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Number System: Level 1 Test -10

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Question 1
Which is the largest?
A
1010
B
(210)5
C
(510)2
D
(45)4
Question 1 Explanation: 
$ \displaystyle \begin{array}{l}{{\left( {{2}^{10}} \right)}^{5}}={{\left( {{2}^{5}} \right)}^{10}}={{\left( 32 \right)}^{10}}\\{{\left( {{5}^{10}} \right)}^{2}}={{\left( {{5}^{2}} \right)}^{10}}={{\left( 25 \right)}^{10}}\\{{\left( {{4}^{5}} \right)}^{4}}={{\left( {{4}^{2}} \right)}^{10}}={{\left( 16 \right)}^{10}}\\and\,\,\,{{10}^{10}}\end{array}$
Question 2
If (236–1) = 68 a 19476735, where a is any digit, then the value of a is
A
1
B
3
C
5
D
7
Question 2 Explanation: 
There is a property in number system that
if n= even number, then (2nn–1), is divisible by 3.
and A number is divisible by 3, if the sum of its digits is divisible by 3
so$ \displaystyle \begin{array}{l}{{2}^{2}}-1=4-1=3\\{{2}^{4}}-1=16-1=15\\{{2}^{6}}-1=64-1=63\\{{2}^{8}}-1=256-1=255\end{array}$
Therefore a= 1
Question 3
Let x be an odd natural number. If x is divided by 6, it leaves a remainder y. if y2 is divided by 4, it leaves remainder of z. Which of the following must be true for z?
A
z = 3
B
z = 5
C
z = 1
D
z is even
Question 3 Explanation: 
$latex \displaystyle \begin{array}{l}x=6,\,\,Q+y\\{{y}^{2}}=4{{Q}_{1}}+z\end{array}$
The value of z may be 1, 2 or 3.
the value of y may be 1, 3, or 5 as if 2 or 4 be the value,
y2 will be exactly divisible by 4.
Therefore z= 1
Question 4
A boy was asked to write 25 × 92 but he wrote 2592. The numerical difference between the two is
A
0
B
1
C
2
D
3
Question 4 Explanation: 
25 × 92 = 81 × 32 = 2592.
Question 5
Which of the following is not the reciprocal of
$ \displaystyle {{\left( \frac{2}{3} \right)}^{4}}$
A
$ \displaystyle {{\left( \frac{3}{2} \right)}^{4}}$
B
$ \displaystyle {{\left( \frac{2}{3} \right)}^{-4}}$
C
$ \displaystyle {{\left( \frac{3}{2} \right)}^{-4}}$
D
$ \displaystyle \frac{{{3}^{4}}}{{{2}^{4}}}$
Question 5 Explanation: 
Reciprocal of
$ \displaystyle {{\left( \frac{2}{3} \right)}^{4}}={{\left( \frac{3}{2} \right)}^{4}}=\frac{{{3}^{4}}}{{{2}^{4}}}$
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