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 Q30
11 marks Standard +0.3
Liquid is poured into a container at a constant rate of 30 cm\(^3\) s\(^{-1}\). At time \(t\) seconds liquid is leaking from the container at a rate of \(\frac{1}{5}V\) cm\(^3\) s\(^{-1}\), where \(V\) cm\(^3\) is the volume of liquid in the container at that time.
  1. Show that $$-15 \frac{dV}{dt} = 2V - 450.$$ [3]
Given that \(V = 1000\) when \(t = 0\),
  1. find the solution of the differential equation, in the form \(V = f(t)\). [7]
  2. Find the limiting value of \(V\) as \(t \to \infty\). [1]
Edexcel C4 Q1
6 marks Standard +0.3
Use integration by parts to find the exact value of \(\int_1^3 x^2 \ln x \, dx\). [6]
Edexcel C4 Q2
12 marks Moderate -0.3
Fluid flows out of a cylindrical tank with constant cross section. At time \(t\) minutes, \(t \geq 0\), the volume of fluid remaining in the tank is \(V\) m\(^3\). The rate at which the fluid flows, in m\(^3\) min\(^{-1}\), is proportional to the square root of \(V\).
  1. Show that the depth \(h\) metres of fluid in the tank satisfies the differential equation $$\frac{dh}{dt} = -k\sqrt{h}, \quad \text{where } k \text{ is a positive constant.}$$ [3]
  2. Show that the general solution of the differential equation may be written as $$h = (A - Bt)^2, \quad \text{where } A \text{ and } B \text{ are constants.}$$ [4] Given that at time \(t = 0\) the depth of fluid in the tank is 1 m, and that 5 minutes later the depth of fluid has reduced to 0.5 m,
  3. find the time, \(T\) minutes, which it takes for the tank to empty. [3]
  4. Find the depth of water in the tank at time \(0.5T\) minutes. [2]
Edexcel C4 Q3
14 marks Standard +0.3
  1. Use the identity for \(\cos(A + B)\) to prove that \(\cos 2A = 2\cos^2 A - 1\). [2]
  2. Use the substitution \(x = 2\sqrt{2} \sin \theta\) to prove that $$\int_2^{\sqrt{6}} \sqrt{(8 - x^2)} \, dx = \frac{1}{3}(\pi + 3\sqrt{3} - 6).$$ [7]
A curve is given by the parametric equations $$x = \sec \theta, \quad y = \ln(1 + \cos 2\theta), \quad 0 \leq \theta < \frac{\pi}{2}.$$
  1. Find an equation of the tangent to the curve at the point where \(\theta = \frac{\pi}{3}\). [5]
Edexcel C4 Q4
11 marks Standard +0.3
\includegraphics{figure_2} Figure 2 shows a sketch of the curve \(C\) with equation \(y = \frac{4}{x - 3}\), \(x \neq 3\). The points \(A\) and \(B\) on the curve have \(x\)-coordinates 3.25 and 5 respectively.
  1. Write down the \(y\)-coordinates of \(A\) and \(B\). [1]
  2. Show that an equation of \(C\) is \(\frac{3y + 4}{y} = 0\), \(y \neq 0\). [1]
The shaded region \(R\) is bounded by \(C\), the \(y\)-axis and the lines through \(A\) and \(B\) parallel to the \(x\)-axis. The region \(R\) is rotated through 360° about the \(y\)-axis to form a solid shape \(S\).
  1. Find the volume of \(S\), giving your answer in the form \(\pi(a + b \ln c)\), where \(a\), \(b\) and \(c\) are integers. [7]
The solid shape \(S\) is used to model a cooling tower. Given that 1 unit on each axis represents 3 metres,
  1. show that the volume of the tower is approximately 15500 m\(^3\). [2]
Edexcel C4 Q5
11 marks Standard +0.3
Relative to a fixed origin \(O\), the point \(A\) has position vector \(3\mathbf{i} + 2\mathbf{j} - \mathbf{k}\), the point \(B\) has position vector \(5\mathbf{i} + \mathbf{j} + \mathbf{k}\), and the point \(C\) has position vector \(7\mathbf{i} - \mathbf{j}\).
  1. Find the cosine of angle \(ABC\). [4]
  2. Find the exact value of the area of triangle \(ABC\). [3]
The point \(D\) has position vector \(7\mathbf{i} + 3\mathbf{k}\).
