Question

If \(M(a, b)=a^{2}+b^{2}+a b ; N(a, b)=a^{2}+b^{2}-a b\). Find the value of \(M(7, N(9,4))\) (a) 5197 (b) 3197 (c) 4197 (d) none of these

Short answer

Answer: The value of M(7, N(9,4)) is 4197.

Step-by-step solution

Find N(9,4)

Substitute the given values of a = 9 and b = 4 in the given expression for N(a, b): N(9, 4) = (9)^2 + (4)^2 - (9)(4) Now, perform the calculations as follows: N(9, 4) = 81 + 16 - 36 = 61

Find M(7, N(9,4))

We have found that N(9, 4) = 61. Now, we will substitute the given value of a = 7 and the calculated value of b = 61 in the given expression for M(a, b): M(7, 61) = 7^2 + 61^2 + 7 * 61 Now, perform the calculations as follows: M(7, 61) = 49 + 3721 + 427 = 49 + 3721 + 427 = 4197

Identify the correct option

From our calculations, we have found that M(7, N(9,4)) = 4197. Looking at the given options, we can see that option (c) matches our calculated value. Therefore, the correct answer is (c) 4197.

Key concepts

Problem Solving

When tackling a new math problem, such as finding the value of a composite function like \( M(7, N(9,4)) \), it's important to break down the task into manageable steps.

Effective problem solving often involves:

• Understanding the problem by carefully reading and identifying what is being asked.
• Breaking down the problem into smaller parts, like calculating \( N(9, 4) \) before \( M(7, 61) \).
• Performing calculations step by step to avoid mistakes, such as performing arithmetic correctly once expressions are fully substituted.
• Checking your work, which includes comparing your answer to given options to ensure it makes sense.

Remember, practicing these steps increases proficiency and helps you become more confident in solving various types of problems.

Algebraic Expressions

Algebraic expressions are the building blocks we use in mathematics to represent numbers and operations. In the problem, expressions like \( M(a, b) = a^2 + b^2 + ab \) and \( N(a, b) = a^2 + b^2 - ab \) represent these relationships.

Algebraic expressions consist of:

• Variables (e.g., \(a\) and \(b\)) which can take on different values.
• Constants, which are specific fixed values like the numbers 9 and 4 in the problem.
• Operators, such as \(+\), \(-\), and \(\times\), which define how the variables and constants interact.

Simplifying and substituting values into these expressions are critical skills that help in crunching the numbers correctly in a real-world problem.

Function Evaluation

Function evaluation involves computing the result of a function given certain input values. Here, this concept is applied by first evaluating \( N(9, 4) \) and using the result as an input to another function \( M \).

Key points in function evaluation include:

• Identifying the function rule like \( M(a, b) = a^2 + b^2 + ab \), which tells us how to process the inputs.
• Substituting the known values into the function. For instance, after finding \( N(9, 4) = 61 \), substitute \( b = 61 \) into \( M(7, b) \).
• Performing accurate calculations. In this case, calculating \( 49 + 3721 + 427 \) to yield 4197.
• Ensuring accuracy by double-checking calculations against possible options or solutions provided with the problem.

Mastering function evaluation opens up the ability to tackle more complex algebraic problems with composure.