Let’s go through the questions in the image you’ve provided:

Question 1:

State whether each pair of gradients represents:

• Parallel lines (when gradients are equal)
• Perpendicular lines (when gradients multiply to -1)
• Neither.

The gradients are as follows:

• a. $m_1 = \frac{1}{4}, m_2 = \frac{1}{4}$
  Since $m_1 = m_2$, the lines are parallel.

• b. $m_1 = 3, m_2 = -\frac{1}{3}$
  Check for perpendicularity: $m_1 \times m_2 = 3 \times \left(-\frac{1}{3}\right) = -1$, so the lines are perpendicular.

• c. $m_1 = 0.5, m_2 = \frac{1}{2}$
  These gradients are equal, so the lines are parallel.

• d. $m_1 = 2, m_2 = \frac{7}{2}$
  Since the gradients are different, and their product is not -1, the lines are neither parallel nor perpendicular.

• e. $m_1 = \frac{3}{10}, m_2 = -0.3$
  Convert $m_2$ into a fraction: $-0.3 = -\frac{3}{10}$. Since $m_1 = -m_2$, the lines are neither parallel nor perpendicular.

• f. $m_1 = 0.5, m_2 = -\frac{6}{5}$
  Check for perpendicularity: $m_1 \times m_2 = 0.5 \times -\frac{6}{5} = -\frac{3}{5} \neq -1$, so the lines are neither parallel nor perpendicular.

Question 2:

Find the gradient of a line that is parallel to a line with the given gradients:

• a. Parallel to 4 → The gradient is $4$.
• b. Parallel to $-2$ → The gradient is $-2$.
• c. Parallel to $\frac{3}{5}$ → The gradient is $\frac{3}{5}$.
• d. Parallel to $-0.2$ → The gradient is $-0.2$.

Would you like further details on any of these solutions? Here are 5 related questions:

1. How do you find if lines are parallel in a coordinate plane?
2. How can you identify perpendicular lines using gradients?
3. What is the relationship between slopes for perpendicular lines?
4. How would you convert a decimal slope to a fraction?
5. How can you derive the slope of a line from its equation in standard form?

Tip: Remember, lines are parallel when their gradients (slopes) are equal, and they are perpendicular when the product of their gradients is $-1$.