(a) Draw a line through point P that is perpendicular to line l.

(b) Write down the mathematical names for two different quadrilaterals with two lines of symmetry and rotational symmetry of order two.

(c) The diagram shows a quadrilateral on a 1 cm² grid.

Find the area of this quadrilateral.

(d) The diagram shows a quadrilateral DEFG and a straight line FGH, with angle DEF=82°, angle EFG=103°, angle FGD=143°, angle DGH=x°, and angle GDE=y°.

1. (i) Write down the mathematical name for angle DEF (82°).
2. (ii) Work out the value of x. Give a geometrical reason for your answer.
3. (iii) Work out the value of y. Give a geometrical reason for your answer.

Show solution

1. (a) Method: Place a set square (or use compasses) to construct a line through P that crosses line l at exactly 90°. With compasses: open to any radius, draw two arcs centred at P crossing l on each side; from those two crossing points, draw two more arcs (same radius, larger than half their separation) that intersect above/below P; the line joining that intersection through P is perpendicular to l.
2. (b) Shapes with exactly two lines of symmetry and rotational symmetry of order 2 include the rectangle (non-square) and the rhombus (non-square).
   Rectangle and Rhombus
3. (c) Using the grid squares (each 1 cm²), split the quadrilateral into simpler shapes (e.g. a triangle and a trapezium) or use the shoelace/counting-squares method to find the enclosed area.
4. (d)(i) An angle between 0° and 90° is called an acute angle
5. (d)(ii) Since FGH is a straight line, angle FGD and angle DGH (=x°) lie on a straight line and sum to 180°: x = 180−143 = 37°, because angles on a straight line sum to 180°.
6. (d)(iii) The interior angles of any quadrilateral sum to 360°: 82+103+143+y = 360 → 328+y=360 → y = 32°, because the angle sum of a quadrilateral is 360°.

Please double-check against the original diagram: part (c) requires reading the exact vertex coordinates of the quadrilateral off the printed grid, which can’t be confirmed precisely from the extracted text alone. Please count grid squares (or use the shoelace formula on the exact vertex coordinates) directly from the PDF to confirm the area before sharing with students.

(a) Fuel prices: Garage A = $1.41 per litre, Garage B = $1.50 per litre.

1. (i) Tiya buys 55 litres of fuel from garage A. Work out the change she receives from $100.
2. (ii) Work out how much cheaper it is to buy 20 litres of fuel from garage A than from garage B.
3. (iii) These are the amounts that 6 people spend on fuel at garage A: $63, $84.50, $72.23, $46, $54.10, $80. Calculate the mean number of litres that they buy.
4. (iv) The cost of fuel at garage B increases from $1.50 to $1.53. Calculate the percentage increase.

(b) The fuel tank of a car is ⅖ full. It takes 39 more litres of fuel to fill the tank. Work out the number of litres of fuel in a full tank.

(c) Use 1 litre = 0.22 gallons.

1. (i) Complete this conversion graph.
2. (ii) Complete this statement: 1 gallon = ………… litres.

(d) A cylindrical tank for storing fuel has radius 1.5 metres and height 8 metres. Calculate the volume of the tank in litres.

(a) In triangle DEF, DE=6 cm and DF=4.8 cm. Using a ruler and compasses only, construct triangle DEF. Leave in your construction arcs. The line EF has been drawn for you.

(b) The grid shows triangle T and six other labelled triangles A–G.

1. (i) Write down the letter of the triangle that is congruent to triangle T.
2. (ii) Write down the letter of the triangle that is similar but not congruent to triangle T.

(c) The diagram shows an isosceles triangle with base 7 cm and base angles 62° each, and perpendicular height h.

1. (i) Show that the perpendicular height, h, is 6.58 cm, correct to 3 significant figures.
2. (ii) Calculate the area of the triangle. Give the units of your answer.
3. (iii) Kalpit tries to arrange some of these triangles to make a regular polygon with centre O. Show that Kalpit cannot make a regular polygon.

