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MATHS Assertion (A) & Reason (B) MCQ

Trigonometric Functions · UP Board (UPMSP) STD 11 Science (UPMSP - English Medium). 5 questions.

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Assertion (A) & Reason (B) MCQ

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5 questions · auto-graded multiple-choice test.

MCQ 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:
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.
  • 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.
  • ✓
    Assertion is correct statement but Reason is wrong statement.
  • D
    Assertion is wrong statement but Reason is correct statement.
Answer
Correct option: C.
Assertion is correct statement but Reason is wrong statement.
$\sin(-690^\circ)=-\sin690^\circ=-\sin(2\times360^\circ-30^\circ)=-(-\sin30^\circ)=\frac{1}{2}$
$\cos(-300^\circ)=\cos300^\circ=\cos(360^\circ-60^\circ)=\cos60^\circ=\frac{1}{2}$
$\cos(-750^\circ)=\cos750^\circ=\cos(2\times360^\circ+30^\circ)=\cos30^\circ=\frac{\sqrt{3}}{2}$
$\sin(-240^\circ)=-\sin240^\circ=-\sin(180^\circ+60^\circ)$
$=(-\sin60^\circ)=\sin60^\circ=\frac{\sqrt{3}}{2}$
$\therefore\sin(-690^\circ)\cos(-300^\circ)+\cos(-750^\circ)\sin(-240^\circ)$
$=\Big(\frac{1}{2}\Big)\Big(\frac{1}{2}\Big)+\Big(\frac{\sqrt{3}}{2}\Big)\Big(\frac{\sqrt{3}}{2}\Big)$
$=\frac{1}{4}+\frac{3}{4}$
$=1.$
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MCQ 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:
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.
  • 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.
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MCQ 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 $\sec\theta+\tan\theta=\text{m},$ 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.
  • 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.
  • ✓
    Assertion is correct statement but Reason is wrong statement.
  • D
    Assertion is wrong statement but Reason is correct statement.
Answer
Correct option: C.
Assertion is correct statement but Reason is wrong statement.
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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MCQ 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:
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.
  • 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.
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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MCQ 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.
  • 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.
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