Let \(S=x: x∈R\) and $\left(\sqrt{3}+\sqrt{2})^{x^2-4}+(\sqrt{3}-\sqrt{2})^{x^2-4}=10\right\}$Then $n(S)$ is equal to
- 2
- 4
- 0
- 6
The Correct Option is B
Approach Solution - 1
The correct answer is (B) : 4
\(\text{Let }(\sqrt3+\sqrt2)^{x^{2}−4}=t\)
\(t+\frac{1}{t}=10\)
\(⇒t=5+2\sqrt6, 5−2\sqrt6\)
\(⇒(\sqrt3+\sqrt2)^{x^{2}−4}=5+2\sqrt6, 5−2\sqrt6\)
\(⇒x^2−4=2, −2\) or \(x^2 =6, 2\)
\(⇒x=±\sqrt2, ±\sqrt6\)
Approach Solution -2
Let \( (\sqrt{3} + \sqrt{2})^{x-4} = t \), then \[ t + \frac{1}{t} = 10 \] Solving for \( t \), we get \[ t = 5 + 2\sqrt{6}, \quad t = 5 - 2\sqrt{6} \] Thus, \[ (\sqrt{3} + \sqrt{2})^{x-4} = 5 + 2\sqrt{6}, \quad 5 - 2\sqrt{6} \] Squaring both sides, \[ x^2 - 4 = 2, -2 \Rightarrow x^2 = 6, 2 \] \[ x = \pm\sqrt{2}, \pm\sqrt{6} \]
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Concepts Used:
Sets
Set is the collection of well defined objects. Sets are represented by capital letters, eg. A={}. Sets are composed of elements which could be numbers, letters, shapes, etc.
Example of set: Set of vowels A={a,e,i,o,u}
Representation of Sets
There are three basic notation or representation of sets are as follows:
Statement Form: The statement representation describes a statement to show what are the elements of a set.
- For example, Set A is the list of the first five odd numbers.
Roster Form: The form in which elements are listed in set. Elements in the set is seperatrd by comma and enclosed within the curly braces.
- For example represent the set of vowels in roster form.
A={a,e,i,o,u}
Set Builder Form:
- The set builder representation has a certain rule or a statement that specifically describes the common feature of all the elements of a set.
- The set builder form uses a vertical bar in its representation, with a text describing the character of the elements of the set.
- For example, A = { k | k is an even number, k ≤ 20}. The statement says, all the elements of set A are even numbers that are less than or equal to 20.
- Sometimes a ":" is used in the place of the "|".