Show solution

1. (a) Method: Open compasses to 6 cm and draw an arc centred at E (since DE=6 cm, point D lies on this arc). Open compasses to 4.8 cm and draw an arc centred at F (since DF=4.8 cm, D also lies on this arc). Where the arcs cross is point D. Join DE and DF, leaving all arcs visible.
2. (b)(i) The triangle congruent to T is the same size and shape (just possibly rotated or translated).
3. (b)(ii) The triangle similar but not congruent to T has the same angles/shape but is a different (larger or smaller) size.
4. (c)(i) Dropping a perpendicular from the apex bisects the base into two 3.5 cm halves. In the resulting right triangle, tan(62°) = h÷3.5, so h = 3.5 × tan(62°) = 3.5 × 1.8807 = 6.5825, which rounds to 6.58 cm ✓
5. (c)(ii) Area = ½ × base × height = ½ × 7 × 6.5825 = 23.0 cm² (3 s.f., using the unrounded height)
6. (c)(iii) The angle at the apex of each triangle (opposite the 7 cm base) = 180−62−62 = 56°. For several of these triangles to fit exactly around point O with no gaps or overlaps, the apex angles must divide exactly into 360°.
   360 ÷ 56 = 6.43 (not a whole number), so the triangles cannot fit exactly around O — Kalpit cannot make a regular polygon.

Please double-check against the original diagram: part (b) requires visually comparing the exact size and orientation of triangles A–G against T on the printed grid, which can’t be confirmed from the extracted text alone. Please verify directly which lettered triangle matches T’s exact dimensions (congruent) and which is a scaled copy (similar) before sharing with students.

(a) A shop sells 58 televisions in one week. The bar chart shows the number sold on five of the days: Monday=3, Tuesday=4, Thursday=4, Saturday=16, Sunday=11.

1. (i) Write down the number of televisions that the shop sells on Monday.
2. (ii) Find the fraction of the televisions that the shop sells on Sunday.
3. (iii) The number sold on the other two days is in the ratio Wednesday:Friday = 2:3. Complete the bar chart.
4. (iv) Write down the mode.

(b) A television has a price of $550. This price is reduced by 4%. Calculate the new price of this television.

(c) The scatter diagram shows the prices of different sized televisions.

Write down the type of correlation shown in the scatter diagram.

(d) Hemang buys two televisions. The probability that a television is faulty is 0.02.

1. (i) Complete the tree diagram.
2. (ii) Find the probability that Hemang buys two faulty televisions.
3. (iii) The shop sells 4150 televisions in one year. Calculate the expected number of faulty televisions.

Show solution

1. (a)(i) Reading directly from the bar chart: 3
2. (a)(ii) Sunday sold 11 out of 58 total: 11/58
3. (a)(iii) Known days total: 3+4+4+16+11 = 38. Remaining for Wednesday+Friday = 58−38 = 20.
   Ratio 2:3 has 5 total parts, so 1 part = 20÷5 = 4. Wednesday = 2×4 = 8, Friday = 3×4 = 12
4. (a)(iv) Full week’s data: 3,4,8,4,12,16,11. The value 4 occurs twice (Tuesday and Thursday) — more than any other value: mode = 4
5. (b) 550 × (1−0.04) = 550 × 0.96 = $528
6. (c) As television size increases, price also increases: positive correlation
7. (d)(i) P(faulty)=0.02 so P(not faulty)=0.98 on every branch (since these probabilities don’t change between televisions): all four missing branches = 0.98, 0.02, 0.98, 0.02 (not faulty first branch = 0.98, then faulty=0.02/not faulty=0.98 on each second-television branch)
8. (d)(ii) P(both faulty) = 0.02 × 0.02 = 0.0004
9. (d)(iii) Expected faulty = 4150 × 0.02 = 83

(a) (i) Complete the table of values for y = −x² + 5x + 7.

(ii) On the grid, draw the graph of y = −x² + 5x + 7 for −1 ≤ x ≤ 6.

(iii)(a) Write down the equation of the line of symmetry of the graph.
(b) The points (−8, −97) and (t, −97) also lie on the graph. Use symmetry to find the value of t.

(b) Write down the gradient of the line y=9x−4.

(c) Write down the equation of a line parallel to y=−5x+19.

(d) The diagram shows line L, passing through (−4,6) and (4,−2).

