Real numbers
Grade Class 10
Question:
<p>If <span class="math-tex">\(\frac{\sqrt{3}-1}{\sqrt{3}+1}=a-b \sqrt{3}\)</span>,then</p>
<p style="display:inline">a = 2, b = 1</p>
<p style="display:inline">a = b = 1</p>
<p style="display:inline">a = 2, b = -1</p>
<p style="display:inline">a = - 2, b = 1</p>
Step-by-Step Solution
Key Concept: Rationalize the denominator by multiplying the numerator and denominator by its conjugate to simplify the expression into a form where rational and irrational parts can be compared.
<p><span class="math-tex">$a-b \sqrt{3} $</span> = <span class="math-tex">$\frac{\sqrt{3}-1}{\sqrt{3}+1} $</span><br />
= <span class="math-tex">$\frac{\sqrt{3}-1}{\sqrt{3}+1} \times \frac{\sqrt{3}-1}{\sqrt{3}-1}$</span><br />
= <span class="math-tex">$\frac{(\sqrt{3}-1)^2}{(\sqrt{3})^2-(1)^2} $</span><br />
=<span class="math-tex">$\frac{(\sqrt{3})^2 + (1)^2-2(1))(\sqrt{3})}{3-1} $</span><br />
=<span class="math-tex">$\frac{3 + 1 -2\sqrt{3}}{2} $</span><br />
= <span class="math-tex">$\frac{4 -2\sqrt{3}}{2} $</span><br />
= <span class="math-tex">$\frac{2(2 -\sqrt{3})}{2} $</span><br />
= 2- <span class="math-tex">$\sqrt{3}$</span><br />
<span class="math-tex">$\Rightarrow$</span> <span class="math-tex">$a-b \sqrt{3} $</span> = 2 - <span class="math-tex">$\sqrt{3}$</span><br />
<span class="math-tex">$\Rightarrow$</span> a = 2, b = 1</p>
Correct Answer: A