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Question of the week - 33 (A quadratic equation is in the form ......)

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EgmatQuantExpert
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Question of the week - 33 (A quadratic equation is in the form ......) [#permalink] New post   25 Jan 2019, 06:20
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e-GMAT Question of the Week #33


A quadratic equation is in the form of \(x^2 – 2px + m = 0\), where m is divisible by 5 and is less than 120. One of the roots of this equation is 7. If p is a prime number and one of the roots of the equation, \(x^2 – 2px + n = 0\) is 12, then what is the value of p + n – m?

    A. 0
    B. 6
    C. 16
    D. 26
    E. 27

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D
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Re: Question of the week - 33 (A quadratic equation is in the form ......) [#permalink] New post   25 Jan 2019, 06:30
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\(x^2–2px+m=0\)
and m is divisible by 5.
And one of the roots = 7.
Put x = 7
49-14p+m = 0
and also, P is prime.
Only number which satisfies the equation is for p = 11
Then 49-154+m = 0
m = 105.

Now, \(x^2–2px+n=0\) has one root as x = 12
Then 144-24p+n = 0
p = 11
Then n = 120

p+n-m = 11+120-105 = 26

D is the answer.
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Re: Question of the week - 33 (A quadratic equation is in the form ......) [#permalink] New post   25 Jan 2019, 08:13
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EgmatQuantExpert wrote:
A quadratic equation is in the form of \(x^2 – 2px + m = 0\), where m is divisible by 5 and is less than 120. One of the roots of this equation is 7. If p is a prime number and one of the roots of the equation, \(x^2 – 2px + n = 0\) is 12, then what is the value of p + n – m?

    A. 0
    B. 6
    C. 16
    D. 26
    E. 27



IMO D

prime numbers will be 2,3,5,11

Multiples of 5 which can give us a 7 => 35,70 and 105

\(x^2 – 2px + m = 0\), here p can be 11 and m will be 105

\(x^2 – 22x + m = 0\)
\(x^2 – 15x -7x + 105 = 0\)
\(x(x – 15) -7(x -15) = 0\)

\(x^2 – 2px + n = 0\), here p can be 11 and n will be 120
\(x^2 – 22x + 120 = 0\)
\(x^2 – 10x -12x +120 = 0\)
\(x(x – 10) -12(x - 10) = 0\)

so values of p + n – m
11 + 120 - 105
Answer D

P.S. I was down to realize that the value was 11(in 2 mins), and when i was writing this solution to ask some help on how to solve this question.

I didn't realize that the factors of 105 would have given me 22.

Always take all the possibilities of the factors
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Re: Question of the week - 33 (A quadratic equation is in the form ......) [#permalink] New post   25 Jan 2019, 10:55
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\(x^2 – 2px + m = 0\)

given one value of x=7
49-14p+m=0
p=2,3,5,7,11,13...
upon doing substitution for values of p only at 11 we get m as a integer which is divisible by 5
so
49-14*11+m=0
m= 105

\(x^2 – 2px + n = 0\)
at x=12
144-24p+n
p=11
n=120
so
p + n – m
11+120-105

26
IMO D


EgmatQuantExpert wrote:
A quadratic equation is in the form of \(x^2 – 2px + m = 0\), where m is divisible by 5 and is less than 120. One of the roots of this equation is 7. If p is a prime number and one of the roots of the equation, \(x^2 – 2px + n = 0\) is 12, then what is the value of p + n – m?

