Pearson Edexcel Level 3 GCE A-Level Mathematics

⚠️ Unofficial Reconstructed Paper Disclaimer:
This is a reconstruction of the official Edexcel A Level Paper 1 2026 based on anecdotal data. Both exact question phrasing and specific mark distributions are professional estimates and may not fully match the official Pearson Edexcel materials. All questions are adjusted to sum exactly to 100 marks.

Instructions & Information

Use black ink or ball-point pen. Answer all questions.
Inexact answers should be given to three significant figures unless otherwise stated.
Calculators must not have the facility for symbolic algebra manipulation, differentiation, or integration.
The total mark for this paper is 100. The marks for each question are shown in brackets.

Question 1

The point P(−4, 5) lies on the curve with equation y = f(x).

State the coordinates of the image of P on the curve with equation:

(a) y = 4f(2x) (2)

(b) y = f(x + 3) − 2 (2)

Question 2

The function f is defined by

f (x) = e 2 x - 1 + 3 x - 7, x \in ℝ

where x is measured in radians.

The equation f(x) = 0 has a single root α.

(a) Show that α lies in the interval [1.1, 1.2]. (2)

(b) Using the iterative formula

x n +1 = ½ (1 + ln(7 - 3 x n))

with x₁ = 1.1, calculate the value of x₂ and hence find the value of α correct to 4 decimal places. (3)

Question 3

The curve C has equation y = x³ − 2x² + 5x − 7.

(a) Find dydx. (2)

(b) Find an equation of the tangent to C at the point where x = 2, giving your answer in the form y = mx + c, where m and c are constants to be found. (3)

Question 4

The function f is defined by f(x) = 3x² − 2, x ∈ ℝ.
The function g is defined by g(x) = 5x − 1x − 4, x ∈ ℝ, x ≠ 4.

(a) State the range of f. (1)

(b) Find the value of gf(2). (2)

Question 5

A helicopter's flight path after take-off is modelled parametrically. The position of the helicopter, t seconds after take-off, is given by:

x = 100 \sqrt t, y = 10 sin(t ² 4), t \geq 0

where x is the horizontal distance in metres, and y is the vertical height above ground level in metres.

(a) State the maximum vertical height of the helicopter above the ground during its flight. (1)

(b) Find the horizontal distance of the helicopter from its take-off point when its vertical height is 5 m for the second time. Give your answer to 1 decimal place. (4)

Question 6

(a) Find ∫ 37x + 1 dx. (2)

(b) Given that p > 2 and

p 2 \int 3 7 x + 1 d x = 67

find the exact value of p. (4)

Question 7

A metal badge is designed in the shape of a semicircle with diameter AC, centre O and radius r. The point B lies on the arc of the semicircle such that angle BOC = θ radians, where 0 < θ < π.

Region R₁ is bounded by the arc BC and the straight chord BC.

Region R₂ is bounded by the straight chords AB and BC, and the arc AC.

Given that the area of region R₁ is exactly twice the area of region R₂:

(a) Show that sin θ + 3θ − 2π = 0. (5)

(b) Using the initial approximation θ₁ = 1.3, apply the Newton-Raphson method once to the equation in part (a) to obtain a second approximation θ₂. Give your answer to 3 decimal places. (2)

Question 8

(a) Write lim_{δx → 0} ∑_{x = 0}³ x e^2x δx as a definite integral. (1)

(b) Use algebraic integration to show that the value of this limit is exactly A e⁶ + B, where A and B are rational constants to be found. (4)

Question 9

(a) Show that

(cos θ + sin θ)(cosec θ - sec θ) = k cot 2 θ

where k is an integer to be found. (2)

(b) Hence solve, for −90° < x < 90°, the equation

5(cos x + sin x)(cosec x - sec x) = 4 cosec²(2 x)

giving your answers to 1 decimal place. (4)

Question 10

The curve C has equation y = 7x√3x² + 4.

The point P lies on C and has x-coordinate 2. The line l is the normal to the curve C at P.

(a) Show that an equation of the line l is 32x + 14y − 113 = 0. (5)

(b) The line l meets the y-axis at the point Q. Find the y-coordinate of Q. (1)

(c) The region R is bounded by the curve C, the normal line l and the y-axis. Using algebraic integration, find the exact area of R. (4)

Question 11

The daily energy consumption, D gigawatt-hours (GWh), in a certain region t hours after midnight is modelled by the equation

D = 30 + 4 cos(π t 12 + 0.2) - 13 sin(π t 12 + 0.2), 0 \leq t \leq 24

(a) Express 4 cos θ − 13 sin θ in the form R cos(θ + α), where R > 0 and 0 < α < π2. Give the exact value of R as a surd, and the value of α in radians to 3 decimal places. (3)

(b) Hence, find the minimum daily energy consumption predicted by this model. (1)

(d) State one reason why this model may not be suitable to predict energy consumption over several weeks. (1)

Question 12

The curve C₁ has equation y = kx, where k > 0 and x ≠ 0.

The curve C₂ has equation y = |px − q|, where p and q are positive constants.

(a) Sketch C₁ and C₂ on the same set of axes. State, in terms of p and q, the coordinates of the points where C₂ meets the coordinate axes. (3)

(b) Given that C₁ and C₂ intersect at exactly three distinct points, find the range of possible values of k in terms of p and q. (5)

Question 13

The third, fourth and fifth terms of an infinite geometric series are:

sin θ, \sqrt 2 cos θ, \sqrt 3 cos θ cot θ

respectively, where 0 < θ < π2.

(a) Show that θ = π6 and find the exact value of the common ratio, r, of this series. (5)

(b) Find the exact sum of the first three terms of this series, giving your answer in the form a + √bc, where a, b and c are integers. (4)

Question 14

A chemical reaction is taking place in a closed laboratory container. The concentration of the chemical, x g ml⁻¹ at time t hours after the reaction starts, is modelled by the differential equation

d x d t = x (A - t)

where A is a positive constant.

Initially, the concentration of the chemical is 0.3 g ml⁻¹.

When t = 4, the concentration is 21 g ml⁻¹.

(a) Solve the differential equation to find an expression for x in terms of t and A. Show that A = 2 + ¼ ln 70. (6)

[For parts (b) and (c), you must use the model equation found in part (a), where A = 2 + ¼ ln 70]

(b) Find the maximum concentration of the chemical predicted by this model. Give your answer to 1 decimal place. (3)

(c) Find the value of T (T > 0) at which the concentration of the chemical falls back to 0.1 g ml⁻¹. Give your answer to 3 decimal places. (2)