SATHEE: Inverse Circular Functions 2 Question 12

Inverse Circular Functions 2 Question 12

12. The number of real solutions of the equation $\sin^{- 1} \sum_{i = 1}^{\infty} x^{i + 1} - x \sum_{i = 1}^{\infty} \frac{x}{2}$

$= \frac{π}{2} - \cos^{- 1} \sum_{i = 1}^{\infty} - \frac{x^{i}}{2} - \sum_{i = 1}^{\infty} (- x)^{i}$ lying in the interval

$- \frac{1}{2}, \frac{1}{2} is$

(Here, the inverse trigonometric functions $\sin^{- 1} x$ and $\cos^{- 1} x$ assume values in $- \frac{π}{2}, \frac{π}{2}$ and $[0, π]$ , respectively.)

(2018 Adv.)

Analytical & Descriptive Question

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Answer:

Correct Answer: 12. (2)

Solution:

We have,

$\begin{matrix} \sin^{- 1} \sum_{i = 1}^{\infty} x^{i + 1} - x \sum_{i = 1}^{\infty} \frac{x^{i}}{2} \\ = \frac{π}{2} - \cos^{- 1} \sum_{i = 1}^{\infty} \frac{- x^{i}}{2} - \sum_{i = 1}^{\infty} (- x)^{i} \\ \Rightarrow \sin^{- 1} \frac{x^{2}}{1 - x} - \frac{x \cdot \frac{x}{2}}{1 - \frac{x}{2}} \\ = \frac{π}{2} - \cos^{- 1} \frac{\frac{- x}{2}}{1 + \frac{x}{2}} - \frac{(- x)}{1 + x} \\ ∵ \sum_{i = 1}^{\infty} x^{i + 1} = x^{2} + x^{3} + x^{4} + \dots = \frac{x^{2}}{1 - x} \end{matrix}$

using sum of infinite terms of GP

$\Rightarrow \sin^{- 1} \frac{x^{2}}{1 - x} - \frac{x^{2}}{2 - x} = \frac{π}{2} - \cos^{- 1} \frac{x}{1 + x} - \frac{x}{2 + x}$

$\Rightarrow \sin^{- 1} \frac{x^{2}}{1 - x} - \frac{x^{2}}{2 - x} = \sin^{- 1} \frac{x}{1 + x} - \frac{x}{2 + x}$

$\begin{aligned} ∵ \sin^{- 1} x = \frac{π}{2} - \cos^{- 1} x \\ \Rightarrow x^{2} \frac{2 - x - 1 + x}{(1 - x) (2 - x)} = x \frac{x^{2}}{(1 + x) (2 + x)} = \frac{x}{1 + x} - \frac{x}{2 + x} \\ \Rightarrow \frac{x}{2 - 3 x + x^{2}} = \frac{1}{2 + 3 x + x^{2}} or x = 0 \\ \Rightarrow x^{3} + 3 x^{2} + 2 x = x^{2} - 3 x + 2 \\ \Rightarrow x^{3} + 2 x^{2} + 5 x - 2 = 0 or x = 0 \end{aligned}$

Let $f (x) = x^{3} + 2 x^{2} + 5 x - 2$

$f^{'} (x) = 3 x^{2} + 4 x + 5$

$f^{'} (x) > 0, \forall x \in R$

$∴ x^{3} + 2 x^{2} + 5 x - 2$ has only one real roots

Therefore, total number of real solution is 2 .

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Inverse Circular Functions 2 Question 12

12. The number of real solutions of the equation $\sin^{- 1} \sum_{i = 1}^{\infty} x^{i + 1} - x \sum_{i = 1}^{\infty} \frac{x}{2}$

Analytical & Descriptive Question

Answer:

इंजीनियरिंग की तैयारी

चिकित्सा तैयारी

एनसीईआरटी पुस्तकें और समाधान

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नमूना मॉक टेस्ट

सदस्यता

एनसीईआरटी व्यायाम वीडियो समाधान

Inverse Circular Functions 2 Question 12

12. The number of real solutions of the equation sin−1⁡∑i=1∞xi+1−x∑i=1∞x2

Analytical & Descriptive Question

Answer:

इंजीनियरिंग की तैयारी

चिकित्सा तैयारी

एनसीईआरटी पुस्तकें और समाधान

महत्वपूर्ण संसाधन

अन्य संसाधन

पाठ्यक्रम

परीक्षा मंच

नमूना मॉक टेस्ट

सदस्यता

एनसीईआरटी व्यायाम वीडियो समाधान

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12. The number of real solutions of the equation $\sin^{- 1} \sum_{i = 1}^{\infty} x^{i + 1} - x \sum_{i = 1}^{\infty} \frac{x}{2}$