Question : The value of $\sqrt{72+\sqrt{72+\sqrt{72+...}}}$ is:
Option 1: 9
Option 2: 8
Option 3: 18
Option 4: 12
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Correct Answer: 9
Solution : Let $ x=\sqrt{72+\sqrt{72+\sqrt{72+...}}}$ $x=\sqrt{72+x}$. By squaring both sides, we get, $⇒ x^{2}=(\sqrt{72+x})^{2}$ $ ⇒x^{2}=72+x$ $ ⇒x^{2}-x-72=0$ $ ⇒x^{2}-9x+8x-72=0$ $⇒x(x-9)+8(x-9)=0$ $⇒(x-9)(x+8)=0$ $\therefore x=9,-8$ Since $x$ can not be equal to –8, the value of $x$ will be 9. Hence, the correct answer is 9.
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Question : The value of $\sqrt{9+2\sqrt{16}+\sqrt[3]{512}}$ is:
Option 1: 6
Option 2: 5
Option 3: $\sqrt[2]{8}$
Option 4: $\sqrt[3]{6}$
Question : If $\sin \theta+\cos \theta=\frac{\sqrt{3}-1}{2 \sqrt{2}}$, then what is the value of $\tan \theta+\cot \theta$?
Option 1: $8(\sqrt{3}-2)$
Option 2: $12(\sqrt{3}-2)$
Option 3: $12(\sqrt{3}+2)$
Option 4: $8(\sqrt{3}+2)$
Question : The value of $\frac{1}{4-\sqrt{15}}-\frac{1}{\sqrt{15}-\sqrt{14}}+\frac{1}{\sqrt{14}-\sqrt{13}}-\frac{1}{\sqrt{13}-\sqrt{12}}+\frac{1}{\sqrt{12}-\sqrt{11}}-\frac{1}{\sqrt{11}-\sqrt{10}}+\frac{1}{\sqrt{10}-3}-\frac{1}{3-\sqrt{8}}$ is:
Option 1: $2-2 \sqrt{2}$
Option 2: $4+2 \sqrt{2}$
Option 3: $4-2 \sqrt{2}$
Option 4: $2+2 \sqrt{2}$
Question : The value of $\sqrt{–\sqrt{3}+\sqrt{3+8\sqrt{7+4\sqrt{3}}}}$ is:
Option 1: 2
Option 2: 4
Option 3: $\pm 2$
Option 4: –2
Question : If $x=\sqrt[3]{2+\sqrt{3}}$, the value of $x^3+\frac{1}{x^3}$ is:
Option 1: 8
Option 2: 9
Option 3: 2
Option 4: 4
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