  1. Show that \(AC\) is perpendicular to \(CD\). [2]
  2. Find the ratio \(AD:DB\). [2]
Edexcel C4 Q6
9 marks Standard +0.3
\includegraphics{figure_2} Figure 2 shows the cross-section of a road tunnel and its concrete surround. The curved section of the tunnel is modelled by the curve with equation \(y = 8\sqrt{\sin \frac{\pi x}{10}}\), in the interval \(0 \leq x \leq 10\). The concrete surround is represented by the shaded area bounded by the curve, the \(x\)-axis and the lines \(x = -2\), \(x = 12\) and \(y = 10\). The units on both axes are metres.
  1. Using this model, copy and complete the table below, giving the values of \(y\) to 2 decimal places.
    \(x\)0246810
    \(y\)06.130
    [2]
The area of the cross-section of the tunnel is given by \(\int_0^{10} y \, dx\).
  1. Estimate this area, using the trapezium rule with all the values from your table. [4]
  2. Deduce an estimate of the cross-sectional area of the concrete surround. [1]
  3. State, with a reason, whether your answer in part (c) over-estimates or under-estimates the true value. [2]
Edexcel C4 Q7
16 marks Standard +0.8
$$\text{f}(x) = \frac{25}{(3 + 2x)^2(1 - x)}, \quad |x| < 1.$$
  1. Express f(x) as a sum of partial fractions. [4]
  2. Hence find \(\int \text{f}(x) \, dx\). [5]
  3. Find the series expansion of f(x) in ascending powers of \(x\) up to and including the term in \(x^2\). Give each coefficient as a simplified fraction. [7]
Edexcel C4 Q1
8 marks Standard +0.3
The curve \(C\) has equation \(5x^2 + 2xy - 3y^2 + 3 = 0\). The point \(P\) on the curve \(C\) has coordinates \((1, 2)\).
  1. Find the gradient of the curve at \(P\). [5]
  2. Find the equation of the normal to the curve \(C\) at \(P\), in the form \(y = ax + b\), where \(a\) and \(b\) are constants. [3]
Edexcel C4 Q2
8 marks Standard +0.3
\includegraphics{figure_1} In Fig. 1, the curve \(C\) has equation \(y = f(x)\), where $$f(x) = x + \frac{2}{x^2}, \quad x > 0.$$ The shaded region is bounded by \(C\), the \(x\)-axis and the lines with equations \(x = 1\) and \(x = 2\). The shaded region is rotated through \(2\pi\) radians about the \(x\)-axis. Using calculus, calculate the volume of the solid generated. Give your answer in the form \(\pi(a + \ln b)\), where \(a\) and \(b\) are constants. [8]
Edexcel C4 Q3
9 marks Moderate -0.3
\includegraphics{figure_2} Figure 2 shows part of the curve with equation $$y = e^x \cos x, \quad 0 \leq x \leq \frac{\pi}{2}.$$ The finite region \(R\) is bounded by the curve and the coordinate axes.
  1. Calculate, to 2 decimal places, the \(y\)-coordinates of the points on the curve where \(x = 0\), \(\frac{\pi}{6}\), \(\frac{\pi}{3}\) and \(\frac{\pi}{2}\). [3]
  2. Using the trapezium rule and all the values calculated in part (a), find an approximation for the area of \(R\). [4]
  3. State, with a reason, whether your approximation underestimates or overestimates the area of \(R\). [2]
Edexcel C4 Q4
10 marks Moderate -0.3
A curve is given parametrically by the equations $$x = 5 \cos t, \quad y = -2 + 4 \sin t, \quad 0 \leq t < 2\pi.$$
  1. Find the coordinates of all the points at which \(C\) intersects the coordinate axes, giving your answers in surd form where appropriate. [4]
  2. Sketch the graph at \(C\). [2]
\(P\) is the point on \(C\) where \(t = \frac{1}{6}\pi\).
  1. Show that the normal to \(C\) at \(P\) has equation $$8\sqrt{3}y = 10x - 25\sqrt{3}.$$ [4]
Edexcel C4 Q5
11 marks Standard +0.3
\includegraphics{figure_1} The curve \(C\) has equation \(y = f(x)\), \(x \in \mathbb{R}\). Figure 1 shows the part of \(C\) for which \(0 \leq x \leq 2\). Given that $$\frac{dy}{dx} = e^x - 2x^2,$$ and that \(C\) has a single maximum, at \(x = k\),
  1. show that \(1.48 < k < 1.49\). [3]
Given also that the point \((0, 5)\) lies on \(C\),
  1. find \(f(x)\). [4]
The finite region \(R\) is bounded by \(C\), the coordinate axes and the line \(x = 2\).
