Algebra Banker

1. (a) Complete the table of values for y = x² − 2x − 3.x: −2, −1, 0, 1, 2, 3, 4y: …, 0, −3, …, −3, 0, … [2]
2. (b) By solving an equation, find the x-coordinates of the points where the curve y = x² − 2x − 3 crosses the x-axis. [2]
3. (c) Write down the coordinates of the turning point of the curve. [1]

1. (a) The table shows some values of y = x³ − 4x + 1.x: −3, −2, −1, 0, 1, 2, 3y: …, 1, …, 1, −2, …, 16Complete the table. [2]
2. (b) Find the gradient of the straight line joining the points on the curve where x = 1 and x = 3. [2]
3. (c) The line y = 2x + 1 meets the curve y = x³ − 4x + 1 at three points. Show that the x-coordinates satisfy x³ − 6x = 0 and find all three values. Give non-integer answers correct to 2 decimal places. [3]

1. (a) Write x² − 6x + 5 in the form (x − a)² − b. [2]
2. (b) Hence write down the coordinates of the turning point of y = x² − 6x + 5. [1]
3. (c) Find the coordinates of the points where the curve crosses the axes. [2]
4. (d) Find the range of values of k for which the equation x² − 6x + 5 = k has two different solutions. [1]

1. (a) Simplify (x² − 9)/(x² + 5x + 6). [3]
2. (b) Write 3/x + 2/(x + 1) as a single fraction in its simplest form. [2]

1. (a) Show that the equation 3/(x − 1) − 2/(x + 1) = 1 can be written as x² − x − 6 = 0. [3]
2. (b) Solve 3/(x − 1) − 2/(x + 1) = 1. [2]

1. (a) Simplify (2x² + 5x − 3)/(4x² − 1). [3]
2. (b) Write 2/(x − 3) − (x + 9)/(x² − 9) as a single fraction in its simplest form. [4]

1. (a) Solve x² − 5x − 14 = 0. [2]
2. (b) Solve 2x² + 7x + 3 = 0. [2]

A rectangular garden has length (x + 4) metres and width (x − 1) metres. Its area is 40 m².

1. (a) Show that x² + 3x − 44 = 0. [2]
2. (b) Solve x² + 3x − 44 = 0. Give your answers correct to 2 decimal places. [3]
3. (c) Find the perimeter of the garden. [2]

Sana cycles 60 km to a lake at an average speed of x km/h. She returns along the same route at an average speed of (x + 5) km/h. The return journey takes 1 hour less than the outward journey.

1. (a) Write down an expression, in terms of x, for the time taken on the outward journey. [1]
2. (b) Show that x² + 5x − 300 = 0. [3]
3. (c) Solve the equation and find the time taken for the outward journey. [3]

1. (a) Expand and simplify 3(2x − 5) − 2(x − 4). [2]
2. (b) Factorise completely 6xy − 9y². [2]
3. (c) Factorise x² − 49. [1]

1. (a) Expand and simplify (x − 1)(x + 2)(x + 3). [3]
2. (b) Factorise completely 2x² − 18. [2]
3. (c) Factorise ax + 2a − 3x − 6. [2]

1. (a) Factorise 6x² − 7x − 20. [2]
2. (b) Show that (n + 3)² − (n − 3)² is a multiple of 12 for every whole number n. [3]
3. (c) Factorise completely 16a⁴ − 81. [2]

The first four terms of a sequence are 7, 11, 15, 19.

1. (a) Write down the next term. [1]
2. (b) Find an expression for the nth term. [2]
3. (c) Explain why 150 is not a term of this sequence. [1]

A pattern of tiles is made. Pattern 1 uses 8 tiles, Pattern 2 uses 13 tiles and Pattern 3 uses 18 tiles. The number of tiles increases by the same amount each time.

1. (a) How many tiles are in Pattern 6? [1]
2. (b) Find an expression for the number of tiles in Pattern n. [2]
3. (c) Which pattern uses exactly 128 tiles? [2]

The first five terms of a sequence are 3, 9, 19, 33, 51.

1. (a) Find an expression for the nth term. [3]
2. (b) Find the 20th term. [1]
3. (c) Find the first term of the sequence that is greater than 1000, and state its position. [2]

f(x) = 3x − 5 and g(x) = x² + 1.

1. (a) Find f(4). [1]
2. (b) Find g(−3). [1]
3. (c) Find f⁻¹(x). [2]

f(x) = (2x + 1)/3 and g(x) = 4 − x.

1. (a) Find fg(2). [2]
2. (b) Find gf(x), giving your answer as a single fraction. [2]
3. (c) Solve f(x) = g(x). [2]
4. (d) Find f⁻¹(x). [2]

f(x) = 5/(x − 2), x ≠ 2, and g(x) = 2x − 1.

1. (a) Find f⁻¹(x). [2]
2. (b) Solve fg(x) = 1. [2]
3. (c) Write down the value of x that cannot be used in fg(x). [1]
4. (d) Solve gf(x) = x. [3]

1. (a) Solve 5x − 7 = 2x + 11. [2]
2. (b) Solve 3(2x − 1) = 4(x + 2). [2]

1. (a) Solve (x + 3)/4 − (2x − 1)/3 = 1. [3]
2. (b) The angles of a triangle are x°, (2x + 15)° and (3x − 21)°. Form an equation in x and solve it to find the size of the largest angle. [3]

Ayesha is 4 times as old as her son Bilal. In 8 years' time, Ayesha will be 2.5 times as old as Bilal.

