Square Roots and Cube Roots - 3



 

17.If a $\displaystyle\frac{{\sqrt 5  + 1}}{{\sqrt 5  - 1}}$ and b = $\displaystyle\frac{{\sqrt 5  - 1}}{{\sqrt 5  + 1}}$, the value of $\left( {\displaystyle\frac{{{a^2} + ab + {b^2}}}{{{a^2} - ab + {b^2}}}} \right) = $
a. $\displaystyle\frac{3}{4}$
b. $\displaystyle\frac{4}{3}$
c. $\displaystyle\frac{3}{5}$
d. $\displaystyle\frac{5}{3}$
Correct Option: B
Explanation:
a = $\displaystyle\frac{{\sqrt 5  + 1}}{{\sqrt 5  - 1}} \times \displaystyle\frac{{\sqrt 5  + 1}}{{\sqrt 5  + 1}} = {\displaystyle\frac{{\left( {\sqrt 5  + 1} \right)}}{{(5 - 1)}}^2}$ =$\displaystyle\frac{{5 + 1 - 2\sqrt 5 }}{4} = \left[ {\displaystyle\frac{{3 - \sqrt 5 }}{2}} \right]$

b = $\displaystyle\frac{{\sqrt 5  - 1}}{{\sqrt 5  + 1}} \times \displaystyle\frac{{\sqrt 5  - 1}}{{\sqrt 5  - 1}} = {\displaystyle\frac{{\left( {\sqrt 5  - 1} \right)}}{{(5 - 1)}}^2}$ =$\displaystyle\frac{{5 + 1 - 2\sqrt 5 }}{4} = \left[ {\displaystyle\frac{{3 - \sqrt 5 }}{2}} \right]$

${a^2} + {b^2} = \displaystyle\frac{{{{\left( {3 + \sqrt 5 } \right)}^2}}}{4} + \displaystyle\frac{{{{\left( {3 - \sqrt 5 } \right)}^2}}}{4}$ =$\displaystyle\frac{{{{\left( {3 + \sqrt 5 } \right)}^2}}}{4} + \displaystyle\frac{{{{\left( {3 - \sqrt 5 } \right)}^2}}}{4}$

=$\displaystyle\frac{{2(9 + 5)}}{4} = 7$

Also, ab = $\displaystyle\frac{{(3 + \sqrt 5 )}}{2}$ $\displaystyle\frac{{3 - \sqrt 5 }}{2} = \displaystyle\frac{{(9 - 5)}}{4} = 1$

$\displaystyle\frac{{{a^2} + ab + {b^2}}}{{{a^2} - ab + {b^2}}} = \displaystyle\frac{{{a^2} + {b^2} + ab}}{{({a^2} + {b^2}) - ab}} = \displaystyle\frac{{7 + 1}}{{7 - 1}}$

=$\displaystyle\frac{8}{6} = \displaystyle\frac{4}{3}$


18.The expression $\left( {2 + \sqrt 2 } \right) + \displaystyle\frac{1}{{\left( {2 + \sqrt 2 } \right)}} + \displaystyle\frac{1}{{\left( {2 - \sqrt 2 } \right)}} = ?$
a. 2
b. $2\sqrt 2 $
c. ${2 - \sqrt 2 }$
d. ${2 + \sqrt 2 }$
Correct Option: A
Explanation:
Given Expression = $\left( {2 + \sqrt 2 } \right) + \displaystyle\frac{1}{{\left( {2 + \sqrt 2 } \right)}} \times \displaystyle\frac{{\left( {2 - \sqrt 2 } \right)}}{{\left( {2 - \sqrt 2 } \right)}}$ + $\displaystyle\frac{1}{{\left( {2 - \sqrt 2 } \right)}} \times \displaystyle\frac{{\left( {2 + \sqrt 2 } \right)}}{{\left( {2 + \sqrt 2 } \right)}}$

=$\left( {2 + \sqrt 2 } \right) + \displaystyle\frac{{\left( {2 - \sqrt 2 } \right)}}{{\left( {4 - 2} \right)}} \times \displaystyle\frac{{\left( {2 + \sqrt 2 } \right)}}{{\left( {4 - 2} \right)}}$