Find the equation of line L in the form y=mx+c.

(e) Make x the subject of the formula y=mx+c.

Show solution

1. (a)(i) Substituting each x-value into −x²+5x+7:

   x	−1	0	1	2	3	4	5	6
   y	1	7	11	13	13	11	7	1

2. (a)(ii) Plot the eight points and join with a smooth curve (an upside-down parabola/dome shape).
3. (a)(iii)(a) The vertex of y=−x²+5x+7 occurs at x = −5÷(2×−1) = 2.5. Line of symmetry: x = 2.5
4. (a)(iii)(b) Since both points share the same y-value, they must be symmetric about x=2.5: t = 2(2.5) − (−8) = 5+8 = 13
5. (b) For y=mx+c, the gradient is m: 9
6. (c) Parallel lines share the same gradient: y = −5x + c (any value of c other than 19, e.g. y = −5x + 1)
7. (d) Using points (−4,6) and (4,−2): gradient = (−2−6)÷(4−(−4)) = −8÷8 = −1.
   The line crosses the y-axis at (0,2): y = −x + 2
8. (e) y=mx+c → y−c=mx → x = (y − c) / m

(a) Town S is 44 km from town R on a bearing of 117°.

1. (i) Using a scale of 1 cm represents 8 km, mark the position of town S.
2. (ii) Anvi cycles the 44 km from R to S. She leaves R at 13 15 and cycles at a speed of 12 km/h. Work out the time she arrives at S.

(b) A tower has a height of 16 metres. When Jai makes a scale drawing of the tower it has a height of 20 cm. Work out the scale Jai uses, giving your answer in the form 1:n.

(c) X, Y and Z are three towns. X is on a bearing of 288° from Y. Z is on a bearing of 018° from Y.

1. (i) Show that angle XYZ is 90°.
2. (ii) XY=6 km and YZ=9.7 km. Calculate XZ.

Show solution

1. (a)(i) Convert 44 km to the scale drawing length: 44÷8 = 5.5 cm. Using a protractor at R, measure 117° clockwise from North, then draw a line 5.5 cm long in that direction and label the end S.
2. (a)(ii) Time = distance÷speed = 44÷12 = 3.6667 hours = 3 hours 40 minutes.
   Arrival time = 13:15 + 3:40 = 16:55
3. (b) Convert to the same units: 16 m = 1600 cm. Scale = 20:1600, dividing both by 20: 1 : 80
4. (c)(i) The angle at Y between the two bearings can be found using: 360° − 288° + 18° = 90° (going from the 288° bearing round through North to the 18° bearing). So angle XYZ = 90° ✓
5. (c)(ii) Since angle XYZ=90°, triangle XYZ is right-angled at Y, so use Pythagoras: XZ² = XY²+YZ² = 6²+9.7² = 36+94.09 = 130.09
   XZ = √130.09 = 11.4 km (3 s.f.)

(a) P=3a+5. Find the value of P when a=2.

(b) Solve these equations:

1. (i) 7x=−42
2. (ii) 9(8x−7)=72

(c) 5⁸ × 5^k = 5⁻²⁴. Find the value of k.

(d) Solve the simultaneous equations: −6x−y=13 and 8x+y=−51.

(e) n is an integer where n > −3 and n ≤ 1. Write down all the possible values of n.

(f) A boy walks for 35 minutes at x metres per minute. He then runs for t minutes at 160 metres per minute. Write down an expression, in terms of x and t, for the total distance, in metres, the boy travels.

(g) A rectangle has sides (x−2) and (x+5).

Find an expression for the area of this rectangle. Give your answer in the form x²+ax+b.

(a) 120 people teach in a university mathematics department. One fifth of the people are professors. 30% of the people are part-time. 11 of the part-time staff are professors.

Work out the number of full-time lecturers.

(b) 𝒰 = {children in a school}, F = {children who like fruit}, V = {children who like vegetables}. 24 children like vegetables but do not like fruit. 8 children do not like fruit and do not like vegetables. n(F∩V)=9. n(F)=3×n(V).

1. (i) Complete the Venn diagram.
2. (ii) Work out n(F∪V).