Question: Expand the following: 1. \((2a + 1)^2\) 2. \((5a + 2b)(4a - 3b)\) and state the coefficient…

Expand the following:

1. \((2a + 1)^2\)

2. \((5a + 2b)(4a - 3b)\) and state the coefficient of \(ab\).

3. Find the one-ninth of a positive difference between 55 and the sum of 12 and 7.

4. Make \(u\) the subject of the formula: \(v = \frac{1}{2}(u + v)\).

5. The formula \(C = 180n + 450\) gives the weekly cost, in euros, of running a household of \(n\) people.

6. Find the weekly cost for 5 people.

7. How many people are there if the weekly cost is €1170?

Solution

Problem 1: Expand the following: \[ (5a + 1)^2 \] Apply the formula \((x + y)^2 = x^2 + 2xy + y^2\): \[ (5a)^2 + 2 \times 5a \times 1 + 1^2 & \] Calculate each term: \[ 25a^2 + 10a + 1 & \] Final Answer: \[ 25a^2 + 10a + 1 \] --- Problem 2: Simplify \((5a + 2b)(4a - 3b)\) and find the coefficient of \(ab\). Apply the distributive property: \[ 5a \times 4a + 5a \times (-3b) + 2b \times 4a + 2b \times (-3b) & \] Multiply each term: \[ 20a^2 - 15ab + 8ab - 6b^2 & \] Combine like terms: \[ 20a^2 - 7ab - 6b^2 & \] Coefficient of \(ab\): \[ -7 \] --- Problem 3: Find the \(n\)th term of an arithmetic sequence between 55 and the sum of 12 and 7. First, calculate the sum of 12 and 7: \[ 12 + 7 = 19 & \] Assuming 55 is the first term (\(a_1\)) and 19 is the second term (\(a_2\)) of the arithmetic sequence, find the common difference \(d\): \[ d = a_2 - a_1 = 19 - 55 = -36 & \] The \(n\)th term of an arithmetic sequence is given by: \[ a_n = a_1 + (n - 1)d & \] Substitute the known values: \[ a_n = 55 + (n - 1)(-36) & \] Simplify: \[ a_n = 55 - 36(n - 1) & \] \[ a_n = 55 - 36n + 36 & \] \[ a_n = 91 - 36n & \] Final Answer: \[ a_n = 91 - 36n \] --- Problem 4: Make \(n\) the subject of the formula \(v = \frac{u + at}{2}\). Multiply both sides by 2: \[ 2v = u + at & \] Subtract \(u\) from both sides: \[ 2v - u = at & \] Divide both sides by \(a\): \[ n = \frac{2v - u}{a} & \] Final Answer: \[ n = \frac{2v - u}{a} \] --- Problem 5: The formula \(C = 180n + 450\) gives the weekly cost (in dollars) of running a household of \(n\) people. Find the weekly cost for 6 people. Substitute \(n = 6\) into the formula: \[ C = 180 \times 6 + 450 & \] Calculate: \[ C = 1080 + 450 & \] \[ C = 1530 & \] Final Answer: \[ \$1530 \] --- Problem 6: How many people are there if the weekly cost is \$1170? Set \(C = 1170\) in the formula: \[ 1170 = 180n + 450 & \] Subtract 450 from both sides: \[ 1170 - 450 = 180n & \] \[ 720 = 180n & \] Divide both sides by 180: \[ n = \frac{720}{180} & \] \[ n = 4 & \] Final Answer: \[ 4 \text{ people} \]