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Basic Maths: Test 52

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Question 1
The simplified value of $\left( 1-\frac{1}{5} \right)\,\left( 1-\frac{1}{6} \right)\,\left( 1-\frac{1}{7} \right)\,......\left( 1-\frac{1}{49} \right)\,\left( 1-\frac{1}{50} \right)$<br> is
A
$\frac{2}{25}$
B
$\frac{1}{25}$
C
$\frac{1}{50}$
D
$\frac{1}{100}$
Question 1 Explanation:
$\begin{align} & =\left( 1-\frac{1}{5} \right)\,\left( 1-\frac{1}{6} \right)\,\left( 1-\frac{1}{7} \right)\,......\left( 1-\frac{1}{49} \right)\,\left( 1-\frac{1}{50} \right) \\ & =\left( \frac{5-1}{5} \right)\,\left( \frac{6-1}{6} \right)\,\left( \frac{7-1}{7} \right)\,......\left( \frac{49-1}{49} \right)\,\left( \frac{50-1}{50} \right) \\ & =\frac{4}{5}\times \frac{5}{6}\times \frac{6}{7}\times ......\times \frac{48}{49}\times \frac{49}{50} \\ & =\frac{4}{50} \\ & =\frac{2}{25} \\ \end{align}$
Question 2
$\frac{125{{\left( 1.3 \right)}^{3}}+1}{{{\left( 6.5 \right)}^{2}}-5.5}$ is equal to
A
2.75
B
9/5
C
4.75
D
7.5
Question 2 Explanation:
$\begin{align} & =\frac{125{{\left( 1.3 \right)}^{3}}+1}{{{\left( 6.5 \right)}^{2}}-5.5} \\ & =\frac{{{\left( 5\times 1.3 \right)}^{3}}+1}{{{\left( 6.5 \right)}^{2}}-6.5\times 1+{{1}^{2}}} \\ & =\frac{{{\left( 6.5 \right)}^{3}}+1}{{{\left( 6.5 \right)}^{2}}-6.5\times 1+{{1}^{2}}} \\ & \left[ {{a}^{3}}+{{b}^{3}}=\left( a+b \right)\left( {{a}^{2}}-ab+{{b}^{2}} \right) \right] \\ & =6.5+1 \\ & =7.5 \\ \end{align}$
Question 3
The value of $\frac{{{\left( 1.987-4.523 \right)}^{2}}+{{\left( 1.987+4.523 \right)}^{2}}}{1.987\times 1.987+4.523\times 4.523}$ is
A
4
B
2
C
2.199
D
3.195
Question 3 Explanation:
Let 1.987=a and 4.523=b
Therefore expression
$\begin{align} & =\frac{{{\left( a-b \right)}^{2}}+{{\left( a+b \right)}^{2}}}{{{a}^{2}}+{{b}^{2}}} \\ & =\frac{2\left( {{a}^{2}}+{{b}^{2}} \right)}{{{a}^{2}}+{{b}^{2}}}=2 \\ \end{align}$
Question 4
$\left( 5555\div 110 \right)+\left( 925\div 75 \right)+\left( 5168\div 51 \right)=?$
A
164.16
B
174.26
C
184.16
D
194.26
Question 4 Explanation:
$\begin{align} & ?=\frac{5555}{110}+\frac{925}{75}+\frac{5168}{51} \\ & =50.5+12.33+101.33 \\ & =164.16 \\ \end{align}$
Question 5
$\sqrt{32041}\times \sqrt{1681}-{{\left( 71 \right)}^{2}}={{\left( ? \right)}^{2}}+{{\left( 23 \right)}^{2}}$(? Is approximately equal to)
A
47
B
42
C
44
D
48
Question 5 Explanation:
$\begin{align} & \sqrt{32041}\times \sqrt{1681}-{{\left( 71 \right)}^{2}}={{\left( ? \right)}^{2}}+{{\left( 23 \right)}^{2}} \\ & \Rightarrow 179\times 41-5041={{?}^{2}}+529 \\ & \Rightarrow 2298={{?}^{2}}+529 \\ & \Rightarrow {{?}^{2}}=2298-529=1769 \\ & \Rightarrow ?=\sqrt{1769}=42.06 \\ \end{align}$
Approximately equals to 42
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