25 O y x C (0, 1) P (3, 5) NOT TO SCALE The diagram shows a circle centre C (0, 1). P (3, 5) is a point on the circumference of the circle. Find the equation of the tangent at P. … [4]

25 4y + 3x = 29 oe final answer 4 M1 for [grad CP] = 5 1 3 0 − − oe M1FT for [grad tangent] = − ¾ or − 1 / their gradient CP M1 for substitution of (3,5) into y = their mx + c oe To earn these marks, you must find the equation of the tangent; marks are awarded for correctly calculating the gradient of CP, then correctly finding the perpendicular gradient for the tangent, and finally substituting the point P into the tangent equation to find the y-intercept.

3 (a) Complete the table for y x x 2 3 1 3 = - - . x 3 - 2 - 1 - 0 1 2 3 y 1 1.5 1 - .3 5 - 3 - 3.5 [1] (b) On the grid, draw the graph of y x x 2 3 1 3 = - - for x 3 3 G G - . 0 – 1 – 2 – 3 – 4 – 5 – 6 6 5 4 3 2 1 1 2 3 – 3 – 2 – 1 x y [3] (c) Use your graph to explain why x x 6 2 6 3 - - = has only one solution. … [2] (d) Line L passes through the points (1, 1) and ( 2 - , 1 - ). (i) On the grid, draw line L. [1] (ii) Work out the gradient of line L. … [2] (iii) Find the x‑coordinates of the points where line L intersects the curve y x x 2 3 1 3 = - - . x = … , x = … , x = … [2]

3(a) –5.5 or – 1 5 2 or –11 2 1 3(b) Correct smooth curve 3 B2FT for 6 or 7 points correctly plotted or B1FT for 4 or 5 points correctly plotted 3(c) Line y = 3 only intersects the graph once oe 2 M1 for 3 3 1 3 2 − −= x x soi or y = 3 soi 3(d)(i) Ruled line through (1, 1) and (–2, –1) 1 3(d)(ii) 2 3 nfww 2 M1 for gradient = 1 1 1 2 + + oe 3(d)(iii) FT reading three x-values where their L intersects their curve 2 B1FT for two correct (a) You earn this mark by correctly calculating the y-value for x = -3. (b) To get these marks, you must draw a smooth curve that accurately passes through at least 6 or 7 (for B2) or 4 or 5 (for B1) of the correctly plotted points from the table. (c) You need to explain that the equation simplifies to y=3, and this horizontal line intersects the drawn graph only once. (d)(i) You earn this mark by drawing a straight, ruled line that passes through both given points. (d)(ii) To get these marks, you must correctly calculate the gradient of line L using the coordinates of the two given points. (d)(iii) You need to read and state the x-coordinates of the three points where your drawn line L intersects your drawn curve.

6 (a) PQR is an isosceles triangle with PR = QR. P is the point (1, 5) and Q is the point (5, 1). Angle PRQ is not a right angle. Find the coordinates for one possible position of R. You may use the grid to help you. y x O ( … , … ) [2] (b) Here are the equations of five curves. y x 2 2 = - y x 2 3 = - y x x 2 8 2 = + - y x x 3 3 = - y x x 3 2 = - Sketches of three of these curves are drawn below. Write the correct equation underneath each sketch. y x O y x O y x O … … … [3] (c) A is the point (-1, -5) and B is the point (3, 3). Find the equation of the line perpendicular to AB which passes through the midpoint of AB. … [5]

6(a) (a, a) with a ≠ 1, 3 or 5 2 SC1 for answer (1, 1) or (5, 5) 6(b) [y =] x2 – 3x [y =] 2 – x2 [y =] x3 – 2 cao 3 B1 for each 6(c) y = 1 1 2 2 − − x oe nfww 5 M1 for [gradient AB =] 3 ( 5) 3 ( 1) −− −− oe M1 for [gradient M =] 1 gradient −their AB B1 for (1, –1) M1 for their (1, –1) substituted into y = their 1 2   −     x + c oe (a) To earn marks, you must provide coordinates (a, a) where 'a' is any value other than 1, 3, or 5, indicating a point on the line y=x that is equidistant from P and Q. (b) To earn marks, you must correctly match each of the three given sketches to its corresponding equation from the list. (c) To earn marks, you must first calculate the gradient of line AB, then find the negative reciprocal for the perpendicular gradient. You also need to find the midpoint of AB. Finally, substitute the midpoint coordinates and the perpendicular gradient into the equation y = mx + c to find 'c' and write the full equation.

7 (a) P is the point ( 5 - , 2), Q is the point (3, 7) and QR 4 6 = - e o . (i) Find the coordinates of the midpoint of PQ. ( … , … ) [1] (ii) Find the coordinates of point R. ( … , … ) [1] (iii) Find QR . QR = … units [2] (b) NOT TO SCALE b a C A N O B OACB is a quadrilateral and N is a point on AB. OA a = and OB b = . OA BC 2 = and BN : NA = 1 : 3. Find, in terms of a and b, in its simplest form (i) AB, AB = … [1] (ii) NC. NC = … [3]

7(a)(i) (–1, 4.5) 1 7(a)(ii) (–1, 13) 1 7(a)(iii) 7.21[1...] 2 M1 for (–4)2 + 62 oe 7(b)(i) b – a 1 7(b)(ii) 1 1 4 4 + a b or ( ) 1 4 + a b 3 M1 for correct vector route along the lines of the diagram B1 for 2 =  BC a soi or for ( ) 1 4 = −  NB their b a soi or ( ) 3 4 = −  NA their a b soi

8 A is the point (-2, 3) and B is the point (4, 5). (a) Find the coordinates of the midpoint of AB. (… , …) [1] (b) Show that the equation of line AB is y x 3 11 = + . [3] (c) Find the equation of the perpendicular bisector of line AB. … [3]

8(a) (1, 4) 1 8(b) Gradient = 5 3 4 2 − −− oe M1 1 5 4 3   = × +     their c oe M1 (4, 5), (–2, 3) or their midpoint substituted into 1 3   = × +     y their x c oe Rearrangement to 11 3 = c and hence showing 3y = x + 11 A1 or expansion to 3y – 15 = x – 4 with completion or expansion to 3y – 9 = x + 2 with completion or expansion to 3y – 12 = x – 1 with completion With no errors or omissions in working 8(c) y = –3x + 7 3 B1FT for gradient = 1 (b) gradient −their M1 for substituting their (1, 4) into y = their (–3)x + c oe (a) To get the mark, correctly calculate the midpoint coordinates using the given points. (b) You need to first calculate the gradient of line AB, then substitute one of the points (or the midpoint) and the gradient into the equation y=mx+c to find c, and finally rearrange the equation to match the given form, showing all steps without errors. (c) To find the equation of the perpendicular bisector, first find the negative reciprocal of the gradient of AB, then use the midpoint of AB and this new gradient to find the y-intercept (c) for the perpendicular bisector's equation.

Cambridge O Level Mathematics (Syllabus D) · 4024

O-Level Mathematics (Syllabus D) (4024) Coordinate geometry past paper questions

Revise O-Level Mathematics (Syllabus D) (4024) — Coordinate geometry with real Cambridge past-paper questions grouped by topic, each shown with its marking scheme. Working one topic at a time targets exactly the skill each question tests.

44 free questions from 2016–2021, each with its marking scheme. Open the full topical bank to filter by subtopic, paper, and year.