GCSE Maths Higher: Paper 3 Preparation Questions

Q1. Solve x2 > 3x + 4

...........................................................

(Total for question = 3 marks)Q2. 

(i)  Sketch the graph of f(x) = x2 − 5x + 10, showing the coordinates of the turning point and the coordinates of any intercepts with the coordinate axes.

(ii)  Hence, or otherwise, determine whether f(x + 2) − 3 = 0 has any real roots. Give reasons for your answer.

 

(Total for question = 6 marks)Q3. The functions f and g are such that

f(x) = 1 − 5x      and      g(x) = 1 + 5x

(a) Show that gf(1) = − 19

(2)(b)  Prove that f−1(x) + g−1(x) = 0 for all values of x.

(3)(Total for question = 5 marks)

Q4. In triangle RPQ,

RP = 8.7 cm PQ = 5.2 cm Angle PRQ = 32°

(a)  Assuming that angle PQR is an acute angle, calculate the area of triangle RPQ. Give your answer correct to 3 significant figures.

...........................................................cm2

(4)

(b)  If you did not know that angle PQR is an acute angle, what effect would this have on your calculation of the area of triangle RPQ?

.............................................................................................................................................

.............................................................................................................................................

.............................................................................................................................................

(1) 

(Total for question = 5 marks)Q5. 

Sami asked 50 people which drinks they liked from tea, coffee and milk.

All 50 people like at least one of the drinks 19 people like all three drinks. 16 people like tea and coffee but do not like milk. 21 people like coffee and milk. 24 people like tea and milk. 40 people like coffee. 1 person likes only milk.

Sami selects at random one of the 50 people.

(a)  Work out the probability that this person likes tea.

...........................................................

(4)

(b)  Given that the person selected at random from the 50 people likes tea, find the probability that this person also likes exactly one other drink.

...........................................................

(2)(Total for question = 6 marks)

Q6. 

Karol runs in a race.

The graph shows her speed, in metres per second, t seconds after the start of the race.

(a)  Calculate an estimate for the gradient of the graph when t = 4       You must show how you get your answer.

...........................................................

(3)

(b)  Describe fully what your answer to part (a) represents.

.............................................................................................................................................

.............................................................................................................................................

(2)(c)  Explain why your answer to part (a) is only an estimate.

.............................................................................................................................................

.............................................................................................................................................

(1) 

(Total for question = 6 marks)Q7. Here is a Venn diagram.

(a)  Write down the numbers that are in set

(i)  A ∪ B ...........................................................(ii)  A ∩ B ..........................................................

(2)One of the numbers in the diagram is chosen at random.

(b)  Find the probability that the number is in set A'

...........................................................

(2) 

(Total for question = 4 marks)

Q8. 

The graph shows information about the velocity, v m/s, of a parachutist t seconds after leaving a plane.

(a)  Work out an estimate for the acceleration of the parachutist at t = 6

........................................................... m/s2

(2)(b)  Work out an estimate for the distance fallen by the parachutist in the first 12 seconds after leaving the plane. Use 3 strips of equal width.

........................................................... m

(3)(Total for question is 5 marks)

Q9. (a)  Show that the equation 3x2 − x3 + 3 = 0 can be rearranged to give

(2)

(b)  Using

find the values of x1, x2 and x3

...........................................................

(3)

(c)  Explain what the values of x1, x2 and x3 represent.

.............................................................................................................................................

.............................................................................................................................................

(1)

 

(Total for question is 6 marks)Q11. Bob asked each of 40 friends how many minutes they took to get to work.

The table shows some information about his results.

Time taken (m minutes) Frequency

0 < m ≤ 10 3 10 < m ≤ 20 8 20 < m ≤ 30 11 30 < m ≤ 40 9 40 < m ≤ 50 9

Work out an estimate for the mean time taken. 

. . . . . . . . . . . . . . . . . . . . . . minutes

(Total for Question is 4 marks)

Q10. The graph of y = f(x) is shown on the grid.

(a)  On the grid above, sketch the graph of y = f(x + 3) (2)

The graph of y = g(x) is shown below.

The graph G is the reflection of y = g(x) in the x-axis.

(b)  Write down an equation of graph G. q ........................................................... (1)

  (Total for question = 3 marks)

Q12. 

Here are the ages, in years, of 15 women at West Ribble Tennis Club.

16,     18,     18,     20,     25,     25,     27,     28,     30,     35,     38,     42,     45,     46,    50

(a)   On the grid, draw a box plot for this information.

(3)

The box plot below shows the distribution of the ages of the men at West Ribble Tennis Club.

*(b)   Use the box plots to compare the distributions of the ages of these women and the distributions of the ages of these men.

.............................................................................................................................................

.............................................................................................................................................

.............................................................................................................................................

.............................................................................................................................................

.............................................................................................................................................

(2)

(Total for Question is 5 marks)

Q13.

Describe fully the single transformation which maps triangle A onto triangle B.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

(Total for Question is 3 marks)Q14.

Describe fully the single transformation that maps triangle A onto triangle B.

(Total for Question is 3 marks)

Q15. The expression x2 − 8x + 6 can be written in the form (x − p)2 + q for all values of x.

(a)  Find the value of p and the value of q.

p = ...........................................................

q = ...........................................................

(3)The graph of y = x2 − 8x + 6 has a minimum point.

(b)  Write down the coordinates of this point.

(............................. , .............................)

(1)  (Total for question = 4 marks)Q16. Solve     x2 –17x –56 = 0 Give your solutions correct to 2 decimal places.

........................................................... (Total for question = 3 marks)

 Q17.(a) (i) Factorise    x2 – 12x + 27

 

 

      ..............................................................................................................................................     (ii) Solve the equation    x2 – 12x + 27 = 0

 

 

      ..............................................................................................................................................(3)

(b) Factorise    y2 – 100

      ..............................................................................................................................................(1)

(Total for Question is 4 marks)