Questions — Edexcel (10514 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 C3 Q29
6 marks Moderate -0.3
  1. Express as a fraction in its simplest form $$\frac{2}{x-3} + \frac{13}{x^2 + 4x - 21}.$$ [3]
  2. Hence solve $$\frac{2}{x-3} + \frac{13}{x^2 + 4x - 21} = 1.$$ [3]
Edexcel C3 Q30
4 marks Moderate -0.3
Prove that $$\frac{1 - \tan^2 \theta}{1 + \tan^2 \theta} = \cos 2\theta.$$ [4]
Edexcel C3 Q31
13 marks Standard +0.3
The functions \(f\) and \(g\) are defined by $$f: x \mapsto |x - a| + a, \quad x \in \mathbb{R},$$ $$g: x \mapsto 4x + a, \quad x \in \mathbb{R},$$ where \(a\) is a positive constant.
  1. On the same diagram, sketch the graphs of \(f\) and \(g\), showing clearly the coordinates of any points at which your graphs meet the axes. [5]
  2. Use algebra to find, in terms of \(a\), the coordinates of the point at which the graphs of \(f\) and \(g\) intersect. [3]
  3. Find an expression for \(fg(x)\). [2]
  4. Solve, for \(x\) in terms of \(a\), the equation $$fg(x) = 3a.$$ [3]
Edexcel C3 Q32
14 marks Standard +0.3
The curve \(C\) has equation \(y = f(x)\), where $$f(x) = 3 \ln x + \frac{1}{x}, \quad x > 0.$$ The point \(P\) is a stationary point on \(C\).
  1. Calculate the \(x\)-coordinate of \(P\). [4]
  2. Show that the \(y\)-coordinate of \(P\) may be expressed in the form \(k - k \ln k\), where \(k\) is a constant to be found. [2]
The point \(Q\) on \(C\) has \(x\)-coordinate \(1\).
  1. Find an equation for the normal to \(C\) at \(Q\). [4]
The normal to \(C\) at \(Q\) meets \(C\) again at the point \(R\).
  1. Show that the \(x\)-coordinate of \(R\)
    1. satisfies the equation \(6 \ln x + x + \frac{2}{x} - 3 = 0\),
    2. lies between \(0.13\) and \(0.14\). [4]
Edexcel C3 Q33
5 marks Moderate -0.3
The function \(f\) is given by \(f: x \mapsto 2 + \frac{3}{x + 2}, x \in \mathbb{R}, x \neq -2\).
  1. Express \(2 + \frac{3}{x + 2}\) as a single fraction. [1]
  2. Find an expression for \(f^{-1}(x)\). [3]
  3. Write down the domain of \(f^{-1}\). [1]
Edexcel C3 Q34
9 marks Standard +0.3
The function \(f\) is even and has domain \(\mathbb{R}\). For \(x \geq 0\), \(f(x) = x^2 - 4ax\), where \(a\) is a positive constant.
  1. In the space below, sketch the curve with equation \(y = f(x)\), showing the coordinates of all the points at which the curve meets the axes. [3]
  2. Find, in terms of \(a\), the value of \(f(2a)\) and the value of \(f(-2a)\). [2]
Given that \(a = 3\),
  1. use algebra to find the values of \(x\) for which \(f(x) = 45\). [4]
Edexcel C3 Q35
10 marks Moderate -0.3
Given that \(y = \log_a x, x > 0\), where \(a\) is a positive constant,
    1. express \(x\) in terms of \(a\) and \(y\), [1]
    2. deduce that \(\ln x = y \ln a\). [1]
  2. Show that \(\frac{dy}{dx} = \frac{1}{x \ln a}\). [2]
The curve \(C\) has equation \(y = \log_{10} x, x > 0\). The point \(A\) on \(C\) has \(x\)-coordinate \(10\). Using the result in part (b),
  1. find an equation for the tangent to \(C\) at \(A\). [4]
The tangent to \(C\) at \(A\) crosses the \(x\)-axis at the point \(B\).
