Practice MCQ Questions and Answer on Square Root and Cube Root

41.

The square root of 123454321 is = ?

• (A) 111111
• (B) 12341
• (C) 11111
• (D) 11211

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 $$\therefore \sqrt {123454321} = 11111$$

42.

The number $${\text{2}}{{\text{5}}^{64}} \times {64^{25}}$$   is the square of a natural number n. The sum of the digits of n is = ?

• (A) 7
• (B) 14
• (C) 21
• (D) 28

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 $$\eqalign{ & \Leftrightarrow {{\text{n}}^2} = {\left( {{\text{25}}} \right)^{64}} \times {\left( {64} \right)^{25}} \cr & \Leftrightarrow {{\text{n}}^2} = {\left( {{{\text{5}}^2}} \right)^{64}} \times {\left( {{2^6}} \right)^{25}} \cr & \Leftrightarrow {{\text{n}}^2} = {5^{128}} \times {2^{150}} \cr & \Leftrightarrow {{\text{n}}^2} = {5^{128}} \times {2^{128}} \times {2^{22}} \cr & \Leftrightarrow n = {5^{64}} \times {2^{64}} \times {2^{11}} \cr & \Leftrightarrow n = {\left( {5 \times 2} \right)^{64}} \times {2^{11}} \cr & \Leftrightarrow n = {10^{64}} \times 2048 \cr} $$ ∴ Sum of digits of n = 2 + 0 + 4 + 8 = 14

43.

$${\left( {\sqrt 3 - \frac{1}{{\sqrt 3 }}} \right)^2}\,{\text{simplifies}}\,{\text{to:}}$$

• (A) $$\frac{3}{4}$$
• (B) $$\frac{4}{{\sqrt 3 }}$$
• (C) $$\frac{4}{3}$$
• (D) None of these

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

44.

The square root of 41209 is equal to = ?

• (A) 103
• (B) 203
• (C) 303
• (D) 403

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 $$\eqalign{ & \,\,\,\,\,2|4\,\,\overline {12} \,\,\overline {09} \,(203 \cr & \,\,\,\,\,\,\,\,\,|4 \cr & \,\,\,\,\,\,\,\,\,| - - - - - - - - \cr & \,\,40|\,\,\,\,\,\,\,\,12 \cr & \,\,\,\,\,\,\,\,\,|\,\,\,\,\,\,\,\,\,\,0 \cr & \,\,\,\,\,\,\,\,\,| - - - - - - - - \cr & \,\,\,\,\,\,\,\,\,|\,\,\,\,\,\,\,\,\,12\,09 \cr & 403|\,\,\,\,\,\,\,\,\,12\,09 \cr & \,\,\,\,\,\,\,\,\,| - - - - - - - - \cr & \,\,\,\,\,\,\,\,\,|\,\,\,\,\,\,\,\,\,\,\,\,x \cr & \,\,\,\,\,\,\,\,\,| - - - - - - - - \, \cr & \therefore \sqrt {41209} = 203 \cr} $$

45.

Determined the value of $$\frac{1}{{\sqrt 1 + \sqrt 2 }}{\text{ + }}$$  $$\frac{1}{{\sqrt 2 + \sqrt 3 }}\, + $$   $$\frac{1}{{\sqrt 3 + \sqrt 4 }}\, + $$   $$...... + $$   $$\frac{1}{{\sqrt {120} + \sqrt {121} }}{\text{ = ?}}$$

• (A) 8
• (B) 10
• (C) $$\sqrt {120} $$
• (D) $$12\sqrt 2 $$

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 Given expressing, $$ = \frac{1}{{\sqrt 1 + \sqrt 2 }}{\text{ + }}\frac{1}{{\sqrt 2 + \sqrt 3 }}$$     $$ + \frac{1}{{\sqrt 3 + \sqrt 4 }}$$   $$ + ...... + $$   $$\frac{1}{{\sqrt {120} + \sqrt {121} }}$$ $$ = \frac{1}{{\sqrt 2 + \sqrt 1 }}{\text{ + }}\frac{1}{{\sqrt 3 + \sqrt 2 }}$$     $$ + \frac{1}{{\sqrt 4 + \sqrt 3 }}$$   $$ + ...... + $$   $$\frac{1}{{\sqrt {121} + \sqrt {120} }}$$ $$ = \frac{1}{{\sqrt 2 + \sqrt 1 }} \times $$   $$\frac{{\sqrt 2 - \sqrt 1 }}{{\sqrt 2 - \sqrt 1 }}{\text{ + }}$$   $$\frac{1}{{\sqrt 3 + \sqrt 2 }} \times $$   $$\frac{{\sqrt 3 - \sqrt 2 }}{{\sqrt 3 - \sqrt 2 }} + $$   $$\frac{1}{{\sqrt 4 + \sqrt 3 }} \times $$   $$\frac{{\sqrt 4 - \sqrt 3 }}{{\sqrt 4 - \sqrt 3 }} + $$   $$...... + $$   $$\frac{1}{{\sqrt {121} + \sqrt {120} }} \times $$   $$\frac{{\sqrt {121} - \sqrt {120} }}{{\sqrt {121} - \sqrt {120} }}$$ $$ = \frac{{\sqrt 2 - \sqrt 1 }}{{2 - 1}} + \frac{{\sqrt 3 - \sqrt 2 }}{{3 - 2}}$$     $$ + \frac{{\sqrt 4 - \sqrt 3 }}{{4 - 3}}$$   $$ + ...... + $$   $$\frac{{\sqrt {121} - \sqrt {120} }}{{121 - 120}}$$ $$ = \sqrt 2 - \sqrt 1 + \sqrt 3 - \sqrt 2 $$     $$ + \sqrt 4 - \sqrt 3 $$   $$ + ...... + $$   $$\sqrt {121} - \sqrt {120} $$ $$ = - 1 + \sqrt {121} $$ $$ = - 1 + 11$$ $$ = 10$$

