| Assertion (A): In \[\Delta PQR\] right angled at Q, \[\text{QR}=\text{3 cm}\] and \[\text{PR}-\text{PQ}=\text{1 cm}\]. The value of \[{{\sin }^{2}}R+\text{cosec}\,\text{R}\]is \[\frac{189}{100}\]. |
| Reason (R): \[{{\sin }^{2}}A={{(\sin A)}^{2}}\] and \[\text{cosec A}={{(\sec A)}^{-1}}\] . |
A) 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 Assertion (A)
C) Assertion (A) is true but Reason (R) is false
D) Assertion (A) is false but Reason (R) is true
Correct Answer: C
Solution :
| [c] Since \[\Delta PQR\] is right angled triangle. |
| From Pythagoras theorem |
| \[P{{R}^{2}}=P{{Q}^{2}}+Q{{R}^{2}}\] |
| \[Q{{R}^{2}}=P{{R}^{2}}-P{{Q}^{2}}\] |
| \[{{(3)}^{2}}=P{{R}^{2}}-P{{Q}^{2}}\] \[[\ \ \ OR=3]\] |
| \[\Rightarrow \ P{{R}^{2}}-P{{Q}^{2}}=9\] |
| \[\Rightarrow \ \ \ \ (PR+PQ)(PR-PQ)=9\ \ [\ \ \ {{a}^{2}}-{{b}^{2}}=(a+b)(a-b)]\] |
| \[\Rightarrow \ \ \ \ (PR+PQ)(1)=9\ \ [\ \ \ PR-PQ=1]\] |
| \[\Rightarrow \ \ \ \ PR+PQ=9\] |
| On solving PR+PQ= 9 and PR-PQ=1 we get PR=5 and PQ=4 |
| \[\therefore \ \ \ \ \ \ \ \ \ \ \ \ \ \ \operatorname{sinR}=\frac{PQ}{PR}=\frac{4}{5}\] |
| So, \[{{\sin }^{2}}R+cosec\ R\] |
| \[={{\left( \frac{4}{5} \right)}^{2}}+\ \frac{1}{4\sqrt{5}}=\frac{16}{25}+\frac{5}{4}=\frac{64+125}{25\times 4}=\frac{189}{100}\] |
| Also, \[\operatorname{cosec} A ={{\left( sin\ A \right)}^{-1}}\] |
| \[\therefore \] Assertion: True: Reason : False. |
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