Find the gradient, equation of the line PQ, and the coordinates of the intersection point A of the perpendicular bisector with PQ.

The gradient of line PQ is $\boxed{\frac{1}{2}}$.

b) Write down the equation of the line PQ.
Using the point-slope form $y - y_1 = m(x - x_1)$ with $m = \frac{1}{2}$ and point $P(-2, -1)$.

The equation of the line PQ is $\boxed{y = \frac{1}{2}x}$.

c) Calculate the perpendicular bisector meets PQ intersects at A.

d) Find the coordinates of A.
Point A is the midpoint of PQ.
The formula for the midpoint $M$ is $M = \left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\right)$.

The coordinates of A are $\boxed{\left(1, \frac{1}{2}\right)}$.

e) Write down the equation of the perpendicular line.
This refers to the perpendicular bisector of PQ.

The gradient of PQ is $m_{PQ} = \frac{1}{2}$.
The gradient of the perpendicular bisector $m_{\perp}$ is the negative reciprocal of $m_{PQ}$.

The equation of the perpendicular bisector is $\boxed{y = -2x + \frac{5}{2}}$.

f) State any two (2) real life applications of Coordinate Geometry.

• Navigation systems (GPS) use coordinates to pinpoint locations and plan routes.
• Engineering and architecture use coordinate geometry for designing structures, mapping out construction sites, and ensuring precise measurements.

QUESTION THREE

a) Define assessment.
Assessment is the systematic process of documenting and using empirical data on knowledge, skills, attitudes, and beliefs to refine programs and improve student learning. It involves gathering information about student performance.

b) State two (2) purposes of assessing learners.

• To monitor student progress and identify areas where they may need additional support or challenge.
• To evaluate the effectiveness of teaching methods and curriculum, providing feedback for instructional improvement.

c) State the difference between mathematics and mathematics education.
Mathematics is the pure academic discipline focused on the study of quantity, structure, space, and change. Mathematics education is the field concerned with the methods, theories, and practices of teaching and learning mathematics.

d) Describe an effective teacher.
An effective teacher creates a positive and engaging learning environment, possesses strong subject matter knowledge, uses diverse instructional strategies, and provides constructive feedback to foster student growth. They are also adaptable and responsive to student needs.

e) State the difference between a Lesson Plan and Schemes of Work.
A Lesson Plan is a detailed, day-to-day guide for a single lesson, outlining specific objectives, activities, materials, and assessment for that particular class period. A Scheme of Work is a broader, long-term plan that outlines the content, learning objectives, and sequence of topics to be covered over an extended period, such as a term or year.

QUESTION FOUR

a) Simplify: $\frac{y^2+4y+3}{2y^2+7y+5}$

The simplified expression is $\boxed{\frac{y+3}{2y+5}}$.

b) Express as a single fraction: $\frac{2k-2t}{15} + \frac{k-t}{3}$

The expression as a single fraction is $\boxed{\frac{7k-7t}{15}}$.

c) Solve: $\frac{x+4}{2} - \frac{2x+1}{3} = 0$

Multiply the entire equation by 6.

The solution is $\boxed{x=10}$.

d) Solve the simultaneous equations: $2x+3y=-4$ and $x-y=3$

The solution is $\boxed{x=1, y=-2}$.

e) Factorise the following expressions;
i. $x+8x+7$
Assuming the expression is $x^2+8x+7$.
We need two numbers that multiply to 7 and add to 8. These numbers are 1 and 7. $x^2+8x+7 = (x+1)(x+7)$ The factorised expression is $\boxed{(x+1)(x+7)}$.

*ii. $6-5b-4b^2$*