Practice MCQ Questions and Answer on Surds and Indices

21.

If $$\sqrt{33}$$  = 5.745, then the value of the following is approximately:

• (A) 0.5223
• (B) 6.32
• (C) 2.035
• (D) 1

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Answer & Solution Answer: Option A No explanation is given for this question Let's Discuss on Board

22.

If a = b^p, b = c^q, c = a^r then pqr is

• (A) 1
• (B) 0
• (C) -1
• (D) abc

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 a = bp and b = cq ∴ c = ar = (bp)r = (b)pr = (cq)pr = cpqr ⇒ pqr = 1

23.

If a = b^p, b = c^q, c = a^r then pqr is

• (A) 1
• (B) 0
• (C) -1
• (D) abc

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 a = bp and b = cq ∴ c = ar = (bp)r = (b)pr = (cq)pr = cpqr ⇒ pqr = 1

24.

$${\left(\frac{x^b}{x^c}\right)}^{\left(b+c-a\right)}.$$   $${\left(\frac{x^c}{x^a}\right)}^{\left(c+a-b\right)}.$$   $${\left(\frac{x^a}{x^b}\right)}^{\left(a+b-c\right)}=?$$

• (A) xabc
• (B) 1
• (C) xab+bc+ca
• (D) xa+b+c

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 $${x^{\left( {b - c} \right)\left( {b + c - a} \right)}}.{x^{\left( {c - a} \right)\left( {c + a - b} \right)}}.{x^{\left( {a - b} \right)\left( {a + b - c} \right)}}$$   $$ = {x^{\left( {b - c} \right)\left( {b + c} \right) - a\left( {b - c} \right)}}.$$    $${x^{\left( {c - a} \right)\left( {c + a} \right) - b\left( {c - a} \right)}}.$$   $${x^{\left( {a - b} \right)\left( {a + b} \right) - c\left( {a - b} \right)}}$$ $$\eqalign{ & = {x^{\left( {{b^2} - {c^2} + {c^2} - {a^2} + {a^2} - {b^2}} \right)}}.{x^{ - a\left( {b - c} \right) - b\left( {c - a} \right) - c\left( {a - b} \right)}} \cr & = \left( {{x^0} \times {x^0}} \right) \cr & = \left( {1 \times 1} \right) \cr & = 1 \cr} $$

25.

(25) × (5) ÷ (125) = 57.52.51.5?

• (A) 8.5
• (B) 13
• (C) 16
• (D) 17.5

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 $$\eqalign{ & {\text{Let}}\,{\left( {25} \right)^{7.5}} \times {\left( 5 \right)^{2.5}} \div {\left( {125} \right)^{1.5}} = {5^x} \cr & {\text{Then}},\,\frac{{{{\left( {{5^2}} \right)}^{7.5}} \times {{\left( 5 \right)}^{2.5}}}}{{{{\left( {{5^3}} \right)}^{1.5}}}} = {5^x} \cr & \Rightarrow \frac{{{5^{\left( {2 \times 7.5} \right)}} \times {5^{2.5}}}}{{{5^{\left( {3 \times 1.5} \right)}}}} = {5^x} \cr & \Rightarrow \frac{{{5^{15}} \times {5^{2.5}}}}{{{5^{4.5}}}} = {5^x} \cr & \Rightarrow {5^x} = {5^{\left( {15 + 2.5 - 4.5} \right)}} \cr & \Rightarrow {5^x} = {5^{13}} \cr & \therefore x = 13 \cr} $$

26.

What are the values of x and y that satisfy the equation, $${\text{2}}^{0.7x}\text{.}{\text{3}}^{-1.25y}\text{ = }\frac{8\sqrt{6}}{27}\text{ ?}$$

• (A) x = 2.5, y = 6
• (B) x = 3, y = 5
• (C) x = 3, y = 4
• (D) x = 5, y = 2

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 $$\eqalign{ & {{\text{2}}^{0.7x}}{\text{.}}{{\text{3}}^{ - 1.25y}}{\text{ = }}\frac{{8\sqrt 6 }}{{27}} \cr & \Leftrightarrow \frac{{{{\text{2}}^{0.7x}}}}{{{{\text{3}}^{ 1.25y}}}}{\text{ = }}\frac{{{2^3}{{.2}^{\frac{1}{2}}}{{.3}^{\frac{1}{2}}}}}{{{3^3}}} \cr & \Leftrightarrow \frac{{{2^{\left( {3 + \frac{1}{2}} \right)}}}}{{{3^{\left( {3 - \frac{1}{2}} \right)}}}} = \frac{{{2^{\frac{7}{2}}}}}{{{2^{\frac{5}{2}}}}} = \frac{{{2^{3.5}}}}{{{3^{2.5}}}} \cr & \therefore 0.7x = 3.5 \Rightarrow x = \frac{{3.5}}{{0.7}}{\text{ = 5}} \cr & {\text{and }}1.25y = 2.5 \cr & \Rightarrow y = \frac{{2.5}}{{1.25}} = 2 \cr} $$

27.

