- This is an assessment test.
- To draw maximum benefit, study the concepts for the topic concerned.
- Kindly take the tests in this series with a pre-defined schedule.

## Algebra Level 3 Test 3

Start

Congratulations - you have completed

*Algebra Level 3 Test 3*.You scored %%SCORE%% out of %%TOTAL%%.You correct answer percentage: %%PERCENTAGE%% .Your performance has been rated as %%RATING%% Your answers are highlighted below.

Question 1 |

There are 8436 steel balls, each with a radius of 1 centimeter, stacked in a pile, with 1 ball on top, 3 balls in the second layer, 6 in the third layer, 10 in the fourth, and so on. The number of horizontal layers in the pile is

A | 34 |

B | 38 |

C | 36 |

D | 32 |

Question 1 Explanation:

Let the number of horizontal layers in the pile is = p

The number of balls in each layer in such a way that the

first layer contains one ball, second layer contains 3 balls,

whereas the third ball contains 6 balls and so on âˆ‘p(p+1)/2 = 8436

Â½ [âˆ‘p

= {p(p+1)(2p+1)}/12 + { p(p+1)}/4 = 8436

= n(n+1)[(2p+1)/12] = 8436

= {p(p+1)(2p+1)}/6 = 8436

On solving this we have p = 36

The number of balls in each layer in such a way that the

first layer contains one ball, second layer contains 3 balls,

whereas the third ball contains 6 balls and so on âˆ‘p(p+1)/2 = 8436

Â½ [âˆ‘p

^{2}+ âˆ‘p] = 8436= {p(p+1)(2p+1)}/12 + { p(p+1)}/4 = 8436

= n(n+1)[(2p+1)/12] = 8436

= {p(p+1)(2p+1)}/6 = 8436

On solving this we have p = 36

Question 2 |

If the product of n positive real numbers is unity, then their sum is necessarily

A | a multiple of n |

B | equal to (n +1/n) |

C | never less than n |

D | a positive integer |

Question 2 Explanation:

The best way to do this is to take some value and verify.

Letâ€™s take the values as 2,1/2,1 so the sum of three of them is 7/2 with n = 3

Letâ€™s take the values as 2,1/2,1/4 so the sum of three of them is 11/4 with n = 3

Therefore options Â A, B and C are ruled out and the correct answer for the question is option number (d)

Letâ€™s take the values as 2,1/2,1 so the sum of three of them is 7/2 with n = 3

Letâ€™s take the values as 2,1/2,1/4 so the sum of three of them is 11/4 with n = 3

Therefore options Â A, B and C are ruled out and the correct answer for the question is option number (d)

Question 3 |

Given that âˆ’1â‰¤ v â‰¤ 1, âˆ’2 â‰¤ u â‰¤ âˆ’0.5 and âˆ’2 â‰¤ z â‰¤ âˆ’0.5 and w = vz /u, then which of the following is necessarily true?

A | âˆ’0.5 â‰¤ w â‰¤1 |

B | âˆ’4 â‰¤ w â‰¤ 4 |

C | âˆ’4 â‰¤ w â‰¤ 2 |

D | âˆ’2 â‰¤ w â‰¤ âˆ’0.5 |

Question 3 Explanation:

U is always negative. Hence, for us to have a minimum value of vz/u, vz should be positive.

Also for the least value, the numerator has to be the maximum positive value and the denominator has to be the smallest negative value.

In other words, vz has to be 2 and u has to be â€“0.5.

Hence the minimum value of vz/u = -2/0.5 = -4.

For us to get the maximum value, vz has to be the smallest negative value and u has to be the highest negative value.

Thus, vz has to be â€“2 and u has to be â€“0.5. Hence the maximum value of vz/u = -2/-0.5 = 4

Also for the least value, the numerator has to be the maximum positive value and the denominator has to be the smallest negative value.

In other words, vz has to be 2 and u has to be â€“0.5.

Hence the minimum value of vz/u = -2/0.5 = -4.

For us to get the maximum value, vz has to be the smallest negative value and u has to be the highest negative value.

Thus, vz has to be â€“2 and u has to be â€“0.5. Hence the maximum value of vz/u = -2/-0.5 = 4

Question 4 |

If x, y, z are distinct positive real numbers the [{x

^{2}(y + z) y^{2}(x+ z) z^{2}(x+ y)}/xyz] would beA | greater than 4. |

B | greater than 5. |

C | greater than 6 |

D | None of the above. |

Question 4 Explanation:

Here x, y, z are distinct positive real number

[{x

z

This can be written as x/y + x/z + y/x + y/z + z/x + z/y

This is also can be modified as

(x/y + y/z) + (y/z + z/y ) + (z/x + x/z)

Here in school days we did a property of numbers that if the numbers are distinct then the value of a/b + b/a > 2

So here the value of the expression must be greater than 2+2+2 > 6

[{x

^{2}(y + z) y^{2}(x+ z)z

^{2}(x+ y)}/xyz]This can be written as x/y + x/z + y/x + y/z + z/x + z/y

This is also can be modified as

(x/y + y/z) + (y/z + z/y ) + (z/x + x/z)

Here in school days we did a property of numbers that if the numbers are distinct then the value of a/b + b/a > 2

So here the value of the expression must be greater than 2+2+2 > 6

Question 5 |

In a certain examination paper, there are n questions. For j = 1,2 â€¦n, there are 2

^{nâ€“j}students who answered j or more questions wrongly. If the total number of wrong answers is 4095, then the value of n isA | 12 |

B | 11 |

C | 10 |

D | 9 |

Question 5 Explanation:

Let us consider that there are only 3 questions.

Therefore the number of students that have done 1 or more questions wrongly are 2

The number of students that have done 2 or more

questions wrongly are 2

Thus total number of answers which are wrong = 4 + 2 + 1 = 7 = 2

But we are given by the wrong answers in our question,

the total number of wrong answers = 4095 = 2

Therefore the number of students that have done 1 or more questions wrongly are 2

^{3â€“1}= 4,The number of students that have done 2 or more

questions wrongly are 2

^{3â€“2}= 2 And the number of students that must have done all 3 wrongly areÂ 2^{3â€“3}= 1.Thus total number of answers which are wrong = 4 + 2 + 1 = 7 = 2

^{3 }â€“ 1 = 2^{n}â€“ 1.But we are given by the wrong answers in our question,

the total number of wrong answers = 4095 = 2

^{12 }â€“ 1. Thus n = 12. Once you are finished, click the button below. Any items you have not completed will be marked incorrect. Get Results

There are 5 questions to complete.

← | List | → |

Return

Shaded items are complete.

1 | 2 | 3 | 4 | 5 |

End |

Return

You have completed

questions

question

Your score is

Correct

Wrong

Partial-Credit

You have not finished your quiz. If you leave this page, your progress will be lost.

Correct Answer

You Selected

Not Attempted

Final Score on Quiz

Attempted Questions Correct

Attempted Questions Wrong

Questions Not Attempted

Total Questions on Quiz

Question Details

Results

Date

Score

Hint

Time allowed

minutes

seconds

Time used

Answer Choice(s) Selected

Question Text

All done

Need more practice!

Keep trying!

Not bad!

Good work!

Perfect!