|
|
Exam-Style Questions.Problems adapted from questions set for previous Mathematics exams. |
1. | IGCSE Core [560] |
Nora and Dora are trying to solve the following simultaneous equations by finding the intersection of their graphs.
$$y+x=6$$ $$y-x=1$$Here are their graphs:
(a) Who has plotted the graphs most correctly?
(b) What is the correct solution?
2. | GCSE Higher [477] |
The diagram below is a sketch of \(y = f(x)\) where \(f(x)\) is a quadratic function.
The graph intersects the x-axis where \(x=-2\) and \(x = 0.5\).
Which of the following is the solution of \(f(x) \le 0\) ?
3. | GCSE Higher [834] |
\(3x^2+9x-12\) can be written in the form \(a(x+b)^2-c\).
Find the values of \(a\), \(b\) and \(c\).
4. | GCSE Higher [650] |
(a) Use the red graphs to solve the simultaneous equations:
$$4x-7y=-47$$ $$3y=-5x$$(b) Use the blue graph to find estimates for the solutions of the quadratic equation:
$$x^2-5x+3=0$$
5. | GCSE Higher [743] |
The volume of the larger red cube is 386cm3 greater than the volume of the smaller blue cube.
The lengths of the edges of the blue cube can be expressed as \(x\) cm.
The lengths of the edges of the red cube can be expressed as \(x+2\) cm.
(a) Show that \( x^2 + 2x - 63 = 0 \).
(b) Find the volume of the blue cube.
6. | GCSE Higher [779] |
The straight line \(y = 3x - 4\) intersects the curve \(y = x^2 + 4x – 7\) at the points \(A\) and \(B\).
Find the coordinates of \(A\) and \(B\).
Give your answers correct to 2 decimal places.
7. | GCSE Higher [802] |
(a) Write \(x^2 - 6x + 5\) in the form \((x-a)^2 + b\).
(b) Write down the coordinates of the turning point of the graph of \(y = x^2 - 6x + 5\).
9. | GCSE Higher [828] |
The table shows some values for \(y=3x^{2}-x^{3}-1\).
| \(x\) | \(-2\) | \(-1\) | \(0\) | \(1\) | \(2\) | \(3\) | \(4\) |
|---|---|---|---|---|---|---|---|
| \(y\) | \(19\) | \(-1\) | \(3\) | \(-1\) |
10. | GCSE Higher [493] |
Estimate the solutions of the following simultaneous equations using their graphs as drawn on the grid below.
$$3x-5y=17$$ $$y=5-\frac{7x}{5}$$
12. | IGCSE Extended [118] |
(a) Show that the equation \(\frac{3}{x+1}+\frac{3x-9}{2}=1\) can be simplified to \(3x^2-8x-5=0\).
(b) Solve the equation \(3x^2-8x-5=0\) showing all of your working and giving answers to three significant figures.
(c) The total surface area of a cone with radius \(x\) and slant height \(8x\) is equal to the area of a circle with radius r. Show that \(r = 3x\).
[The curved surface area, \(A\), of a cone with radius \(r\) and slant height \(l\) is \(A=\pi rl\).]
13. | GCSE Higher [134] |
A rectangular sheet of paper can be cut into two identical rectangular pieces in two different ways, either by cutting along line A or by cutting along line B.
When the original sheet of paper is cut along line A, the perimeter of each of the two pieces is 56 cm.
When the original sheet of paper is cut along line B, the perimeter of each of the two pieces is 64 cm.
What is the perimeter of the original sheet of paper?
14. | GCSE Higher [181] |
Show that you understand equations and inequalities by answering the following:
(a) Solve \(5x^2=80\)
(b) Solve \(8x + 2 \gt x + 7\)
(c) Write down the largest integer that satisfies \(8x - 2 \lt 25\)
(d) Solve the following pair of equations
$$3x + 5y = 21$$ $$8x - 5y = 1$$15. | GCSE Higher [799] |
In this question all measurements are in centimetres.
The height of a triangle is \(h\) and the height of a rectangle is \((h + 4)\).
The length of the base of the triangle is \(x\) and the length of the rectangle is \((x + 2)\).
The area of the triangle is \(21 \text{ cm}^2\) and the area of the rectangle is \(90 \text{ cm}^2\).
(a) Write down an expression, in terms of \(x\), for the height of the rectangle.
(b) Show that \(x^2 - 10x + 21 = 0\).
(c) By factorising and solving \(x^2 - 10x + 21 = 0\), find the two possible heights of the triangle.
Image not drawn to scale, nor to the proportions of the triangle and rectangle mentioned in the question. In fact, this image is for decoration only and has nothing to do with the question
16. | GCSE Higher [559] |
Draw the graph of \(y = 2x^2 + 3x - 7\) for \( -3.5 \le x \le 2\). Draw suitable straight lines to find approximate solutions (to one decimal place) of the equations:
(a) \(2x^2 +3x - 9 = 0\)
(b) \(2x^2 +4x - 7 = 0\)
(c) \(2x^2 - x -5 = 3 - x\)
17. | IB Studies [91] |
A red rug has a width of \(x-3\) cm and a length of \(4x\) cm.
(a) Write down an ex
The area of the rug is 3240 cm2.
(b) Calculate the value of \(x\).
(c) Hence, write down the value of the length and of the width of the rug in centimetres.
18. | GCSE Higher [482] |
In the diagram below, which is not drawn to scale, all dimensions are in centimetres and all angles are multiples of 90o. If the shaded area is 698cm2, work out the value of \(x\).
19. | GCSE Higher [500] |
Given that:
$$ x^2 : (5x + 3) = 1 : 3 $$find the possible values of \(x\).
20. | GCSE Higher [275] |
The area of triangle ABC (not drawn to scale) is
$$ \frac{35 \sqrt{3}}{4} m^2$$If AB = \(x+2\) metres and AC = \(2x+1 \) metres, find the value of \(x\).
21. | GCSE Higher [547] |
Solve algebraically the simultaneous equations to find the solution where \(-10 \le x \le 10\).
$$ 3x^2 - y^2 = 11 $$ $$ 5x + 3y = 27 $$22. | GCSE Higher [321] |
Two numbers are chosen so that the sum of their squares is 25.
If those numbers are represented by \(x\) and \(y\) they will also satisfy the equation:
$$y-3x=13$$Use an algebraic method to find two possible values of \(x\) and \(y\) .
23. | GCSE Higher [178] |
The prices of two watches are in the ratio \(a:b\).
When the prices are both increased by £10, the ratio becomes \(5 : 7\).
When the prices are both reduced by £10, the ratio becomes \(1 : 3\).
Express the ratio \(a:b\) in its lowest terms.
If you would like space on the right of the question to write out the solution try this Thinning Feature. It will collapse the text into the left half of your screen but large diagrams will remain unchanged.
The exam-style questions appearing on this site are based on those set in previous examinations (or sample assessment papers for future examinations) by the major examination boards. The wording, diagrams and figures used in these questions have been changed from the originals so that students can have fresh, relevant problem solving practice even if they have previously worked through the related exam paper.
The solutions to the questions on this website are only available to those who have a Transum Subscription.
Exam-Style Questions Main Page
To search the entire Transum website use the search box in the grey area below.
Do you have any comments about these exam-style questions? It is always useful to receive feedback and helps make this free resource even more useful for those learning Mathematics anywhere in the world. Click here to enter your comments.