=$\left( {2 + \sqrt 2 } \right) + \displaystyle\frac{1}{2}\left( {2 - \sqrt 2 } \right) - \displaystyle\frac{1}{2}\left( {2 + \sqrt 2 } \right) = 2$


19.$\sqrt {\displaystyle\frac{{0.081 \times 0.324 \times 4.624}}{{15.625 \times 0.0289 \times 72.9 \times 64}}}  = ?$
a. 24
b. 2.4
c. 0.024
d. None
Correct Option: C
Explanation:
Given Expression : $\sqrt {\displaystyle\frac{{81 \times 324 \times 4624}}{{15625 \times 289 \times 729 \times 64}}} $ = $\left[ {\displaystyle\frac{{9 \times 18 \times 68}}{{125 \times 17 \times 27 \times 8}}} \right] = \displaystyle\frac{3}{{125}} = 0.024$

20.$\sqrt {\displaystyle\frac{{0.081 \times 0.484}}{{0.0064 \times 6.25}}} $ is equal to
a. 9
b. 0.9
c. 99
d. 0.99
Correct Option: D
Explanation:

Sum of decimal places in the numerator and denominator under the radical sign being the same, we remove the decimal places:
Given Exp. = $\sqrt {\displaystyle\frac{{81 \times 484}}{{64 \times 625}}} $ = $\displaystyle\frac{{9 \times 22}}{{8 \times 25}} = 0.99$

21.if $\displaystyle\frac{x}{{\sqrt {2.25} }} = 550$, then the value of x is 
a. 825
b. 82.5
c. 3666.66
d. 2
Correct Option: A
Explanation :
$\displaystyle\frac{x}{{\sqrt {2.25} }} = 550 \Rightarrow \displaystyle\frac{x}{{1.5}} = 550 \Rightarrow$ x = $(550 \times 1.5) \Rightarrow$ x = $\left[ {\displaystyle\frac{{550 \times 1.5}}{{10}}} \right] = 825$

22.$\sqrt {81}  + \sqrt {0.81}  = 10.09 - ?$
a. 1.19
b. 0.19
c. 1
d. 0.19
Correct Option: D
Explanation :
Let $\sqrt {81}  + \sqrt {0.81}  = 10.09 - x \Rightarrow \sqrt {81}  + \sqrt {\displaystyle\frac{{81}}{{100}}}  = 10.09 - x$
x = 10.09- (9 + 0.9) = 0.19

23.$\sqrt {\displaystyle\frac{{32.4}}{?}}  = 2$
a. 9
b. 0.9
c. 0.09
d. None
Correct Option: D
Explanation:
Let $\sqrt {\displaystyle\frac{{32.4}}{x}}  = 2$ Then $\displaystyle\frac{{32.4}}{x} = 4$
or 4x = 32.4 or x = 8.1

24. $\displaystyle\frac{{\sqrt {32}  + \sqrt {48} }}{{\sqrt 8  + \sqrt {12} }} = ?$
a. $\sqrt 2 $
b. 2
c. 4
d. 8
Correct Option: B
Explanation:
$\displaystyle\frac{{\sqrt {32}  + \sqrt {48} }}{{\sqrt 8  + \sqrt {12} }} = \displaystyle\frac{{\sqrt {16 \times 2}  + \sqrt {16 \times 3} }}{{\sqrt {4 \times 2}  + \sqrt {4 \times 3} }}$ = $\displaystyle\frac{{4\left( {\sqrt 2  + \sqrt 3 } \right)}}{{2\left( {\sqrt 2  + \sqrt 3 } \right)}} = 2$


25. If $\sqrt {1 + \displaystyle\frac{x}{{144}}}  = \displaystyle\frac{{13}}{{12}}$ then 'x' is equal to
a. 1
b. 12
c. 13
d. 25
Correct Option: D
Explanation:
$\sqrt {1 + \displaystyle\frac{x}{{144}}}  = \displaystyle\frac{{13}}{{12}} \Rightarrow 1 + \displaystyle\frac{x}{{144}} = \displaystyle\frac{{169}}{{144}}$
$\displaystyle\frac{x}{{144}} = \left[ {\displaystyle\frac{{160}}{{144}} - 1} \right]$ or $\displaystyle\frac{x}{{144}}$ = $\displaystyle\frac{{25}}{{144}}$ or x = 25