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New Algebra Set!!!

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New Algebra Set!!! [#permalink] New post 18 Mar 2013, 06:56
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The next set of medium/hard PS algebra questions. I'll post OA's with detailed explanations after some discussion. Please, post your solutions along with the answers.

1. If x=\sqrt[4]{x^3+6x^2}, then the sum of all possible solutions for x is:

A. -2
B. 0
C. 1
D. 3
E. 5

Solution: new-algebra-set-149349-60.html#p1200948


2. The equation x^2 + ax - b = 0 has equal roots, and one of the roots of the equation x^2 + ax + 15 = 0 is 3. What is the value of b?

A. -64
B. -16
C. -15
D. -1/16
E. -1/64

Solution: new-algebra-set-149349-60.html#p1200950


3. If a and b are positive numbers, such that a^2 + b^2 = m and a^2 - b^2 = n, then ab in terms of m and n equals to:

A. \frac{\sqrt{m-n}}{2}
B. \frac{\sqrt{mn}}{2}
C. \frac{\sqrt{m^2-n^2}}{2}
D. \frac{\sqrt{n^2-m^2}}{2}
E. \frac{\sqrt{m^2+n^2}}{2}

Solution: new-algebra-set-149349-60.html#p1200956


4. What is the maximum value of -3x^2 + 12x -2y^2 - 12y - 39 ?

A. -39
B. -9
C. 0
D. 9
E. 39

Solution: new-algebra-set-149349-60.html#p1200962


5. If x^2 + 2x -15 = -m, where x is an integer from -10 and 10, inclusive, what is the probability that m is greater than zero?

A. 2/7
B. 1/3
C. 7/20
D. 2/5
E. 3/7

Solution: new-algebra-set-149349-60.html#p1200970


6. If mn does not equal to zero, and m^2n^2 + mn = 12, then m could be:

I. -4/n
II. 2/n
III. 3/n

A. I only
B. II only
C. III only
D. I and II only
E. I and III only

Solution: new-algebra-set-149349-60.html#p1200973


7. If x^4 = 29x^2 - 100, then which of the following is NOT a product of three possible values of x?

I. -50
II. 25
III. 50

A. I only
B. II only
C. III only
D. I and II only
E. I and III only

Solution: new-algebra-set-149349-60.html#p1200975


8. If m is a negative integer and m^3 + 380 = 381m , then what is the value of m?

A. -21
B. -20
C. -19
D. -1
E. None of the above

Solution: new-algebra-set-149349-60.html#p1200980


9. If x=(\sqrt{5}-\sqrt{7})^2, then the best approximation of x is:

A. 0
B. 1
C. 2
D. 3
E. 4

Solution: new-algebra-set-149349-60.html#p1200982


10. If f(x) = 2x - 1 and g(x) = x^2, then what is the product of all values of n for which f(n^2)=g(n+12) ?

A. -145
B. -24
C. 24
D. 145
E. None of the above

Solution: new-algebra-set-149349-80.html#p1200987


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Re: New Algebra Set!!! [#permalink] New post 26 Mar 2013, 04:03
1. If x=\sqrt[4]{x^3+6x^2}, then the sum of all possible solutions for x is:

A. -2
B. 0
C. 1
D. 3
E. 5


Kudos points for each correct solution!!![/quote]

X^4 = X^3 + 6x^2.Solving this we get teh values of x as 0, 3 , -2.
But X cannot be negative as X is the positive fourth root of an expression(\sqrt{Expression}>= 0)
So X = 0 and 3
Sum of the values = 0+3 = 3
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Re: New Algebra Set!!! [#permalink] New post 26 Mar 2013, 04:12
2. The equation x^2 + ax - b = 0 has equal roots, and one of the roots of the equation x^2 + ax + 15 = 0 is 3. What is the value of b?

