Question
Download Solution PDFयदि \(a + \dfrac{\pi}{2} < 2 \tan^{-1} x + 3 \cot^{-1} x < b\) है, तो 'a' और 'b' का मान क्रमशः क्या है?
Answer (Detailed Solution Below)
Detailed Solution
Download Solution PDFसंकल्पना:
\(\rm \tan^{-1}x+\cot^{-1}x=\frac{\pi}{2}\)
0 < cot-1 x < π
गणना:
0 < cot-1 x < π
\( \rm \Rightarrow \pi+0<2\left(\frac{\pi}{2}\right)+\cot ^{-1} x<\pi+\pi \)
\(\because \tan ^{-1} x+\cot ^{-1} x=\frac{π}{2}\)
\(\rm \Rightarrow π<2\left(\tan ^{-1} x+\cot ^{-1} x\right)+\cot ^{-1} x<2 π\)
\( \rm \Rightarrow \pi+0<2\left(\frac{\pi}{2}\right)+\cot ^{-1} x<\pi+\pi \) ---(1)
After comparison,
\( a+\frac{π}{2}\) < 2 tan-1 x + 3 cot-1 x < b
⇒ \(\rm a=\frac{\pi}{2} \)
\(\rm b=2 \pi \)
अतः विकल्प (2) सही है।
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