Assertion (A) & Reason (B) MCQ

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 : The area of the smaller region bounded by the ellipse $\frac{\text{x}^2}{9}+\frac{\text{y}^2}{4}=1$ the line $\frac{\text{x}}{3}+\frac{\text{y}}{2}=1$ is $\frac{3}{2}(\pi-2)\text{ sq.units}$
Reason : Formula to calculate the area of the smaller region bounded by the ellipse $\frac{\text{x}^2}{\text{a}^2}+\frac{\text{y}^2}{\text{b}^2}=1$ and the line $\frac{\text{x}}{\text{a}}+\frac{\text{y}}{\text{b}}=1$ is $\frac{\text{ab}}{4}(\pi-2) \text{ sq.units}$

✓
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.

Clearly, reason is correct statement. Now, we have, equation of ellipse
$\frac{\text{x}^2}{9}+\frac{\text{y}^2}{4}=1$ the line $\frac{\text{x}}{3}+\frac{\text{y}}{2}=1$
$\therefore$ Here, $\text{a}=3, \text{ b}=3$
$\therefore$ Required area $=\frac{\text{ab}}{4}(\pi-2)$
$=\frac{3\times2}{4}(\pi-2)=\frac{3}{2}(\pi-2)\text{ sq.units}$

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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 : The area of the region bounded by the curve $y^2 = 4x$ and the line $x = 3$ is $8\sqrt{3} \text{sq. units.}$
Reason : The area of the region bounded by the curve $x^2 = 4y$ and the line $x = 4y - 2$ is $\frac{9}{8} \text{sq. units}.$

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.

Asseration : We have, $y^2= 4x$ and $x = 3$
$\therefore$ Required area
$=2\int_{0}^{3}|\text{y}|\text{dx}=2\int_{0}^{3}2\sqrt{\text{x}}\text{ dx}$
$=\bigg[\frac{\text{x}^\frac{3}{2}}{\frac{3}{2}}\bigg]^3_0$
$=\frac{8}{3}(3\sqrt{3})=8\sqrt{3}\text{ sq.units}$

Reason : We have, $\text{x}^2=4\text{y}$
$\Rightarrow\text{y}=\frac{\text{x}^2}{4}$
and $\text{x}=4\text{y}-2$
$\Rightarrow\text{y}=\frac{\text{x}+2}{4}$
The point of intersection of given curves are $\text{A}(2, 1)$ and $\text{B}\big(-1,\frac{1}{4}\big)$

$\therefore$ Required area $=\int_{-1}^{2}\big(\frac{\text{x}+2}{4}\big)\text{dx}-\int_{-1}^{2}\frac{\text{x}^2}{4}\text{dx}$
$=\frac{1}{4}\big[\frac{\text{x}^2}{2}+2\text{x}]^2\frac{1}{4}\big[\frac{\text{x}^3}{3}\big]^2_1$
$\frac{1}{4}\big(6+\frac{3}{2}\big)-\frac{1}{12}\times9=\frac{15}{8}-\frac{3}{4}=\frac{9}{8}\text{ sq. units}$

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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 area bounded by the curves $\text{y}^2 = 4\text{a}^2(\text{x} - 1) $ and lines $\text{x}=1$ and $\text{y}=4 a$ is $\frac{8\text{a}}{3}\text{sq.units}$
Reason : The area enclosed between the parabola $\text{y}=\text{x}^2-\text{x}+2$ and the line $\text{y}=\text{x+2}$ is $\frac{4}{3}\text{ sq.units}$

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.

Answer

Correct option: D.

Assertion is wrong statement but Reason is correct statement.

Assertion : On solving $y^2 = 4a^2(x - 1)$ and $y = 4a$, we get $x = 5$

$\therefore$ Required area $=\int_{1}^{5}(4\text{a}-2\text{a}\sqrt{\text{x}-1})\text{dx}$
$=\bigg[4\text{ax}-2\text{a}\frac{(\text{x-1})^\frac{3}{2}}{\frac{3}{2}}\bigg]^5_1=\frac{16\text{a}}{3}\text{ sq.units}$
Reason : Given, parabola $y = x^2 - x + 2$ and the line $y = x + 2$ intersects each other at points $(0, 2)$ and $(2, 4).$

$\therefore$ Required area $=\int_{0}^{2}\big[(\text{x}+2)-(\text{x}^2-\text{x}+2)\big]\text{dx}$
$=\int_{0}^{2}(-\text{x}^2+2\text{x})\text{dx}=\big[\frac{\text{-x}^3}{3}+\text{x}^2\big]^2_0=-\frac{8}{3}+4=\frac{4}{3}\text{ sq.units}$

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MATHS Assertion (A) & Reason (B) MCQ

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