Year 10

Year 10 Paper 5: Equations, Graphs and Sequences

Covers surds and indices, simultaneous equations and inequalities, functions and their graphs, coordinate geometry, and sequences.

19 questions - 80 marks - calculator allowed

Year 10 here means a typical teaching order, not a syllabus rule. No exam board defines what belongs to Year 10, and schools sequence the course differently. Check it against your own scheme of work before using it to decide what a class has covered.

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Questions

Question 1 [2 marks]

Coordinate Geometry

Find the gradient of the straight line joining the points A(2, 3) and B(6, 11).

Question 2 [2 marks]

Simultaneous Equations and Inequalities

Solve the simultaneous equations 2x + y = 11 and x - y = 1.

Question 3 [3 marks]

Coordinate Geometry

Find the equation of the straight line that passes through the point (-2, 5) and has gradient -4.

Give your answer in the form y = mx + c.

Question 4 [3 marks]

Functions and their Graphs

The function g is defined by g(x) = 2x^2 - 5 for all real x.

Find the values of x for which g(x) = 45.

Question 5 [3 marks]

Surds and Indices

Rationalise the denominator of 6/sqrt(3), giving your answer in its simplest form.

Question 6 [4 marks]

Sequences

An arithmetic sequence has first term 15 and common difference -4.

Find the number of terms needed for the sequence to first fall below -50.

Question 7 [4 marks]

Surds and Indices

Simplify (2x^3y^-2)^3 / (4x^-1y), giving your answer in the form kx^ay^b.

Question 8 [4 marks]

Coordinate Geometry

Line L1 has equation y = -3x + 4.

Line L2 is perpendicular to L1 and passes through the point (6, 1).

Find the equation of L2, giving your answer in the form y = mx + c.

Question 9 [4 marks]

Surds and Indices

Solve 4^(3x - 1) = 8^(x + 2), giving the exact value of x.

Question 10 [4 marks]

Simultaneous Equations and Inequalities

Solve the quadratic inequality x^2 + 2x - 24 >= 0, giving your answer as two inequalities.

Question 11 [5 marks]

Sequences

The nth term of a sequence is given by Un = an^2 + b, where a and b are constants.

Given that U2 = 11 and U4 = 35, find the values of a and b.

Question 12 [5 marks]

Surds and Indices

Rationalise the denominator of 12/sqrt(18), giving your answer in its simplest form.

Question 13 [5 marks]

Functions and their Graphs

f(x) = x^2 - 2 and g(x) = 3x + 1.

Find fg(x), giving your answer in its simplest form, and hence find fg(-1).

Question 14 [5 marks]

Simultaneous Equations and Inequalities

Solve the inequality 4(2x - 3) < 3(3x + 4), giving your answer in the form x > k.

Question 15 [5 marks]

Functions and their Graphs

The function h is defined by h(x) = x^2 + 8x + 19 for all real x.

Express h(x) in the form (x + a)^2 + b, where a and b are integers.

Hence state the minimum value of h(x) and the value of x at which it occurs.

Question 16 [6 marks]

Surds and Indices

Solve 9^(n - 1) = 27^(2 - n) x 3^n, giving the exact value of n.

Question 17 [5 marks]

Simultaneous Equations and Inequalities

A rectangle has length (x + 3) cm and width (x - 1) cm, where x > 1.

The area of the rectangle is greater than 45 cm^2.

Form and solve an inequality to find the range of possible values of x.

Question 18 [6 marks]

Sequences

The sum of the first n terms of an arithmetic series is given by Sn = (n/2)(3n - 1).

Find the first term and the common difference of the series, and find the value of n for which Sn = 330.

Question 19 [5 marks]

Simultaneous Equations and Inequalities

A rectangle has length (2x + 1) cm and width (x - 3) cm, where x > 3.

The area of the rectangle is less than 60 cm^2.

Form and solve an inequality to find the range of possible values of x.

Model solutions

Mark scheme for Question 1 [2 marks]
Question 1[2 marks]
Answer or working	Marks
using gradient = (y2 - y1)/(x2 - x1)	M1
gradient = 2	A1

Mark scheme for Question 2 [2 marks]
Question 2[2 marks]
Answer or working	Marks
eliminating one variable correctly	M1
x = 4 and y = 3	A1
Final answer: x = 4, y = 3

Mark scheme for Question 3 [3 marks]
Question 3[3 marks]
Answer or working	Marks
using y - 5 = -4(x + 2)	M1
expanding to y = -4x - 8 + 5	M1
y = -4x - 3	A1

Mark scheme for Question 4 [3 marks]
Question 4[3 marks]
Answer or working	Marks
forming the equation 2x^2 - 5 = 45	M1
rearranging to x^2 = 25	M1
x = 5 or x = -5	A1

Mark scheme for Question 5 [3 marks]
Question 5[3 marks]
Answer or working	Marks
multiplying numerator and denominator by sqrt(3)	M1
simplifying to 6sqrt(3)/3	M1
2sqrt(3)	A1