46.

If $$\sqrt {\left( {x - 1} \right)\left( {y + 2} \right)} = 7,$$     x and y being positive whole numbers, then the values of x and y respectively are ?

• (A) 8, 5
• (B) 15, 12
• (C) 22, 19
• (D) None of these

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 $$\eqalign{ & \Leftrightarrow \sqrt {\left( {x - 1} \right)\left( {y + 2} \right)} = 7 \cr & \Leftrightarrow \left( {x - 1} \right)\left( {y + 2} \right) = {\left( 7 \right)^2} \cr & \Leftrightarrow \left( {x - 1} \right) = 7{\text{ and}}\left( {y + 2} \right) = 7 \cr & \Leftrightarrow x = 8{\text{ and }}y = 5 \cr} $$

47.

$${1.5^2} \times \sqrt {0.0225} = ?$$

• (A) 0.0375
• (B) 0.3375
• (C) 3.275
• (D) 32.75

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 $$\eqalign{ & = {1.5^2} \times \sqrt {0.0225} \cr & = {1.5^2} \times \sqrt {\frac{{225}}{{10000}}} \cr & = 2.25 \times \frac{{15}}{{100}} \cr & = 2.25 \times 0.15 \cr & = 0.3375 \cr} $$

48.

$$99 \times 21 - \root 3 \of ? = 1968$$

• (A) 1367631
• (B) 111
• (C) 1366731
• (D) 1367

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 $$\eqalign{ & {\text{Let,}} \cr & {\text{ }}99 \times 21 - \root 3 \of x = 1968 \cr & {\text{Then,}} \cr & \Leftrightarrow 2079 - \root 3 \of x = 1968 \cr & \Leftrightarrow \root 3 \of x = 2079 - 1968 \cr & \Leftrightarrow \root 3 \of x = 111 \cr & \Leftrightarrow x = {\left( {111} \right)^3} \cr & \Leftrightarrow x = 1367631 \cr} $$

49.

What is the least number which should be subtracted 0.000326 in order to make it a perfect square = ?

• (A) 0.000002
• (B) 0.000004
• (C) 0.02
• (D) 0.04

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 $$\eqalign{ & 0.000326 = \frac{{326}}{{{{10}^6}}} \cr & \,\,\,\,1|\overline 3 \,\,\overline {26} \,\,(18 \cr & \,\,\,\,\,\,\,|\,\,1 \cr & \,\,\,\,\,\,\,| - - - - - - \cr & 28|\,\,\,2\,26 \cr & \,\,\,\,\,\,\,|\,\,\,2\,24 \cr & \,\,\,\,\,\,\,| - - - - - - \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,2 \cr} $$ ∴ Required number to be subtracted $$\eqalign{ & = \frac{2}{{{{10}^6}}} \cr & = 0.000002 \cr} $$

50.

If the product of four consecutive natural numbers increased by a natural number p, is a perfect square, then the value of p is = ?

• (A) 1
• (B) 2
• (C) 4
• (D) 8

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 $$\eqalign{ & {\text{We have,}} \cr & 1 \times 2 \times 3 \times 4 = 24 \cr & {\text{And }}24 + 1 = 25\left[ {25 = {5^2}} \right] \cr & 2 \times 3 \times 4 \times 5 = 120{\text{ }} \cr & {\text{And 1}}20 + 1 = 121\left[ {121 = {{11}^2}} \right] \cr & 3 \times 4 \times 5 \times 6 = 360{\text{ }} \cr & {\text{And }}360 + 1 = 361\left[ {361 = {{19}^2}} \right] \cr & 4 \times 5 \times 6 \times 7 = 840{\text{ }} \cr & {\text{And }}840 + 1 = 841\left[ {841 = {{29}^2}} \right] \cr & \therefore p = 1 \cr} $$