Cambridge CIE IGCSE Past Papers and solutions.

Questions and Worked Solutions for IGCSE 2020 0580/04 Specimen Paper 4.

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IGCSE 2020 0580/04 Specimen Paper (pdf)

- (a) Kristian and Stephanie share some money in the ratio 3 : 2.

Kristian receives $72.

(i) Work out how much Stephanie receives.

(ii) Kristian spends 45% of his $72 on a computer game.

Calculate the price of the computer game.

(iii) Kristian also buys a meal for $8.40 .

Calculate the fraction of the $72 Kristian has left after buying the computer game and the meal.

Give your answer in its lowest terms.

(iv) Stephanie buys a book in a sale for $19.20 .

This sale price is after a reduction of 20%.

Calculate the original price of the book.

(b) Boris invests $550 at a rate of 2% per year simple interest.

Calculate the value of the investment at the end of 10 years.

(c) Marlene invests $550 at a rate of 1.9% per year compound interest.

Calculate the value of the investment at the end of 10 years.

(d) Hans invests $550 at a rate of x% per year compound interest.

At the end of 10 years, the value of the investment is $638.30, correct to the nearest cent.

Find the value of x. - (a) 200 students estimate the volume, Vm
^{3}, of a classroom.

The cumulative frequency diagram shows their results.

Use the graph to find an estimate of

(i) the median,

(ii) the interquartile range,

(iii) the 60th percentile,

(iv) the number of students who estimate that the volume is greater than 300m^{3}

(b) The 200 students also estimate the total area, Am^{2}, of the windows in the classroom.

The table shows their results.

(i) Calculate an estimate of the mean.

You must show all your working.

(ii) Complete the histogram to show the information in the table.

(iii) Two students are chosen at random from those students that estimated the area of the windows to be more than 100m^{2}.

Find the probability that one of the two students estimates the area to be greater than 150m^{2}and the other student estimates the area to be 150m^{2}or less.

- f(x) = 20/x + x, x ≠ 0

(a) Complete the table.

(b) On the grid, draw the graph of y = f(x) for –10 ≤ x ≤ –1.6 and 1.6 ≤ x ≤ 10.

(c) Using your graph, solve the equation f(x) = 11.

(d) k is a prime number and f(x) = k has no solutions.

Find the possible values of k.

(e) The gradient of the graph of y = f(x) at the point (2, 12) is −4.

Write down the coordinates of the other point on the graph of y = f(x) where the gradient is −4.

(f) (i) The equation f(x) = x^{2}can be written as x^{3}+ px^{2}+ q = 0.

Show that p = −1 and q = −20.

(ii) On the grid opposite, draw the graph of y = x^{2}for –4 ≤ x ≤ 4.

(iii) Using your graphs, solve the equation x^{3}– x^{2}– 20 = 0.

(iv) The diagram shows a sketch of the graph of y = x^{3}– x^{2}– 20.

P is the point (n, 0).

Write down the value of n. - (a) (i) Draw the reflection of triangle T in the line x = 0.

(ii) Draw the rotation of triangle T about (–2, –1) through 90° clockwise.

(b) Describe fully the single transformation that maps triangle T onto triangle U. - (a) The perimeter of the rectangle is 80cm.

The area of the rectangle is Acm^{2}.

(i) Show that x^{2}– 40x + A = 0.

(ii) When A = 300, solve the equation x^{2}– 40x + A = 0 by factorising.

(iii) When A = 200, solve the equation x^{2}– 40x + A = 0 using the quadratic formula.

Show all your working and give your answers correct to 2 decimal places.

(b) A car completes a 200km journey at an average speed of x km/h.

The car completes the return journey of 200 km at an average speed of (x + 10)km/h.

(i) Show that the difference between the time taken for each of the two journeys is 2000/(x(x+10)) hours.

(ii) Find the difference between the time taken for each of the two journeys when x = 80.

Give your answer in minutes and seconds. - OPQR is a rectangle and O is the origin.

M is the midpoint of RQ and PT : TQ = 2 : 1.

OP = p and OR = r.

(a) Find, in terms of p and/or r, in its simplest form

(i) MQ

(ii) MT

(iii) OT

(b) RQ and OT are extended and meet at U.

Find the position vector of U in terms of p and r.

Give your answer in its simplest form.

(c) Find the positive value of k. - f(x) = 2x + 1

g(x) = x^{2}+ 4

h(x) = 2x

(a) Solve the equation f(x) = g(1).

(b) Find f^{−1}(x).

(c) Find gf(x) in its simplest form.

(d) Solve the equation h^{−1}(x) = 0.5.

(e) 1/h(x)= 2^{kx}

Write down the value of k. - The grid shows the graph of y = cos x for 0° ≤ x ≤ 360°.

(a) Solve the equation 3cos x = 1 for 0° ≤ x ≤ 360°.

Give your answers correct to 1 decimal place.

(b) On the same grid, sketch the graph of y = sin x for 0° ≤ x ≤ 360°. - The diagram shows a trapezium ABCD.

AB is parallel to DC.

AB = 55m, BD = 70m, angle ABD = 40° and angle BCD = 32°.

(a) Calculate AD.

(b) Calculate BC.

(c) Calculate the area of ABCD.

(d) Calculate the shortest distance from A to BD. - (a) Show that the volume of a metal sphere of radius 15cm is 14140cm
^{3}, correct to 4 significant figures.

(The volume, V, of a sphere with radius r is V = 4/3 πr^{3})

(b) (i) The sphere is placed inside an empty cylindrical tank of radius 25cm and height 60cm.

The tank is filled with water.

Calculate the volume of water needed to fill the tank.

(ii) The sphere is removed from the tank.

Calculate the depth, d, of water in the tank.

(c) The diagram below shows a solid circular cone and a solid sphere.

The cone has radius 5x cm and height 12x cm.

The sphere has radius r cm.

The cone has the same total surface area as the sphere.

Show that r^{2}= 45/2 x^{2}

(The curved surface area, A, of a cone with radius r and slant height l is A = πrl.)

(The surface area, A, of a sphere with radius r is A = 4πr^{2}) - A curve has equation y = x
^{3}– 6x^{2}+ 16.

(a) Find the coordinates of the two turning points.

(b) Determine whether each of the turning points is a maximum or a minimum.

Give reasons for your answers.

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