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 C2 Q3
8 marks Moderate -0.3
\includegraphics{figure_2} Figure 2 shows the curves with equations \(y = 7 - 2x - 3x^2\) and \(y = \frac{2}{x}\). The two curves intersect at the points \(P\), \(Q\) and \(R\).
  1. Show that the \(x\)-coordinates of \(P\), \(Q\) and \(R\) satisfy the equation $$3x^3 + 2x^2 - 7x + 2 = 0.$$ [2] Given that \(P\) has coordinates \((-2, -1)\),
  2. find the coordinates of \(Q\) and \(R\). [6]
Edexcel C2 Q4
9 marks Moderate -0.8
  1. Expand \((1 + x)^4\) in ascending powers of \(x\). [2]
  2. Using your expansion, express each of the following in the form \(a + b\sqrt{2}\), where \(a\) and \(b\) are integers.
    1. \((1 + \sqrt{2})^4\)
    2. \((1 - \sqrt{2})^8\) [7]
Edexcel C2 Q5
9 marks Moderate -0.3
  1. Describe fully a single transformation that maps the graph of \(y = 3^x\) onto the graph of \(y = (\frac{1}{3})^x\). [1]
  2. Sketch on the same diagram the curves \(y = (\frac{1}{3})^x\) and \(y = 2(3^x)\), showing the coordinates of any points where each curve crosses the coordinate axes. [3]
The curves \(y = (\frac{1}{3})^x\) and \(y = 2(3^x)\) intersect at the point \(P\).
  1. Find the \(x\)-coordinate of \(P\) to 2 decimal places and show that the \(y\)-coordinate of \(P\) is \(\sqrt{2}\). [5]
Edexcel C2 Q6
9 marks Standard +0.3
A curve has the equation $$y = x^3 + ax^2 - 15x + b,$$ where \(a\) and \(b\) are constants. Given that the curve is stationary at the point \((-1, 12)\),
  1. find the values of \(a\) and \(b\), [6]
  2. find the coordinates of the other stationary point of the curve. [3]
Edexcel C2 Q7
9 marks Moderate -0.3
\includegraphics{figure_3} Figure 3 shows part of the curve \(y = \text{f}(x)\) where $$\text{f}(x) = \frac{1 - 8x^3}{x^2}, \quad x \neq 0.$$
  1. Solve the equation \(\text{f}(x) = 0\). [3]
  2. Find \(\int \text{f}(x) \, dx\). [3]
  3. Find the area of the shaded region bounded by the curve \(y = \text{f}(x)\), the \(x\)-axis and the line \(x = 2\). [3]
Edexcel C2 Q8
10 marks Standard +0.3
  1. Given that \(\sin \theta = 2 - \sqrt{2}\), find the value of \(\cos^2 \theta\) in the form \(a + b\sqrt{2}\) where \(a\) and \(b\) are integers. [3]
  2. Find, in terms of \(\pi\), all values of \(x\) in the interval \(0 \leq x < \pi\) for which $$\cos(2x - \frac{\pi}{6}) = \frac{1}{2}.$$ [7]
Edexcel C2 Q9
12 marks Standard +0.3
The second and fifth terms of a geometric series are \(-48\) and \(6\) respectively.
  1. Find the first term and the common ratio of the series. [5]
  2. Find the sum to infinity of the series. [2]
  3. Show that the difference between the sum of the first \(n\) terms of the series and its sum to infinity is given by \(2^{6-n}\). [5]
Edexcel C2 Q1
4 marks Easy -1.2
A geometric series has first term 75 and second term \(-15\).
  1. Find the common ratio of the series. [2]
  2. Find the sum to infinity of the series. [2]
Edexcel C2 Q2
5 marks Moderate -0.3
A circle has the equation $$x^2 + y^2 + 8x - 4y + k = 0,$$ where \(k\) is a constant.
  1. Find the coordinates of the centre of the circle. [2]
Given that the \(x\)-axis is a tangent to the circle,
  1. find the value of \(k\). [3]
Edexcel C2 Q3
6 marks Standard +0.8
\includegraphics{figure_1} Figure 1 shows a circle of radius \(r\) and centre \(O\) in which \(AD\) is a diameter. The points \(B\) and \(C\) lie on the circle such that \(OB\) and \(OC\) are arcs of circles of radius \(r\) with centres \(A\) and \(D\) respectively. Show that the area of the shaded region \(OBC\) is \(\frac{1}{6}r^2(3\sqrt{3} - \pi)\). [6]
Edexcel C2 Q4
6 marks Moderate -0.3
  1. Sketch on the same diagram the graphs of \(y = \sin 2x\) and \(y = \tan \frac{x}{2}\) for \(x\) in the interval \(0 \leq x \leq 360°\). [4]
  2. Hence state how many solutions exist to the equation $$\sin 2x = \tan \frac{x}{2},$$ for \(x\) in the interval \(0 \leq x \leq 360°\) and give a reason for your answer. [2]
Edexcel C2 Q5
7 marks Moderate -0.3
  1. Find the value of \(a\) such that $$\log_a 27 = 3 + \log_a 8.$$ [3]
  2. Solve the equation $$2^{x+3} = 6^{x-1},$$ giving your answer to 3 significant figures. [4]
Edexcel C2 Q6
9 marks Moderate -0.8
