A) 8p | B) pq |

C) pq+27 | D) -p |

Explanation:

A perfect cube will have prime factors that are in groups of 3; for example 125 has the prime factors 5 x 5 x 5 , and 64 x 125 will also be a cube because its factors will be 4 x 4 x 4 x 5 x 5 x 5

Consider the answer choices in turn.

8 is the cube of 2, and p is a cube, and so the product will also be a cube.

pq will also be a cube as shown above.

pq is a cube and so is 27, but their sum need not be a cube. Consider the case where p =1 and q = 8, the sum of pq and 27 will be 35 which has factors 5 x 7 and is not a cube.

-p will be a cube.

Since the difference between p and q is raised to the power of 6, this expression will be a cube (with cube root = difference squared).

A) 168 | B) 196 |

C) 222 | D) 256 |

A) 0.241 | B) 1.732 |

C) 4 | D) All of the above |

Explanation:

Any number which can be expressed as a fraction of two integers like P & Q as P/Q where Q is not equal to zero.

Every integer is a rational number since Q can be 1.

Hence, in the given options, 4 can be expressed as a simple fraction as 4/1. And all other options cannot be expressed as fractions.

Hence, 4 is a rational number in the given options.

A) i | B) 1 |

C) -i | D) -1 |

Explanation:

We know that,

${\mathrm{i}}^{2}=-1\phantom{\rule{0ex}{0ex}}{\mathrm{i}}^{3}={\mathrm{i}}^{2}\mathrm{x}\mathrm{i}=-1\mathrm{x}\mathrm{i}=-\mathrm{i}\phantom{\rule{0ex}{0ex}}{\mathrm{i}}^{4}={\mathrm{i}}^{2}\mathrm{x}{\mathrm{i}}^{2}=-1\mathrm{x}-1=1\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\mathrm{Hence},{\mathbf{i}}^{\mathbf{233}\mathbf{}}\mathbf{}\mathbf{=}{\mathbf{i}}^{\mathbf{4}\mathbf{}\mathbf{x}\mathbf{}\mathbf{58}\mathbf{}\mathbf{+}\mathbf{}\mathbf{1}}\mathbf{}\mathbf{=}\mathbf{}{\mathbf{i}}^{\mathbf{232}}\mathbf{}\mathbf{x}\mathbf{}{\mathbf{i}}^{\mathbf{1}}\mathbf{}\mathbf{=}\mathbf{}\mathbf{1}\mathbf{}\mathbf{x}\mathbf{}\mathbf{i}\mathbf{}\mathbf{=}\mathbf{}\mathbf{i}$

A) 1 | B) 0 |

C) -1 | D) Infinity |

Explanation:

The mutiplicative inverse of a number is nothing but a reciprocal of a number.

Now, the product of a number and its multiplicative inverse is always equal to **1**.

**For example :**

Let the number be 15

Multiplicative inverse of 15 = 1/15

The product of a number and its multiplicative inverse is = **15 x 1/15 = 1.**

A) 3.571 | B) 35.71 |

C) 0.351 | D) 0.0357 |

Explanation:

Here we have 25 divided by 7.

25 will not go directly in 7

Hence, we get the result in decimals.

**25/7 = 3.571.**

A) 5 | B) 11 |

C) 21 | D) 37 |

Explanation:

**A prime number** is a whole number greater than 1 whose only factors are 1 and itself.

Factors of **5** are **1, 5**

Factors of **11** are **1, 11**

Factors of **21** are **1, 3, 7, 21**

Factors of **37** are **1, 37.**

Hence, according to the definition of a prime number, **21 is not a prime number** as it has more than two factors.

The Multiples of 4 are **4, 8, 12, 16, 20, 24, 28, 32, 36, 40 **upto 40. Mutiples of 4 means which can be divided by 4 leaving remainder '0'.

**Common Multiples of 4 & 6** are **12, 24, 36, 48, 60 **upto 60.

A) 31 | B) 33 |

C) 29 | D) 27 |

Explanation:

Let the three consecutive odd numbers be **x, x+2, x+4**

**Then,**

**x + x + 2 + x + 4 = 93**

**=> **3x + 6 = 93

=> 3x = 87

=> x = 29 => **29, 31, 33 are three consecutive odd numbers.**

Therefore, the middle number is **31.**