Algebra worksheet — GCSE Higher

1.A square has vertices at (−2, 3), (−2, −3) and (4, −3). Write down the coordinates of the fourth vertex.

2.A student attempts to prove that the sum of any three consecutive integers is a multiple of 3. Line 1: Let the three consecutive integers be n, n + 1 and n + 2. Line 2: Their sum is n + (n + 1) + (n + 2) = 3n + 2. Line 3: 3n + 2 leaves a remainder of 2 when divided by 3, so it is not a multiple of 3. Line 4: So the sum of three consecutive integers is not always a multiple of 3. Which line contains the FIRST error?

3.The point (3, 4) lies on the circle x² + y² = 25, which has centre (0, 0). Work out the gradient of the tangent to the circle at (3, 4).

4.A geometric sequence has first term 3 and common ratio √3. Work out the position of the first term of the sequence that exceeds 30.

5.A rope of length L metres is cut into 6 equal pieces, and 4 metres is then removed from one piece. Write an expression, in metres, for the length of that piece after the cut.

6.A circle has equation x² + y² = 49. Work out the coordinates of the point(s) on the circle where the tangent is horizontal.

7.The graph of y = f(x) has a maximum turning point at (−1, 6). Write down the coordinates of the maximum turning point of the graph of y = f(x − 3).

8.The line y = 3x + 6 meets the x-axis at one point. Work out the coordinates of that point.

9.A point has coordinates (x, y), where x + y = 0 and x is not 0. In which two quadrants could this point lie?

10.f(x) = 2x + 1. Work out the value of x for which f⁻¹(x) = 5.

11.The nth term of a sequence is n² + 2n − 4. Work out the 7th term.

12.A straight line passes through the points (1, 3) and (2, 5). Work out the equation of the line.

13.A walker's distance-time graph is described as follows: she walks 3 km in the first 15 minutes at a steady speed, rests for 10 minutes, then walks a further 3 km in the next 20 minutes, also at a steady speed. Work out her speed, in km/h, during the first 15 minutes.

14.Work out the maximum value of y = sin x, and the smallest positive value of x, in degrees, at which it occurs.

15.A car's journey is represented by a velocity-time graph. The total distance travelled, found from the area under the graph, is 350 metres, and the total time for the journey is 25 seconds. Work out the car's average speed for the whole journey, in m/s.