Questions — Edexcel C4 (386 questions)

Browse by module
All questions AEA AS Paper 1 AS Paper 2 AS Pure C1 C12 C2 C3 C34 C4 CP AS CP1 CP2 D1 D2 F1 F2 F3 FD1 FD1 AS FD2 FD2 AS FM1 FM1 AS FM2 FM2 AS FP1 FP1 AS FP2 FP2 AS FP3 FS1 FS1 AS FS2 FS2 AS Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Extra Pure Further Mechanics Further Mechanics A AS Further Mechanics AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics Further Paper 4 Further Pure Core Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Pure with Technology Further Statistics Further Statistics A AS Further Statistics AS Further Statistics B AS Further Statistics Major Further Statistics Minor Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 H240/01 H240/02 H240/03 M1 M2 M3 M4 M5 P1 P2 P3 P4 PMT Mocks PURE Paper 1 Paper 2 Paper 3 Pre-U 9794/1 Pre-U 9794/2 Pre-U 9794/3 Pre-U 9795 Pre-U 9795/1 Pre-U 9795/2 S1 S2 S3 S4 Unit 1 Unit 2 Unit 3 Unit 4
Browse by board
AQA AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Paper 1 Further Paper 2 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics M1 M2 M3 Paper 1 Paper 2 Paper 3 S1 S2 S3 CAIE FP1 FP2 Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 4 M1 M2 P1 P2 P3 S1 S2 Edexcel AEA AS Paper 1 AS Paper 2 C1 C12 C2 C3 C34 C4 CP AS CP1 CP2 D1 D2 F1 F2 F3 FD1 FD1 AS FD2 FD2 AS FM1 FM1 AS FM2 FM2 AS FP1 FP1 AS FP2 FP2 AS FP3 FS1 FS1 AS FS2 FS2 AS M1 M2 M3 M4 M5 P1 P2 P3 P4 PMT Mocks PURE Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 OCR AS Pure C1 C2 C3 C4 D1 D2 FD1 AS FM1 AS FP1 FP1 AS FP2 FP3 FS1 AS Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Mechanics Further Mechanics AS Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Statistics Further Statistics AS H240/01 H240/02 H240/03 M1 M2 M3 M4 PURE S1 S2 S3 S4 OCR MEI AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further Extra Pure Further Mechanics A AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Pure Core Further Pure Core AS Further Pure with Technology Further Statistics A AS Further Statistics B AS Further Statistics Major Further Statistics Minor M1 M2 M3 M4 Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 Pre-U Pre-U 9794/1 Pre-U 9794/2 Pre-U 9794/3 Pre-U 9795 Pre-U 9795/1 Pre-U 9795/2 WJEC Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 Unit 1 Unit 2 Unit 3 Unit 4
Sort by: Default | Easiest first | Hardest first
Edexcel C4 Q2
8 marks Standard +0.3
A curve has the equation $$x^2 + 3xy - 2y^2 + 17 = 0.$$
  1. Find an expression for \(\frac{dy}{dx}\) in terms of \(x\) and \(y\). [5]
  2. Find an equation for the normal to the curve at the point \((3, -2)\). [3]
Edexcel C4 Q3
10 marks Standard +0.3
  1. Find the values of the constants \(A\), \(B\), \(C\) and \(D\) such that $$\frac{2x^3 - 5x^2 + 6}{x^2 - 3x} \equiv Ax + B + \frac{C}{x} + \frac{D}{x-3}.$$ [5]
  2. Evaluate $$\int_1^2 \frac{2x^3 - 5x^2 + 6}{x^2 - 3x} \, dx,$$ giving your answer in the form \(p + q \ln 2\), where \(p\) and \(q\) are integers. [5]
Edexcel C4 Q4
12 marks Standard +0.3
A mathematician is selling goods at a car boot sale. She believes that the rate at which she makes sales depends on the length of time since the start of the sale, \(t\) hours, and the total value of sales she has made up to that time, £\(x\). She uses the model $$\frac{dx}{dt} = \frac{k(5-t)}{x},$$ where \(k\) is a constant. Given that after two hours she has made sales of £96 in total,
  1. solve the differential equation and show that she made £72 in the first hour of the sale. [8]
The mathematician believes that is it not worth staying at the sale once she is making sales at a rate of less than £10 per hour.
