A Level Paper 1: Pure Foundations
Covers proof and algebraic methods, coordinate geometry, sequences and series with the binomial expansion, trigonometry, exponentials and logarithms, and differentiation.
Questions
Question 1 [2 marks]
Exponentials and Logarithms
Without using a calculator, solve the equation 2^x = 1/8, giving the exact value of x.
Question 2 [3 marks]
Sequences, Series and the Binomial Expansion
The first three terms of a geometric sequence are 8, 12, 18.
Find the common ratio and the 6th term.
Question 3 [3 marks]
Coordinate Geometry
A line passes through the point (2, -5) and is parallel to the line 3x - y + 4 = 0.
Find the equation of the line in the form y = mx + c.
Question 4 [3 marks]
Trigonometry
Without using a calculator, given that sin(theta) = 3/5 where theta is acute, find the exact values of cos(theta) and tan(theta).
Question 5 [3 marks]
Differentiation
Without using a calculator, find dy/dx for y = e^(3x) - 4x.
Question 6 [3 marks]
Proof and Algebraic Methods
By finding a suitable counter-example, show that the statement "n^2 + n + 1 is a prime number for every positive integer n" is false.
Question 7 [4 marks]
Exponentials and Logarithms
The number of bacteria in a culture is modelled by N = 200 e^(0.15t), where t is the time in hours after the culture is started.
Find the number of bacteria after 10 hours, and find the time taken for the number of bacteria to reach 1000, giving your answer to 1 decimal place.
Question 8 [4 marks]
Differentiation
A curve has equation y = (4x - 1)/(x + 2).
Find dy/dx using the quotient rule, and find the gradient of the curve at the point where x = 1.
Question 9 [4 marks]
Exponentials and Logarithms
Given that log_a(5) = p and log_a(2) = q, express log_a(20) in terms of p and q.
Question 10 [5 marks]
Differentiation
Without using a calculator, differentiate y = (5x^2 - 3x)^4 with respect to x, using the chain rule.
Question 11 [5 marks]
Differentiation
A curve has parametric equations x = t^2, y = t^3 - 3t.
Find dy/dx in terms of t, and find the gradient of the curve at the point where t = 2.
Question 12 [5 marks]
Exponentials and Logarithms
Solve the equation 5^x = 3^(x + 1), giving your answer to 3 significant figures.
Question 13 [5 marks]
Sequences, Series and the Binomial Expansion
In the binomial expansion of (2 + kx)^5, the coefficient of the x^2 term is 720.
Find the possible value(s) of k.
Question 14 [5 marks]
Exponentials and Logarithms
Without using a calculator, solve the equation (log_2(x))^2 - 5 log_2(x) + 6 = 0, giving both values of x as exact values.
Question 15 [6 marks]
Differentiation
A curve has equation y = 2x^3 - 3x^2 - 12x + 5.
Find the coordinates of the two stationary points on the curve, and determine the nature of each.
Model solutions
| Question 1[2 marks] | |
|---|---|
| Answer or working | Marks |
| writing 1/8 as 2^-3 | M1 |
| x = -3 | A1 |
| Question 2[3 marks] | |
|---|---|
| Answer or working | Marks |
| common ratio r = 12/8 = 1.5 | M1 |
| using the 6th term = a*r^5 | M1 |
| 6th term = 60.75 | A1 |
| Final answer: r = 1.5, 6th term = 60.75 | |
| Question 3[3 marks] | |
|---|---|
| Answer or working | Marks |
| finding the gradient of the given line as 3 (from y = 3x + 4) | M1 |
| forming the equation y - (-5) = 3(x - 2) | M1 |
| y = 3x - 11 | A1 |
| Question 4[3 marks] | |
|---|---|
| Answer or working | Marks |
| using Pythagoras' theorem (a 3-4-5 triangle) to find the third side = 4 | M1 |
| cos(theta) = 4/5 | A1 |
| tan(theta) = 3/4 | A1 |