The value of $$\frac{1}{1+\sqrt{2}+\sqrt{3}}+$$   $$\frac{1}{1-\sqrt{2}+\sqrt{3}}$$   is = ?

• (A) $$\sqrt{2}$$
• (B) $$\sqrt{3}$$
• (C) 1
• (D) $$4\left(\sqrt{3}+\sqrt{2}\right)$$

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 $$\eqalign{ & \frac{1}{{1 + \sqrt 2 + \sqrt 3 }} + \frac{1}{{1 - \sqrt 2 + \sqrt 3 }} \cr & = \frac{1}{{1 + \sqrt 3 + \sqrt 2 }} + \frac{1}{{1 + \sqrt 3 - \sqrt 2 }} \cr & = \frac{{1 + \sqrt 3 - \sqrt 2 + 1 + \sqrt 3 + \sqrt 2 }}{{{{\left( {1 + \sqrt 3 } \right)}^2} - {{\left( {\sqrt 2 } \right)}^2}}} \cr & = \frac{{2 + 2\sqrt 3 }}{{4 + 2\sqrt 3 - 2}} \cr & = \frac{{2 + 2\sqrt 3 }}{{2 + 2\sqrt 3 }} \cr & = 1 \cr} $$

28.

Given $$\sqrt{2}$$ = 1.414, the value of $$\sqrt{8}$$ $$+$$ $$\text{2}\sqrt{32}$$ $$-$$ $$3\sqrt{128}$$ $$+$$ $$\text{4}\sqrt{50}$$  is = ?

• (A) 8.484
• (B) 8.526
• (C) 8.426
• (D) 8.876

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 $$\eqalign{ & \sqrt 2 = 1.414 \cr & \Rightarrow \sqrt 8 {\text{ + 2}}\sqrt {32} - 3\sqrt {128} {\text{ + 4}}\sqrt {50} \cr & \Rightarrow 2\sqrt 2 + 2 \times 4\sqrt 2 - 3 \times 8\sqrt 2 + 4 \times 5\sqrt 2 \cr & \Rightarrow 2\sqrt 2 + 8\sqrt 2 - 24\sqrt 2 + 20\sqrt 2 \cr & \Rightarrow 6\sqrt 2 \cr & \Rightarrow 6 \times 1.414 \cr & \Rightarrow 8.484{\text{ }} \cr} $$

29.

The least one among $$\text{2}\sqrt{3}\text{,}$$  $$\text{2}\sqrt[4]{5}\text{,}$$  $$\sqrt{8}\text{,}$$  $$\text{3}\sqrt{2}$$  is = ?

• (A) $$\text{2}\sqrt{3}$$
• (B) $$\text{2}\sqrt[4]{5}$$
• (C) $$\sqrt{8}$$
• (D) $$\text{3}\sqrt{2}$$

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 $$2\sqrt 3 = {\left( {4 \times 3} \right)^{\frac{1}{2}}} \to {12^{\frac{1}{2}}} \to {12^{\frac{2}{4}}} \to \root 4 \of {144} $$ $$2\root 4 \of 5 = \root 4 \of {\left( {5 \times 16} \right)} \to {80^{\frac{1}{4}}} \to \root 4 \of {80} $$ $$\sqrt 8 = {8^{\frac{1}{2}}} \to {8^{\frac{1}{2}}} \to \boxed{\root 4 \of {64} }\,{\text{smallest}}$$ $$3\sqrt 2 = \sqrt {18} \to {18^{\frac{1}{2}}} \to {18^{\frac{2}{4}}} \to \root 4 \of {324} $$ $$\sqrt 8 \,{\text{ is answer}}$$

30.

What is the value of $$\sqrt{121}+\sqrt{12321}+\sqrt{1234321}+\sqrt{123454321}?$$

• (A) 12345
• (B) 123456
• (C) 12344
• (D) 123454

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 $$\eqalign{ & \sqrt {121} + \sqrt {12321} + \sqrt {1234321} + \sqrt {123454321} \cr & = 11 + 111 + 1111 + 11111 \cr & = 12344 \cr} $$