Algebra and graphs — IGCSE Mathematics (0580) questions by topic

A ticket costs $18. Write down an expression, in dollars, for the cost of t tickets.

These are the first four terms of a sequence: 33, 26, 19, 12. Write down the term-to-term rule for this sequence.

Ky cycles from his office to a meeting and back again. The travel graph shows his time at the meeting and his journey back. How far is the meeting from his office?

Solve. 5x + 8 = 3x - 2

Simplify. 6a + 4b - a - 5b

Solve the simultaneous equations. 2x + 5y = 5 3x + 4y = 11

Complete the table of values for y = (x + 3)(x - 2). x -4 -3 -2 -1 0 1 2 3 --- ---- ---- ---- ---- --- --- --- --- y 6 -4 -4

Beth thinks of a positive number, n. She squares n then subtracts 55. The answer is 9. Work out the value of n.

Factorise. 9x - 6xy

Solve the simultaneous equations. 5x + 2y = 3 3x + 4y = 27

Line A has equation y = 3x + 1. Line B has equation y = 3x - 1. Draw a ring around the description that is correct. - Line A intersects line B - Line A has a steeper gradient than…

Write down the inequality represented on the number line. The number line runs from -6 to 5. There is a filled (solid) circle at -4 and an open (unfilled) circle at 3, with the…

Simplify. y^4 × y^6

In this question, all lengths are in centimetres. The triangle has sides 2x + 1, 9 and x + 3. The rectangle has dimensions (2x - 1) by x. The perimeter of the triangle is equal to…

g = h/3 - 8 Rearrange the formula to make h the subject.

Solve the simultaneous equations. 8x + 5y = 4 2x - y = 10

Simplify. 7c - 5d + c + 3d

The diagram shows the graph of y = f(x) and the point P(-2, 11). The graph is a downward (n-shaped) parabola with a maximum near the y-axis at about y = 4, drawn on a grid with x…

f(x) = 3x - 4 g(x) = 4x + 1 Find f(-2).

Expand and simplify. (x + 4)(x - 3)(3x + 2)

One day, Anya runs 12 km at a speed of x km/h. The next day she walks 10 km at a speed of (x - 4) km/h. Write down an expression, in terms of x, for the time she spends running.

Solve. 8x + 7 = 39

These are the first 4 terms of a sequence. 11 8 5 2 Find the next term of this sequence.

The diagram shows the graph of y = 2/x - 1, drawn on axes with x from -8 to 8 and y from -5 to 5. The curve has two branches (a reciprocal-type curve shifted down by 1). Write…

y is inversely proportional to the square root of x, i.e. y proportional to 1/sqrt(x). When x = 9, y = 2. Find the value of y when x = 36.

The diagram (NOT TO SCALE) shows the graph of y = 3x - x^3. The graph crosses the x-axis at A, at O and at B. The turning points of the graph are at P (a minimum in the third…

The line y = 7x + 3 intersects the curve y = x^2 + 5x - 12 at the points A and B. Find the coordinates of A and B.

Solve. 5x^2 = 12 − 17x

Solve the simultaneous equations. 4x − 5y = 13 3x − 2y = 8

Make t the subject of the formula. 2 = m(1 − t) / (pt)

f(x) = 2x + 5, g(x) = x − 4, h(x) = 5^x. Find f(3).

Solve. 3y = 18

T = 3(5P - 8) + 4 Find the value of T when P = 12.

Represent the inequality -3 < x <= 2 on the number line. (A number line marked with integer values from -5 to 5 is provided.)

Complete the table of values for y = x^2 - x - 5. x -3 -2 -1 0 1 2 3 4 --- --- --- --- --- --- --- --- --- y 1 -3 -3 1 (Find the missing y values at x = -3, x = 0, x = 1 and x =…

2, e, f, 20, … A linear sequence has a first term of 2 and a fourth term of 20. The term-to-term rule for this sequence is add k. Work out the values of e, f and k.

P = 6a + 5b Find the value of b when P = 25 and a = 3.

Solve. x + 3 = 3

Simplify. (w^3)^10

In this question, all lengths are in centimetres. The diagram shows a right-angled triangle with sides 2x − 5, 2x + 2 and 4x − 9. Write down an expression, in terms of x, for the…

Tim drives home from work. The travel graph shows his journey. Write down the time Tim leaves work.

Solve. 4p + 11 = 25

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

Complete the table of values for y = 20/x. x 1 2 3 4 5 --- --- --- --- --- --- y 20 10 4

Solve. 4c - 9 = 13

These are the first 5 terms of a sequence. 1 8 27 64 125 Find the 10th term of this sequence.

Expand. 5x^2(3x - 2)

Make x the subject of this formula. A = w^2 + 5x^2

The table shows some values for y = 5x^2 - x^3 - 4. x -2 -1 0 1 2 3 4 5 --- ---- ---- --- --- --- --- --- --- y 24 -4 8 14 -4 Complete the table.

2x^2 + 12x - 2 can be written in the form a(x + b)^2 - c. Find the values of a, b and c.

y = mx + c Rearrange the formula to make m the subject.

