A) 1009 | B) 991 |

C) 2000 | D) 1000 |

Explanation:

Given total number of rats = 999919

999919 = 1000000 - 81 = $\left({1000}^{2}-{9}^{2}\right)$ = (1000+9)(1000-9) = 1009 x 991.

Since there were more rats than there were cats, 991 cats killed 1009 rats each.

Let the required number be 'X'

**Condition 1** :

$\frac{X}{X}=2X\phantom{\rule{0ex}{0ex}}=2{X}^{2}=X\phantom{\rule{0ex}{0ex}}=\mathbf{}\mathit{X}\mathbf{}\mathbf{=}\mathbf{}\frac{\mathbf{1}}{\mathbf{2}}$

**Condition 2 :**

**$\mathrm{X}\times \mathrm{X}=\frac{\mathrm{X}}{2}\phantom{\rule{0ex}{0ex}}=\mathbf{X}\mathbf{}\mathbf{=}\mathbf{}\frac{\mathbf{1}}{\mathbf{2}}$**

**Therefore, **$\left(\frac{\mathbf{1}}{\mathbf{2}}\right)$ is the number which becomes double when it is divided by itself and becomes half when it is multiplied by itself.

A) 4! | B) 5! |

C) 6! | D) Can't be determined |

Explanation:

NUMERICAL has 9 positions in which 2, 4, 6, 8 are even positions.

And it contains 5 consonents i.e, N, M, R, C & L. Hence this cannot be done as 5 letters cannot be placed in 4 positions.

Therefore, Can't be determined.

A) 1/8 | B) 0 |

C) 1/4 | D) 1 |

Explanation:

Non collision can happen only when all the cars move in same direction.

That is either all cars moving in clockwise direction or all cars moving in anti clockwise direction.

P(all cars not colliding)

= P(all cars moving in the same direction)

= P(all cars moving in clockwise direction) + P(all cars moving in anticlockwise direction)

= [p(A --> C) x p(C --> B) x p(B --> A)] + [p(A --> B) x p(B --> C) x p(C --> A)]

= [1/2 x 1/2 x 1/2] + [1/2 x 1/2 x 1/2]

= 1/8 + 1/8

= 1/4

A) 42 | B) 142 |

C) 119 | D) 21 |

Let the two digits be x & y

From given data in the question, we get

x + y = 3(x - y)

x + y = 3x - 3y

=> 2x = 4y

x/y = 2/1

For having the 2:1, numbers satisfying are

12, 21, 24, 42, 36, 63, 48, 84

From this 36 is eleminated because it is a perfect square and no others are prime or perfect squares.

Hence, there exists 7 such 2-digit numbers => 12, 21, 24, 42, 63, 48, 84.

A) A to G : 2, 3, 7, 6, 5, 1, 4 | B) A to G : 1, 4, 5, 6, 2, 7, 3 |

C) A to G : 5, 4, 7, 6, 3, 1, 2 | D) None |

Explanation:

This can be done in two ways

1. A to G : 2, 3, 7, 6, 5, 1, 4

2. A to G : 4, 3, 7, 5, 6, 2, 1

A) 93714 | B) 42789 |

C) 36671 | D) 93747 |

Explanation:

From the hints,

Only two options for 1st and 2nd digits =>** 4,2 and 9,3** since it is single digits

Sum of second and third digit = 10

If 2 => 3rd digit = 8

If 3 => 3rd digit = 7

4th digit = 2nd digit + 1 => (2+1 =3) or (3+1 = 4)

Sum of all digits = 30

Let 5th digit be x

=> 1st possibility = 4 + 2 + 8 + 3 + x = 30 => x = 13 (Not possible since it has 2 digits)

=> 2nd possibility = 9 + 3 + 7 + 4 + x = 30 => x = 7

Therefore, the PIN is **93747 **