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New Algebra Set!!!

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Joined: 02 Sep 2009
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18 Mar 2013, 07:56
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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

Kudos points for each correct solution!!!
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18 Mar 2013, 08:04
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Please suggest on what category would you like the next set to be. Thank you!
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18 Mar 2013, 08:06
1
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Edited

1. x^4 = x^3 + 6x^2
=> x^2 (x^2 - x - 6) = 0
The roots of x^2 - x - 6 are -2 and 3, but -2 cannot be the value of x. So 3 and 0 are the only possible roots.
=> Sum of all possible solutions = 3.

Option D
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Last edited by GyanOne on 18 Mar 2013, 13:34, edited 3 times in total.

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18 Mar 2013, 08:11
1
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Expert's post
GyanOne wrote:
1. x^4 = x^3 + 6x^2
=> x^2 (x^2 - x - 6) = 0
Sum of roots of x^2 - x - 6 = 0 is 1 and the only other solution is x=0
=> Sum of all possible solutions = 1.

Option C

Some questions are tricky!!!
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18 Mar 2013, 08:17
1
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2. Let one of the roots be x, then other root is also x.
Sum of roots = 2x = -a
Product of roots = x^2 = - b

For other equation one root is 3 and product of the roots is 15, so other root is 5.
Sum of roots = 8 = -a
or a = -8.
From above 2x = 8 or x = 4,
b = - x^2 = -16.

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18 Mar 2013, 08:22
4
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3. a^2 + b^ 2 = m
a^2 - b^2 = n
Solving both the equations( adding them, and then subtracting them ):
2a^2 = m + n
2b^2 = m - n.
a = ((m+n)/2)^(1/2)
b = ((m-n)/2)^(1/2)

ab = ((m^2 - n^2)^(1/2))/2

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18 Mar 2013, 08:23
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4. -3x^2 + 12x -2y^2 - 12y - 39 = 3(-x^2 + 4x - 13) + 2(-y^2-6y)

Now -x^2 +4x - 13 has its maximum value at x = -4/-2 = 2 and -y^2 - 6y has its maximum value at y=6/-2 = -3
Therefore max value of the expression
= 3(-4 + 8 -13) + 2 (-9+18) = -27 + 18 = -9

Should be B
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Last edited by GyanOne on 18 Mar 2013, 10:13, edited 1 time in total.

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18 Mar 2013, 08:34
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9. x = (sqrt(5) - sqrt(7))^2
= 5 + 7 - 2sqrt(35)
=~ 12 - 2*6 = 0

Option A
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Manager
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18 Mar 2013, 08:40
2
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6. M^2 * n^2 + mn = 12
mn( mn + 1 ) = 12
mn = 3 or mn = -4 ( 3 * 4 = 12; -4 * -3 = 12 )
So m = 3/n or m = -4/n

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18 Mar 2013, 09:02
8. If m is a negative integer and m^3 + 380 = 381m , then what is the value of m?

m^3 - 381m + 380 = 0
Sum of roots = 0
Product of roots = -380

Therefore 19,1,-20 is definitely a possible solution set. Sum of roots = 0, Product of roots = -380

B is therefore the right answer.
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Manager
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18 Mar 2013, 09:03
4
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8. m^3 + 380 = 381m
m^3 + 380 = 380m + m
m^3 -m = 380m - 380
m(m-1)(m+1) = 380 (m-1)
m(m+1) = 380 ( -20 * -19; m is negative )
m = -20.

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18 Mar 2013, 09:04
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
$$x=\sqrt[4]{x^3+6x^2}$$
=>$$x^4=x^3+6x^2$$
=>$$x^4-x^3-6x^2=0$$

Sum of roots for $$ax^n+bx^(n-1)+cx^(n-2)+... + constant$$ = $$-b/a$$
In the given prob : b= -1, a=1

Sum of roots = $$-b/a = -(-1)/1 = 1$$
C
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18 Mar 2013, 09:08
1
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10.
2x^2 - 1 = (x+12)^2
2x^2 - 1 = x^2 + 24x +144
x^2 - 24x - 145 = 0
Product of roots = -145

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18 Mar 2013, 09:12
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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

$$f(n^2) = 2n^2 -1 g(n+12) = (n+12)^2 =>2n^2 -1 = (n+12)^2 =>2n^2 -1 = n^2 + 24n + 144 => n^2 - 24n - 145 = 0$$

Product of all n = product of roots of the above equation.
For ax^2 + bx + c = 0 => Product of roors = c/a
Product of roots = -145/1 = -145
Ans = A
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18 Mar 2013, 09:26
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7. If$$x^4 = 29x^2 - 100$$, then which of the following is NOT a product of two possible values of x?