| Final answer: cos(theta) = 4/5, tan(theta) = 3/4 | |
| Question 5[3 marks] | |
|---|---|
| Answer or working | Marks |
| using the chain rule to differentiate e^(3x) | M1 |
| the derivative of e^(3x) is 3e^(3x) | A1 |
| dy/dx = 3e^(3x) - 4 | A1 |
| Question 6[3 marks] | |
|---|---|
| Answer or working | Marks |
| testing successive values of n (e.g. n = 1, 2, 3) and finding each gives a prime | M1 |
| testing n = 4 to obtain n^2 + n + 1 = 21 | M1 |
| identifying 21 = 3 x 7 is not prime, so the statement is false | A1 |
| Final answer: False: n = 4 gives n^2 + n + 1 = 21 = 3 x 7, which is not prime. | |
| Question 7[4 marks] | |
|---|---|
| Answer or working | Marks |
| substituting t = 10 into the model | M1 |
| 896 bacteria (nearest whole number) | A1 |
| setting 200 e^(0.15t) = 1000 and taking logarithms | M1 |
| t = 10.7 hours (1 dp) | A1 |
| Final answer: 896 bacteria after 10 hours; t = 10.7 hours | |
| Question 8[4 marks] | |
|---|---|
| Answer or working | Marks |
| applying the quotient rule with u' = 4 and v' = 1 | M1 |
| dy/dx = 9/(x + 2)^2 after simplification | A1 |
| substituting x = 1 | M1 |
| gradient = 1 | A1 |
| Final answer: dy/dx = 9/(x + 2)^2; gradient at x = 1 is 1 | |
| Question 9[4 marks] | |
|---|---|
| Answer or working | Marks |
| writing 20 as 4 x 5 = 2^2 x 5 | M1 |
| using the multiplication law log_a(2^2 x 5) = log_a(2^2) + log_a(5) | M1 |
| using the power law log_a(2^2) = 2 log_a(2) = 2q | A1 |
| combining to give log_a(20) = p + 2q | A1 |
| Final answer: log_a(20) = p + 2q | |
| Question 10[5 marks] | |
|---|---|
| Answer or working | Marks |
| identifying the inner function u = 5x^2 - 3x and outer function u^4 | M1 |
| differentiating the outer function: 4u^3 | M1 |
| differentiating the inner function: du/dx = 10x - 3 | A1 |
| applying the chain rule dy/dx = 4u^3 x du/dx | M1 |
| dy/dx = 4(5x^2 - 3x)^3(10x - 3) | A1 |
| Final answer: dy/dx = 4(5x^2 - 3x)^3 (10x - 3) | |
| Question 11[5 marks] | |
|---|---|
| Answer or working | Marks |
| finding dx/dt = 2t | M1 |
| finding dy/dt = 3t^2 - 3 | M1 |
| dy/dx = (3t^2 - 3)/(2t) | A1 |
| substituting t = 2 | M1 |
| the gradient = 9/4 | A1 |
| Final answer: dy/dx = (3t^2 - 3)/(2t); gradient at t = 2 is 9/4 | |
| Question 12[5 marks] | |
|---|---|
| Answer or working | Marks |
| taking logarithms of both sides | M1 |
| using the power law to give x ln(5) = (x + 1) ln(3) | M1 |
| rearranging to x(ln(5) - ln(3)) = ln(3) | A1 |
| x = ln(3) / (ln(5) - ln(3)) (unrounded) | A1 |
| x = 2.15 (3 sf) | A1 |
| Question 13[5 marks] | |
|---|---|
| Answer or working | Marks |
| identifying the general term C(5,2)(2)^3(kx)^2 | M1 |
| simplifying this term to 80k^2 x^2 | M1 |
| forming the equation 80k^2 = 720 | A1 |
| solving to give k^2 = 9 | M1 |
| k = 3 or k = -3 | A1 |
| Question 14[5 marks] | |
|---|---|
| Answer or working | Marks |
| substituting y = log_2(x) to obtain the quadratic y^2 - 5y + 6 = 0 | M1 |
| factorising to (y - 2)(y - 3) = 0 | A1 |
| y = 2 or y = 3 | A1 |
| converting back using x = 2^y | M1 |
| x = 4 or x = 8 | A1 |
| Question 15[6 marks] | |
|---|---|
| Answer or working | Marks |
| differentiating to dy/dx = 6x^2 - 6x - 12 | M1 |
| factorising/solving to x = 2 and x = -1 | M1 |
| the y-coordinates (2, -15) and (-1, 12) | A1 |
| finding the second derivative d^2y/dx^2 = 12x - 6 | M1 |
| at x = 2, d^2y/dx^2 = 18 > 0, so a minimum | A1 |
| at x = -1, d^2y/dx^2 = -18 < 0, so a maximum | A1 |
| Final answer: (2, -15) is a minimum; (-1, 12) is a maximum | |