  1. Expand \((2 + x)^4\) in ascending powers of \(x\), simplifying each coefficient. [4]
  2. Find the integers \(A\), \(B\) and \(C\) such that $$(2 + x)^4 + (2 - x)^4 = A + Bx^2 + Cx^4.$$ [2]
  3. Find the real values of \(x\) for which $$(2 + x)^4 + (2 - x)^4 = 136.$$ [3]
Edexcel C2 Q7
11 marks Moderate -0.3
$$f(x) = 2x^3 - 5x^2 + x + 2.$$
  1. Show that \((x - 2)\) is a factor of \(f(x)\). [2]
  2. Fully factorise \(f(x)\). [4]
  3. Solve the equation \(f(x) = 0\). [1]
  4. Find the values of \(\theta\) in the interval \(0 \leq \theta \leq 2\pi\) for which $$2\sin^3 \theta - 5\sin^2 \theta + \sin \theta + 2 = 0,$$ giving your answers in terms of \(\pi\). [4]
Edexcel C2 Q8
13 marks Standard +0.3
The curve \(C\) has the equation $$y = 3 - x^{\frac{1}{2}} - 2x^{-\frac{1}{2}}, \quad x > 0.$$
  1. Find the coordinates of the points where \(C\) crosses the \(x\)-axis. [4]
  2. Find the exact coordinates of the stationary point of \(C\). [5]
  3. Determine the nature of the stationary point. [2]
  4. Sketch the curve \(C\). [2]
Edexcel C2 Q9
14 marks Standard +0.3
\includegraphics{figure_2} Figure 2 shows the curve \(C\) with equation \(y = 3x - 4\sqrt{x} + 2\) and the tangent to \(C\) at the point \(A\). Given that \(A\) has \(x\)-coordinate 4,
  1. show that the tangent to \(C\) at \(A\) has the equation \(y = 2x - 2\). [6]
The shaded region is bounded by \(C\), the tangent to \(C\) at \(A\) and the positive coordinate axes.
  1. Find the area of the shaded region. [8]
Edexcel C3 Q1
4 marks Moderate -0.5
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 C3 Q2
6 marks Moderate -0.3
The root of the equation f(x) = 0, where $$f(x) = x + \ln 2x - 4$$ is to be estimated using the iterative formula \(x_{n+1} = 4 - \ln 2x_n\), with \(x_0 = 2.4\).
  1. Showing your values of \(x_1, x_2, x_3, \ldots\), obtain the value, to 3 decimal places, of the root. [4]
  2. By considering the change of sign of f(x) in a suitable interval, justify the accuracy of your answer to part (a). [2]
Edexcel C3 Q3
8 marks Moderate -0.3
The function f is defined by $$f: x \mapsto |2x - a|, \quad x \in \mathbb{R}$$ where \(a\) is a positive constant.
  1. Sketch the graph of \(y = f(x)\), showing the coordinates of the points where the graph cuts the axes. [2]
  2. On a separate diagram, sketch the graph of \(y = f(2x)\), showing the coordinates of the points where the graph cuts the axes. [2]
  3. Given that a solution of the equation f(x) = \(\frac{1}{2}x\) is \(x = 4\), find the two possible values of \(a\). [4]
Edexcel C3 Q4
6 marks Moderate -0.3
Prove that $$\frac{1 - \tan^2 \theta}{1 + \tan^2 \theta} = \cos 2\theta$$ [6]
Edexcel C3 Q5
7 marks Standard +0.3
Express \(\frac{3}{x^2 + 2x} + \frac{x - 4}{x^2 - 4}\) as a single fraction in its simplest form. [7]
Edexcel C3 Q6
10 marks Standard +0.3
The function f, defined for \(x \in \mathbb{R}, x > 0\), is such that $$f'(x) = x^2 - 2 + \frac{1}{x^2}$$
  1. Find the value of f''(x) at \(x = 4\). [3]
  2. Given that f(3) = 0, find f(x). [4]
  3. Prove that f is an increasing function. [3]
Edexcel C3 Q7
10 marks Moderate -0.3
$$f(x) = \frac{2}{x - 1} - \frac{6}{(x - 1)(2x + 1)}, \quad x > 1$$
  1. Prove that f(x) = \(\frac{4}{2x + 1}\). [4]
  2. Find the range of f. [2]
  3. Find \(f^{-1}(x)\). [3]
  4. Find the range of \(f^{-1}(x)\). [1]
Edexcel C3 Q8
13 marks Standard +0.2
The function f is given by $$f: x \mapsto \ln(3x - 6), \quad x \in \mathbb{R}, \quad x > 2$$
  1. Find \(f^{-1}(x)\). [3]
  2. Write down the domain of \(f^{-1}\) and the range of \(f^{-1}\). [2]
  3. Find, to 3 significant figures, the value of \(x\) for which f(x) = 3. [2]
The function g is given by $$g: x \mapsto \ln|3x - 6|, \quad x \in \mathbb{R}, \quad x \neq 2$$
  1. Sketch the graph of \(y = g(x)\). [3]
  2. Find the exact coordinates of all the points at which the graph of \(y = g(x)\) meets the coordinate axes. [3]
Edexcel C3 Q1
9 marks Moderate -0.3
The function f is given by $$f : x \alpha \frac{x}{x^2-1} - \frac{1}{x+1}, \quad x > 1.$$
  1. Show that \(\text{f}(x) = \frac{1}{(x-1)(x+1)}\). [3]
  2. Find the range of f. [2]
The function g is given by $$g : x \alpha \frac{2}{x}, \quad x > 0.$$
  1. Solve gf(x) = 70. [4]