I. -50
II. 25
III. 100

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

$$x^4 = 29x^2 - 100$$
$$=> x^4-29x^2+100 = 0$$
$$=> x^4 - 25x^2 - 4x^2 + 100=0$$
$$=> x^2(x^2-25) - 4(x^2-25) = 0$$
$$=>(x^2 - 4)(x^2-25) = 0$$
=> x = +2,-2,+5,-5

None of -50, 25 or 100 is possible with product of two possible values of x.
what I am missing ?
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18 Mar 2013, 09:32
ConnectTheDots wrote:
7. If$$x^4 = 29x^2 - 100$$, then which of the following is NOT a product of two possible values of x?

I. -50
II. 25
III. 100

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

$$x^4 = 29x^2 - 100$$
$$=> x^4-29x^2+100 = 0$$
$$=> x^4 - 25x^2 - 4x^2 + 100=0$$
$$=> x^2(x^2-25) - 4(x^2-25) = 0$$
$$=>(x^2 - 4)(x^2-25) = 0$$
=> x = +2,-2,+5,-5

None of -50, 25 or 100 is possible with product of two possible values of x.
what I am missing ?

Edited typos. Thank you.
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18 Mar 2013, 09:33
3
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1)$$x^4=x^3+6x^2$$
$$x^4-x^3-6x^2=0$$
$$x^2(x^2-x-6)=0$$
$$x^2=0 (1) x = 0$$
$$x^2-x-6=0 (2) x = 3 (3) x = -2$$
$$0+3-2=1$$
C

2)The equation x^2 + ax - b = 0 has equal roots
$$a^2+4b=0$$
one of the roots of the equation x^2 + ax + 15 = 0 is 3
$$(-a+-\sqrt{a^2-4*15})/2=3$$
$$+-\sqrt{a^2-60}=6+a$$
$$(\sqrt{a^2-60})^2=(6+a)^2$$
$$-96=12a$$
$$a=-8$$

$$a^2+4b=0$$
$$64+4b=0$$
$$b=-16$$
B

3)We can use some numbers
$$2^2+1^2=5=m 2^2-1^2=5=n$$
$$ab=2$$

$$\sqrt{m^2-n^2}/2 = \sqrt{25-9}/2 = 2 = ab$$
C

4) -3x^2 + 12x -2y^2 - 12y - 39 MAX
$$-3x^2 + 12x$$ has max in (2,12)
$$-2y^2 - 12y - 39$$ has max in (-6,-3)
-3x^2 + 12x -2y^2 - 12y - 39 MAX = 12 - 3 =9
D
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18 Mar 2013, 09:36
1
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Expert's post
Zarrolou wrote:
1)$$x^4=x^3+6x^2$$
$$x^4-x^3-6x^2=0$$
$$x^2(x^2-x-6)=0$$
$$x^2=0 (1) x = 0$$
$$x^2-x-6=0 (2) x = 3 (3) x = -2$$
$$0+3-2=1$$
C

2)The equation x^2 + ax - b = 0 has equal roots
$$a^2+4b=0$$
one of the roots of the equation x^2 + ax + 15 = 0 is 3
$$(-a+-\sqrt{a^2-4*15})/2=3$$
$$+-\sqrt{a^2-60}=6+a$$
$$(\sqrt{a^2-60})^2=(6+a)^2$$
$$-96=12a$$
$$a=-8$$

$$a^2+4b=0$$
$$64+4b=0$$
$$b=-16$$
B

3)We can use some numbers
$$2^2+1^2=5=m 2^2-1^2=5=n$$
$$ab=2$$

$$\sqrt{m^2-n^2}/2 = \sqrt{25-9}/2 = 2 = ab$$
C

4) -3x^2 + 12x -2y^2 - 12y - 39 MAX
$$-3x^2 + 12x$$ has max in (2,12)
$$-2y^2 - 12y - 39$$ has max in (-6,-3)
-3x^2 + 12x -2y^2 - 12y - 39 MAX = 12 - 3 =9
D

Better to post each solution as a separate post, since I cannot give more than 1 kudos point for one post.
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18 Mar 2013, 09:37
5. I have taken a longer approach, there should be a better solution.
x^2 + 2x -15 = -m
x^2 + 5x - 3x - 15 = -m
(x+5)(x-3) = -m
x will have integer values, since m is also an integer.
Putting positive values for x such as ( 1, 2....) we will get values for m which lies between -10 and 10, they are -9, 0 and 7.
We will get same result for negative values of x, the equation is a parabola, it will be symmetric
Hence the probability is 1/3

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18 Mar 2013, 09:42
2
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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

$$m^2n^2 + mn = 12$$
$$mn(mn+1) = 12$$
mn = 3
$$3(3+1)=12$$
OR
mn = -4
$$-4(-4+1)=12$$

$$m= 3/n$$
OR
$$m=-4/n$$
E
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Re: New Algebra Set!!!   [#permalink] 18 Mar 2013, 09:42

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