  1. Find the exact \(x\)-coordinate of \(B\). [2]
Edexcel C3 Q36
12 marks Standard +0.3
    1. Express \((12 \cos \theta - 5 \sin \theta)\) in the form \(R \cos (\theta + \alpha)\), where \(R > 0\) and \(0 < \alpha < 90°\). [4]
    2. Hence solve the equation $$12 \cos \theta - 5 \sin \theta = 4,$$ for \(0 < \theta < 90°\), giving your answer to 1 decimal place. [3]
  2. Solve $$8 \cot \theta - 3 \tan \theta = 2,$$ for \(0 < \theta < 90°\), giving your answer to 1 decimal place. [5]
Edexcel C3 Q37
4 marks Moderate -0.8
Express as a single fraction in its simplest form $$\frac{x^2 - 8x + 15}{x^2 - 9} \times \frac{2x^2 + 6x}{(x - 5)^2}.$$ [4]
Edexcel P4 2024 June Q1
5 marks Standard +0.3
In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable. Find $$\int_0^{\pi/6} x \cos 3x \, dx$$ giving your answer in simplest form. [5]
Edexcel P4 2024 June Q2
6 marks Moderate -0.8
With respect to a fixed origin, \(O\), the point \(A\) has position vector $$\overrightarrow{OA} = \begin{pmatrix} 7 \\ 2 \\ -5 \end{pmatrix}$$ Given that $$\overrightarrow{AB} = \begin{pmatrix} -2 \\ 4 \\ 3 \end{pmatrix}$$
  1. find the coordinates of the point \(B\). [2]
The point \(C\) has position vector $$\overrightarrow{OC} = \begin{pmatrix} a \\ 5 \\ -1 \end{pmatrix}$$ where \(a\) is a constant. Given that \(\overrightarrow{OC}\) is perpendicular to \(\overrightarrow{BC}\)
  1. find the possible values of \(a\). [4]
Edexcel P4 2024 June Q3
7 marks Standard +0.3
The curve \(C\) is defined by the equation $$8x^3 - 3y^2 + 2xy = 9$$ Find an equation of the normal to \(C\) at the point \((2, 5)\), giving your answer in the form \(ax + by + c = 0\), where \(a\), \(b\) and \(c\) are integers. [7]
Edexcel P4 2024 June Q4
6 marks Standard +0.3
\includegraphics{figure_1} Figure 1 shows a sketch of a segment \(PQRP\) of a circle with centre \(O\) and radius \(5\) cm. Given that • angle \(PQR\) is \(\theta\) radians • \(\theta\) is increasing, from \(0\) to \(\pi\), at a constant rate of \(0.1\) radians per second • the area of the segment \(PQRP\) is \(A\) cm²
  1. show that $$\frac{dA}{d\theta} = K(1 - \cos \theta)$$ where \(K\) is a constant to be found. [2]
  2. Find, in cm²s⁻¹, the rate of increase of the area of the segment when \(\theta = \frac{\pi}{3}\) [4]
Edexcel P4 2024 June Q5
13 marks Standard +0.3
\includegraphics{figure_2} Figure 2 shows a sketch of the curve defined by the parametric equations $$x = t^2 + 2t \quad y = \frac{2}{t(3-t)} \quad a \leq t \leq b$$ where \(a\) and \(b\) are constants. The ends of the curve lie on the line with equation \(y = 1\)
  1. Find the value of \(a\) and the value of \(b\) [2]
The region \(R\), shown shaded in Figure 2, is bounded by the curve and the line with equation \(y = 1\)
  1. Show that the area of region \(R\) is given by $$M - k \int_a^b \frac{t+1}{t(3-t)} dt$$ where \(M\) and \(k\) are constants to be found. [5]
