text{dy}}{\text{dx}}=\text{y}\ ($Using $-(i))$
$\therefore$ Order is $1.$
View full question & answer→MCQ 41 Mark
Directions : In these questions, a statement of Assertion is followed by a statement of Reason is given. Choose the correct answer out of the following choices :
Assertion : $\text{x}\sin\text{x}\frac{\text{dy}}{\text{dx}}+(\text{x}+\text{x}\cos\text{x}+\sin \text{x}) \text{y}=\sin\text{xy},$
$(\frac{\pi}{2}) =1-\frac{2}{\pi}\Rightarrow \lim\limits_{\text{x}\rightarrow0}\text{y(x)}=\frac{1}{3}.$
Reason : The differential equation is linear with integrating factor $\text{x}(1-\cos\text{x})$
- ✓
Assertion and Reason both are correct statements and Reason is the correct explanation of Assertion.
- B
Assertion and Reason both are correct statements but Reason is not the correct explanation of Assertion.
- C
Assertion is correct statement but Reason is wrong statement.
- D
Assertion is wrong statement but Reason is correct statement.
AnswerCorrect option: A. Assertion and Reason both are correct statements and Reason is the correct explanation of Assertion.
$\frac{\text{dy}}{\text{dx}}+\bigg(\frac{1}{\sin\text{x}}+\cot\text{x}+\frac{1}{2}\bigg)^\text{y}=\frac{1}{\text{x}}$
$\text{I.F}=\text{exp}\int\bigg(\frac{1}{\sin\text{x}}+\cot\text{x}+\frac{1}{\text{x}}\bigg)\text{dx}$
$=\text{exp In}\bigg(\text{x}\tan\frac{\text{x}}{2}\sin\text{x}\bigg)$
$=\text{x}\tan\frac{\text{x}}{2}\times2\sin\frac{\text{x}}{2}\cos\frac{\text{x}}{2}=\text{x}(1-\cos\text{x})$
$\therefore$ Solution is, $\text{yx}(1-\cos\text{x})=\int\frac{1}{\text{x}}\text{x}(1-\cos\text{x})\text{dx}$
$=\text{x}-\sin\text{x+c}$
View full question & answer→MCQ 51 Mark
Directions: In these questions, a statement of Assertion is followed by a statement of Reason is given.Choose the correct answer out of the following choices:
Assertion: The differential equation of all circles in a plane must be of order $3.$
Reason: If three points are non$-$collinear, then only one circle passes through these points.
- A
Assertion and Reason both are correct statements and Reason is the correct explanation of Assertion.
- ✓
Assertion and Reason both are correct statements but Reason is not the correct explanation of Assertion.
- C
Assertion is correct statement but Reason is wrong statement.
- D
Assertion is wrong statement but Reason is correct statement.
AnswerCorrect option: B. Assertion and Reason both are correct statements but Reason is not the correct explanation of Assertion.
Let $x^2 + y^2 + 2gx + 2fy + c = 0$
Here, in this equation, there are three constants.
$\therefore$ Order $= 3$
Reason is also correct.
View full question & answer→MCQ 61 Mark
Directions : In these questions, a statement of Assertion is followed by a statement of Reason is given. Choose the correct answer out of the following choices :
Assertion $:\text{y}=\text{a}\sin\text{ x}+\text{b }\cos \text{x}$ isa general solution of $\text{" y ”}+ \text{y}= 0.$
Reason $:\text{y}=\text{a}\sin\text{ x}+\text{b }\cos \text{x}$ is a trigonometric function.
- A
Assertion and Reason both are correct statements and Reason is the correct explanation of Assertion.
- ✓
Assertion and Reason both are correct statements but Reason is not the correct explanation of Assertion.
- C
Assertion is correct statement but Reason is wrong statement.
- D
Assertion is wrong statement but Reason is correct statement.
AnswerCorrect option: B. Assertion and Reason both are correct statements but Reason is not the correct explanation of Assertion.
$\because\text{ y}=\text{a}\sin\text{ x}+\text{b }\cos \text{x}$
$\therefore\text{ y}=\text{a}\cos\text{ x}-\text{b }\sin \text{x}$
$\Rightarrow\text{y"}=-\text{a}\sin\text{x}-\text{b}\cos\text{x}=-\text{y}$
$\Rightarrow\text{y''}+\text{y}=0$
View full question & answer→MCQ 71 Mark
Assertion $(A) :$ If $\frac{d y}{d x}+x y=x^3 y^3, x>0, y \geq 0$ and $y(0)=1$, then $y(1)=\frac{1}{\sqrt{2}}$.
Reason $(R) :$ The differential equation is linear with integrating factor $e^x$.
