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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MCQ 61 Mark

Assertion $(A): \int_2^8 \frac{\sqrt{10-x}}{\sqrt{x}+\sqrt{10-x}} d x=3$
Reason $(R): \int_a^b f(x) d x=\int_a^b f(a+b-x) d x$

✓
Both assertion $(A)$ and reason $(R)$ are true and reason $(R)$ is the correct explanation of assertion $(A)$.
B
Both assertion $(A)$ and reason $(R)$ are true, but reason $(R)$ is not the correct explanation of the assertion $(A)$.
C
Assertion $(A)$ is true and reason $(R)$ is false.
D
Assertion $(A)$ is false, but reason $(R)$ is true.

Answer

Correct option: A.

Both assertion $(A)$ and reason $(R)$ are true and reason $(R)$ is the correct explanation of assertion $(A)$.

Let $I=\int_2^8 \frac{\sqrt{10-x}}{\sqrt{x}+\sqrt{10-x}} d x$
$=\int_2^8 \frac{\sqrt{10-(10-x)}}{\sqrt{10-x}+\sqrt{10-(10-x)}} d x\left(\because \int_a^b f(x) d x=\frac{b}{a} f(a+b-x) d x\right)$
$=\int_2^8 \frac{\sqrt{x}}{\sqrt{10-x}+\sqrt{x}} d x$
Adding $(i)$ and $(ii),$ we get
$2 I=\int_2^8 \frac{\sqrt{10-x}+\sqrt{x}}{\sqrt{x}+\sqrt{10-x}} d x=\int_2^8 1 d x=[x]_2^8$
$\Rightarrow I =\frac{1}{2}(8-2)=\frac{6}{2}=3$
Hence, both assertion $(A)$ and reason $(R)$ are true and reason $(R)$ is the correct explanation of assertion $(A).$

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MCQ 71 Mark

Assertion (A) : The value of$\int_{-3}^3\left(a x^5+b x^3+c x+k\right) d x,$ where $a, b, c, k$ are constants, depends on only $k$.
Reason (R) : $\int_{-a}^a f(x) d x=0$, if $f(-x)=-f(x)$ i.e., $f$ is an odd function.

✓
Both (A) and (R) are true and (R) is the correct explanation of (A).
B
Both (A) and (R) are true but (R) is not the correct explanation of (A).
C
(A) is true but (R) is false.
D
(A) is false but (R) is true.

Answer

Correct option: A.

Both (A) and (R) are true and (R) is the correct explanation of (A).

(a) : Clearly, Reason is true.
Let $I=\int_{-3}^3\left(a x^5+b x^3+c x+k\right) d x$
$
=a \int_{-3}^3 x^5 d x+b \int_{-3}^3 x^3 d x+c \int_{-3}^3 x d x+k \int_{-3}^3 1 d x
$
Since, $x^5, x^3, x$ are odd function
$
\therefore \quad I=0+0+0+k[x]_{-3}^3=6 k \text {, }
$
which is dependent only on $k$.

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MCQ 81 Mark

Assertion (A) : $\int_0^{2 \pi} \sin ^3 x d x=0$
Reason (R) : $\sin ^3 x$ is an odd function.

A
Both (A) and (R) are true and (R) is the correct explanation of (A).
✓
Both (A) and (R) are true but (R) is not the correct explanation of (A).
C
(A) is true but (R) is false.
D
(A) is false but (R) is true.

Answer

Correct option: B.

Both (A) and (R) are true but (R) is not the correct explanation of (A).

(b) : Let $I=\int_0^{2 \pi} \sin ^3 x d x=\int_0^{2 \pi}\left(1-\cos ^2 x\right) \sin x d x$
Putting $\cos x=t \Rightarrow \sin x d x=-d t$
When $x=0, t=1$ and $x=2 \pi, t=1$
$\therefore \quad I=\int_1^1\left(1-t^2\right)(-d t)=0$

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MCQ 91 Mark

Assertion $(A) : I=\int_0^1 \frac{d x}{\sqrt[3]{1+x^3}}=\int_0^{2^{-1 / 3}} \frac{d t}{1-t^3}$
Reason $(R) :$ The integrand of the integral $I$ becomes rational by the substitution $t=\frac{x}{\sqrt[3]{1+x^3}}$.