    1. Write \(\frac{t+1}{t(3-t)}\) in partial fractions.
    2. Use algebraic integration to find the exact area of \(R\), giving your answer in simplest form. [6]
Edexcel P4 2024 June Q6
10 marks Standard +0.3
With respect to a fixed origin \(O\), the line \(l_1\) is given by the equation $$\mathbf{r} = \mathbf{i} + 2\mathbf{j} + 5\mathbf{k} + \lambda(8\mathbf{i} - \mathbf{j} + 4\mathbf{k})$$ where \(\lambda\) is a scalar parameter. The point \(A\) lies on \(l_1\) Given that \(|\overrightarrow{OA}| = 5\sqrt{10}\)
  1. show that at \(A\) the parameter \(\lambda\) satisfies $$81\lambda^2 + 52\lambda - 220 = 0$$ [3]
Hence
    1. show that one possible position vector for \(A\) is \(-15\mathbf{i} + 4\mathbf{j} - 3\mathbf{k}\)
    2. find the other possible position vector for \(A\). [3]
The line \(l_2\) is parallel to \(l_1\) and passes through \(O\). Given that • \(\overrightarrow{OA} = -15\mathbf{i} + 4\mathbf{j} - 3\mathbf{k}\) • point \(B\) lies on \(l_2\) where \(|\overrightarrow{OB}| = 4\sqrt{10}\)
  1. find the area of triangle \(OAB\), giving your answer to one decimal place. [4]
Edexcel P4 2024 June Q7
11 marks Standard +0.3
The current, \(x\) amps, at time \(t\) seconds after a switch is closed in a particular electric circuit is modelled by the equation $$\frac{dx}{dt} = k - 3x$$ where \(k\) is a constant. Initially there is zero current in the circuit.
  1. Solve the differential equation to find an equation, in terms of \(k\), for the current in the circuit at time \(t\) seconds. Give your answer in the form \(x = f(t)\). [6]
Given that in the long term the current in the circuit approaches \(7\) amps,
  1. find the value of \(k\). [2]
  2. Hence find the time in seconds it takes for the current to reach \(5\) amps, giving your answer to \(2\) significant figures. [3]
Edexcel P4 2024 June Q8
8 marks Standard +0.8
$$f(x) = (8 - 3x)^{\frac{4}{3}} \quad 0 < x < \frac{8}{3}$$
  1. Show that the binomial expansion of \(f(x)\) in ascending powers of \(x\) up to and including the term in \(x^3\) is $$A - 8x + \frac{x^2}{2} + Bx^3 + ...$$ where \(A\) and \(B\) are constants to be found. [4]
  2. Use proof by contradiction to prove that the curve with equation $$y = 8 + 8x - \frac{15}{2}x^2$$ does not intersect the curve with equation $$y = A - 8x + \frac{x^2}{2} + Bx^3 \quad 0 < x < \frac{8}{3}$$ where \(A\) and \(B\) are the constants found in part (a). (Solutions relying on calculator technology are not acceptable.) [4]
Edexcel P4 2024 June Q9
9 marks Challenging +1.2
\includegraphics{figure_3} The curve \(C\), shown in Figure 3, has equation $$y = \frac{x^{-\frac{1}{4}}}{\sqrt{1+x}\left(\arctan\sqrt{x}\right)}$$ The region \(R\), shown shaded in Figure 3, is bounded by \(C\), the line with equation \(x = 3\), the \(x\)-axis and the line with equation \(x = \frac{1}{3}\) The region \(R\) is rotated through \(360°\) about the \(x\)-axis to form a solid. Using the substitution \(\tan u = \sqrt{x}\)
  1. show that the volume \(V\) of the solid formed is given by $$k \int_a^b \frac{1}{u^2} du$$ where \(k\), \(a\) and \(b\) are constants to be found. [6]
  2. Hence, using algebraic integration, find the value of \(V\) in simplest form. [3]
Edexcel P4 2022 October Q1
3 marks Moderate -0.8
