# Problems on Numbers for Exam

Q11. 86 X 86 - 14 X 14

(a) 7020

(b) 7156

(c) 7200

(d) 7400

**Answer:** (c) 7200

=86 X 86 - 14 X 14

=(86)^{2} - (14)^{2}

=(86+14) - (86-14) Formula [a^{2} - b^{2}=(a+b)(a-b)]

=100 X 72

=7200

(251 X 251 X 251) - (69 X 69 X 69) |

(251 X 251 + 251 X 69 + 69 X 69) |

(a) 180

(b) 182

(c) 285

(d) 1081

**Answer:** (b) 182

(251 X 251 X 251) - (69 X 69 X 69) |

(251 X 251 + 251 X 69 + 69 X 69) |

(251)^{3} - (69)^{3} |

(251)^{2} + (251 X 69) + (69)^{2} |

(251 - 69) (251^{2} + 251 X 69 + 69^{2}) |

(251^{2} + 251 X 69 + 69^{2}) |

=251 - 69

=182

Formula a^{3} - b^{3}=(a-b)(a^{2}+ab+b^{2})

(407 + 315)^{2} - (407 - 315)^{2} |

407 X 315 |

(a) 41

(b) 401

(c) 4

(d) 104

**Answer:** (c) 4

(407 + 315)^{2} - (407 - 315)^{2} |

407 X 315 |

4 X 407 X 315 |

407 X 315 |

=4

Formula (a+b)^{2} - (a-b)^{2} = 4ab

(215 + 102)^{2} + (215 - 102)^{2} |

215 X 215 + 102 X 102 |

(a) 2

(b) 12

(c) 102

(d) 215

**Answer:** (a) 2

(215 + 102)^{2} + (215 - 102)^{2} |

215 X 215 + 102 X 102 |

2(215^{2} + 102^{2}) |

215^{2} + 102^{2} |

=2

Q15. 1 + 2 + 3 + ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ + 20 = ?

(a) 20

(b) 120

(c) 201

(d) 210

**Answer:** (d) 210

=1 + 2 + 3 + ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ + 20

=20 (20 + 1) ÷ 2

=10 X 21

=210

Formula 1 + 2 + 3 + ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ + n = n(n + 1) ÷ 2

Q16. 3 + 4 + 5 + ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ + 20 = ?

(a) 205

(b) 206

(c) 207

(d) 208

**Answer:** (c) 207

=3 + 4 + 5 + ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ + 20

=1 + 2 + 3 + ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ + 20 - (1 + 2)

=[20 (20 + 1) ÷ 2] - 3

=10 X 21 - 3

=210 -3

=207

Formula 1 + 2 + 3 + ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ + n = n(n + 1) ÷ 2

Q17. 1^{2} + 2^{2} + ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ + 10^{2} = ?

(a) 285

(b) 385

(c) 485

(d) 585

**Answer:** (b) 385

=1^{2} + 2^{2} + ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ + 10^{2}

=[10 (10 + 1) (2 X 10 + 1)] ÷ 6

=(10 X 11 X 21) ÷ 6

=385

Formula 1^{2} + 2^{2} + ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ + n^{2} = [n(n+1)(2n+1) ÷ 6

Q18. 3^{2} + 4^{2} + 5^{2} + ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ + 10^{2} = ?

(a) 380

(b) 382

(c) 384

(d) 386

**Answer:** (a) 380

=3^{2} + 4^{2} + 5^{2} + ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ + 10^{2}

=1^{2} + 2^{2} + 3^{2} + ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ + 10^{2} - (1^{2} + 2^{2})

=[10 (10 + 1) (2 X 10 + 1) ÷ 6] - (1+4)

=[10 X 11 X 21 ÷ 6] - 5

=385 - 5

=380

Q19. 1^{3} + 2^{3} + 3^{3} + ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ + 10^{3} = ?

(a) 2025

(b) 3025

(c) 3052

(d) 3205

**Answer:** (b) 3025

=1^{3} + 2^{3} + 3^{3} + ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ + 10^{3}

10^{2}(10 + 1)^{2} |

4 |

100 X 121 |

4 |

=3025

Formula [1^{3} + 2^{3} + ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ + n^{3}] =[n^{2}(n + 1)^{2}] ÷ 4

Q20. The difference between the place value and face value of 5 in the number 835422 is

(a) 4995

(b) 5000

(c) 5005

(d) 5995

**Answer:** (a) 4995

Place value of 5 = 5000

Face value of 5 = 5

Diference = 4995