12 questions · 100 marks · ≈ 2 hours
Calculator allowed · Answer all questions · Show all your working
A = 23 × 32 × 5 × 7 B = 2 × 34 × 52 × 11
Find the highest common factor (HCF) of A and B.
Find the lowest common multiple (LCM) of A and B.
Find the least number that A must be multiplied by to give a square number. C = 3 × 10205 × 5 × 10205
Work out the value of C, giving your answer as a number in standard form.
The points with coordinates (2, 1), (5, 1), (3, 3) and (6, 3) are the vertices of parallelogram A.
On the grid, draw and label parallelogram A. ⎛ 2⎞ Parallelogram A is transformed to parallelogram B under the translation −5⎠
On the grid, draw and label parallelogram B. Parallelogram B is transformed to parallelogram C by a reflection in the line with equation x = 2
On the grid, draw and label parallelogram C. Parallelogram C is transformed to parallelogram D by a rotation through 90° anticlockwise about the point with coordinates (–5, 0)
On the grid, draw and label parallelogram D.
Diagram NOT accurately drawn O B 25° C A 15 cm Figure 1 In Figure 1, A and B are two points on a circle with centre O. The line AC is the tangent to the circle at A and OBC is a straight line. AC = 15 cm and ∠OCA = 25°
Calculate the area, in cm² to 3 significant figures, of the circle. The region inside triangle OAC but outside the circle is shown shaded in Figure 1
Calculate the area, in cm² to 3 significant figures, of this shaded region.
In a race, Akaash ran a distance of (8x2 – 105) metres in (6x + 1) seconds. His average speed for the race was x m/s.
Show that x satisfies the equation 2x2 – x – 105 = 0
Hence calculate the distance, in metres, that Akaash ran in the race. Show clear algebraic working.
When Joe goes to school in winter, the probability that he wears a coat is If he wears a coat, the probability that he wears a scarf is If he does not wear a coat, the probability that he wears a scarf is
Complete the probability tree diagram. Scarf Coat No scarf Scarf No coat No scarf On a day Joe goes to school in winter, calculate the probability that
he is not wearing a coat and is not wearing a scarf …………… , ……………
he is wearing a coat or he is wearing a scarf but he is not wearing both a coat and a scarf. On a day Joe goes to school in winter, if he is wearing a coat and a scarf then the probability that he is also wearing a hat is
Calculate the probability, that on a day Joe goes to school in winter, he is not wearing all three of a coat, a scarf and a hat.
f and g are two functions such that f : x ↦ x² + x – 6 where x ≥ − 2 x − 24 g : x ↦ 3 − 2x
State the value of x that must be excluded from any domain of g
Find the value of a and the value of b such that f (x) = (x + a)² + b
Hence write down the range of f
Find the value of c for which g–1(0) = c Given that f (x) = g (x)
show that 2x3 – x² – 13x – 6 = 0 Show clear algebraic working.
Use the factor theorem to show that (2x + 1) is a factor of 2x3 – x² – 13x – 6 The curve with equation y = f (x) intersects the curve with equation y = g (x) at the point A with coordinates ( p, q), where p > −
Find the coordinates of A.
Diagram NOT accurately drawn 2h cm 4 cm Figure 2 Figure 2 shows a right circular cylinder cut from a solid sphere of radius 4 cm. The height of the cylinder is 2h cm. The volume of the cylinder is V cm³
Show that V = 32πh – 2π
Using calculus, find the stationary value, to 3 significant figures, of V. Show your working clearly. ⎡⎣ Volume of cylinder = πr 2 h ⎤⎦ Part of the curve with equation V = 32πh – 2πh3 is drawn on the grid. V 160 150 140 130 120 110 100 O h 0.5 1 1.5 2 2.5 3 3.5 4
Figure not shown in this reconstruction — the printed paper includes a diagram (not drawn accurately). Use the text description to work through it.
Find an estimate, to one decimal place, of each of the two values of h for which the volume of the cylinder is 100
…………… cm³
C x metres D Diagram NOT accurately drawn A B y metres F E Figure 3 Figure 3 shows the plan of a floor ABCDEF in which all the corners are right angles and CD = x metres AF = y metres The length of DE is twice the length of CD. The length of FE is twice the length of AF. The perimeter of the floor plan is 68 metres.
Find and simplify an equation in terms of x and y for this information. The area of the floor plan is 248
…………… m²
Show that 2y 2 + 2x2 – xy = 248
Find the possible lengths of AB. Show clear algebraic working.
A=⎜ B= −1 2 2 −2 The transformation with matrix A is equivalent to the transformation with matrix B followed by the transformation with matrix C, where C is a 2 × 2 matrix. Find matrix C. ⎡ ⎛ a b⎞ 1 ⎛ d − b⎞ ⎤ ⎢The inverse of matrix ⎜ is ⎥ ⎣ ⎝ c d ad − bc −c a
Triangle ABC has sides AB = 10 cm, BC = 7 cm and CA = 6 cm.
Calculate the area, in cm² to 3 significant figures, of triangle ABC. C Diagram NOT accurately drawn 6 cm 7 cm A B 10 cm Figure 4 Figure 4 shows triangle ABC and a circle inside the triangle so that each side of the triangle is a tangent to the circle.
Calculate the radius, in cm to 2 significant figures, of the circle. ⎡Cosine rule: = + − 2bc cos A⎤ ⎢ ⎥ ⎢ 1 ⎥ Area of triangle = ab sin C ⎢⎣ 2 ⎦⎥
Find the range of values of x for which both 2x + 3(4 – 3x) < 8x and 6x – 5 ≤ (2x – 3)² Show clear algebraic working.
A company produces bird food made with sunflower seeds, corn and millet. The weights of sunflower seeds, corn and millet in each bag of the bird food are in the ratios sunflower seeds : corn : millet = 8 : 3 : 1 The total weight of the bird food in each bag is 30 kg.
Calculate the weight, in kg, of corn in each bag of the bird food. Tom buys bags of the bird food from the company and sells them in his shop. He sells each bag for $15.04 He makes a profit of 17.5% on each bag he sells.
Calculate, in $, how much Tom pays the company for each bag of the bird food he buys. The bags are filled with the bird food at a rate of 530 grams per second.
Calculate the number of bags of the bird food that are filled completely in 6 hours. Sunflower seeds cost £150.75 for 30 kg from farmer A. Sunflower seeds cost $162.35 for 25 kg from farmer B. Using an exchange rate of £1 = $1.315
find which of the two farmers, A or B, sells sunflower seeds at the cheaper cost per kilogram. You must give a reason for your answer.