The number of elements in the set S= {x∈R : 2cos\((\frac{x_2+x}{6})\)=\(4^x+4^{-x}\)}
- 1
- 3
- 0
- infinite
The Correct Option is A
Approach Solution - 1
To find the number of elements in the set \( S = \{ x \in \mathbb{R} : 2 \cos \left( \frac{x_2 + x}{6} \right) = 4^x + 4^{-x} \} \), we need to analyze and solve the given equation for \( x \).
The equation we need to solve is:
\(2 \cos \left( \frac{x_2 + x}{6} \right) = 4^x + 4^{-x}\)
To simplify, let's analyze both sides of the equation:
The left-hand side, \( 2 \cos \left( \frac{x_2 + x}{6} \right) \), is bound between -2 and 2, because the range of the cosine function is \([-1, 1]\).
The right-hand side, \( 4^x + 4^{-x} \), using the property of exponents and recognizing this as an exponential function, is always greater than or equal to 2, since:
\(4^x + 4^{-x} = 2 + 2 \left( \frac{4^x - 4^{-x}}{2} \right)^2 \geq 2\) using the identity \( a + \frac{1}{a} \geq 2 \) for \( a > 0 \).
The only possible value for \( 4^x + 4^{-x} \) which matches the maximum possible value of \( 2 \cos \left( \frac{x_2 + x}{6} \right) \) is 2. This implies:
\( 4^x + 4^{-x} = 2 \)
This equality holds only when \( 4^x = 4^{-x} = 1 \). Therefore, \( x \) must satisfy:
\(4^x = 1 \Rightarrow 4^x = 4^0 \Rightarrow x = 0\).
Let's verify: For \( x = 0 \), the original condition becomes:
\( 2 \cos \left( \frac{x_2 + 0}{6} \right) = 1 + 1 = 2 \)
At \( x = 0 \), the equation holds since \( 2 \cos(0) = 2 \).
Therefore, the only solution is \( x = 0 \).
The number of elements in the set \( S \) is thus 1.
Approach Solution -2
S={x∈R:2cos\((\frac{x_2+x}{6})=4^x+4^{-x}\)}
L.H.S. is less than or equal to 2 and RHS is greater than or equal to 2.
So equality holds only if LHS = RHS = 2
R.H.S. is 2 when x = 0
and at x = 0, LHS is also 2.
So, only one solution exist.
Therefore, the correct option is (A): 1.
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Concepts Used:
Sets
In mathematics, a set is a well-defined collection of objects. Sets are named and demonstrated using capital letter. In the set theory, the elements that a set comprises can be any sort of thing: people, numbers, letters of the alphabet, shapes, variables, etc.
Read More: Set Theory
Elements of a Set:
The items existing in a set are commonly known to be either elements or members of a set. The elements of a set are bounded in curly brackets separated by commas.
Read Also: Set Operation
Cardinal Number of a Set:
The cardinal number, cardinality, or order of a set indicates the total number of elements in the set.
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