Solve the simultaneous equations. You must show all your working. 2w - 3y = 11 3w + y = 11

A group of 12 adults and 9 children travel on a bus. The cost of an adult ticket is $n. The cost of a child ticket is $(n - 10). The total cost of the tickets is $277.50. Find the…

Factorise. 5x - 10 - ax + 2a

f(x) = x + 1 g(x) = 5 - 2x h(x) = 2^x Find f(-3).

Simplify. 3t^5 × 5t^3

Ahmed walks 2 km at a speed of x km/h. He then walks a further 3 km at a speed of (x + 1) km/h. The total time he takes to walk the 5 km is 1 1/4 hours. Show that 5x^2 - 15x - 8 =…

Factorise. 28xy - 12x

Simplify. 5x^5 y x 7x^3 y^2

Expand and simplify. 2x - x(5 - x^2)

The table shows three sequences. Complete the table (find the 6th term and the nth term for each sequence). 1st term 2nd term 3rd term 4th term 5th term 6th term nth term --- ---…

m is inversely proportional to the square root of (t + 2). m = 0.5 when t = 23. Find m when t = 98.

A grid is provided with x-axis from -4 to 5 and y-axis from -2 to 7. By shading the unwanted regions of the grid, draw and label the region R which satisfies these three…

Simplify 3a − 5b − a − 6b.

The cost of hiring a bicycle, $C, for y hours is given by the formula C = 12 + 3.5y. Maria pays $36.50 to hire this bicycle. Work out the number of hours she hires the bicycle for.

Factorise completely 8x² − 20x.

The nth term of a sequence is 10 − n². Write down the first three terms of this sequence.

At a cinema, an adult ticket costs $a and a child ticket costs $c. Farah buys 3 adult tickets and 4 child tickets for $38.50. Complete the equation: 3a + 4c = ...

x is an integer where x ⩾ −3 and x < 3. Write down all the possible values of x.

Simplify 18x¹² ÷ 3x³.

Simplify d⁸ ÷ d².

Expand and simplify (x − 5)(x + 8).

In this question, both lengths are in centimetres. The diagram shows two lines AB and CD. The length of AB is 10x − 12. The length of CD is 2x + 3. Line AB is 3 times as long as…

The nth term of a sequence is 10 − n². Write down the first three terms of this sequence.

y is directly proportional to the square of (x + 3). When x = 2, y = 5. Find y when x = 1.

Simplify (81x¹²)^(3/4).

The table shows some values for y = 3x² − 2x − 1 for x = −1, −0.5, 0, 0.5, 1, 1.5. Complete the table (the missing y-values).

Solve 30/x = 6.

One solution of the equation ax² + b = 181 is x = 8. a and b are both positive integers greater than 1. Find the value of b.

f(x) = 5x + 2. Find f⁻¹(x).

Make x the subject of the formula c = 3x/(2x − 5).

m is inversely proportional to the square of (t + 2). m = 0.64 when t = 3. Find m when t = 8.

Write 5/(3x + 2) + 4/(2x − 1) as a single fraction in its simplest form.

v = u − 9.8t. Find the value of v when u = 4 and t = −7.

Simplify d⁸ ÷ d².

The distance–time graph shows Kai's journey from home to the office (home at 0 km, office at 65 km; departs 10 00, arrives 12 00). Calculate the average speed, in km/h, for Kai's…

Rearrange the formula to make m the subject: R = 2(m − k)/m.

y is inversely proportional to the cube root of (x + 5). When x = 3, y = 12. Find y when x = 22.

Solve the equation x² + 5x − 7 = 0. You must show all your working and give your answers correct to 2 decimal places.

f(x) = 6x − 7, g(x) = x⁻³. Find f(x + 2). Give your answer in its simplest form.

Simplify (2x² + 5x − 12)/(4x² − 9).

These are the first four terms of a sequence: 2.75, 6, 11.25, 20. The nth term of this sequence is (1/4)n³ + an² + bn. Calculate the value of a and the value of b.

Complete the table of values for y = 5 + 3x − x² for x = −2, −1, 0, 1, 2, 3, 4, 5.

The travel graph shows Diego's journey by bicycle from his home to the swimming pool (6 km in 40 min, 10 00 to 10 40). Calculate Diego's speed for his journey from his home to the…

50 Australian dollars = 540 South African rands. On the grid, draw a conversion graph between Australian dollars and South African rands.

T = 5P + 3Q. Find the value of T when P = 6 and Q = 8.

These are the first four terms of a sequence: 2, 8, 14, 20. Write down the next term.

A boat sails from A to B. The travel graph shows this journey (B at 6 km). Write down the time that the boat leaves A.

The grid shows the first three diagrams in a sequence made using sticks. On the grid, draw Diagram 4.

Factorise fully 27y² − 3.

A tailor makes x dresses and y shirts in one week. He makes at least 4 dresses; no more than 7 shirts; less than 14 dresses and shirts altogether; the number of shirts is more…

f(x) = x − 4, g(x) = 2x + 5, h(x) = 3ˣ. Find f(−3).