  1. Verify that at 3 hours and 5 minutes after the start of the sale, she should have already left. [4]
Edexcel C4 Q5
12 marks Standard +0.3
Relative to a fixed origin, two lines have the equations $$\mathbf{r} = \begin{pmatrix} 4 \\ 1 \\ 1 \end{pmatrix} + s \begin{pmatrix} 1 \\ 4 \\ 5 \end{pmatrix}$$ and $$\mathbf{r} = \begin{pmatrix} -3 \\ 1 \\ -6 \end{pmatrix} + t \begin{pmatrix} 3 \\ a \\ b \end{pmatrix},$$ where \(a\) and \(b\) are constants and \(s\) and \(t\) are scalar parameters. Given that the two lines are perpendicular,
  1. find a linear relationship between \(a\) and \(b\). [2]
Given also that the two lines intersect,
  1. find the values of \(a\) and \(b\), [8]
  2. find the coordinates of the point where they intersect. [2]
Edexcel C4 Q6
13 marks Standard +0.8
\includegraphics{figure_6} Figure 1 shows the curve with equation \(y = x\sqrt{1-x}\), \(0 \leq x \leq 1\).
  1. Use the substitution \(u^2 = 1 - x\) to show that the area of the region bounded by the curve and the \(x\)-axis is \(\frac{8}{15}\). [8]
  2. Find, in terms of \(\pi\), the volume of the solid formed when the region bounded by the curve and the \(x\)-axis is rotated through \(360°\) about the \(x\)-axis. [5]
Edexcel C4 Q7
14 marks Standard +0.3
A curve has parametric equations $$x = 3 \cos^2 t, \quad y = \sin 2t, \quad 0 \leq t < \pi.$$
  1. Show that \(\frac{dy}{dx} = -\frac{2}{3} \cot 2t\). [4]
  2. Find the coordinates of the points where the tangent to the curve is parallel to the \(x\)-axis. [3]
  3. Show that the tangent to the curve at the point where \(t = \frac{\pi}{6}\) has the equation $$2x + 3\sqrt{3} y = 9.$$ [3]
  4. Find a cartesian equation for the curve in the form \(y^2 = \text{f}(x)\). [4]
Edexcel C4 Q1
8 marks Standard +0.8
A curve has the equation $$2x^2 + xy - y^2 + 18 = 0.$$ Find the coordinates of the points where the tangent to the curve is parallel to the \(x\)-axis. [8]
Edexcel C4 Q2
8 marks Standard +0.3
Use the substitution \(x = 2\tan u\) to show that $$\int_0^2 \frac{x^2}{x^2 + 4} \, dx = \frac{1}{2}(4 - \pi).$$ [8]
Edexcel C4 Q3
9 marks Standard +0.3
  1. Show that \((1 + \frac{1}{24})^{-\frac{1}{2}} = k\sqrt{6}\), where \(k\) is rational. [2]
  2. Expand \((1 + \frac{1}{4}x)^{-\frac{1}{2}}\), \(|x| < 2\), in ascending powers of \(x\) up to and including the term in \(x^3\), simplifying each coefficient. [4]
  3. Use your answer to part \((b)\) with \(x = \frac{1}{6}\) to find an approximate value for \(\sqrt{6}\), giving your answer to 5 decimal places. [3]
Edexcel C4 Q4
9 marks Standard +0.3
Relative to a fixed origin, two lines have the equations $$\mathbf{r} = (7\mathbf{i} - 4\mathbf{k}) + s(4\mathbf{i} - 3\mathbf{j} + \mathbf{k}),$$ and $$\mathbf{r} = (-7\mathbf{i} + \mathbf{j} + 8\mathbf{k}) + t(-3\mathbf{i} + 2\mathbf{k}),$$ where \(s\) and \(t\) are scalar parameters.
