If \(P^2-QR=10\) ,\(Q^2+PR=10\) ,\(R^2+PQ=10\)  and \(R\neq QR \neq Q\), what is the value of \(P+Q^2+R^2\) ?

Question: If \(P^2-QR=10\), \(Q^2+PR=10\), \(R^2+PQ=10\) and \(R\neq QR \neq Q\), what is the value of \(P+Q^2+R^2\) ?

1. 10
2. 15
3. 20
4. 25
5. 30

This topic is a part of GMAT Quantitative reasoning section of GMAT. This question has been taken from the book "GMAT Quantitative Review" published in the year 2022. GMAT Quant section consists of a total of 31 questions. To solve GMAT Problem Solving questions a student must have knowledge about a good amount of qualitative skills. The GMAT quant topics in the problem-solving part require calculative mathematical problems that should be solved with proper mathematical knowledge.

Answer:

Write down all the original equations and give them numbering:

\(P^2-QR=10\)━ (1)

\(Q^2+PR=10\)━ (2)

\(R^2+PQ=10\)━ (3)

Now to solve the equations, subtract equation (2) from equation (3), we will get:

\(R^2+PQ-(Q^2+PR)=10-10\)

\(R^2-Q^2\)+ PQ - PR = 0

Further solving this equation, we will take the common values out from the brackets as:

\(R^2-Q^2\)+ P (Q – R) = 0 ━ (4)

As we know that:

\(a^2-b^2=(a+b)(a-b)\)

Put this formula in the equation (4), we will get:

(R + Q) (R – Q) + P (Q – R) = 0

This will become: (R + Q) (R – Q) - P (R - Q) = 0

(R + Q) (R – Q) = P (R - Q)

After solving this equation, we will get:

R + Q = P ━ (5)

Now add all the equation (1), (2), (3), we will get that :

\(P^2-QR+Q^2+PR+R^2+PQ=10+10+10\)

Rearranging these terms:

\(P^2+Q^2+R^2-QR+PR+PQ=30\)

\(P^2+Q^2+R^2-QR+P(R+Q)=30\)

Now put the value of (R + Q) from equation (5) to the above equation:

We will get: \(P^2+Q^2+R^2-QR+P(P)=30\)

This will become as: \(P^2+Q^2+R^2-QR+P^2=30\)

From equation (1), we will get the value of \(P^2-QR\)

We will put this value in the above final equation and we will get that:

\(P^2+Q^2+R^2-10=30\)

This will become: \(P^2+Q^2+R^2=20\)

Correct Answer: C

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