Year 11 Paper 4: Higher Algebra and Functions
Covers indices, surds and standard form, linear algebra, quadratics and simultaneous equations, functions and rates of change, sequences and graphs, and statistics and probability.
Year 11 here means a typical teaching order, not a syllabus rule. No exam board defines what belongs to Year 11, and schools sequence the course differently. Check it against your own scheme of work before using it to decide what a class has covered.
Questions
Question 1 [3 marks]
Quadratics and Simultaneous Equations
Factorise x^2 + 2x - 35.
Question 2 [3 marks]
Linear Equations and Inequalities
Expand and simplify 3(2x - 5) + 4(x + 6).
Question 3 [3 marks]
Indices, Surds and Standard Form
Simplify sqrt(8) x sqrt(18), giving your answer as an integer.
Question 4 [4 marks]
Sequences and Graphs
The straight line L has equation y = 2x - 7.
Find the equation of the line parallel to L that passes through (3, 4).
Question 5 [4 marks]
Statistics and Probability
A bag contains 4 red counters and 6 blue counters.
A counter is drawn at random, its colour noted, and then replaced. A second counter is then drawn at random.
Find the probability that both counters are the same colour.
Question 6 [5 marks]
Functions and Rates of Change
The function h is defined by h(x) = 3/(x - 2) for x is not equal to 2.
Find h^-1(x), and hence find h^-1(1).
Question 7 [4 marks]
Statistics and Probability
120 students were asked whether they play a musical instrument.
Of the 65 boys, 28 play an instrument. Of the 55 girls, 31 play an instrument.
A student is chosen at random from all 120 students. Find the probability that the student is a girl who does not play an instrument.
Question 8 [5 marks]
Linear Equations and Inequalities
At a cinema, 3 adult tickets and 2 child tickets cost 34 pounds in total.
2 adult tickets and 5 child tickets cost 41 pounds in total.
Find the cost of one adult ticket and the cost of one child ticket.
Question 9 [5 marks]
Quadratics and Simultaneous Equations
Complete the square for x^2 - 10x + 34.
Hence write down the minimum value of x^2 - 10x + 34.
Question 10 [6 marks]
Linear Equations and Inequalities
Solve the simultaneous equations
3x + 4y = 26
5x - 3y = 24
Question 11 [6 marks]
Sequences and Graphs
The first five terms of a quadratic sequence are 6, 15, 28, 45, 66.
Find an expression for the nth term, then use it to find the 10th term.
Question 12 [6 marks]
Functions and Rates of Change
A curve has equation y = x^3 - 4x^2 + 5.
Find the equation of the tangent to the curve at the point where x = 3.
Question 13 [6 marks]
Linear Equations and Inequalities
A cylinder has total surface area S, base radius r and height h, connected by the formula S = 2 pi r^2 + 2 pi r h.
Make r the subject of the formula, using the quadratic formula. Give your answer in terms of S, h and pi.