1. (a) Taking Bilal's age now as b years, form an equation in b. [2]
2. (b) Solve your equation to find their ages now. [2]
3. (c) How many years ago was Ayesha 7 times as old as Bilal? [2]

1. (a) v = u + at. Find v when u = 12, a = −3 and t = 2.5. [2]
2. (b) Make t the subject of v = u + at. [2]
3. (c) s = ut + ½at². Find s when u = 4, t = 6 and a = 2. [2]

The time T seconds for one swing of a pendulum of length L metres is given by T = 2π√(L/g), where g = 9.8.

1. (a) Find T when L = 1.5. Give your answer correct to 3 significant figures. [2]
2. (b) Make L the subject of the formula. [3]
3. (c) Find L when T = 3. Give your answer correct to 3 significant figures. [1]

1. (a) Make x the subject of y = (3x + 2)/(x − 4). [3]
2. (b) p = 2q² − r. Find the possible values of q when p = 72 and r = −22. [2]
3. (c) Make a the subject of m = (a² − b)/c. [2]

A speed–time graph for a runner is made of straight lines joining the points (0, 0), (10, 20), (40, 20) and (50, 0), where time is in seconds and speed in m/s.

1. (a) Find the acceleration during the first 10 seconds. [1]
2. (b) Find the total distance travelled. [2]
3. (c) Find the deceleration during the last 10 seconds. [1]

Imran leaves home at 09 00 and cycles 12 km to a shop, arriving at 09 45. He stays at the shop for 30 minutes and then cycles home at a constant speed, arriving at 11 15. On a distance–time graph his journey is made of straight lines.

1. (a) Find his speed on the way to the shop, in km/h. [2]
2. (b) Find his speed on the way home, in km/h. [1]
3. (c) Find his average speed for the whole time he was away from home, in km/h. [2]

A car is travelling at 8 m/s at time t = 0. It accelerates uniformly to V m/s at t = 12 seconds, travels at V m/s until t = 30 seconds, then decelerates uniformly to rest at t = 36 seconds. The total distance travelled is 696 m.

1. (a) Show that 48 + 27V = 696 and find V. [3]
2. (b) Find the acceleration during the first 12 seconds. Give your answer correct to 3 significant figures. [2]
3. (c) Find the time when the speed of the car first reaches 20 m/s. [2]

1. (a) Solve 3x − 4 < 11. [2]
2. (b) List the integers n that satisfy −2 ≤ 2n + 1 < 9. [2]

A region R is defined by the inequalities y ≥ 1, x + y ≤ 5 and y ≤ 2x.

1. (a) List all the points (x, y) with integer coordinates that lie in R (including its boundary). [3]
2. (b) Find the greatest value of 2x + 3y for these points. [1]

Line L1 passes through (0, 6) and (6, 0). Line L2 passes through (0, 0) and (4, 8). Line L3 is x = 1. The three lines form a triangle T.

1. (a) Find the equations of L1 and L2. [2]
2. (b) Find the coordinates of the three vertices of T. [2]
3. (c) Write down three inequalities that define the region inside T. [2]
4. (d) Find the number of points with integer coordinates that lie in T, including its sides, and the area of T. [2]

1. (a) Solve the simultaneous equations3x + 2y = 16x − 2y = 0 [3]

At a stationery shop, 3 pens and 2 notebooks cost $5.40. 5 pens and 4 notebooks cost $9.80.

1. (a) Taking the cost of a pen as p cents and a notebook as n cents, write down two equations. [2]
2. (b) Solve your equations to find the cost of one pen and one notebook. [3]

1. (a) Solve the simultaneous equationsy = x² − 3x + 1y = 2x − 3 [4]
2. (b) Find the distance between the two points of intersection. Give your answer in the form k√5. [2]

1. (a) Simplify x⁵ × x³. [1]
2. (b) Simplify (y⁴)³. [1]
3. (c) Simplify 12a⁶ ÷ 3a². [2]

1. (a) Find the value of 27^(2/3). [1]
2. (b) Find the value of 16^(−1/2). [1]
3. (c) Simplify (3x²y)³. [2]
4. (d) Solve 2^(x + 1) = 32. [2]

1. (a) Simplify (8x⁶)^(2/3). [2]
2. (b) Simplify (25a⁴b⁻²)^(−1/2). [2]
3. (c) Solve 9^x = 27^(x − 1). [2]
4. (d) Find n when 4ⁿ × 2³ = 2¹¹. [2]

y is directly proportional to x. When x = 3, y = 12.

1. (a) Find an equation connecting y and x. [2]
2. (b) Find y when x = 7. [1]
3. (c) Find x when y = 50. [1]

y is inversely proportional to x². When x = 4, y = 5.

1. (a) Find an equation connecting y and x. [2]
2. (b) Find y when x = 2.5. [1]
3. (c) Find the positive value of x when y = 20. [2]

The air resistance F newtons on a cyclist is proportional to the square of her speed v m/s. When v = 12, F = 360.

1. (a) Find a formula for F in terms of v. [2]
2. (b) Find F when v = 20, and find v when F = 490. [3]
3. (c) Her speed increases by 20%. Find the percentage increase in the air resistance. [2]