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Number of real solution of the equation sin (ex) = 5x + 5-x is |
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Answer» Correct Answer - Option 1 : 0 Concept: -1 ≤ sin x ≤ 1
AM, GM, HM Formulas If A is the arithmetic mean of numbers a and b and is given by ⇔ \({\rm{A}} = \frac{{{\rm{a\;}} + {\rm{\;b}}}}{2}\) If G is the geometric mean of the numbers a and b and is given by ⇔ \({\rm{G}} = \sqrt {{\rm{ab}}} \) If H is the Harmonic mean of numbers a and b and is given by ⇔ \({\rm{H}} = \frac{{2{\rm{ab}}}}{{{\rm{a}} + {\rm{b}}}}\) Relation between AM, GM and HM
Calculations: Given, the equation is sin (ex) = 5x + 5-x Consider, LHS = sin (ex) < 1 As we know AM ≥ GM \(\rm \Rightarrow \frac{5^x + 5^{-x}}{2} \geq (5^x \times 5^{-x})^{1/2}\) ∴ 5x + 5-x ≥ 2 RHS = 5x + 5-x ≥ 2 Here, LHS \(\neq\) RHS ⇒The equations sin (ex) = 5x + 5-x have no solution. This means that no matter what value is plugged in for the variable, you will ALWAYS get a contradiction. Hence, Number of real solution of the equation sin (ex) = 5x + 5-x is zero. |
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