A. -64
B. -16
C. -15
D. -1/16
E. -1/64
Kudos points for each correct solution!!![/quote]

Since 3 is the root of the equation x^2 + ax + 15 = 0, therefore 3 must satisfy the equation x^2 + ax + 15 = 0 .
On putting x=3 in the equation x^2 + ax + 15 = 0 , we get a= -8
On putting the value of a in the original equation x^2 + ax - b = 0, we get x^2 - 8x - b = 0. As per question x^2 - 8x - b = 0 has equal roots so Discriminant(D) = b^2 -4ac = 0 = (-8)^2 - 4.1.(-b)= 0. From this we get the value of b as -16
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Re: New Algebra Set!!! [#permalink] New post 26 Mar 2013, 04:17
3. If a and b are positive numbers, such that a^2 + b^2 = m and a^2 - b^2 = n, then ab in terms of m and n equals to:

A. \frac{\sqrt{m-n}}{2}
B. \frac{\sqrt{mn}}{2}
C. \frac{\sqrt{m^2-n^2}}{2}
D. \frac{\sqrt{n^2-m^2}}{2}
E. \frac{\sqrt{m^2+n^2}}{2}
Kudos points for each correct solution!!![/quote]

Adding the given two equations we get 2a^2 = (m+n)/2
Subtracting the given two equations we get 2b^2 = (m-n)/2

Multiplying above two equations we get 4a^2b^2 = (m^2 - n^2) => ab = \frac{\sqrt{m^2-n^2}}{2}
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Re: New Algebra Set!!! [#permalink] New post 26 Mar 2013, 04:29
4. What is the maximum value of -3x^2 + 12x -2y^2 - 12y - 39 ?

A. -39
B. -9
C. 0
D. 9
E. 39
Kudos points for each correct solution!!![/quote]

Rearranging the given equation we get
-3(x^2 - 4x + 4) -2(y^2 + 6y +9) - 9 = -3(x -2)^2 - 2(y +3)^2 -9
So, we need to maximize the value of -3(x -2)^2 - 2(y +3)^2 -9

Since, the maximum value of -3(x -2)^2 and - 2(y +3)^2 is zero when x= 2 and y = -3, then the maximum value of the whole expression is 0 + 0 -9 = -9
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Re: New Algebra Set!!! [#permalink] New post 26 Mar 2013, 04:39
5. If x^2 + 2x -15 = -m, where x is an integer from -10 and 10, inclusive, what is the probability that m is greater than zero?

A. 2/7
B. 1/3
C. 7/20
D. 2/5
E. 3/7
Kudos points for each correct solution!!![/quote]


On Re-arranging the given equation we get m = - x^2 - 2x +15

For m to be positive - x^2 - 2x +15 should be greater then 0. Solving we get (x + 5)(x - 3)<0. Since x is an integer we get the values of x as -4 , -3, -2, -1, 0, 1, 2 (total 7 values)

Given that X is an integer from -10 and 10, inclusive (21 values)

So, the probability is 7/21=1/3.

Answer: B.
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Re: New Algebra Set!!! [#permalink] New post 26 Mar 2013, 04:44
6. If mn does not equal to zero, and m^2n^2 + mn = 12, then m could be:

I. -4/n
II. 2/n
III. 3/n

A. I only
B. II only
C. III only
D. I and II only
E. I and III only
Kudos points for each correct solution!!![/quote]

Rearranging we get m^2n^2 + mn - 12 = 0 => (mn + 4)(mn - 3)= 0 =>mn = -4 or mn = 3 => m= -4/n or 3/n
E. I and III only
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Re: New Algebra Set!!! [#permalink] New post 26 Mar 2013, 04:55
7. If x^4 = 29x^2 - 100, then which of the following is NOT a product of three possible values of x?

I. -50
II. 25
III. 50

A. I only
B. II only
C. III only
D. I and II only
E. I and III only
Kudos points for each correct solution!!![/quote]

Rearranging the equation we get x^4 - 29x^2 + 100 = 0 => (x^2 - 25)(x^ - 4) =0 => x= +5, -5, +2, -2
From these values of x we can get -50 (-5 * +5 * +2), +50(-5 * +5 * -2) but we can't get 25

B. II only
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Re: New Algebra Set!!! [#permalink] New post 26 Mar 2013, 05:17
8. If m is a negative integer and m^3 + 380 = 381m , then what is the value of m?

