Assertion (A) & Reason (B) MCQ

Question 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: 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.

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.
Assertion is correct statement but Reason is wrong statement.
Assertion is wrong statement but Reason is correct statement.

Answer

Assertion is wrong statement but Reason is correct statement.

Solution:
$\because \text{y}=(\text{c}_1\text{e}^{\text{c}2}+\text{c}_3\text{e}^{\text{c}4})\text{e}^\text{x}=\text{ce}^\text{x}$
$\therefore\frac{\text{dy}}{\text{dx}}=\text{ce}^\text{x}\Rightarrow\frac{\text{dy}}{\text{dx}}=\text{y}$ (Using-(i))
$\therefore$ Order is 1.

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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: 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.

Answer

Correct 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} $
$\Rightarrow\text{ze}^{-\text{x}^2}=(\text{x}^2+1)\text{e}^{\text{-x}^2}+\text{C}\Rightarrow\text{z}=\text{x}^2+1+\text{C}\text{e}^{\text{x}^2}$
$\therefore\frac{1}{\text{y}^2}=\text{x}^2+1+\text{C}\text{e}^{\text{x}^2}$
$\because\text{y}(0)=1\Rightarrow\text{c}=0$
$\therefore\text{y}^2=\frac{1}{\text{x}^2+1}\Rightarrow\text{y}=\frac{1}{\sqrt{\text{x}^2+1}}\Rightarrow\text{y}(1)=\frac{1}{\sqrt{2}}$

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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: 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.

Answer

Correct 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, $\text{y} = \text{Ax}$
$\Rightarrow\frac{\text{dy}}{\text{dx}}=\text{A}\Rightarrow\frac{\text{dy}}{\text{dx}}=\frac{\text{y}}{\text{x}}$
$\Rightarrow\text{x}\frac{\text{dy}}{\text{dx}}=\text{y}$

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Question 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.
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.
Assertion is wrong statement but Reason is correct statement.

Answer

Assertion and Reason both are correct statements and Reason is the correct explanation of Assertion.

Solution:
$\frac{\text{dy}}{\text{dx}}+\bigg(\frac{1}{\sin\text{x}}+\cot\text{x}+\frac{1}{2}\bigg)^\text{y}=\frac{1}{\text{x}}$
$\text{I.F}=\text{exp}\int\bigg(\frac{1}{\sin\text{x}}+\cot\text{x}+\frac{1}{\text{x}}\bigg)\text{dx}$
$=\text{exp In}\bigg(\text{x}\tan\frac{\text{x}}{2}\sin\text{x}\bigg)$
$=\text{x}\tan\frac{\text{x}}{2}\times2\sin\frac{\text{x}}{2}\cos\frac{\text{x}}{2}=\text{x}(1-\cos\text{x})$
$\therefore$ Solution is, $\text{yx}(1-\cos\text{x})=\int\frac{1}{\text{x}}\text{x}(1-\cos\text{x})\text{dx}$
$=\text{x}-\sin\text{x+c}$

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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.

Answer

Correct 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.

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Question 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.

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.
Assertion is correct statement but Reason is wrong statement.
Assertion is wrong statement but Reason is correct statement.

Answer

Assertion and Reason both are correct statements but Reason is not the correct explanation of Assertion.

Solution:
$\because\text{ y}=\text{a}\sin\text{ x}+\text{b }\cos \text{x}$
$\therefore\text{ y}=\text{a}\cos\text{ x}-\text{b }\sin \text{x}$
$\Rightarrow\text{y}"=-\text{a}\sin\text{x}-\text{b}\cos\text{x}=-\text{y}$
$\Rightarrow\text{y}''+\text{y}=0$

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MATHS Assertion (A) & Reason (B) MCQ

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