Assertion (A) & Reason (B) MCQ

MCQ 11 Mark

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 $\text{I}=\int_{0}^{2\pi}\sin^3\text{x}\text{ dx}=\int_{0}^{2\pi}(1-\cos^2\text{x})\sin\text{x dx}$
Putting $\cos\text{x}=\text{t}$
$\Rightarrow\sin\text{x dx}=-\text{dt}$
When $\text{x}=0,\text{t}=1$ and $\text{x}=2\pi,\text{t}=1$
$\therefore\text{I}=\int_{1}^{1}(1-\text{t}^2(-\text{dt}))=0$

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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 value of $\int_{0}^{\frac{\pi}{0}}\sin^6\text{xdx}=\frac{5\pi}{16}$
Reason: If $\text{n}$ is even, then $\int_{0}^{\frac{\pi}{0}}\sin^\text{n}\text{xdx}$ equals.
$\frac{\text{n-1}}{\text{n}}\frac{\text{n}-3}{\text{n}-2}\frac{\text{n-5}}{\text{n-4}}...\frac{1}{2}\frac{\pi}{2}$

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.

Reason is obvious.
$\therefore \int_{0}^{\frac{\pi}{2}}\sin^6\text{x dx}=\frac{5}{6}\times\frac{3}{4}\times\frac{1}{2}\times\frac{\pi}{2}=\frac{5\pi}{32}$
$\therefore$ Assertion is false.

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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 function $F(x)$ satisfies $\text{F(x}+\pi)=\text{F}\text{(x)}$ for all real $\text{x}$
Reason: $\text{Sin}^2(\text{x}+\pi)=\sin^2\text{x}$ for all real $\text{x}$

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.
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.

$\text{F(x)}=\int\sin^2\text{x}\text{dx}=\int\frac{1}{2}(1-\cos2\text{x})\text{dx}$
$\frac{\text{x}}{2}-\frac{\sin^2\text{x}}{4}+\text{c}$
$\because\text{F}(\text{x}+\pi)-\text{F(x)}=\frac{\pi}{2}\neq0$
$\therefore$ Assertion is false.
$\sin^2(\text{x}+\pi)=(-\sin\text{x})^2=\sin^2\text{x}$
$\therefore$ Reason is true.

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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 : $\int\sin3\text{x}\cos5\text{x}\text{ dx}=\frac{-\cos8\text{x}}{16}+\frac{\cos2\text{x}}{4}+\text{C}$
Reason : $2\cos\text{A}\sin\text{B}=\sin(\text{A+B})-\sin(\text{A-B})$

✓
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.

We have, $\int\sin3\text{x}\cos5\text{x}\text{ dx}$
$=\frac{1}{2}\int2\cos5\text{x}\sin3\text{x}\text{dx}$
$=\frac{1}{2}\int(\sin8\text{x}-\sin1\text{x})\text{dx}=\frac{1}{2}[\int\sin8\text{x}\text{dx}-\int\sin2\text{x}\text{ dx}]$
$\frac{1}{2}\big[\frac{-\cos8\text{x}}{8}\big]-\big[\frac{-\cos2\text{x}}{2}\big]+\text{c}=\frac{-\cos8\text{x}}{16}+\frac{\cos2\text{x}}{4}=\text{c}$

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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 : $\text{I}=\int_{0}^{1}\frac{\text{dx}}{3\sqrt{1+\text{x}^3}}=\int_{0}^{{2}^\frac{-1}{3}}\frac{\text{dt}}{1-\text{t}^3}$
Reason : The integrand of the integral I becomes rational by the substitution $\text{t}=\frac{\text{x}}{3\sqrt{1+\text{x}^3}}$

✓
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 $\text{t}=\frac{\text{x}}{3\sqrt{1+\text{x}^3}}$
$\Rightarrow\text{dt}=\frac{\text{dx}}{(1+\text{x}^3)^\frac{4}{3}}$
$\therefore(1+\text{x}^3)\text{t}^3=\text{x}^3$
$\Rightarrow\text{t}^3+\text{x}^3\text{t}^3=\text{x}^3$
$\Rightarrow\text{t}^3=\text{x}^3(1-\text{t}^3)$
$\Rightarrow\text{x}^3=\frac{\text{t}^3}{1-\text{t}^3}$
$\Rightarrow1+\text{x}^3=\frac{1}{1-\text{t}^3}$
when $\text{x}=0,\text{t}=0$ and $\text{x}=1,\text{t}=2^\frac{-1}{3}$
$\Rightarrow\text{I}=\int_{0}^{{2}^\frac{-1}{3}}\frac{\text{dt}}{1-\text{t}^3}$

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Maths Assertion (A) & Reason (B) MCQ

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