- A
Both $(A)$ and $(R)$ are true and $(R)$ is the correct explanation of $(A).$
- B
Both $(A)$ and $(R)$ are true but $(R)$ is not the correct explanation of $(A).$
- ✓
$(A)$ is true but $(R)$ is false.
- D
$(A)$ is false but $(R)$ is true.
AnswerCorrect option: C. $(A)$ is true but $(R)$ is false.
$\frac{1}{y^3} \frac{d y}{d x}+\frac{x}{y^2}=x^3$
Put $\frac{1}{y^2}=z$
$\Rightarrow-\frac{2}{y^3} d y=d z$
$\therefore \frac{d z}{d x}-2 x z=-2 x^3,$
which is a linear differential equation with $I.F. =e^{-x^2}$
$\therefore$ Solution, $z e^{-x^2}=-\int e^{-x^2} 2 x^3 d x+C$
$\Rightarrow z e^{-x^2}=\left(x^2+1\right) e^{-x^2}+C$
$\Rightarrow z=x^2+1+C e^{x^2}$
$\therefore \frac{1}{y^2}=x^2+1+C e^{x^2}$
$\because y(0)=1$
$\Rightarrow C=0$
$\therefore y^2=\frac{1}{x^2+1}$
$\Rightarrow y=\frac{1}{\sqrt{x^2+1}}$
$\Rightarrow y(1)=\frac{1}{\sqrt{2}}$
View full question & answer→MCQ 81 Mark
Assertion $(A) : x \sin x \frac{d y}{d x}+(x+x \cos x+\sin x)$ $y=\sin x, y\left(\frac{\pi}{2}\right)=1-\frac{2}{\pi} \Rightarrow y=\frac{x-\sin x}{x(1-\cos x)}$
Reason $(R) :$ The differential equation is linear with integrating factor $x(1-\cos x)$.
- ✓
Both $(A)$ and $(R)$ are true and $(R)$ is the correct explanation of $(A).$
- B
Both $(A)$ and $(R)$ are true but $(R)$ is not the correct explanation of $(A).$
- C
$(A)$ is true but $(R)$ is false.
- D
$(A)$ is false but $(R)$ is true.
AnswerCorrect option: A. Both $(A)$ and $(R)$ are true and $(R)$ is the correct explanation of $(A).$
$\frac{d y}{d x}+\left(\frac{1}{\sin x}+\cot x+\frac{1}{x}\right) y=\frac{1}{x z}$
$\text { I.F. }=\exp \int\left(\frac{1}{\sin x}+\cot x+\frac{1}{x}\right) d x=\exp \ln x(1-\cos x)$
$=x(1-\cos x)$
$ \therefore$ Solution is, $y x(1-\cos x)=\int \frac{1}{x} \cdot x(1-\cos x) d x+C$
$y\left(\frac{\pi}{2}\right)=1-\frac{2}{\pi}$
$\Rightarrow c=0 x-\sin x+c$
$\therefore y=\frac{x-\sin x}{x(1-\cos x)}$
View full question & answer→MCQ 91 Mark
Assertion (A) : ' $x$ ' is not an integrating factor for the differential equation $x \frac{d y}{d x}+2 y=e^x$.
Reason (R) : $x\left(x \frac{d y}{d x}+2 y\right)=\frac{d}{d x}\left(x^2 y\right)$.
- A
Both (A) and (R) are true and (R) is the correct explanation of (A).
- ✓
Both (A) and (R) are true but (R) is not the correct explanation of (A).
- C
(A) is true but (R) is false.
- D
(A) is false but (R) is true.
AnswerCorrect option: B. Both (A) and (R) are true but (R) is not the correct explanation of (A).
(b) : $\frac{d y}{d x}+\frac{2}{x} y=\frac{e^x}{x}$
$
\text { I.F. }=e^{\int \frac{2}{x} d x}=e^{2 \log x}=e^{\log x^2}=x^2
$
$\Rightarrow$ Assertion is correct.
Now, $\frac{d}{d x}\left(x^2 y\right)=x^2 \frac{d y}{d x}+y \cdot 2 x=x\left(x \frac{d y}{d x}+2 y\right)$
$\Rightarrow$ Reason is correct.
View full question & answer→MCQ 101 Mark
Assertion (A) : Integrating factor of $\left(1+x^2\right) \frac{d y}{d x}+x y=\frac{1}{2 x}$ is given by $\sqrt{1+x^2}$.
Reason (R) : Integrating factor of $\frac{d y}{d x}+P(x) \cdot y=Q(x)$ is $e^{\int P(x) d x}$.
- ✓
Both (A) and (R) are true and (R) is the correct explanation of (A).
