Assertion (A) & Reason (B) MCQ

Question 11 Mark

Directions: In the following questions, the Assertions (A) and Reason(s) (R) have been put forward. Read both the statements carefully and choose the correct alternative from the following:
If $\text{A} + \text{B} + \text{C} = 180^\circ$, then
Assertion: $\cos^{2}\frac{\text{A}}{2}+\cos^{2}\frac{\text{B}}{2}-\cos^{2}\frac{\text{C}}{2}$
$=2\cos\frac{\text{A}}{2}\cos\frac{\text{B}}{2}\sin\frac{\text{C}}{2}.$
Reason: $\cos\text{C}+\cos\text{D}=2\cos\Big(\frac{\text{C}+\text{D}}{2}\Big)\cos\Big(\frac{\text{C}+\text{D}}{2}\Big).$

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.

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Question 21 Mark

Directions: In the following questions, the Assertions (A) and Reason(s) (R) have been put forward. Read both the statements carefully and choose the correct alternative from the following:
Let $\sec\theta+\tan\theta=\text{m},$ where 0 < m < 1.
Assertion: $\sec\theta=\frac{\text{m}^{2}+1}{2\text{m}}$ and $\sin\theta=\frac{\text{m}^{2}-1}{\text{m}^{2}+1}.$
Reason: $\theta$ lies in the third quadrant.

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 correct statement but Reason is wrong statement.

Solution:

Given $\sec\theta+\tan\theta=\text{m},0 < \text{m} < 1$

Also, $\sec^{2}\theta-\tan^{2}\theta=1$

Dividing (ii) by (i), we get $\sec\theta-\tan\theta=\frac{1}{\text{m}}$

Note that $\frac{1}{\text{m}}>1$

$(\therefore 0 < \text{m} < 1)$

Adding (i) and (iii), we get

$=\sec\theta=\frac{\text{m}^2+1}{2\text{m}}>0.$

and subtracting (iii) from (i), we get

$=\tan\theta=\frac{\text{m}^2-1}{2\text{m}}<0.$

As $\sec\theta>0$ and $\tan\theta<0,$

$\therefore\theta$ lies in the fourth quadrant.

Also, $\sec\theta=\tan\theta\cos\theta=\frac{\tan\theta}{\sec\theta}=\frac{\text{m}^2-1}{\text{m}^2+1}.$

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Question 31 Mark

Directions: In the following questions, the Assertions (A) and Reason(s) (R) have been put forward. Read both the statements carefully and choose the correct alternative from the following:
Let $\alpha$ be a real number lying between 0 and $\frac{\pi}{2}$ and n be a positive integer.
Assertion: $\tan\alpha+2\tan2\alpha+2^{2}\tan2^{2}\alpha+...+2^{\text{n-1}}\tan2^{\text{n}-1}\alpha+2^{\text{n}}\cot2^{\text{n}}\alpha=\cot\alpha.$
Reason: $\cot\alpha-\tan\alpha=2\cot2\alpha.$

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:

Now, $\cot\alpha=\tan\alpha=\frac{1}{\tan\alpha}-\tan\alpha=\frac{1\tan^{2}\alpha}{\tan\alpha}$

$=2\Big(\frac{1-\tan^{2}\alpha}{2\tan\alpha}\Big)=2\cot2\alpha$

From here, we get $\tan\alpha=\cot\alpha-2\cot2\alpha$

Making repeated use of this identity, we shall obtain

$\tan\alpha+2\tan2\alpha+2^{2}\tan2^{2}\alpha+...+2^{\text{n-1}}\tan2^{\text{n}-1}\alpha+2^{\text{n}}\cot2^{\text{n}}\alpha$

$=(\cot\alpha-2\cot2\alpha)+2(\cot2\alpha-2\cot2^{2}\alpha)+2^{2}(\cot2^{2}\alpha-2\cot2^{3}\alpha)\\+...+2^{\text{n-1}}(\cot2^{\text{n-1}}\alpha-2\cot2^{\text{n}}\alpha)+2^{\text{n}}\cot2^{\text{n}}\alpha=\cot\alpha$

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Question 41 Mark

Directions: In the following questions, the Assertions (A) and Reason(s) (R) have been put forward. Read both the statements carefully and choose the correct alternative from the following:
Assertion: The value of $\sin(-690^\circ)\cos(-300^\circ)+\cos(-750^\circ)\sin(-240^\circ)=1.$
Reason: The values of $\sin$ and $\cos$ is negative in third and fourth quadrant respectively.

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 correct statement but Reason is wrong statement.

Solution:

$\sin(-690^\circ)=-\sin690^\circ=-\sin(2\times360^\circ-30^\circ)=-(-\sin30^\circ)=\frac{1}{2}$

$\cos(-300^\circ)=\cos300^\circ=\cos(360^\circ-60^\circ)=\cos60^\circ=\frac{1}{2}$

$\cos(-750^\circ)=\cos750^\circ=\cos(2\times360^\circ+30^\circ)=\cos30^\circ=\frac{\sqrt{3}}{2}$

$\sin(-240^\circ)=-\sin240^\circ=-\sin(180^\circ+60^\circ)$

$=(-\sin60^\circ)=\sin60^\circ=\frac{\sqrt{3}}{2}$

$\therefore\sin(-690^\circ)\cos(-300^\circ)+\cos(-750^\circ)\sin(-240^\circ)$

$=\Big(\frac{1}{2}\Big)\Big(\frac{1}{2}\Big)+\Big(\frac{\sqrt{3}}{2}\Big)\Big(\frac{\sqrt{3}}{2}\Big)=\frac{1}{4}+\frac{3}{4}=1.$

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Question 51 Mark

Directions: In the following questions, the Assertions (A) and Reason(s) (R) have been put forward. Read both the statements carefully and choose the correct alternative from the following:
Assertion: The value of $\theta=\frac{\pi}{3}$ or $\frac{2\pi}{3},$ when $\theta$ lies between $(0,2\pi)$ and $\sin^{2}\theta=\frac{3}{4}.$
Reason: $\sin\theta$ is positive in the first and second quadrant.

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.

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MATHS Assertion (A) & Reason (B) MCQ

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