✓
Both $(A)$ and $(R)$ are true and $(R)$ is the correct explanation of $(A).$
B
Both $(A)$ and $(R)$ are true but $(R)$ is not the correct explanation of $(A).$
C
$(A)$ is true but $(R)$ is false.
D
$(A)$ is false but $(R)$ is true.

Answer

Correct option: A.

Both $(A)$ and $(R)$ are true and $(R)$ is the correct explanation of $(A).$

Let $t=\frac{x}{\sqrt[3]{1+x^3}} \Rightarrow d t=\frac{d x}{\left(1+x^3\right)^{\frac{4}{3}}}$
$\therefore \left(1+x^3\right) t^3=x^3$
$\Rightarrow t^3+x^3 t^3=x^3$
$\Rightarrow t^3=x^3\left(1-t^3\right)$
$\Rightarrow x^3=\frac{t^3}{1-t^3}$
$\Rightarrow 1+x^3=\frac{1}{1-t^3}$
When $x=0, t=0$ and $x=1, t=2^{-1 / 3}$
$\Rightarrow I=\int_0^{2^{-1 / 3}} \frac{d t}{1-t^3}$

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MCQ 101 Mark

Let $F(x)$ be an indefinite integral of $\sin ^2 x$.
Assertion (A) : The function $F(x)$ satisfies $F(x+\pi)=F(x)$ for all real $x$.
Reason (R) : $\sin ^2(x+\pi)=\sin ^2 x$ for all real $x$.

A
Both (A) and (R) are true and (R) is the correct explanation of (A).
B
Both (A) and (R) are true but (R) is not the correct explanation of (A).
C
(A) is true but (R) is false.
D
(A) is false but (R) is true.

Answer

$
\begin{aligned}
(d): F(x) & =\int \sin ^2 x d x=\int \frac{1}{2}(1-\cos 2 x) d x \\
& =\frac{x}{2}-\frac{\sin 2 x}{4}+C
\end{aligned}
$
$
\because \quad F(x+\pi)-F(x)=\frac{\pi}{2} \neq 0
$
$\therefore \quad$ Assertion is false.
$
\sin ^2(x+\pi)=(-\sin x)^2=\sin ^2 x
$
$\therefore \quad$ Reason is true.

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MCQ 111 Mark

Assertion $(A) :\int \sin 3 x \cos 5 x d x=\frac{-\cos 8 x}{16}+\frac{\cos 2 x}{4}+C$
Reason $(R) :2 \cos A \sin B=\sin (A+B)-\sin (A-B)$

✓
Both $(A)$ and $(R)$ are true and $(R)$ is the correct explanation of $(A).$
B
Both $(A)$ and $(R)$ are true but $(R)$ is not the correct explanation of $(A).$
C
$(A)$ is true but $(R)$ is false.
D
$(A)$ is false but $(R)$ is true.

Answer

Correct option: A.

Both $(A)$ and $(R)$ are true and $(R)$ is the correct explanation of $(A).$

We have, $\int \sin 3 x \cos 5 x d x$$=\frac{1}{2} \int 2 \cos 5 x \sin 3 x\ d x$
$=\frac{1}{2} \int(\sin 8 x-\sin 2 x) d x=\frac{1}{2}\left[\int \sin 8 x\ d x-\int \sin 2 x\ d x\right]$
$=\frac{1}{2}\left[\frac{-\cos 8 x}{8}\right]-\frac{1}{2}\left[\frac{-\cos 2 x}{2}\right]+C$
$=\frac{-\cos 8 x}{16}+\frac{\cos 2 x}{4}+C$
$\therefore $ Both assertion and reason are true and reason is the correct explanation of assertion. $\text{ZXZ}$

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MATHS Assertion (A) & Reason (B) MCQ

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