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

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Question of the week - 33 (A quadratic equation is in the form ......)  [#permalink]

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25 Jan 2019, 06:20
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Difficulty:

55% (hard)

Question Stats:

55% (03:25) correct 45% (03:56) wrong based on 22 sessions

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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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Re: Question of the week - 33 (A quadratic equation is in the form ......)  [#permalink]

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25 Jan 2019, 06:30
$$x^2–2px+m=0$$
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

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Re: Question of the week - 33 (A quadratic equation is in the form ......)  [#permalink]

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25 Jan 2019, 08:13
1
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

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]

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25 Jan 2019, 10:55
1
$$x^2 – 2px + m = 0$$

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]

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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]

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30 Jan 2019, 00:43

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.

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Re: Question of the week - 33 (A quadratic equation is in the form ......)  [#permalink]

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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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Joined: 12 Sep 2017
Posts: 302
Re: Question of the week - 33 (A quadratic equation is in the form ......)  [#permalink]

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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.

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?

Kind regards!
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Re: Question of the week - 33 (A quadratic equation is in the form ......)  [#permalink]

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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]

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01 Feb 2019, 02:39
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?

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

Regards,
Sandeep
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Re: Question of the week - 33 (A quadratic equation is in the form ......)   [#permalink] 01 Feb 2019, 02:39
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