Algebra worksheet — GCSE Higher

1.The nth term of a sequence is n² − 2n + 5. Work out the 10th term.

2.The equation x² − 4x + k = 0 has two different real solutions. Work out the range of values of k.

3.A packet contains s sweets. The sweets are shared equally between f friends. Write an expression for the number of sweets each friend receives.

4.The simultaneous equations kx + 2y = 4 and 3x + y = 5 have no solution. Work out the value of k.

5.A car's speed increases steadily while it accelerates. Its speed v m/s after t seconds is given by v = u + at, where u is the starting speed in m/s and a is the acceleration in m/s². A car starts at u = 4 m/s and accelerates at a = 2 m/s² until it reaches v = 20 m/s. Work out t, the time taken in seconds.

6.A taxi firm charges a fixed fee of £3.50 plus £2.20 per mile. Work out the total cost of a journey of 6 miles.

7.Expand and simplify 4(2x − 1) − 3(x + 2) − 5x

8.Solve the simultaneous equations 4x + 3y = 25 and 4x − y = 1. Work out the value of y.

9.Which of these statements is an identity?

10.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?

11.Write down the first four terms of the sequence with nth term 6n − 5.

12.Work out the coordinates of the reflection of (6, −3) in the y-axis.

13.A cyclist's speed rises from rest to a maximum — quickly at first, then more and more slowly — so her speed-time graph is a curve that is concave down (its gradient decreases as time goes on). A trapezium estimate is made for the distance travelled during this phase, using two points on the curve. Is the trapezium estimate an overestimate or an underestimate of the true distance, and why?

14.A scout leader is 44 years old and one of the scouts is 16 years old. Work out how many years it will be until the leader is exactly twice as old as the scout.

15.To solve 6x − 4 = 2x + 20, Yusuf's first step is to subtract 2x from both sides. Work out what equation this gives.