Model solutions
| Question 1[3 marks] | |
|---|---|
| Answer or working | Marks |
| factors of -35 that sum to 2 | M1 |
| identifying 7 and -5 | M1 |
| (x + 7)(x - 5) | A1 |
| Question 2[3 marks] | |
|---|---|
| Answer or working | Marks |
| expanding to 6x - 15 + 4x + 24 | M1 |
| collecting like x terms and constants | M1 |
| 10x + 9 | A1 |
| Question 3[3 marks] | |
|---|---|
| Answer or working | Marks |
| multiplying to get sqrt(8 x 18) = sqrt(144) | M1 |
| evaluating 8 x 18 = 144 | M1 |
| 12 | A1 |
| Question 4[4 marks] | |
|---|---|
| Answer or working | Marks |
| identifying gradient 2 from a parallel line | M1 |
| using y = 2x + c | M1 |
| substituting (3, 4) to get 4 = 6 + c | M1 |
| y = 2x - 2 | A1 |
| Question 5[4 marks] | |
|---|---|
| Answer or working | Marks |
| P(red) = 4/10 and P(blue) = 6/10, unchanged after replacement | M1 |
| P(both red) = 4/10 x 4/10 = 16/100 | M1 |
| P(both blue) = 6/10 x 6/10 = 36/100 | M1 |
| 52/100, or 13/25 | A1 |
| Final answer: 13/25 | |
| Question 6[5 marks] | |
|---|---|
| Answer or working | Marks |
| writing x = 3/(y - 2), swapping x and y | M1 |
| multiplying both sides by (y - 2) to get x(y - 2) = 3 | M1 |
| expanding and rearranging to make y the subject | M1 |
| h^-1(x) = (3 + 2x)/x | A1 |
| h^-1(1) = 5 | A1 |
| Final answer: h^-1(x) = (3 + 2x)/x; h^-1(1) = 5 | |
| Question 7[4 marks] | |
|---|---|
| Answer or working | Marks |
| finding the number of girls who do not play an instrument: 55 - 31 = 24 | M1 |
| identifying the total number of students is 120 | M1 |
| forming the probability 24/120 | M1 |
| 1/5 | A1 |
| Question 8[5 marks] | |
|---|---|
| Answer or working | Marks |
| forming the equations 3a + 2c = 34 and 2a + 5c = 41 | M1 |
| multiplying the first equation by 5 and the second by 2 to align the c terms | M1 |
| subtracting to eliminate c, giving 11a = 88 | M1 |
| a = 8 (adult ticket = 8 pounds) | A1 |
| c = 5 (child ticket = 5 pounds) | A1 |
| Final answer: adult ticket = 8 pounds, child ticket = 5 pounds | |
| Question 9[5 marks] | |
|---|---|
| Answer or working | Marks |
| starting (x - 5)^2 | M1 |
| (x - 5)^2 - 25 + 34 | M1 |
| (x - 5)^2 + 9 | A1 |
| recognising the square term is at least zero | M1 |
| minimum value 9 | A1 |
| Final answer: (x - 5)^2 + 9, minimum 9 | |
| Question 10[6 marks] | |
|---|---|
| Answer or working | Marks |
| multiplying the first equation by 3 to get 9x + 12y = 78 | M1 |
| multiplying the second equation by 4 to get 20x - 12y = 96 | M1 |
| adding the two equations to eliminate y | M1 |
| 29x = 174 | M1 |
| x = 6 | A1 |
| y = 2 | A1 |
| Final answer: x = 6, y = 2 | |
| Question 11[6 marks] | |
|---|---|
| Answer or working | Marks |
| first differences 9, 13, 17, 21 | M1 |
| second difference 4, giving 2n^2 | M1 |
| comparing 2n^2 with the sequence to find the linear part 3n + 1 | M1 |
| nth term = 2n^2 + 3n + 1 | A1 |
| substituting n = 10 | M1 |
| 231 | A1 |
| Final answer: 2n^2 + 3n + 1; 10th term = 231 | |
| Question 12[6 marks] | |
|---|---|
| Answer or working | Marks |
| differentiating to get dy/dx = 3x^2 - 8x | M1 |
| substituting x = 3 into dy/dx to find the gradient | M1 |
| gradient = 3 | A1 |
| substituting x = 3 into y = x^3 - 4x^2 + 5 to find the y-coordinate | M1 |
| y = -4, giving the point (3, -4) | A1 |
| y = 3x - 13 | A1 |
| Question 13[6 marks] | |
|---|---|
| Answer or working | Marks |
| rearranging to 2 pi r^2 + 2 pi h r - S = 0 | M1 |
| identifying a = 2 pi, b = 2 pi h and c = -S for the quadratic formula | M1 |
| substituting into r = (-b +/- sqrt(b^2 - 4ac)) / (2a) | M1 |
| simplifying b^2 - 4ac to 4 pi^2 h^2 + 8 pi S | M1 |
| r = (-2 pi h +/- sqrt(4 pi^2 h^2 + 8 pi S)) / (4 pi) | A1 |
| taking the positive square root, since r must be positive | A1 |
| Final answer: r = (-2 pi h + sqrt(4 pi^2 h^2 + 8 pi S)) / (4 pi) | |