A curve \(C\) has parametric equations $$x = \frac{t}{t-3}, \quad y = \frac{1}{t} + 2, \quad t \in \mathbb{R}, \quad t > 3$$ Show that all points on \(C\) lie on the curve with Cartesian equation $$y = \frac{ax - 1}{bx}$$ where \(a\) and \(b\) are constants to be found. [3]
Edexcel P4 2022 October Q2
7 marks Moderate -0.3
  1. Express \(\frac{3x}{(2x-1)(x-2)}\) in partial fraction form. [3]
  2. Hence show that $$\int_5^{25} \frac{3x}{(2x-1)(x-2)} \, dx = \ln k$$ where \(k\) is a fully simplified fraction to be found. (Solutions relying entirely on calculator technology are not acceptable.) [4]
Edexcel P4 2022 October Q3
5 marks Standard +0.3
\includegraphics{figure_1} Figure 1 shows a sketch of triangle \(PQR\). Given that • \(\overrightarrow{PQ} = 2\mathbf{i} - 3\mathbf{j} + 4\mathbf{k}\) • \(\overrightarrow{PR} = 8\mathbf{i} - 5\mathbf{j} + 3\mathbf{k}\)
  1. Find \(\overrightarrow{RQ}\) [2]
  2. Find the size of angle \(PQR\), in degrees, to three significant figures. [3]
Edexcel P4 2022 October Q4
8 marks Standard +0.3
$$g(x) = \frac{1}{\sqrt{4-x^2}}$$
  1. Find, in ascending powers of \(x\), the first four non-zero terms of the binomial expansion of \(g(x)\). Give each coefficient in simplest form. [5]
  2. State the range of values of \(x\) for which this expansion is valid. [1]
  3. Use the expansion from part (a) to find a fully simplified rational approximation for \(\sqrt{3}\) Show your working and make your method clear. [2]
Edexcel P4 2022 October Q5
6 marks Challenging +1.2
In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable. \includegraphics{figure_2} Figure 2 shows a sketch of part of the curve with equation $$y = \frac{12\sqrt{x}}{(2x^2 + 3)^3}$$ The region \(R\), shown shaded in Figure 2, is bounded by the curve, the line with equation \(x = \frac{1}{\sqrt{2}}\), the \(x\)-axis and the line with equation \(x = k\). This region is rotated through \(360°\) about the \(x\)-axis to form a solid of revolution. Given that the volume of this solid is \(\frac{713\pi}{648}\), use algebraic integration to find the exact value of the constant \(k\). [6]
Edexcel P4 2022 October Q6
8 marks Standard +0.3
\includegraphics{figure_3} Figure 3 shows a sketch of the curve \(C\) with parametric equations $$x = 1 + 3\tan t, \quad y = 2\cos 2t, \quad -\frac{\pi}{6} \leq t \leq \frac{\pi}{3}$$ The curve crosses the \(x\)-axis at point \(P\), as shown in Figure 3.
  1. Find the equation of the tangent to \(C\) at \(P\), writing your answer in the form \(y = mx + c\), where \(m\) and \(c\) are constants to be found. [5]
The curve \(C\) has equation \(y = f(x)\), where \(f\) is a function with domain \(\left[k, 1 + 3\sqrt{3}\right]\)
  1. Find the exact value of the constant \(k\). [1]
  2. Find the range of \(f\). [2]
Edexcel P4 2022 October Q7
12 marks Standard +0.3
In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.
  1. Use the substitution \(u = e^x - 3\) to show that $$\int_{\ln 5}^{\ln 7} \frac{4e^{3x}}{e^x - 3} \, dx = a + b \ln 2$$ where \(a\) and \(b\) are constants to be found. [7]
  2. Show, by integration, that $$\int 3e^x \cos 2x \, dx = pe^x \sin 2x + qe^x \cos 2x + c$$ where \(p\) and \(q\) are constants to be found and \(c\) is an arbitrary constant. [5]