  1. Show that the two lines intersect and find the position vector of the point where they meet. [5]
  2. Find, in degrees to 1 decimal place, the acute angle between the lines. [4]
Edexcel C4 Q5
11 marks Standard +0.3
A curve has parametric equations $$x = \frac{t}{2-t}, \quad y = \frac{1}{1+t}, \quad -1 < t < 2.$$
  1. Show that \(\frac{dy}{dx} = -\frac{1}{2}\left(\frac{2-t}{1+t}\right)^2\). [4]
  2. Find an equation for the normal to the curve at the point where \(t = 1\). [3]
  3. Show that the cartesian equation of the curve can be written in the form $$y = \frac{1+x}{1+3x}.$$ [4]
Edexcel C4 Q6
13 marks Standard +0.8
  1. Find \(\int \tan^2 x \, dx\). [3]
  2. Show that $$\int \tan x \, dx = \ln|\sec x| + c,$$ where \(c\) is an arbitrary constant. [4]
\includegraphics{figure_1} Figure 1 shows part of the curve with equation \(y = x^2 \tan x\). The shaded region bounded by the curve, the \(x\)-axis and the line \(x = \frac{\pi}{3}\) is rotated through \(2\pi\) radians about the \(x\)-axis.
  1. Show that the volume of the solid formed is \(\frac{1}{18}\pi^2(6\sqrt{3} - \pi) - \pi \ln 2\). [6]
Edexcel C4 Q7
17 marks Standard +0.8
\includegraphics{figure_2} Figure 2 shows a hemispherical bowl of radius 5 cm. The bowl is filled with water but the water leaks from a hole at the base of the bowl. At time \(t\) minutes, the depth of water is \(h\) cm and the volume of water in the bowl is \(V\) cm³, where $$V = \frac{1}{3}\pi h^2(15 - h).$$ In a model it is assumed that the rate at which the volume of water in the bowl decreases is proportional to \(V\).
  1. Show that $$\frac{dh}{dt} = -\frac{kh(15-h)}{3(10-h)},$$ where \(k\) is a positive constant. [5]
  2. Express \(\frac{3(10-h)}{h(15-h)}\) in partial fractions. [3]
Given that when \(t = 0\), \(h = 5\),
  1. show that $$h^2(15-h) = 250e^{-kt}.$$ [6]
Given also that when \(t = 2\), \(h = 4\),
  1. find the value of \(k\) to 3 significant figures. [3]
Edexcel C4 Q1
5 marks Easy -1.2
  1. Expand \((1 + 4x)^5\) in ascending powers of \(x\) up to and including the term in \(x^5\), simplifying each coefficient. [4]
  2. State the set of values of \(x\) for which your expansion is valid. [1]
Edexcel C4 Q2
6 marks Moderate -0.3
Use the substitution \(u = 1 + \sin x\) to find the value of $$\int_0^{\frac{\pi}{4}} \cos x (1 + \sin x)^3 \, dx.$$ [6]
Edexcel C4 Q3
8 marks Moderate -0.3
  1. Express \(\frac{x+11}{(x+4)(x-3)}\) as a sum of partial fractions. [3]
  2. Evaluate $$\int_0^2 \frac{x+11}{(x+4)(x-3)} \, dx,$$ giving your answer in the form \(\ln k\), where \(k\) is an exact simplified fraction. [5]
Edexcel C4 Q4
8 marks Challenging +1.2
\includegraphics{figure_1} Figure 1 shows the curve with equation \(y = 2\sin x + \cosec x\), \(0 < x < \pi\). The shaded region bounded by the curve, the \(x\)-axis and the lines \(x = \frac{\pi}{6}\) and \(x = \frac{\pi}{2}\) is rotated through \(360°\) about the \(x\)-axis. Show that the volume of the solid formed is \(\frac{1}{2}\pi(4\pi + 3\sqrt{3})\). [8]
Edexcel C4 Q5
8 marks Standard +0.3
A curve has the equation $$x^2 - 3xy - y^2 = 12.$$
