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## Algebra Level 2 Test 5

Congratulations - you have completed Algebra Level 2 Test 5.You scored %%SCORE%% out of %%TOTAL%%.You correct answer percentage: %%PERCENTAGE%% .Your performance has been rated as %%RATING%%
 Question 1
If log7log5Â {âˆš(x + 5) + âˆšx}= 0
 A 1 B 0 C 2 D None of these
Question 1 Explanation:
log7 log5 { âˆš(x + 5) + âˆšxÂ }= 0
log âˆš(x + 5)+ âˆšxÂ = 70 = 1,
or
âˆš(x + 5)+ âˆšxÂ = 51 = 5.
Therefore 2âˆšxÂ = 0. or
x = 0.
 Question 2
If a + b + c = 0, where a â‰  b â‰  c, then {a2/(2a2 +bc)} + {b2/(2b2 +ac)} +{c2/(2c2 +ab)}
 A zero B 1 C â€“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 â‰  b â‰  c, we can find the value of the expression.
Assuming a = 1, b = -1 and c = 0 , we find that Â {a2/(2a2 +bc)} + {b2/(2b2 +ac)} +{c2/(2c2 +ab)}
Â½ + Â½ + 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 âˆšxy.
So from the question 2xy /(x + y)/âˆšxy = 12/13,
Further solving this
Squaring both sides we get
Therefore 2 x3 y3 / (x + y) 2 = 144/169.
But we want the values for x : y so
= 2xy/(x + y) = 12/13
= 2âˆšxy / (x + y ) âˆšxy
= {(x + y ) âˆšxy /Â  2âˆšxy}
For further solving we have to get back in our school days for the concept of component and dividendo
2âˆšx/2âˆšy = 6/4 = x / y = 9/4 or y / x = 4/9
 Question 4
If one root of x2 + px + 12= is Â 4, while the equation x2 + px + q = has equal roots, then the value of q is
 A 49/4 B 4/49 C 4 D Â¼
Question 4 Explanation:
If one root of x2 + px + 12 = 0 is 4, then
42 + 4p + 12 = 0,
From thr above expression
p = -7.
Put this value in the second given equation x2 â€“ 7x + q = 0 has equal roots.
Therefore If the roots are equal then b2 Â = 4ac
4q = 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:
Letâ€™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.
 Question 6
What is the value of m which satisfies 3m2 â€“ 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:
3m2 â€“ 21m + 30 < 0
Dividing by 3
Therefore m2 â€“ 7m + 10 < 0,
NowÂ  on solving this quadratic equation
(m â€“ 5)(m â€“ 2) < 0.
Now there are two cases
Either (m â€“ 5) < 0 and (m â€“ 2) > 0 or (m â€“ 2) < 0 and (m â€“5) > 0.
Hence, either m < 5 and m > 2, i.e. 2< m < 5 or m < 2 and m > 5.
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