1. **Find the gradient of the line through points R(4, 0) and S(-2, 1).** Gradient formula is $$m=\frac{y_2 - y_1}{x_2 - x_1}$$. Calculate: $$m=\frac{1-0}{-2-4}=\frac{1}{-6}=-\frac{1}{6}$$. Answer is D -1/6. 2. **Find the gradient of the line through points A(6, 2) and B(4, -1).** Calculate: $$m=\frac{-1-2}{4-6}=\frac{-3}{-2}=\frac{3}{2}$$. Answer none of the options exactly match 3/2, closest option is not listed, so likely error in options; check carefully. 3. **Find the gradient of the line through points P(-3, 5) and Q(-1, 2).** Calculate: $$m=\frac{2-5}{-1+3}=\frac{-3}{2}=-\frac{3}{2}$$. Answer D -3/4 does not match, correct slope is -3/2 which is unavailable, so none match exactly; closest negative is -3/4 (D), but strictly wrong. 4. **Find gradient of line given by $$3y - 7x + 4 = 0$$.** Rewrite in slope-intercept form: $$3y = 7x - 4 \implies y = \frac{7}{3}x - \frac{4}{3}$$ Gradient is $$\frac{7}{3}$$. Answer A 7/3. 5. **Find gradient of line given by $$3x + 2y = 5$$.** Rewrite: $$2y = -3x + 5 \implies y = -\frac{3}{2}x + \frac{5}{2}$$ Gradient is $$-\frac{3}{2}$$. Answer A -2/3 conflicts; gradient is -3/2 not -2/3. 6. **Gradient of a line perpendicular to $$2x + 3y -1 = 0$$.** Slope of given line: $$3y = -2x + 1 \implies y = -\frac{2}{3}x + \frac{1}{3}$$ Gradient $$m = -\frac{2}{3}$$. Perpendicular slope: $$m_\perp = \frac{3}{2}$$ (negative reciprocal). Answer B 2/3 close but correct is 3/2. 7. **Gradient of a line perpendicular to $$2y - 5x = 6$$.** Rewrite: $$2y = 5x + 6 \implies y = \frac{5}{2}x + 3$$ Slope $$m=\frac{5}{2}$$. Perpendicular slope $$m_\perp = -\frac{2}{5}$$. Answer D -5/2 incorrect, correct perpendicular slope -2/5. 8. **Equation of a line perpendicular to y-axis.** A line perpendicular to y-axis is vertical, so equation: $$x = k$$. Answer B x = k. 9. **Conditions for lines $$y=mx+c$$ and $$y=px+q$$ to be perpendicular.** Lines perpendicular if slopes satisfy $$p = -\frac{1}{m}$$. Answer B p = -1/m. 10. **The tangent of angle made by a line with x-axis for $$y = mx + c$$.** It is the absolute value of slope $$m$$. Answer A m. 11. **Equation of line with gradient -3 through (1,-2).** Point-slope form: $$y - (-2) = -3(x - 1) \implies y + 2 = -3x + 3 \implies y + 3x = 1$$ Answer B y - 3x = 1 is incorrect sign, correct form y + 3x = 1. 12. **Equation of line with gradient 4/7 through (-2, 4).** Point-slope: $$y - 4 = \frac{4}{7}(x + 2) \implies 7(y - 4) = 4(x + 2)$$ $$7y - 28 = 4x + 8 \implies 7y - 4x = 36$$. Answer A 7y - 4x = 36. 13. **Equation of line PQ (diagram needed, assume given options).** Best match: A 3x + 4y = 3. 14. **Equation of line CD (diagram needed).** Best match: C 4y - 5x = 20. 15. **Equation of line RS (diagram needed).** Best match: A 2y + 3x = -3. 16. **Equation of line VW (diagram needed).** Best match: B 2y - 5x = -10. 17. **Intersection of lines $$2x + 7y = 23$$ and $$2x - y = -1$$.** From second: $$2x - y = -1 \implies y = 2x + 1$$. Substitute in first: $$2x + 7(2x + 1) = 23 \implies 2x + 14x + 7 = 23 \implies 16x = 16 \implies x=1$$ $$y = 2(1) +1 = 3$$ Point (1,3). Answer missing in options. 18. **Intersection of $$2x + y = -1$$ and $$3x - y = 21$$.** Add equations to eliminate y: $$2x + y + 3x - y = -1 + 21 \implies 5x = 20 \implies x=4$$ From first: $$2(4) + y = -1 \implies 8 + y = -1 \implies y = -9$$ Point (4,-9). Answer C. 19. **Equation of the shown line (diagram needed).** Likely A y - x = 5. 20. **Equation of the shown line (diagram needed).** Likely A 2y - 5x = -18.