Algebra & graphs Banker

Here is a table of values for y = 2x² − 5.x: −2, −1, 0, 1, 2y: 3, ……, −5, ……, 3

1. (a) Complete the table. [2]
2. (b) The graph of y = 2x² − 5 is drawn for −2 ≤ x ≤ 2.Write down the equation of its line of symmetry. [1]
3. (c) Use algebra to find the values of x where the graph crosses the x-axis. Give your answers correct to 2 decimal places. [2]

Here is a table of values for y = x² − 2x − 3.x: −2, −1, 0, 1, 2, 3, 4y: 5, ……, −3, ……, −3, ……, 5

1. (a) Complete the table. [2]
2. (b) The graph of y = x² − 2x − 3 is drawn for −2 ≤ x ≤ 4.Write down the solutions of x² − 2x − 3 = 0. [1]
3. (c) By drawing a suitable straight line on the graph, solve x² − 2x − 3 = 2. Give your answers correct to 1 decimal place. [2]

The graph of y = x³ − 3x + 1 is drawn for −2 ≤ x ≤ 2.Table of values: x = −2, −1, 0, 1, 2 gives y = −1, 3, 1, −1, 3.

1. (a) Use the table to explain why the equation x³ − 3x + 1 = 0 has three solutions between x = −2 and x = 2. [2]
2. (b) The largest solution lies between two consecutive integers. Complete the statement: …… < x < …… [1]
3. (c) Find the equation of the straight line that must be drawn on the graph to solve x³ − 5x + 2 = 0. [2]

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

1. (a) Expand and simplify (2x − 3)(x + 4). [2]
2. (b) Factorise x² − 7x + 12. [2]
3. (c) Factorise 16a² − 25b². [2]

1. (a) Factorise 3x² + 10x − 8. [2]
2. (b) Factorise completely ax + ay − 2bx − 2by. [2]
3. (c) Expand and simplify (x + 2)(x − 1)(2x + 3). [3]

Here are the first four terms of a sequence.7 11 15 19

1. (a) Write down the next term. [1]
2. (b) Find an expression for the nth term. [2]
3. (c) Show that 143 is a term of the sequence and state which term it is. [2]

1. (a) Here are the first five terms of a sequence.4 9 18 31 48Find an expression for the nth term. [3]
2. (b) Find the nth term of the sequence 3, 6, 12, 24, … [2]

1. (a) Solve 5x − 7 = 3x + 9. [2]
2. (b) Solve (x + 4)/3 = 5. [2]

1. (a) The angles of a triangle are x°, (2x + 15)° and (3x − 9)°.Write down an equation in x and solve it. [3]
2. (b) Find the size of the largest angle. [1]

f(x) = 3x − 2 g(x) = x² + 1

1. (a) Find f(4). [1]
2. (b) Find gf(1). [2]
3. (c) Find f⁻¹(x). [2]
4. (d) Find fg(x), giving your answer in its simplest form. [2]

f(x) = 2x + 5 g(x) = 1/(x − 1), x ≠ 1 h(x) = 3ˣ

1. (a) Find h(−2). [1]
2. (b) Find g⁻¹(x). [2]
3. (c) Solve fg(x) = 7. [3]
4. (d) Find ff(x) in its simplest form. [2]

1. (a) Solve the simultaneous equations.3x + 2y = 165x − 4y = 1Show all your working. [4]

1. (a) Solve the simultaneous equations.y = x² − 4x + 1y = 2x − 7 [5]
2. (b) Describe what your answer to part (a) shows about the line y = 2x − 7 and the curve y = x² − 4x + 1. [1]

1. (a) v = u + atFind v when u = 12, a = −3 and t = 4. [2]
2. (b) Rearrange v = u + at to make t the subject. [2]

1. (a) Make x the subject of y = (3x + 2)/(x − 5). [4]
2. (b) The time period of a pendulum is T = 2π√(L/g).Make L the subject of the formula. [3]

A curve has equation y = 2x³ − 6x² + 5.

1. (a) Find dy/dx. [2]
2. (b) Find the gradient of the curve at the point where x = 3. [1]
3. (c) Find the coordinates of the two stationary points. [3]
4. (d) Determine whether each stationary point is a maximum or a minimum. Show how you decide. [2]

A curve has equation y = x² − 5x + 2.

1. (a) Find the equation of the tangent to the curve at the point where x = 4. [4]
2. (b) Find the coordinates of the point on the curve where the gradient is −1. [2]

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

1. (a) Show that the equation 4/(x + 1) + 3/(x − 2) = 2 can be written as 2x² − 9x + 1 = 0. [3]
2. (b) Solve 2x² − 9x + 1 = 0. Give your answers correct to 2 decimal places. [3]

1. (a) Solve 3x² − 5x − 4 = 0. Show all your working and give your answers correct to 2 decimal places. [4]

1. (a) Write x² − 8x + 5 in the form (x − a)² − b. [2]
2. (b) Write down the coordinates of the minimum point of y = x² − 8x + 5. [1]
3. (c) Solve x² − 8x + 5 = 0, giving your answers in surd form. [2]

1. (a) Solve 7 − 2x > 15. [2]
2. (b) List all the integers n that satisfy −3 ≤ 2n + 1 < 9. [2]

A region R is defined by the three inequalities y ≥ 1, x + y ≤ 6 and y ≤ 2x.

1. (a) Find the coordinates of the three vertices of R. [3]
2. (b) Find the greatest value of 3x + y for points in R. [2]
3. (c) How many points with integer coordinates lie inside R or on its boundary? [2]

A car starts from rest and accelerates uniformly to 18 m/s in 12 seconds. It then travels at a constant 18 m/s for 30 seconds. It then slows down uniformly and stops after a further 8 seconds.(The speed–time graph is made of three straight-line sections.)

1. (a) Calculate the acceleration during the first 12 seconds. [1]
2. (b) Calculate the total distance travelled. [3]
3. (c) Calculate the average speed for the whole journey. [1]

Sana leaves home at 09 15 and cycles 12 km to a library at a constant speed, arriving at 09 55. She stays at the library for 25 minutes. She then cycles home at a constant speed of 16 km/h.(Her distance–time graph is made of three straight-line sections.)

1. (a) Calculate her speed, in km/h, on the way to the library. [1]
2. (b) Find the time she arrives home. [2]
3. (c) Calculate her average speed for the whole time she is away from home, including the stop. Give your answer correct to 3 significant figures. [2]

1. (a) Simplify x⁵ × x³. [1]
2. (b) Simplify (3a²)³. [2]
3. (c) Simplify 12y⁷ ÷ 4y². [1]

1. (a) Simplify (16x⁸)^(3/4). [2]
2. (b) Simplify (27p⁶q⁻³)^(−1/3). [2]
3. (c) Solve 2^(x + 3) = 32^(x − 1). [3]

1. (a) y is inversely proportional to the square of x. When x = 2, y = 18.Find a formula for y in terms of x. [2]
2. (b) Find y when x = 3. [1]
3. (c) Find the positive value of x when y = 0.5. [2]

1. (a) Sketch the graph of y = (x − 2)(x + 3).On your sketch, show the coordinates of the points where the graph crosses the axes. [3]
2. (b) Find the coordinates of the turning point of the graph. [2]