M.C.Q (1 Marks)

MCQ 1511 Mark

Choose the correct answer from the given four options.
The value of $\cot\Big[\cos^{-1}\Big(\frac{7}{25}\Big)\Big]$ is:

A
$\frac{25}{24}$
B
$\frac{25}{7}$
C
$\frac{24}{25}$
✓
$\frac{7}{24}$

Answer

Correct option: D.

$\frac{7}{24}$

$\cot\Big[\cos^{-1}\Big(\frac{7}{25}\Big)\Big]$

$=\cot\Big(\cot^{-1}\frac{7}{24}\Big)$

$=\frac{7}{24}$

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MCQ 1521 Mark

The value of $\sin ^{ -1 }{ \left( \cos { \frac { 53\pi }{ 5 } } \right) }=\sin ^{ -1 }{ \left( \cos { \frac { 50\pi+3\pi }{ 5 } } \right) }:$

A
$ \frac {- \pi }{ 1 }$
B
$ \frac {- \pi }{ 7 }$
C
$ \frac { \pi }{ 10 }$
✓
$ \frac {- \pi }{ 10 }$

Answer

Correct option: D.

$ \frac {- \pi }{ 10 }$

We have: $\sin ^{ -1 }{ \left( \cos { \frac { 53\pi }{ 5 } } \right) }=\sin ^{ -1 }{ \left( \cos { \frac { 50\pi+3\pi }{ 5 } } \right) }$

$ =\sin ^{ -1 }{ \left( \cos \left({ \frac { 50\pi}{5}+\frac{3\pi }{ 5 } }\right) \right) }$

$ =\sin ^{ -1 }{ \left( \cos \left({ 10\pi+\frac{3\pi }{ 5 } }\right) \right) }$

$ =\sin ^{ -1 }{ \left( \cos \left({ \frac{3\pi }{ 5 } }\right) \right) }, [\because \cos(2\text{n}\pi+\theta)=\cos\theta, n\in \text{Z}]$

$ =\sin ^{ -1 }{ \left( \sin \left({ \frac{\pi}{2}-\frac{3\pi }{ 5 } }\right) \right) },[∵\sin(2π−θ)=\cosθ]$

$ =\sin ^{ -1 }{ \left( \sin \left(-\frac{\pi}{10 }\right) \right) } =−\frac{\pi}{10}$

Note $ \sin^{-1}(\sin \theta)=θ \text{ if} -\frac{\pi}{2}\le \theta\le \frac{\pi}{2}$

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MATHS M.C.Q (1 Marks)