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

View full question & answer→

Question 21 Mark

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:
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 $\text{x}=2\pi,\text{t}=1$
$\therefore\text{I}=\int_{1}^{1}(1-\text{t}^2(-\text{dt}))=0$

View full question & answer→

Question 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 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.
$\frac{\text{n-1}}{\text{n}}\frac{\text{n}-3}{\text{n}-2}\frac{\text{n-5}}{\text{n-4}}...\frac{1}{2}\frac{\pi}{2}$

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:
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.

View full question & answer→

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: $\text{I}=\int_{0}^{1}\frac{\text{dx}}{3\sqrt{1+\text{x}^3}}=\int_{0}^{{2}^\frac{-1}{3}}\frac{\text{dt}}{1-\text{t}^3}$
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.
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:
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 $\text{x}=1,\text{t}=2^\frac{-1}{3}$
$\Rightarrow\text{I}=\int_{0}^{{2}^\frac{-1}{3}}\frac{\text{dt}}{1-\text{t}^3}$

View full question & answer→

Question 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 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 $\text{x}$

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

View full question & answer→

MATHS Assertion (A) & Reason (B) MCQ

Test yourself on this topic