Mark scheme for Question 6 [4 marks]
Question 6[4 marks]
Answer or working	Marks
using Un = a + (n - 1)d to give Un = 19 - 4n	M1
setting up the inequality 19 - 4n < -50	M1
solving to give n > 17.25	M1
n = 18 (the 18th term is the first to fall below -50)	A1

Mark scheme for Question 7 [4 marks]
Question 7[4 marks]
Answer or working	Marks
cubing to give 8x^9y^-6	M1
dividing the coefficients to give 2	M1
subtracting the indices to give x^10 and y^-7	M1
2x^10y^-7	A1
Final answer: 2x^10y^-7 (equivalently 2x^10/y^7)

Mark scheme for Question 8 [4 marks]
Question 8[4 marks]
Answer or working	Marks
identifying the gradient of L1 as -3	M1
using the perpendicular gradient rule to find gradient 1/3	M1
using y - 1 = (1/3)(x - 6)	M1
y = x/3 - 1	A1

Mark scheme for Question 9 [4 marks]
Question 9[4 marks]
Answer or working	Marks
writing 4 as 2^2 and 8 as 2^3	M1
writing both sides with base 2: 2^(6x - 2) = 2^(3x + 6)	M1
equating exponents 6x - 2 = 3x + 6	M1
x = 8/3	A1

Mark scheme for Question 10 [4 marks]
Question 10[4 marks]
Answer or working	Marks
factorising x^2 + 2x - 24 as (x + 6)(x - 4)	M1
identifying the critical values x = -6 and x = 4	M1
reasoning about the sign of the quadratic outside the roots	M1
x <= -6 or x >= 4	A1

Mark scheme for Question 11 [5 marks]
Question 11[5 marks]
Answer or working	Marks
forming the equation 4a + b = 11 from U2	M1
forming the equation 16a + b = 35 from U4	M1
subtracting the equations to eliminate b, giving 12a = 24	M1
a = 2	A1
b = 3	A1
Final answer: a = 2, b = 3 (so Un = 2n^2 + 3)

Mark scheme for Question 12 [5 marks]
Question 12[5 marks]
Answer or working	Marks
writing sqrt(18) = 3sqrt(2)	M1
simplifying to 4/sqrt(2)	M1
multiplying numerator and denominator by sqrt(2)	M1
simplifying the numerator to 4sqrt(2)	M1
2sqrt(2)	A1

Mark scheme for Question 13 [5 marks]
Question 13[5 marks]
Answer or working	Marks
substituting g(x) into f	M1
writing (3x + 1)^2 - 2	M1
fg(x) = 9x^2 + 6x - 1	A1
substituting x = -1 into fg(x)	M1
fg(-1) = 2	A1
Final answer: fg(x) = 9x^2 + 6x - 1; fg(-1) = 2

Mark scheme for Question 14 [5 marks]
Question 14[5 marks]
Answer or working	Marks
expanding the left side to 8x - 12	M1
expanding the right side to 9x + 12	M1
collecting x terms to give x on one side	M1
collecting number terms on the other side	M1
x > -24	A1

Mark scheme for Question 15 [5 marks]
Question 15[5 marks]
Answer or working	Marks
attempting to complete the square on x^2 + 8x	M1
(x + 4)^2 as the squared term	M1
(x + 4)^2 + 3	A1
identifying the minimum value as 3	B1
stating this occurs at x = -4	B1
Final answer: (x + 4)^2 + 3; minimum value 3 at x = -4

Mark scheme for Question 16 [6 marks]
Question 16[6 marks]
Answer or working	Marks
writing 9^(n - 1) as 3^(2n - 2)	M1
writing 27^(2 - n) as 3^(6 - 3n)	M1
combining 3^(6 - 3n) x 3^n to give 3^(6 - 2n)	M1
equating exponents 2n - 2 = 6 - 2n	M1
rearranging to 4n = 8	M1
n = 2	A1

Mark scheme for Question 17 [5 marks]
Question 17[5 marks]
Answer or working	Marks
forming the area expression (x + 3)(x - 1)	M1
expanding to x^2 + 2x - 3	M1
forming the inequality x^2 + 2x - 48 > 0	M1
factorising as (x + 8)(x - 6) > 0 and identifying the critical values -8 and 6	M1
x > 6	A1

Mark scheme for Question 18 [6 marks]
Question 18[6 marks]
Answer or working	Marks
using S1 = a to find a = 1	M1
finding S2 and using T2 = S2 - S1 to find the common difference d = 3	M1
a = 1 and d = 3	A1
forming the equation (n/2)(3n - 1) = 330	M1
rearranging to 3n^2 - n - 660 = 0 and solving, e.g. using the quadratic formula	M1
n = 15	A1
Final answer: first term = 1; common difference = 3; n = 15

Mark scheme for Question 19 [5 marks]
Question 19[5 marks]
Answer or working	Marks
forming the area expression (2x + 1)(x - 3)	M1
expanding to 2x^2 - 5x - 3	M1
forming the inequality 2x^2 - 5x - 63 < 0	M1
factorising as (x - 7)(2x + 9) < 0 and identifying the critical values -4.5 and 7	M1
3 < x < 7 (combining with the domain restriction)	A1
Final answer: 3 < x < 7