1. It is given that \( a=2 n(n-3 k) \). (i) Find \( a \) when \( n=0.5 \) and \( k=-1.2 \). (ii) Express \( k \) in terms of \( a \) and \( n \). (ii)

Certainly! Let's tackle each part of the problem step-by-step. --- **Given:** \[ a = 2n(n - 3k) \] --- ### **Part (i): Find \( a \) when \( n = 0.5 \) and \( k = -1.2 \).** **Step 1: Substitute the given values into the equation.** \[ a = 2 \times 0.5 \times (0.5 - 3 \times (-1.2)) \] **Step 2: Calculate the expression inside the parentheses.** \[ 0.5 - 3 \times (-1.2) = 0.5 + 3.6 = 4.1 \] **Step 3: Multiply the terms together.** \[ a = 2 \times 0.5 \times 4.1 \] \[ a = 1 \times 4.1 \] \[ a = 4.1 \] **Answer:** \[ a = 4.1 \] --- ### **Part (ii): Express \( k \) in terms of \( a \) and \( n \).** We start with the original equation: \[ a = 2n(n - 3k) \] **Step 1: Expand the right-hand side (if necessary).** However, in this case, it's more straightforward to solve for \( k \) directly. **Step 2: Divide both sides by \( 2n \) to isolate \( (n - 3k) \).** \[ \frac{a}{2n} = n - 3k \] **Step 3: Rearrange the equation to solve for \( k \).** \[ n - \frac{a}{2n} = 3k \] **Step 4: Divide both sides by 3 to solve for \( k \).** \[ k = \frac{n}{3} - \frac{a}{6n} \] **Alternatively, combining the terms:** \[ k = \frac{2n^2 - a}{6n} \] **Answer:** \[ k = \frac{2n^2 - a}{6n} \] *Or equivalently:* \[ k = \frac{n}{3} - \frac{a}{6n} \] --- **Summary:** - **(i)** When \( n = 0.5 \) and \( k = -1.2 \), the value of \( a \) is **4.1**. - **(ii)** \( k \) expressed in terms of \( a \) and \( n \) is: \[ k = \frac{2n^2 - a}{6n} \quad \text{or} \quad k = \frac{n}{3} - \frac{a}{6n} \] If you have any further questions or need additional clarification, feel free to ask!