  1. Find an expression for \(\frac{dy}{dx}\) in terms of \(x\) and \(y\). [5]
  2. Find an equation for the tangent to the curve at the point \((2, -2)\). [3]
Edexcel C4 Q6
10 marks Standard +0.3
Relative to a fixed origin, \(O\), the points \(A\) and \(B\) have position vectors \(\begin{pmatrix} 1 \\ 5 \\ -1 \end{pmatrix}\) and \(\begin{pmatrix} 6 \\ 3 \\ -6 \end{pmatrix}\) respectively. Find, in exact, simplified form,
  1. the cosine of \(\angle AOB\), [4]
  2. the area of triangle \(OAB\), [4]
  3. the shortest distance from \(A\) to the line \(OB\). [2]
Edexcel C4 Q7
14 marks Standard +0.8
A curve has parametric equations $$x = t(t - 1), \quad y = \frac{4t}{1-t}, \quad t \neq 1.$$
  1. Find \(\frac{dy}{dx}\) in terms of \(t\). [4]
The point \(P\) on the curve has parameter \(t = -1\).
  1. Show that the tangent to the curve at \(P\) has the equation $$x + 3y + 4 = 0.$$ [3]
The tangent to the curve at \(P\) meets the curve again at the point \(Q\).
  1. Find the coordinates of \(Q\). [7]
Edexcel C4 Q8
16 marks Standard +0.3
An entomologist is studying the population of insects in a colony. Initially there are 300 insects in the colony and in a model, the entomologist assumes that the population, \(P\), at time \(t\) weeks satisfies the differential equation $$\frac{dP}{dt} = kP,$$ where \(k\) is a constant.
  1. Find an expression for \(P\) in terms of \(k\) and \(t\). [5]
Given that after one week there are 360 insects in the colony,
  1. find the value of \(k\) to 3 significant figures. [2]
Given also that after two and three weeks there are 440 and 600 insects respectively,
  1. comment on suitability of the model. [2]
An alternative model assumes that $$\frac{dP}{dt} = P(0.4 - 0.25\cos 0.5t).$$
  1. Using the initial data, \(P = 300\) when \(t = 0\), solve this differential equation. [4]
  2. Compare the suitability of the two models. [3]
Edexcel C4 Q1
6 marks Standard +0.3
The region bounded by the curve \(y = x^2 - 2x\) and the \(x\)-axis is rotated through \(2\pi\) radians about the \(x\)-axis. Find the volume of the solid formed, giving your answer in terms of \(\pi\). [6]
Edexcel C4 Q2
6 marks Standard +0.3
Use the substitution \(u = 1 - x^2\) to find $$\int \frac{1}{1-x^2} \, dx.$$ [6]
Edexcel C4 Q3
8 marks Standard +0.3
A curve has the equation $$2 \sin 2x - \tan y = 0.$$
  1. Find an expression for \(\frac{dy}{dx}\) in its simplest form in terms of \(x\) and \(y\). [5]
  2. Show that the tangent to the curve at the point \(\left(\frac{\pi}{6}, \frac{\pi}{3}\right)\) has the equation $$y = \frac{1}{2}x + \frac{\pi}{4}.$$ [3]
Edexcel C4 Q4
9 marks Standard +0.3
\includegraphics{figure_1} Figure 1 shows the curve with parametric equations $$x = a\sqrt{t}, \quad y = at(1-t), \quad t \geq 0,$$ where \(a\) is a positive constant.
  1. Find \(\frac{dy}{dx}\) in terms of \(t\). [3]
The curve meets the \(x\)-axis at the origin, \(O\), and at the point \(A\). The tangent to the curve at \(A\) meets the \(y\)-axis at the point \(B\) as shown.
  1. Show that the area of triangle \(OAB\) is \(a^2\). [6]