Year 10

Year 10 Paper 6: Mixed Review

Brings together algebraic manipulation, surds and indices, simultaneous equations and inequalities, functions and their graphs, and sequences in one longer paper.

24 questions - 100 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]

Simultaneous Equations and Inequalities

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

Question 2 [2 marks]

Sequences

An arithmetic sequence has first term 5 and common difference 4.

Find the 12th term of the sequence.

Question 3 [2 marks]

Surds and Indices

Simplify sqrt(50) - sqrt(18), giving your answer in the form k*sqrt(2).

Question 4 [3 marks]

Algebraic Manipulation

Expand and simplify (x - 5)(x + 5) - (x - 3)^2.

Question 5 [3 marks]

Functions and their Graphs

The function g is defined by g(x) = x^2 - 4 for all real x.

Find the values of x for which g(x) = 12.

Question 6 [3 marks]

Algebraic Manipulation

Factorise fully 3x^2 + 12x + 9.

Question 7 [3 marks]

Surds and Indices

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

Question 8 [4 marks]

Algebraic Manipulation

Expand and simplify (2x - 3)(x - 1) + (x + 2)^2.

Question 9 [4 marks]

Functions and their Graphs

f(x) = (4x - 3)/5 for all real x.

Find f^-1(x), the inverse function of f.

Question 10 [4 marks]

Simultaneous Equations and Inequalities

Solve algebraically the simultaneous equations y = 2x - 3 and x^2 + y^2 = 18.

Question 11 [4 marks]

Surds and Indices

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

Question 12 [4 marks]

Algebraic Manipulation

Write as a single fraction in its simplest form: 3/(x + 1) + 2/(x - 2).

Question 13 [4 marks]

Simultaneous Equations and Inequalities

Solve the quadratic inequality x^2 - 5x - 14 <= 0, giving your answer as a single inequality.

Question 14 [4 marks]

Sequences

The first three terms of an arithmetic sequence are 4k - 3, 6k + 1, and 10k - 3, where k is a constant.

Find the value of k, and find the common difference of the sequence.

Question 15 [5 marks]

Surds and Indices

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

Question 16 [5 marks]

Simultaneous Equations and Inequalities

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

Question 17 [5 marks]

Surds and Indices

Rationalise the denominator of 20/sqrt(32), giving your answer in its simplest form.

Question 18 [5 marks]

Functions and their Graphs

The function h is defined by h(x) = x^2 - 6x + 11 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 19 [5 marks]

Surds and Indices

Show that (sqrt(7) - 1)/(sqrt(7) + 1) can be written in the form a + b*sqrt(7), where a and b are rational numbers to be found.

Question 20 [5 marks]

Algebraic Manipulation

Solve the equation 4/(x - 1) - 2/(x + 2) = 1.

Show that your equation reduces to a quadratic before solving it, and give both solutions.

Question 21 [6 marks]

Functions and their Graphs

f(x) = 3/(x + 1) - 2 for x not equal to -1.

Find f^-1(x), the inverse function of f, stating the value excluded from its domain.

Question 22 [6 marks]

Simultaneous Equations and Inequalities

Solve the quadratic inequality 2x^2 + 3x - 20 > 0, giving your answer as two inequalities.

Question 23 [6 marks]

Surds and Indices

Solve 16^(n + 1) = 8^(2n - 1) x 2^n, giving the exact value of n.

Question 24 [6 marks]

Algebraic Manipulation

Write as a single fraction in its simplest form: (2x + 1)/(x^2 - 16) - 3/(x + 4).

Model solutions

Mark scheme for Question 1 [2 marks]
Question 1[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 2 [2 marks]
Question 2[2 marks]
Answer or working	Marks
using the formula a + (n - 1)d with n = 12	M1
T12 = 49	A1
Final answer: 49

Mark scheme for Question 3 [2 marks]
Question 3[2 marks]
Answer or working	Marks
writing sqrt(50) = 5sqrt(2) and sqrt(18) = 3sqrt(2)	M1
2sqrt(2)	A1

Mark scheme for Question 4 [3 marks]
Question 4[3 marks]
Answer or working	Marks
expanding (x - 5)(x + 5) to x^2 - 25	M1
expanding (x - 3)^2 to x^2 - 6x + 9	M1
6x - 34	A1

Mark scheme for Question 5 [3 marks]
Question 5[3 marks]
Answer or working	Marks
forming the equation x^2 - 4 = 12	M1
rearranging to x^2 = 16	M1
x = 4 or x = -4	A1

Mark scheme for Question 6 [3 marks]
Question 6[3 marks]
Answer or working	Marks
taking out the common factor of 3	M1
factorising x^2 + 4x + 3 into two brackets	M1
3(x + 1)(x + 3)	A1

Mark scheme for Question 7 [3 marks]
Question 7[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 8 [4 marks]
Question 8[4 marks]
Answer or working	Marks
expanding (2x - 3)(x - 1) to give 2x^2 - 5x + 3	M1
expanding (x + 2)^2 to give x^2 + 4x + 4	M1
combining like terms	M1
3x^2 - x + 7	A1

Mark scheme for Question 9 [4 marks]
Question 9[4 marks]
Answer or working	Marks
multiplying both sides of y = (4x - 3)/5 by 5 to give 5y = 4x - 3	M1
rearranging to isolate the x term, giving 4x = 5y + 3	M1
dividing by 4 to make x the subject, giving x = (5y + 3)/4	M1
swapping x and y to give f^-1(x) = (5x + 3)/4	A1
Final answer: f^-1(x) = (5x + 3)/4

