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Basic Maths: Test 34

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Question 1
\[\frac{1}{60}+\frac{1}{90}+\frac{1}{126}+\frac{1}{168}+\frac{1}{216}+\frac{1}{270}+\frac{1}{330}+\frac{1}{396}\]
is equal to:
A
$\frac{1}{18}$
B
$\frac{1}{12}$
C
$\frac{1}{6}$
D
$\frac{1}{10}$
Question 1 Explanation:
\[\begin{align} & =\frac{1}{60}+\frac{1}{90}+\frac{1}{126}+\frac{1}{168}+\frac{1}{216}+\frac{1}{270}+\frac{1}{330}+\frac{1}{396} \\ & =\frac{1}{3}\left[ \frac{1}{4\times 5}+\frac{1}{5\times 6}+\frac{1}{6\times 7}+.....+\frac{1}{11\times 12} \right] \\ & =\frac{1}{3}\left[ \frac{1}{4}-\frac{1}{5}+\frac{1}{5}-\frac{1}{6}+\frac{1}{6}-\frac{1}{7}+....+\frac{1}{11}-\frac{1}{12} \right] \\ & =\frac{1}{12}-\frac{1}{36}=\frac{3-1}{36}=\frac{2}{36}=\frac{1}{18} \\ \end{align}\]
Question 2
Simplify
$\frac{10}{21}-\left[ \frac{3}{7}+\left\{ \frac{1}{7}+\left( \frac{1}{3}-\overline{\frac{10}{35}+\frac{1}{7}} \right) \right\} \right]$
A
0
B
1
C
$\frac{9}{20}$
D
$\frac{9}{10}$
Question 2 Explanation:
$\begin{align} & \frac{10}{21}-\left[ \frac{3}{7}+\left\{ \frac{1}{7}+\left( \frac{1}{3}-\overline{\frac{10}{35}+\frac{1}{7}} \right) \right\} \right] \\ & =\frac{10}{21}-\left[ \frac{3}{7}+\left\{ \frac{1}{7}+\left( \frac{1}{3}-\frac{3}{7} \right) \right\} \right] \\ & =\frac{10}{21}-\left[ \frac{3}{7}+\left\{ \frac{1}{7}+\left( -\frac{2}{21} \right) \right\} \right] \\ & =\frac{10}{21}-\left[ \frac{3}{7}+\left\{ \frac{1}{21} \right\} \right] \\ & \frac{10}{21}-\left[ \frac{3}{7}+\frac{1}{21} \right]=\frac{10}{21}-\frac{10}{21}=0 \\ \end{align}$
Question 3
Simplify
\[\frac{0.6\overline{8}\div 6.2}{3.2\overline{13}-0.7\overline{68}}\,\]
is equal to:
A
0.454
B
0.045
C
0.063
D
0.054
Question 3 Explanation:
$\begin{align} & =\frac{0.6\overline{8}\div 6.2}{3.2\overline{13}-0.7\overline{68}}\,=\frac{\frac{68-6}{90}\div 6.2}{3\frac{211}{990}-\frac{768-7}{990}} \\ & =\frac{\frac{62}{90}\div 6.2}{3\frac{211}{990}-\frac{761}{990}}\,\,\,\,\,\,\,\,\,=\,\,\frac{\frac{62}{90}\div 6.2}{\frac{2420}{990}} \\ & \\ & =\frac{62}{90\times 6.2}\times \frac{990}{2420}\,\,\,\,=\frac{11}{242}=0.045 \\ \end{align}$
Question 4
The value of
$\sqrt{\frac{{{\left( 5.2 \right)}^{2}}+{{\left( 52.2 \right)}^{2}}+{{\left( 522.2 \right)}^{2}}}{{{\left( 0.52 \right)}^{2}}+{{\left( 5.22 \right)}^{2}}+{{\left( 52.22 \right)}^{2}}}}$is
A
0.1
B
1.1
C
10
D
100
Question 4 Explanation:
$\begin{align} & \sqrt{\frac{{{\left( 5.2 \right)}^{2}}+{{\left( 52.2 \right)}^{2}}+{{\left( 522.2 \right)}^{2}}}{{{\left( 0.52 \right)}^{2}}+{{\left( 5.22 \right)}^{2}}+{{\left( 52.22 \right)}^{2}}}} \\ & =\sqrt{\frac{{{\left( 10\times 0.52 \right)}^{2}}+{{\left( 10\times 5.22 \right)}^{2}}+{{\left( 10\times 52.22 \right)}^{2}}}{{{\left( 0.52 \right)}^{2}}+{{\left( 5.22 \right)}^{2}}+{{\left( 52.22 \right)}^{2}}}} \\ & =\sqrt{100}=10 \\ \end{align}$
Question 5
For what value of *, statement
$\left[ \frac{\left( * \right)}{29}\times \frac{\left( * \right)}{261} \right]=1$is correct?
A
261
B
87
C
63
D
21
Question 5 Explanation:
$\begin{align} & \left[ \frac{\left( H \right)}{29}\times \frac{\left( H \right)}{261} \right]=1 \\ & \Rightarrow {{\left( H \right)}^{2}}=29\times 261 \\ & \Rightarrow H=\sqrt{29\times 261}=87 \\ \end{align}$
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