A. -21
B. -20
C. -19
D. -1
E. None of the above

on re-arranging we get m^3 -381m = -380 => m(m^2 - 381)= 380
On checking the options and hit and trial, we get the values of m to be 1 and -20. But since m is negative so m = -20

B. -20
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Re: New Algebra Set!!! [#permalink] New post 26 Mar 2013, 05:29
9. If x=(\sqrt{5}-\sqrt{7})^2, then the best approximation of x is:

A. 0
B. 1
C. 2
D. 3
E. 4

x=(\sqrt{5}-\sqrt{7})^2

x= 5 + 7 - 2. \sqrt{35}
\sqrt{35} ~ 6

x = 5+7 - 2*6 = 12 - 12 = 0
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Re: New Algebra Set!!! [#permalink] New post 26 Mar 2013, 05:34
10. If f(x) = 2x - 1 and g(x) = x^2, then what is the product of all values of n for which f(n^2)=g(n+12) ?

A. -145
B. -24
C. 24
D. 145
E. None of the above

f(n^2) = 2.n^2 - 1
g(n + 12) = (n + 12)^2
f(n^2) = g(n + 12) => 2.n^2 - 1 = (n + 12)^2 => 2.n^2 -1 = n^2 + 144 +24.n => n^2 -24n -145 = 0
From the above equation we get n = -5 and 29
So product of values of n = -5 * 29 = -145

A. -145
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Re: New Algebra Set!!! [#permalink] New post 28 Mar 2013, 00:23
Expert's post
1. If x=\sqrt[4]{x^3+6x^2}, then the sum of all possible solutions for x is:

A. -2
B. 0
C. 1
D. 3
E. 5

Solution: new-algebra-set-149349-60.html#p1200948

x^4=x^3 + 6x^2
x^4 - x^3 - 6x^2 = 0 ------------> x^2 (x^2 - x - 6) = 0 -----------> x^2 (x-3)(x+2) = 0
x=0 or 3 or -2. With x = -2 original equation does not hold true so possible values for X are 3 and 0 Hence their sum is 3
Choice D.


2. The equation x^2 + ax - b = 0 has equal roots, and one of the roots of the equation x^2 + ax + 15 = 0 is 3. What is the value of b?

A. -64
B. -16
C. -15
D. -1/16
E. -1/64

Solution: new-algebra-set-149349-60.html#p1200950

x^2 + ax - b = 0 that means a^2 - 4(-b) = 0 ------> a^2 + 4b = 0 -------------Equation I
x^2 + ax + 15 has 3 as a root, so 9 + 3a + 15 = 0 -------> 3a + 24 = 0 --------> a = -8
We will put the value of a in equation I -----------> a^2 + 4b = 0 --------> 64 + 4b = 0 ------> b = -16 ----> Choice B is the answer.
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Re: New Algebra Set!!! [#permalink] New post 28 Mar 2013, 00:37
Expert's post
3. If a and b are positive numbers, such that a^2 + b^2 = m and a^2 - b^2 = n, then ab in terms of m and n equals to:

A. \frac{\sqrt{m-n}}{2}
B. \frac{\sqrt{mn}}{2}
C. \frac{\sqrt{m^2-n^2}}{2}
D. \frac{\sqrt{n^2-m^2}}{2}
E. \frac{\sqrt{m^2+n^2}}{2}

Solution: new-algebra-set-149349-60.html#p1200956

a^2 + b^2 = m
a^2 - b^2 = n
------------------------
2a^2 = m+n ----------> a^2 = (m+n)/2 --------> a = sq.root (m+n) / sq.root 2
similarly b = sq.root (m-n) / sq.root 2
So ab = (sq.root (m+n) / sq.root 2)(sq.root (m-n) / sq.root 2) -------> ab = sq.root(m^2 - n^2)/2---------------Using (a+b)(a-b) = a^2 - b^2
Choice C



4. What is the maximum value of -3x^2 + 12x -2y^2 - 12y - 39 ?

A. -39
B. -9
C. 0
D. 9
E. 39

Solution: new-algebra-set-149349-60.html#p1200962
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Re: New Algebra Set!!! [#permalink] New post 28 Mar 2013, 00:49
Expert's post
5. If x^2 + 2x -15 = -m, where x is an integer from -10 and 10, inclusive, what is the probability that m is greater than zero?