    A. 0
    B. 6
    C. 16
    D. 26
    E. 27

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Re: Question of the week - 33 (A quadratic equation is in the form ......) [#permalink] New post   25 Jan 2019, 20:24
This is doable, but a very tricky question. Considering m could take 35, 70, and 105 was important, as the value of "m" was not 35, or 70, it was 105.
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Re: Question of the week - 33 (A quadratic equation is in the form ......) [#permalink] New post   30 Jan 2019, 00:43
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Solution


Given:
    • A quadratic equation, \(x^2 – 2px + m\) = 0
      o m is divisible by 5, and
      o m < 120
      o 7 is one root of the equation
      o p is a prime number
    • 12 is one root of the equation, \(x^2 – 2px + n = 0\)

To find:
    • The value of p + n – m

Approach and Working:
In the quadratic equation, \(x^2 – 2px + m = 0\), let us assume that the other root is “b”
    • Sum of the roots = 7 + b = 2p
      o Implies, 7 + b must be even, that is b must be odd

    • Product of the roots = 7b = m = a multiple of 5
      o Implies, b is a multiple of 5

    • Thus, b is an odd multiple of 5
    • And, we are given that m < 120
      o Implies, 7 * m = 7 * 5k < 120
      o \(k < \frac{24}{7}\)
      o Thus, k = 1 or 3

    • If k = 1, then b = 5
      o In this case, 2p = 7 + 5 = 12
      o p = 6, which is not prime

    • If k = 3, then b = 15
      o In this case, 2p = 7 + 15 = 22
      o p = 11, which is a prime number

    • So, p = 11, m = 7 * 15 = 105, and n = 12 * 10 = 120

Therefore, p + n – m = 11 + 120 – 105 = 26

Hence the correct answer is Option D.

Answer: D

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Re: Question of the week - 33 (A quadratic equation is in the form ......) [#permalink] New post   31 Jan 2019, 08:27
Hello Archit3110 !

I was reading your approach and I liked it, I just have a question.

Why can we equal x to one of the roots? Would it be the same result if we equate x to the other unknown root?

I am a bit confused with that:

"given one value of x=7
49-14p+m=0"

Kind regards!
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Re: Question of the week - 33 (A quadratic equation is in the form ......) [#permalink] New post   31 Jan 2019, 08:31
EgmatQuantExpert wrote:

Solution


Given:
    • A quadratic equation, \(x^2 – 2px + m\) = 0
      o m is divisible by 5, and
      o m < 120
      o 7 is one root of the equation
      o p is a prime number
    • 12 is one root of the equation, \(x^2 – 2px + n = 0\)

To find:
    • The value of p + n – m

Approach and Working:
In the quadratic equation, \(x^2 – 2px + m = 0\), let us assume that the other root is “b”
    • Sum of the roots = 7 + b = 2p
      o Implies, 7 + b must be even, that is b must be odd

    • Product of the roots = 7b = m = a multiple of 5
      o Implies, b is a multiple of 5

    • Thus, b is an odd multiple of 5
    • And, we are given that m < 120
      o Implies, 7 * m = 7 * 5k < 120
      o \(k < \frac{24}{7}\)
      o Thus, k = 1 or 3

    • If k = 1, then b = 5
      o In this case, 2p = 7 + 5 = 12
      o p = 6, which is not prime

    • If k = 3, then b = 15
      o In this case, 2p = 7 + 15 = 22
      o p = 11, which is a prime number

    • So, p = 11, m = 7 * 15 = 105, and n = 12 * 10 = 120

Therefore, p + n – m = 11 + 120 – 105 = 26

Hence the correct answer is Option D.

Answer: D



Hello EgmatQuantExpert

I was doing the same approach as you but I had a problem with the following:

If its given that one of the roots is 7, why it doesn't mean that x = -7 because if one root is 7, that would mean that:

(x -7) Isn't it? So the sum of the root -2px shouldn't be (-7) + b?

Could you please help me with this?

Thank you in advance!

Kind regards!
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Re: Question of the week - 33 (A quadratic equation is in the form ......) [#permalink] New post   31 Jan 2019, 08:58
jfranciscocuencag wrote:
Hello Archit3110 !

I was reading your approach and I liked it, I just have a question.

Why can we equal x to one of the roots? Would it be the same result if we equate x to the other unknown root?