Mark scheme for Question 10 [4 marks]
Question 10[4 marks]
Answer or working	Marks
substituting y = 2x - 3 into the second equation	M1
expanding and simplifying to 5x^2 - 12x - 9 = 0	M1
solving the quadratic (e.g. using the quadratic formula) to find x = 3 or x = -0.6	M1
(x, y) = (3, 3) and (-0.6, -4.2)	A1
Final answer: (x, y) = (3, 3) or (-0.6, -4.2)

Mark scheme for Question 11 [4 marks]
Question 11[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 12 [4 marks]
Question 12[4 marks]
Answer or working	Marks
using the common denominator (x + 1)(x - 2)	M1
writing 3(x - 2) + 2(x + 1) as the numerator	M1
simplifying the numerator to 5x - 4	A1
(5x - 4)/((x + 1)(x - 2))	A1

Mark scheme for Question 13 [4 marks]
Question 13[4 marks]
Answer or working	Marks
factorising x^2 - 5x - 14 as (x - 7)(x + 2)	M1
identifying the critical values x = -2 and x = 7	M1
reasoning about the sign of the quadratic between the roots	M1
-2 <= x <= 7	A1

Mark scheme for Question 14 [4 marks]
Question 14[4 marks]
Answer or working	Marks
setting up the equation (T2 - T1) = (T3 - T2)	M1
forming 2k + 4 = 4k - 4	M1
k = 4	A1
common difference = 12	A1
Final answer: k = 4; common difference = 12

Mark scheme for Question 15 [5 marks]
Question 15[5 marks]
Answer or working	Marks
cubing to give 8x^9y^-3	M1
multiplying coefficients: 8 x 1 = 8	M1
adding the indices of x: 9 + (-4) = 5	M1
adding the indices of y: -3 + 2 = -1	M1
8x^5y^-1	A1
Final answer: 8x^5y^-1 (equivalently 8x^5/y)

Mark scheme for Question 16 [5 marks]
Question 16[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 17 [5 marks]
Question 17[5 marks]
Answer or working	Marks
writing sqrt(32) = 4sqrt(2)	M1
simplifying to 5/sqrt(2)	M1
multiplying numerator and denominator by sqrt(2)	M1
simplifying the numerator to 5sqrt(2)	M1
(5/2)sqrt(2)	A1
Final answer: 5sqrt(2)/2 (equivalently 2.5sqrt(2))

Mark scheme for Question 18 [5 marks]
Question 18[5 marks]
Answer or working	Marks
attempting to complete the square on x^2 - 6x	M1
(x - 3)^2 as the squared term	M1
(x - 3)^2 + 2	A1
identifying the minimum value as 2	B1
stating this occurs at x = 3	B1
Final answer: (x - 3)^2 + 2; minimum value 2 at x = 3

Mark scheme for Question 19 [5 marks]
Question 19[5 marks]
Answer or working	Marks
multiplying numerator and denominator by (sqrt(7) - 1)	M1
the denominator simplifying to 7 - 1 = 6	M1
expanding the numerator (sqrt(7) - 1)^2	M1
the numerator simplifying to 8 - 2sqrt(7)	A1
a = 4/3 and b = -1/3	A1
Final answer: a = 4/3, b = -1/3, so 4/3 - (1/3)sqrt(7)

Mark scheme for Question 20 [5 marks]
Question 20[5 marks]
Answer or working	Marks
multiplying both sides by (x - 1)(x + 2)	M1
expanding to 4(x + 2) - 2(x - 1) = (x - 1)(x + 2)	M1
rearranging to x^2 - x - 12 = 0	M1
factorising as (x - 4)(x + 3) = 0	M1
x = 4 or x = -3	A1

Mark scheme for Question 21 [6 marks]
Question 21[6 marks]
Answer or working	Marks
adding 2 to both sides to give y + 2 = 3/(x + 1)	M1
taking reciprocals of both sides to give 1/(y + 2) = (x + 1)/3	M1
multiplying both sides by 3 to give 3/(y + 2) = x + 1	M1
subtracting 1 to give x = 3/(y + 2) - 1	M1
swapping x and y to give f^-1(x) = 3/(x + 2) - 1	A1
stating x = -2 is excluded from the domain of f^-1	B1
Final answer: f^-1(x) = 3/(x + 2) - 1, x not equal to -2

Mark scheme for Question 22 [6 marks]
Question 22[6 marks]
Answer or working	Marks
factorising 2x^2 + 3x - 20 as (2x - 5)(x + 4)	M1
identifying the critical values x = -4 and x = 2.5	M1
sketching or reasoning about the shape of the (positive) parabola	M1
reasoning that the expression is positive outside the roots	M1
x < -4	A1
x > 2.5	A1
Final answer: x < -4 or x > 2.5

Mark scheme for Question 23 [6 marks]
Question 23[6 marks]
Answer or working	Marks
writing 16^(n + 1) as 2^(4n + 4)	M1
writing 8^(2n - 1) as 2^(6n - 3)	M1
combining 2^(6n - 3) x 2^n to give 2^(7n - 3)	M1
equating exponents 4n + 4 = 7n - 3	M1
rearranging to 3n = 7	M1
n = 7/3	A1

Mark scheme for Question 24 [6 marks]
Question 24[6 marks]
Answer or working	Marks
factorising x^2 - 16 as (x - 4)(x + 4)	M1
identifying the common denominator (x - 4)(x + 4)	M1
writing 3/(x + 4) as 3(x - 4)/((x - 4)(x + 4))	M1
combining the numerators to give (2x + 1) - 3(x - 4)	M1
simplifying the numerator to 13 - x	A1
(13 - x)/((x - 4)(x + 4))	A1