  1. Use integration to find the exact area of \(R\). [4]
Edexcel C4 Q6
8 marks Standard +0.3
When \((1 + ax)^n\) is expanded as a series in ascending powers of \(x\), the coefficients of \(x\) and \(x^2\) are \(-6\) and \(27\) respectively.
  1. Find the value of \(a\) and the value of \(n\). [5]
  2. Find the coefficient of \(x^3\). [2]
  3. State the set of values of \(x\) for which the expansion is valid. [1]
Edexcel C4 Q7
12 marks Standard +0.3
Two submarines are travelling in straight lines through the ocean. Relative to a fixed origin, the vector equations of the two lines, \(l_1\) and \(l_2\), along which they travel are \begin{align} \mathbf{r} &= 3\mathbf{i} + 4\mathbf{j} - 5\mathbf{k} + \lambda(\mathbf{i} - 2\mathbf{j} + 2\mathbf{k})
\text{and} \quad \mathbf{r} &= 9\mathbf{i} + \mathbf{j} - 2\mathbf{k} + \mu (4\mathbf{i} + \mathbf{j} - \mathbf{k}), \end{align} where \(\lambda\) and \(\mu\) are scalars.
  1. Show that the submarines are moving in perpendicular directions. [2]
  2. Given that \(l_1\) and \(l_2\) intersect at the point \(A\), find the position vector of \(A\). [5]
The point \(B\) has position vector \(10\mathbf{j} - 11\mathbf{k}\).
  1. Show that only one of the submarines passes through the point \(B\). [3]
  2. Given that 1 unit on each coordinate axis represents 100 m, find, in km, the distance \(AB\). [2]
Edexcel C4 Q8
13 marks Standard +0.8
In a chemical reaction two substances combine to form a third substance. At time \(t\), \(t \geq 0\), the concentration of this third substance is \(x\) and the reaction is modelled by the differential equation $$\frac{dx}{dt} = k(1 - 2x)(1 - 4x), \text{ where } k \text{ is a positive constant.}$$
  1. Solve this differential equation and hence show that $$\ln \left| \frac{1 - 2x}{1 - 4x} \right| = 2kt + c, \text{ where } c \text{ is an arbitrary constant.}$$ [7]
  2. Given that \(x = 0\) when \(t = 0\), find an expression for \(x\) in terms of \(k\) and \(t\). [4]
  3. Find the limiting value of the concentration \(x\) as \(t\) becomes very large. [2]
Edexcel C4 Q1
8 marks Moderate -0.3
  1. Express \(1.5 \sin 2x + 2 \cos 2x\) in the form \(R \sin (2x + \alpha)\), where \(R > 0\) and \(0 < \alpha < \frac{1}{2}\pi\), giving your values of \(R\) and \(\alpha\) to 3 decimal places where appropriate. [4]
  2. Express \(3 \sin x \cos x + 4 \cos^2 x\) in the form \(a \cos 2x + b \sin 2x + c\), where \(a\), \(b\) and \(c\) are constants to be found. [2]
  3. Hence, using your answer to part (a), deduce the maximum value of \(3 \sin x \cos x + 4 \cos^2 x\). [2]
Edexcel C4 Q2
9 marks Standard +0.3
\includegraphics{figure_1} Figure 1 shows part of the curve with equation \(y = f(x)\), where $$f(x) = \frac{x^2 + 1}{(1 + x)(3 - x)}, \quad 0 \leq x < 3.$$
  1. Given that \(f(x) = A + \frac{B}{1 + x} + \frac{C}{3 - x}\), find the values of the constants \(A\), \(B\) and \(C\). [4]
The finite region \(R\), shown in Fig. 1, is bounded by the curve with equation \(y = f(x)\), the \(x\)-axis, the \(y\)-axis and the line \(x = 2\).
  1. Find the area of \(R\), giving your answer in the form \(p + q \ln r\), where \(p\), \(q\) and \(r\) are rational constants to be found. [5]
Edexcel C4 Q3
12 marks Moderate -0.8
A student tests the accuracy of the trapezium rule by evaluating \(I\), where $$I = \int_{0.5}^{1.5} \left(\frac{3}{x} + x^4\right) dx.$$
  1. Complete the student's table, giving values to 2 decimal places where appropriate.
    \(x\)0.50.7511.251.5
    \(\frac{3}{x} + x^4\)6.064.32
    [2]
  2. Use the trapezium rule, with all the values from your table, to calculate an estimate for the value of \(I\). [4]
  3. Use integration to calculate the exact value of \(I\). [4]
  4. Verify that the answer obtained by the trapezium rule is within 3\% of the exact value. [2]
Edexcel C4 Q4
10 marks Standard +0.8
\includegraphics{figure_1} Figure 1 shows a cross-section \(R\) of a dam. The line \(AC\) is the vertical face of the dam, \(AB\) is the horizontal base and the curve \(BC\) is the profile. Taking \(x\) and \(y\) to be the horizontal and vertical axes, then \(A\), \(B\) and \(C\) have coordinates \((0, 0)\), \((3\pi^2, 0)\) and \((0, 30)\) respectively. The area of the cross-section is to be calculated. Initially the profile \(BC\) is approximated by a straight line.
