Edexcel Jan 2019 IGCSE 4MA1/1HR

Edexcel Jan 2019 IGCSE, 4MA1/1HR (pdf)

• Show Step-by-Step Solutions

1. Show that
   1 2/3 + 2 3/4 = 4 5/12
   
2. There are 60 children in a club.
   In the club, the ratio of the number of girls to the number of boys is 3:1
   3/5 of the girls play a musical instrument.
   4/5 of the boys play a musical instrument.
   What fraction of the 60 children play a musical instrument?
   
3. In triangle PQR,
   S is the point on PR such that angle RSQ = 90°
   RQ = 14 cm
   RS = 10 cm
   SP = 5 cm
   Work out the length of PQ
   
4. a, a, b and 40 are four numbers.
   a is the least number.
   40 is the greatest number.
   The range of the four numbers is 14
   The median of the four numbers is 30
   Work out the value of a and the value of b

5. The Shanghai Maglev Train takes 8 minutes to travel a distance of 30.5 kilometres.
   Work out the average speed of the train.
   Give your answer in kilometres per hour
   
6. The diagram shows the triangle PQR
   In the diagram, all the angles are in degrees.
   RP = RQ
   Find the value of y.
   Show clear algebraic working
   
7. The diagram shows two water towers in Kuwait.
   The real height of tower A is 187 m.
   The real height of tower B is 147 m.
   Ahmed makes a scale model of bot towers.
   The height of tower A on the scale model is 90 cm.
   Work out the height of tower B on the scale model.
   Give your answer correct to the nearest centimetre.
   
8. Solve the simultaneous equations
   4x + 2y = 9
   x - 4y = 9
   Show clear algebraic working.
   
9. N = 480 x 10⁹
   (a) Write N as a number in standard form.
   (b) Write N as a product of powers of its prime factors.
   Show your working clearly
   (c) Find the largest factor of N that is an odd number
   
10. The shape, shown shaded in the diagram, is the region between two semicircles.
    The diameter of the outer semicircle is 12 cm.
    The shape has constant thickness 2 cm.
    Calculate the area of the shape.
    Give your answer as a multiple of π
    
11. There are 12 boys and 8 girls in a class.
    The boys and the girls have some coins.
    The mean number of coins that the boys have is 5.5
    The girls have a total of 18 coins.
    Work out the mean number of coins the 20 children have
    Here are the first four terms of a sequence of fractions.
    1/1 2/3 3/5 4/7
    The numerators of the fractions form the sequence of whole numbers 1 2 3 4 …
    The denominators of the fractions form the sequence of odd numbers 1 3 5 7 …
    (a) Write down an expression, in terms of n, for the nth term of this sequence of fractions
    (b) Using algebra, prove that when the square of any odd number is divided by 4 the remainder is 1
    
12. A curve C has equation y = x³ – x² – 8x + 12
    (a) Find dy/dx
    The curve C has two turning points.
    (b) Work out the x coordinates of the two turning points.
    Show your working clearly
    (c) Show that the x-axis is a tangent to the curve C.
    
13. The cumulative frequency diagram gives information about the values, in thousands of euros, of 120 apartments in 2015
    (a) Find an estimate for the number of these apartments with a value of 80 thousand euros or less in 2015
    The table gives information about the values, in thousands of euros, of the same 120 apartments in 2018
    (b) On the grid opposite, draw a cumulative frequency diagram for this information.
    (c) Find an estimate for the increase in the median value for these apartments from 2015 to 2018
    
14. (a) Simplify (3x²y⁵)⁴
    (b) Expand and simplify 4n(n – 3)(n + 5)
    (c) Factorise 4c² – 9d²
    (d) Simplify fully (x² - 7x + 12)/(4x - x²)
    
15. There are 12 beads in a bag.
    7 of the beads are red.
    3 of the beads are green.
    2 of the beads are yellow.
    Lucy takes at random a bead from the bag and keeps it.
    Then Julian takes at random a bead from the bag.
    (a) Work out the probability that they each take a yellow bead
    (b) Work out the probability that the beads they take are not the same colour
    
16. Here are a solid sphere and a solid cylinder.
    The radius of the sphere is r cm.
    The radius of the cylinder is r cm.
    The height of the cylinder is 2r cm,
    The total surface area of the cylinder is kπ cm²
    (a) Find an expression for k in terms of r.
    (b) Show that the ratio
    total surface area of the cylinder:total surface area of the sphere
    is the same as the ratio
    volume of the cylinder:volume of the sphere
    
17. Show that √8/(√8 - 2) can be written in the form n + √n , where n is an integer.
    Show your working clearly
    
18. B, C, D and E are points on a circle.
    AB is the tangent at B to the circle.
    AB is parallel to ED.
    Angle ABE = 73°
    Work out the size of angle DCE.
    Give a reason for each stage of your working
    
19. Here is a cube ABCDEFGH
    M is the midpoint of the edge GH.
    Find the size of the angle between the line MA and the plane ABCD.
    Give your answer correct to 1 decimal place.
    
20. Here is a triangle XYZ
    The perimeter of the triangle is k cm.
    Given that x = y − 1
    find the value of k.
    Show your working clearly
    
21. ABCDEF is a regular hexagon.
    ABX and DCX are straight lines.
    AB = a
    BC = b
    Find EX in terms of a and b.
    Give your answer in its simplest form
    
22. The function f is defined as f(x) = for x > 0 and where k is a positive number.
    (a) Find the value of p for which f⁻¹(p) = k
    The function g is defined as g(x) = x² for x > 0
    (b) Given that gf(a) = k for k > 1
    find an expression for a in terms of k.

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