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## Algebra Level 2 Test 5

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*Algebra Level 2 Test 5*.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 |

If log

_{7}log_{5}{√(x + 5) + √x}= 0A | 1 |

B | 0 |

C | 2 |

D | None of these |

Question 1 Explanation:

log

log √(x + 5)+ √x = 7

or

√(x + 5)+ √x = 5

Therefore 2√x = 0. or

x = 0.

_{7}log_{5}{ √(x + 5) + √x }= 0log √(x + 5)+ √x = 7

^{0}= 1,or

√(x + 5)+ √x = 5

^{1 }= 5.Therefore 2√x = 0. or

x = 0.

Question 2 |

If a + b + c = 0, where a ≠ b ≠ c, then {a

^{2}/(2a^{2}+bc)} + {b^{2}/(2b^{2}+ac)} +{c^{2}/(2c^{2}+ab)}A | zero |

B | 1 |

C | –1 |

D | abc |

Question 2 Explanation:

To solve this question, we use the help of values.

Assuming such values of a, b & c so that a + b + c =0

and a ≠ b ≠ c, we can find the value of the expression.

Assuming a = 1, b = -1 and c = 0 , we find that {a

½ + ½ + 0 = 1.

Hence we get our answer without any sweat.

Assuming such values of a, b & c so that a + b + c =0

and a ≠ b ≠ c, we can find the value of the expression.

Assuming a = 1, b = -1 and c = 0 , we find that {a

^{2}/(2a^{2}+bc)} + {b^{2}/(2b^{2}+ac)} +{c^{2}/(2c^{2}+ab)}½ + ½ + 0 = 1.

Hence we get our answer without any sweat.

Question 3 |

If the harmonic mean between two positive numbers is to their geometric mean as 12 : 13; then the numbers could be in the ratio

A | 12 : 13 |

B | 1/12 : 1/13 |

C | 4 : 9 |

D | 2 : 3 |

Question 3 Explanation:

Since the harmonic mean of two numbers x and y is 2xy/(x + y) and the geometric mean is √xy.

So from the question 2xy /(x + y)/√xy = 12/13,

Further solving this

Squaring both sides we get

Therefore 2 x

But we want the values for x : y so

= 2xy/(x + y) = 12/13

= 2√xy / (x + y ) √xy

= {(x + y ) √xy / 2√xy}

For further solving we have to get back in our school days for the concept of component and dividendo

2√x/2√y = 6/4 = x / y = 9/4 or y / x = 4/9

So from the question 2xy /(x + y)/√xy = 12/13,

Further solving this

Squaring both sides we get

Therefore 2 x

^{3}y^{3}/ (x + y)^{2}= 144/169.But we want the values for x : y so

= 2xy/(x + y) = 12/13

= 2√xy / (x + y ) √xy

= {(x + y ) √xy / 2√xy}

For further solving we have to get back in our school days for the concept of component and dividendo

2√x/2√y = 6/4 = x / y = 9/4 or y / x = 4/9

Question 4 |

If one root of x

^{2}+ px + 12= is 4, while the equation x^{2}+ px + q = has equal roots, then the value of q isA | 49/4 |

B | 4/49 |

C | 4 |

D | ¼ |

Question 4 Explanation:

If one root of x

4

From thr above expression

p = -7.

Put this value in the second given equation x

Therefore If the roots are equal then b

4q = 49

q = 49/4

^{2}+ px + 12 = 0 is 4, then4

^{2 }+ 4p + 12 = 0,From thr above expression

p = -7.

Put this value in the second given equation x

^{2}– 7x + q = 0 has equal roots.Therefore If the roots are equal then b

^{2 }= 4ac4q = 49

q = 49/4

Question 5 |

Fourth term of an arithmetic progression is 8. What is the sum of the first 7 terms of the arithmetic progression?

A | 7 |

B | 64 |

C | 56 |

D | Cannot be determined |

Question 5 Explanation:

Let’s take an example to understand a trick that can help us solve this question.

Let us suppose we have a series 1 + 2 + 3 and the middle term is 2.

We are asked for the sum of this series so a simple shortcut that can be used is that:

sum of series = middle term x number of terms

i.e the sum of the supposed series is 2 x 3 = 6

Using the above, we are given by the fourth term which is the middle term of the series. So the sum of the series is 8 x 7 = 56.

Let us suppose we have a series 1 + 2 + 3 and the middle term is 2.

We are asked for the sum of this series so a simple shortcut that can be used is that:

sum of series = middle term x number of terms

i.e the sum of the supposed series is 2 x 3 = 6

Using the above, we are given by the fourth term which is the middle term of the series. So the sum of the series is 8 x 7 = 56.

Question 6 |

What is the value of m which satisfies 3m

^{2}– 21m + 30 < 0?A | m < 2 or m > 5 |

B | m > 2 c. |

C | 2 < m < 5; |

D | Both a and c |

Question 6 Explanation:

3m

Dividing by 3

Therefore m

Now on solving this quadratic equation

(m – 5)(m – 2) < 0.

Now there are two cases

Either (m – 5) < 0 and (m – 2) > 0 or (m – 2) < 0 and (m –5) > 0.

Hence, either m < 5 and m > 2, i.e. 2< m < 5 or m < 2 and m > 5.

^{2}– 21m + 30 < 0Dividing by 3

Therefore m

^{2}– 7m + 10 < 0,Now on solving this quadratic equation

(m – 5)(m – 2) < 0.

Now there are two cases

Either (m – 5) < 0 and (m – 2) > 0 or (m – 2) < 0 and (m –5) > 0.

Hence, either m < 5 and m > 2, i.e. 2< m < 5 or m < 2 and m > 5.

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