Pearson Edexcel International GCSE (9–1) · MATHEMATICS A · Unit
Unit 9
105 practice questions
Topics covered
Practice questions
91–100 of 105
Question 91
7 marks · Easy
Answer and see the solution →A function is defined by . The graph of is shown below.
Graph of \( y = x^2 \ ) (a)[2 marks]Write down the equation of the curve obtained when the graph of is reflected in the y-axis.
(b)[2 marks]The point on has coordinates (2, 4). The point is the image of under a stretch, scale factor 3, parallel to the y-axis.
Work out the coordinates of point after this transformation.
(c)[3 marks]A cuboid has length 5 cm, width 2 cm, and height 4 cm. One diagonal passes from one vertex (corner) to the opposite vertex through the interior of the cuboid.
Using Pythagoras' theorem in 3D, calculate the length of this diagonal.
Question 92
7 marks · Easy
Answer and see the solution →A straight road runs up a hill and is shown on a diagram below from the side. The road runs from point A at (0, 1) to point B at (5, 7), where the coordinates are given in metres.
(a)[2 marks]Find the gradient of the road from A to B.
(b)[5 marks]A signpost is placed vertically at B and is 3 m tall. Calculate the angle between the signpost at B and the sloping road AB. Give your answer in degrees to 1 decimal place.
Question 93
8 marks · Easy
Answer and see the solution →A box is in the shape of a cuboid with dimensions 12 cm by 5 cm by 9 cm. It is used to store packets whose heights (in cm) are given below.
Packet height distribution Height, h (cm) Frequency 6 < h ≤ 7 5 7 < h ≤ 8 7 8 < h ≤ 9 4 (a)[3 marks]Work out the length of the diagonal from one corner of the cuboid to the opposite corner. Give your answer correct to 3 significant figures.
(b)[3 marks]Draw a histogram to show the distribution of packet heights. You must calculate the frequency density for each class.
(c)[2 marks]State which class interval has the highest frequency density and what this means about the distribution of packet heights.
Question 94
8 marks · Easy
Answer and see the solution →A regular square-based pyramid stands on ABCD, where each side of the base is 10 cm. The vertex, V, is 14 cm above the centre of the base. The masses (in g) of stones used to make several pyramids are grouped into classes as shown.
Masses of stone pyramids Mass, m (g) Frequency 200 < m ≤ 300 4 300 < m ≤ 500 9 500 < m ≤ 800 7 (a)[3 marks]Calculate the slant height of the pyramid (from the centre of one base edge to the vertex). Give your answer to 3 significant figures.
(b)[2 marks]Calculate the frequency density for each class in the mass table.
(c)[3 marks]Estimate the proportion of pyramids with mass greater than 500 g.
Question 95
8 marks · Easy
Answer and see the solution →A solid cuboid has sides 15 cm, 8 cm, and 6 cm. Its dimensions are measured to the nearest cm. A shop packs chocolates of two sizes. The company checks the distributions of their box heights with histograms, shown below.
Cuboid ABCDEFGH with AB = 15, BC = 8, BF = 6 Histogram showing heights of Small and Large boxes (frequency densities in arbitrary units) (a)[2 marks]Using the cuboid shown, explain how you would find the length of the main body diagonal from one corner to the farthest corner. State the formula in terms of the side lengths.
(b)[2 marks]Calculate the length of the body diagonal of the cuboid. Give your answer to 3 significant figures.
(c)[4 marks]Interpret the histogram above. Compare the number and spread of shorter versus taller boxes. What does the distribution tell you about the two products?
Question 96
9 marks · Normal
Answer and see the solution →(a) Rafik earns $135,000 per annum in the country of Kalculus (tax bands: 20% on the first $50,000, 40% on the rest).
(a)[3 marks]Calculate Rafik's annual income tax bill.
(b)[3 marks]The equation of a straight line is given by . Rafik's income after tax, as a function of his pre-tax salary , is linear above $50,000.
Write down an equation for the income after tax for all , using the figures from part (a).
(c)[3 marks]On a set of axes, sketch the graph of against for using your answer from (b). Clearly indicate the point at which the tax band changes.
Question 97
9 marks · Normal
Answer and see the solution →A supermarket sells two products in bulk and uses a mathematical model for pricing.
(a)[3 marks]Toffee A: 1.35 kg for $9.99. Toffee B: 2.1 kg for $15.82.
Work out the cost per kg for each and decide which is better value for money.
(b)[3 marks]The cost per kg, , for a pack weighing kg with total price dollars is given by .
If a special almond pack has weight and price dollars, show, by algebraic reasoning, that the cost per kg increases as decreases.
(c)[3 marks]A third product, caramel fudge, is priced so that for a pack of kg, the price is dollars.
The company claims it is never better value per kg than Toffee B for packs up to 4 kg. Is their claim correct? Justify your answer with calculations and reasoning.
Question 98
9 marks · Normal
Answer and see the solution →A worker in Kalculus has an annual salary given by dollars. The after-tax salary can be written as a function of , based on the income tax rates:
Kalculus Income Tax Bands Band Tax Rate $0 – $50,000 20% >$50,000 40% (on excess) (a)[3 marks]Write the piecewise function for after-tax salary , for all .
(b)[4 marks]A company wishes to set annual after-tax salaries between $40,000 and $80,000 for its staff.
If the company can only offer salaries in increments of $5,000, determine all possible gross salaries that would give after-tax salaries in the required range.
(c)[2 marks]Explain, using algebraic reasoning, why there are more possible gross salaries in the upper range ($>50,000) compared to the lower range ($\leq 50,000), for the same after-tax interval.
Question 99
10 marks · Normal
Answer and see the solution →In a country, Sergio earns an annual salary plus a bonus. The bonus is taxed at a higher rate than the salary. The total income tax Sergio pays is £8,600 in one year.
(a)[4 marks]Sergio receives a salary of £30,000 per year. He also receives a bonus of £12,000 in that year. The first £30,000 of his income is taxed at 20%, and any amount above £30,000 is taxed at 40%. Work out how much income tax Sergio should pay in total for the year.
(b)[6 marks]Let x be the salary (in £) and y be the bonus (in £) that produces exactly £8,600 tax for this tax system, given that x + y = £41,500. Using the rules above and your answer from part (a), show how to set up and solve a pair of simultaneous equations for x and y, and find the values of x and y.
Question 100
9 marks · Normal
Answer and see the solution →Amira is planning a trip and needs to exchange some money. In her country, £1 = $1.35. She exchanges £480 for dollars at this rate. After her trip, she has $216 left and wants to change this back to pounds at the same rate.
(a)[4 marks]Calculate how many dollars Amira receives for her £480. Then, work out how many pounds she will receive when changing $216 back into pounds at the same rate.
(b)[5 marks]Amira notices that the total number of pounds she gets back for her $216 is not in the same ratio as the pounds to dollars in the original exchange. Prove algebraically that, if no commission or rounding is involved and the rate is the same both ways, exchanging money and exchanging back always gives the original currency value (neglecting leftover cents or pence).