A. 2/7
B. 1/3
C. 7/20
D. 2/5
E. 3/7

Solution: new-algebra-set-149349-60.html#p1200970

there are 21 possible values of x so does m has 21 possible values
from x = -4 to x = 2, m gives negative value; Total 7 values
so the probability m being negative is 7/21----> 1/3 ----> choice B


6. If mn does not equal to zero, and m^2n^2 + mn = 12, then m could be:

I. -4/n
II. 2/n
III. 3/n

A. I only
B. II only
C. III only
D. I and II only
E. I and III only

Solution: new-algebra-set-149349-60.html#p1200973

m^2n^2 + mn = 12 ------> mn(mn+1) = 12 ----------> here mn and mn+1 are consecutive integers. So 12 is basically a product of 2 consecutive intergers
those integers must be 3 and 4 or -3 and -4
so possible values for mn are mn = 3 and mn+1 = 4 --------> m = (3/n)
mn = -4 and mn + 1 =-3 -------> m =-(4/n)
Choice E
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Re: New Algebra Set!!! [#permalink] New post 28 Mar 2013, 03:05
Expert's post
7. If x^4 = 29x^2 - 100, then which of the following is NOT a product of three possible values of x?

I. -50
II. 25
III. 50

A. I only
B. II only
C. III only
D. I and II only
E. I and III only

Solution: new-algebra-set-149349-60.html#p1200975

x^4 - 29x^2 + 100 = 0 --------> let X^2 be y -------> y^2 - 29y + 100 = 0 ------> y^2 - 25y - 4y + 100 = 0 --------> (y-25)(y-4)=0
y=25 -----> x^2 = 25 -----> x= 5 or -5
y=4-------> x^2 = 4 ------> x = 2 or -2

Possible values of x are 5, -5, 2, and -2
Out of options 25 can never be a product of any 3 values of x, so Choice B is the answer



8. If m is a negative integer and m^3 + 380 = 381m , then what is the value of m?

A. -21
B. -20
C. -19
D. -1
E. None of the above

Solution: new-algebra-set-149349-60.html#p1200980

This was a bit complex one
m^3 - 381m = - 380 ------> m(m^2 - 381) = - 380
Since -380 is the product of m(m^2 - 381), any one of either m or (m^2 - 381) must be negative.
Since we are given that m is a negative integer, then m^2 - 381 must be positive
So m^2 - 381 > 0 -------> m^2 > 381 ---------> |m| > 19.5 ----approx.
so m > 19.5-------if m is positive and m < 19.5 ------ if m is negative
We know that m is negative so m must equal to -20 : choice B
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Re: New Algebra Set!!! [#permalink] New post 28 Mar 2013, 03:10
Expert's post
9. If x=(\sqrt{5}-\sqrt{7})^2, then the best approximation of x is:

A. 0
B. 1
C. 2
D. 3
E. 4

Solution: new-algebra-set-149349-60.html#p1200982

x=(sq.root 5 - sq.root 7)^2 --------> x = 5 + 7 -2(sq.root 35) ---------using (a-b)^2 = a^2 + b^2 - 2ab
12 - 2(sq.root 35) ---------> assuming sq.root 35 = 6 ----------> 12-12 = 0
Choice A


10. If f(x) = 2x - 1 and g(x) = x^2, then what is the product of all values of n for which f(n^2)=g(n+12) ?

A. -145
B. -24
C. 24
D. 145
E. None of the above

Solution: new-algebra-set-149349-80.html#p1200987
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Re: New Algebra Set!!! [#permalink] New post 29 Mar 2013, 05:54
Bunuel wrote:
SOLUTIONs:

1. If x=\sqrt[4]{x^3+6x^2}, then the sum of all possible solutions for x is:

A. -2
B. 0
C. 1
D. 3
E. 5

Take the given expression to the 4th power: x^4=x^3+6x^2;

Re-arrange and factor out x^2: x^2(x^2-x-6)=0;

Factorize: x^2(x-3)(x+2)=0;

So, the roots are x=0, x=3 and x=-2. But x cannot be negative as it equals to the even (4th) root of some expression (\sqrt{expression}\geq{0}), thus only two solution are valid x=0 and x=3.

The sum of all possible solutions for x is 0+3=3.

Answer: D.



