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## Number System: Level 1 Test -10

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Question 1 |

Which is the largest?

10 ^{10} | |

(2 ^{10})^{5} | |

(5 ^{10})^{2} | |

(4 ^{5})^{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 (2

^{36}–1) = 68 a 19476735, where a is any digit, then the value of a is1 | |

3 | |

5 | |

7 |

Question 2 Explanation:

There is a property in number system that

if n= even number, then (2

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

if n= even number, then (2

^{n}n–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 y

^{2}is divided by 4, it leaves remainder of z. Which of the following must be true for z?z = 3 | |

z = 5 | |

z = 1 | |

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,

y

Therefore z= 1

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,

y

^{2}will be exactly divisible by 4.Therefore z= 1

Question 4 |

A boy was asked to write 2

^{5}× 9^{2}but he wrote 2592. The numerical difference between the two is 0 | |

1 | |

2 | |

3 |

Question 4 Explanation:

2

^{5}× 9^{2}= 81 × 32 = 2592.Question 5 |

Which of the following is not the reciprocal of

$ \displaystyle {{\left( \frac{2}{3} \right)}^{4}}$

$ \displaystyle {{\left( \frac{2}{3} \right)}^{4}}$

$ \displaystyle {{\left( \frac{3}{2} \right)}^{4}}$ | |

$ \displaystyle {{\left( \frac{2}{3} \right)}^{-4}}$ | |

$ \displaystyle {{\left( \frac{3}{2} \right)}^{-4}}$ | |

$ \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}}}$

$ \displaystyle {{\left( \frac{2}{3} \right)}^{4}}={{\left( \frac{3}{2} \right)}^{4}}=\frac{{{3}^{4}}}{{{2}^{4}}}$

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