- B
Both (A) and (R) are true but (R) is not the correct explanation of (A).
- C
(A) is true but (R) is false.
- D
(A) is false but (R) is true.
AnswerCorrect option: A. Both (A) and (R) are true and (R) is the correct explanation of (A).
(a) : Clearly, reason is true.
Now, $\frac{d y}{d x}\left(1+x^2\right)+x y=\frac{1}{2 x}$
$\Rightarrow \quad \frac{d y}{d x}+\frac{x}{1+x^2} y=\frac{1}{2 x\left(1+x^2\right)}$ which is of the form
$\frac{d y}{d x}+P(x) y=Q(x)$, where $P(x)=\frac{x}{1+x^2}, Q(x)=\frac{1}{2 x\left(1+x^2\right)}$
I.F. $=e^{\int P(x) d x}=e^{\int \frac{x}{1+x^2} d x}=e^{\int \frac{1}{2}\left(\frac{2 x}{1+x^2}\right) d x}=e^{\frac{1}{2} \log \left(1+x^2\right)}$
$=e^{\log \left(1+x^2\right)^{1 / 2}}=\sqrt{1+x^2}$
$\therefore \quad$ Assertion is true; Reason is true.
View full question & answer→MCQ 111 Mark
Assertion (A) : $y=a \sin x+b \cos x$ is a general solution of $y^{\prime \prime}+y=0$.
Reason (R): $y=a \sin x+b \cos x$ is a trigonometric function.
- A
Both (A) and (R) are true and (R) is the correct explanation of (A).
- ✓
Both (A) and (R) are true but (R) is not the correct explanation of (A).
- C
(A) is true but (R) is false.
- D
(A) is false but (R) is true.
AnswerCorrect option: B. Both (A) and (R) are true but (R) is not the correct explanation of (A).
(b) : $\because y=a \sin x+b \cos x$ ...(i)
$
\begin{array}{ll}
\therefore & y^{\prime}=a \cos x-b \sin x \\
\Rightarrow & y^{\prime \prime}=-a \sin x-b \cos x=-y [Using (i)] \\
\Rightarrow & y^{\prime \prime}+y=0
\end{array}
$
View full question & answer→MCQ 121 Mark
Assertion (A) : Order of the differential equation whose solution is $y=c_1 e^{x+c_2}+c_3 e^{x+c_4}$ is 4.
Reason (R) : Order of the differential equation is equal to the number of independent arbitrary constants mentioned in the solution of the differential equation.
- A
Both (A) and (R) are true and (R) is the correct explanation of (A).
- B
Both (A) and (R) are true but (R) is not the correct explanation of (A).
- C
(A) is true but (R) is false.
- ✓
(A) is false but (R) is true.
AnswerCorrect option: D. (A) is false but (R) is true.
(d): $\because y=\left(c_1 e^{c_2}+c_3 e^{c_4}\right) e^x=c e^x$ ...(i)
Solution of differential equation containing the arbitrary constant.
$\therefore \quad$ Order is 1.
View full question & answer→MCQ 131 Mark
Assertion (A) : A differential equation of the form $y f(x y) d x+x g(x y) d y=0$ can be converted into homogeneous differential equation by substituting $x y=t$
Reason (R) : A differential equation is called homogeneous if $f(\lambda x, \lambda y)=\lambda^0 f(x, y)$.
- A
Both (A) and (R) are true and (R) is the correct explanation of (A).
- B
Both (A) and (R) are true but (R) is not the correct explanation of (A).
- C
(A) is true but (R) is false.
- ✓
(A) is false but (R) is true.
AnswerCorrect option: D. (A) is false but (R) is true.
(d) : Clearly, reason is true.
As, $x y=t$
$
\therefore \quad x d y / d x+y=\frac{d t}{d x} \Rightarrow \frac{x d y}{d x}=\frac{d t}{d x}-\frac{t}{x}
$
Now, given differential equation is
$
\begin{aligned}
& y f(x y) d x+x g(x y) d y=0 \\
\Rightarrow & \frac{t}{x} f(t)+g(t)\left(\frac{d t}{d x}-\frac{t}{x}\right)=0 \\
\Rightarrow & \frac{t}{x} f(t)+g(t) \frac{d t}{d x}-\frac{t}{x} g(t)=0 \\
\Rightarrow & t\left[\frac{f(t)-g(t)}{x}\right]+g(t) \frac{d t}{d x}=0 \\
\Rightarrow & \frac{d x}{x}+\frac{g(t)}{t[f(t)-g(t)]} d t=0,
\end{aligned}
$
which is variable separable form
$\therefore \quad$ It is not homogeneous differential equation.
$\therefore \quad$ Assertion is false; Reason is true.
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