The diagram shows a right-angled triangle ABC, with the right angle at A. AC = (x + 3) cm and AB = (2x + 5) cm. NOT TO SCALE. The area of the triangle is 60 cm². Show that 2x² +…

A sequence has nth term n/(2n + 3). Find the first three terms of this sequence. Give your answers as fractions.

Show that the equation y = (x − 4)(x + 1)(x − 2) can be written as y = x³ − 5x² + 2x + 8.

Simplify (3x²y⁴)³.

The diagram shows the graph of a function. Put a ring around the word which correctly identifies the type of function: reciprocal, quadratic, cubic, exponential, linear.

A shop sells shirts for $x and jackets for $(x + 27). The shop sells 4 shirts and 3 jackets for a total of $194.75. Write down and solve an equation to find the cost of one shirt.

The table shows some values for y = 2x − 3. x −2 −1 0 0.5 1 1.5 2 2.5 --- --- --- --- --- --- --- --- --- y −2.75 −1.58 −0.17 1 2.66 Complete the table.

Complete the statement: When x = ..........., x + 3 = 8.

Factorise completely. 42mk − 35m

Simplify. n⁵ × n

Asha works in a café. Her wage is calculated using the formula wage = hourly rate × number of hours + bonus. Her hourly rate is $11.52. One week Asha works 25 hours and receives a…

These are the first four terms in a sequence: −3, 4, 11, 18. Find the next term.

v = u + at. Find the value of v when u = 30, a = −2 and t = 7.

Expand and simplify. 2(t + w) + 3(w − t)

Factorise completely. 15v² − 3v

Solve the equation. 5x + 7 = 9x − 3

A bar of chocolate costs $3 and a bag of sweets costs $5. Write down an expression for the total cost, in dollars, of x bars of chocolate and y bags of sweets.

Solve the simultaneous equations. You must show all your working. 3x + 5y = 23 and 6x − 4y = 11.

Complete the statement: When x = ..........., x + 3 = 8.

Factorise completely. 42mk − 35m

Simplify. n⁵ × n

Solve. 4(2x − 3) ≥ 43 + 3x

By shading the unwanted regions of the grid, draw and label the region R which satisfies these inequalities: y 1, x ≤ 2, y ≥ x + 2.

v = u + at. Find the value of v when u = 30, a = −2 and t = 7.

Expand and simplify. 2(t + w) + 3(w − t)

The diagram shows a speed–time graph for 16 seconds of a car journey. The car travels at 10 m/s, constant until 12 seconds, then decelerates uniformly to 0 m/s at 16 seconds. NOT…

3^(3p) × 3^(2p) = 729. Find the value of p.

y = 2w² − x. Rearrange the formula to make w the subject.

Find the nth term of the sequence: 11, 8, 5, 2, −1, …

The line y = x + 1 intersects the curve y = x² + x − 3 at two points. Find the coordinates of the two points.

x is inversely proportional to the square root of w. When w = 16, x = 3. Find x in terms of w.

P = (2wy²)/3. Find the positive value of y when P = 108 and w = 8.

The graph of y = 2x + 1 is drawn on the grid. By shading the unwanted regions of the grid, find and label the region R which satisfies these inequalities: y ≥ 2x + 1, y ≥ 1, 4x +…

T = √(3d − e). Rearrange the formula to make d the subject.

(x + a)(x + 2)(2x + 3) is equivalent to 2x³ + bx² + cx − 18. Find the value of each of a, b and c.

Find the coordinates of the point where the line 4x + y = 9 intersects the curve y + x² = 5. You must show all your working.

Simplify. 5a + 3b + 2a − 4b

Complete the table of values for y = x² − 4x − 2. x −2 −1 0 1 2 3 4 5 --- --- --- --- --- --- --- --- --- y 3 −2 −5 −5 −2 3

Simplify. a + 4a − 3a

These are the first four diagrams in a sequence, made using dots and lines. Complete the table. Diagram 1 2 3 4 --- --- --- --- --- Number of small squares 2 4 6 Number of dots 6…

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

s = (1/2)at². Find the value of s when a = 9.8 and t = 20.

f(x) = (3x + 1)(x + 5)(x − 4), g(x) = 2x − 3, h(x) = 4^(2x − 1). Find f(0).

In a shop the cost of a fiction book is $x and the cost of a reference book is $(x + 2). The cost of 11 fiction books is the same as the cost of 10 reference books. Find the value…

f(x) = 1 − 3x, g(x) = (x − 1)², h(x) = 3/x (x ≠ 0). Find g(3).

f(x) = 5x − 3, g(x) = 64^x, h(x) = 2/(x + 1) (x ≠ −1). Find f(2).

Expand and simplify. 4(2x − 1) − 6(3 − x)

The diagram shows a sketch of y = 18 + 5x − 2x². The curve crosses the x-axis at A and C, and the y-axis at B. NOT TO SCALE. Find the coordinates of the points A, B and C.