I am a bit confused with that:

"given one value of x=7
49-14p+m=0"

Kind regards!


jfranciscocuencag
its given in the question that one of the roots of equation is 7 .. so value of x has been substituted as 7.. by roots in quadratic eqn it means that upon doing substitution of the value of x we would get value "= 0"
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Re: Question of the week - 33 (A quadratic equation is in the form ......) [#permalink] New post   01 Feb 2019, 02:39
Expert Reply
jfranciscocuencag wrote:

Hello EgmatQuantExpert

I was doing the same approach as you but I had a problem with the following:

If its given that one of the roots is 7, why it doesn't mean that x = -7 because if one root is 7, that would mean that:

(x -7) Isn't it? So the sum of the root -2px shouldn't be (-7) + b?

Could you please help me with this?

Thank you in advance!

Kind regards!


Hi,

    If x - 7 is a factor of the quadratic equation, then we can write (x - 7) = 0, which implies, x = 7

    Similarly, if x = 7 is a root of a quadratic equation, then (x - 7) is a factor of the quadratic equation

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Sandeep
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Re: Question of the week - 33 (A quadratic equation is in the form ......) [#permalink] New post   21 Aug 2020, 13:47
Hi

Let me try explaining this...

Roots are x-intercepts or when the equation is zero. When the question says that one of the roots is 7, it means that when x is 7, f(x)=0. So you can plug-in 7 in place of x in the equation.

Here is my twist to Archit's method...same result though. After plugging in 7 for x, you can get the 2 equations (144-24p+n=0 and 49-14p+m=0).

Equate the 2 equations and you will get n-m=10p-95. Then simply solve p+(n-m).

p+n-m = p+10p-95 = 11p-95.

Since p is a prime number, try a few values...and you will get D

jfranciscocuencag wrote:
Hello Archit3110 !

I was reading your approach and I liked it, I just have a question.

Why can we equal x to one of the roots? Would it be the same result if we equate x to the other unknown root?

I am a bit confused with that:

"given one value of x=7
49-14p+m=0"

Kind regards!
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Question of the week - 33 (A quadratic equation is in the form ......) [#permalink] New post   11 Apr 2024, 20:04
egmat , egmatquantexpert
EgmatQuantExpert
I followed this approach.
Since 7 is a root,

we can write , 7^2 - 14*p +m = 0
so 14*p - m = 49 

similarly , 24*p - n = 144 by putting 12 in the equation.

Subtracting  , 14*p - m - 24*p + n = -95
n-m -10P + 11p =  -95 + 11p ( adding 11p on both sides)
n-m + p = -95 + 11p

p is a prime number and the result has to be positive. ( see...no option is negative) ; so the final result has to be positive)
hence the multiple of 11 should be bigger than 95. 
Hence , out of 2,3,5,7, 11 , try with 11.

The ans is = n -m +p = -95 + 121 = 26
EgmatQuantExpert wrote:

Solution


Given:
    • A quadratic equation, \(x^2 – 2px + m\) = 0
      o m is divisible by 5, and
      o m < 120
      o 7 is one root of the equation
      o p is a prime number
    • 12 is one root of the equation, \(x^2 – 2px + n = 0\)

To find:
    • The value of p + n – m

Approach and Working:
In the quadratic equation, \(x^2 – 2px + m = 0\), let us assume that the other root is “b”
    • Sum of the roots = 7 + b = 2p
      o Implies, 7 + b must be even, that is b must be odd

    • Product of the roots = 7b = m = a multiple of 5
      o Implies, b is a multiple of 5

    • Thus, b is an odd multiple of 5
    • And, we are given that m < 120
      o Implies, 7 * m = 7 * 5k < 120
      o \(k < \frac{24}{7}\)
      o Thus, k = 1 or 3

    • If k = 1, then b = 5
      o In this case, 2p = 7 + 5 = 12
      o p = 6, which is not prime

    • If k = 3, then b = 15
      o In this case, 2p = 7 + 15 = 22
      o p = 11, which is a prime number

    • So, p = 11, m = 7 * 15 = 105, and n = 12 * 10 = 120

Therefore, p + n – m = 11 + 120 – 105 = 26

Hence the correct answer is Option D.

Answer: D


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