Hi Bunuel,

Could you please explain why -2 is not a solution? Is it because x is a non-negative number? Because if I plug in the value of -2 to the equation x^4=x^3+6x^2 I get -2=\sqrt[4]{16} which can be true isn't it, as \sqrt[4]{16}=|2|

Also could you explain why even root of an expression cannot be negative when we have an expression which is a square, because we know that \sqrt{{x^2}}=|x|?

Thank you
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Re: New Algebra Set!!! [#permalink] New post 11 Apr 2013, 14:31
Bunuel wrote:
6. If mn does not equal to zero, and m^2n^2 + mn = 12, then m could be:

I. -4/n
II. 2/n
III. 3/n


A. I only
B. II only
C. III only
D. I and II only
E. I and III only

Re-arrange: (mn)^2 + mn - 12=0.

Factorize for mn: (mn+4)(mn-3)=0. Thus mn=-4 or mn=3.

So, we have that m=-\frac{4}{n} or m=\frac{3}{n}.

Answer: E.


I still cannot get how did you rearrange the given expression, bunuel . Could you please elaborate more ? Thanks in advance :)
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Re: New Algebra Set!!! [#permalink] New post 12 Apr 2013, 03:29
Bunuel wrote:

m^2n^2 + mn = 12 --> m^2n^2 + mn -12=0 --> (mn)^2 + mn - 12=0 --> say mn=x, so we have x^2+x-12=0 --> (x+4)(x-3)=0 --> x=-4 or x=3 --> mn=-4 or mn=3.

Solving and Factoring Quadratics:
http://www.purplemath.com/modules/solvquad.htm
http://www.purplemath.com/modules/factquad.htm

Hope it helps.


ohhhhhhhhhhhhhhhhhhhhh I thought that m is raised to the power 2n and the whole bracket raised to the power 2 ...that is what made me confused . Thanks a million , Bunuel :)
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Re: New Algebra Set!!! [#permalink] New post 12 Apr 2013, 04:25
Narenn wrote:
nave81 wrote:
Bunuel wrote:
SOLUTIONs:

1. If x=\sqrt[4]{x^3+6x^2}, then the sum of all possible solutions for x is:

A. -2
B. 0
C. 1
D. 3
E. 5

Take the given expression to the 4th power: x^4=x^3+6x^2;

Re-arrange and factor out x^2: x^2(x^2-x-6)=0;

Factorize: x^2(x-3)(x+2)=0;

So, the roots are x=0, x=3 and x=-2. But x cannot be negative as it equals to the even (4th) root of some expression (\sqrt{expression}\geq{0}), thus only two solution are valid x=0 and x=3.

The sum of all possible solutions for x is 0+3=3.

Answer: D.



Hi Bunuel,

Could you please explain why -2 is not a solution? Is it because x is a non-negative number? Because if I plug in the value of -2 to the equation x^4=x^3+6x^2 I get -2=\sqrt[4]{16} which can be true isn't it, as \sqrt[4]{16}=|2|

Also could you explain why even root of an expression cannot be negative when we have an expression which is a square, because we know that \sqrt{{x^2}}=|x|?

Thank you


Note the Important Difference.

On the GMAT if X^2 = 4 then x = +/- 2 or |x| = 2 However if x= sq.root(4) then x has to be positive i.e. 2 and it can not take negative value.
when we plug -2 as the value of x in the equation we would get -2 = 4th root of 16 -----> -2 = 2 This is because fourth root of 16 is 2 and not -2

The rule is even root of a number can not be negative on the GMAT

Regards,

Abhijit.


Hey Abhijit,

This is a very interesting point that you have made here. This statement that -2 is the fourth root of 16 is admissible under normal circumstances but is not in a GMAT question. Can you mention a source which would list out such peculiar rules for GMAT quants?

Thanks in advance

Anshuman
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Re: New Algebra Set!!! [#permalink] New post 13 Apr 2013, 08:01
Expert's post
Hi Anshuman,

For this you need to go thru some strategy books of GMAT Quant.
In My Opinion, two important sources are,
1) Manhattan GMAT Quant strategy guides 4th or 5th Ed
2) GMAT Club Math Book - This is free and you can download it from this website free of cost (Check BB's profile)
Furthermore you can check my signature. There i have attached my articles prepared on some peculiar topics. You may find them useful.

Thanks,

Abhijit
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Re: New Algebra Set!!!   [#permalink] 13 Apr 2013, 08:01
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