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: 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.
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:
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.
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.

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:
Asseration: Thearea bounded bythe parabola $y^{2 }= 4ax$ and the line $x = a$ and $x = 4a$ is $\frac{56\text{a}^2}{3}\text{ sq.units}$
Reason: The area bounded by the curves $y = 3x$ and $y = x^{2 }$ is $9.5\text{ sq.units}$

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

Answer

Assertion:

Required area $=2\int_{\text{a}}^{4\text{a}}\sqrt{4\text{ax}}\text{ dx}$
$=4\sqrt{a}\bigg[\frac{\text{x}^\frac{3}{2}}{\frac{3}{2}}\bigg]^\text{4a}_\text{a}$
$=\frac{8}{3}\sqrt{a}(8\text{a}^\frac{3}{2}-\text{a}^\frac{3}{2}\big)$
$=\frac{56\text{a}^2}{3}\text{ sq.units}$
Reason: The intersection points of given curves are $(0, 0)$ and $(3, 9).$
$\therefore$ Required area $=\int_{0}^{3}(3\text{x}-\text{x}^2)\text{dx}$
$=\bigg[\frac{3\text{x}^2}{2}-\frac{\text{(x)}^3}{3}\bigg]^3_0$
$=\frac{27}{6}$
$=4.5\text{ sq.units}$

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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: 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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