  1. Find an estimate for the area of the cross-section \(R\) using this approximation. [1]
The profile \(BC\) is actually described by the parametric equations. $$x = 16t^2 - \pi^2, \quad y = 30 \sin 2t, \quad \frac{\pi}{4} \leq t \leq \frac{\pi}{2}.$$
  1. Find the exact area of the cross-section \(R\). [7]
  2. Calculate the percentage error in the estimate of the area of the cross-section \(R\) that you found in part (a). [2]
Edexcel C4 Q5
10 marks Moderate -0.3
  1. Prove that, when \(x = \frac{1}{15}\), the value of \((1 + 5x)^{-\frac{1}{3}}\) is exactly equal to \(\sin 60°\). [3]
  2. Expand \((1 + 5x)^{-\frac{1}{3}}\), \(|x| < 0.2\), in ascending powers of \(x\) up to and including the term in \(x^3\), simplifying each term. [4]
  3. Use your answer to part (b) to find an approximation for \(\sin 60°\). [2]
  4. Find the difference between the exact value of \(\sin 60°\) and the approximation in part (c). [1]
Edexcel C4 Q6
11 marks Standard +0.3
  1. Use integration by parts to show that $$\int_0^{\frac{\pi}{4}} x \sec^2 x \, dx = \frac{1}{4}\pi - \frac{1}{2} \ln 2.$$ [6]
\includegraphics{figure_1} The finite region \(R\), bounded by the equation \(y = x^{\frac{1}{2}} \sec x\), the line \(x = \frac{\pi}{4}\) and the \(x\)-axis is shown in Fig. 1. The region \(R\) is rotated through \(2\pi\) radians about the \(x\)-axis.
  1. Find the volume of the solid of revolution generated. [2]
  2. Find the gradient of the curve with equation \(y = x^{\frac{1}{2}} \sec x\) at the point where \(x = \frac{\pi}{4}\). [3]
Edexcel C4 Q7
12 marks Standard +0.3
\includegraphics{figure_3} The curve \(C\) with equation \(y = 2e^x + 5\) meets the \(y\)-axis at the point \(M\), as shown in Fig. 3.
  1. Find the equation of the normal to \(C\) at \(M\) in the form \(ax + by = c\), where \(a\), \(b\) and \(c\) are integers. [4]
This normal to \(C\) at \(M\) crosses the \(x\)-axis at the point \(N(n, 0)\).
  1. Show that \(n = 14\). [1]
The point \(P(\ln 4, 13)\) lies on \(C\). The finite region \(R\) is bounded by \(C\), the axes and the line \(PN\), as shown in Fig. 3.
  1. Find the area of \(R\), giving your answers in the form \(p + q \ln 2\), where \(p\) and \(q\) are integers to be found. [7]
Edexcel C4 Q8
13 marks Standard +0.3
Referred to an origin \(O\), the points \(A\), \(B\) and \(C\) have position vectors \((\mathbf{9i} - \mathbf{2j} + \mathbf{k})\), \((\mathbf{6i} + \mathbf{2j} + \mathbf{6k})\) and \((\mathbf{3i} + p\mathbf{j} + q\mathbf{k})\) respectively, where \(p\) and \(q\) are constants.
  1. Find, in vector form, an equation of the line \(l\) which passes through \(A\) and \(B\). [2]
Given that \(C\) lies on \(l\),
  1. find the value of \(p\) and the value of \(q\), [2]
  2. calculate, in degrees, the acute angle between \(OC\) and \(AB\). [3]
The point \(D\) lies on \(AB\) and is such that \(OD\) is perpendicular to \(AB\).
  1. Find the position vector of \(D\). [6]
Edexcel C4 Q1
6 marks Moderate -0.3
  1. Find the binomial expansion of \((2 - 3x)^{-3}\) in ascending powers of \(x\) up to and including the term in \(x^3\), simplifying each coefficient. [5]
  2. State the set of values of \(x\) for which your expansion is valid. [1]