Questions — CAIE P3 (1110 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
CAIE P3 2024 June Q10
11 marks Standard +0.3
The equations of two straight lines are $$\mathbf{r} = \mathbf{i} + \mathbf{j} + 2a\mathbf{k} + \lambda(3\mathbf{i} + 4\mathbf{j} + a\mathbf{k}) \quad \text{and} \quad \mathbf{r} = -3\mathbf{i} - \mathbf{j} + 4\mathbf{k} + \mu(-\mathbf{i} + 2\mathbf{j} + 2\mathbf{k}),$$ where \(a\) is a constant.
  1. Given that the acute angle between the directions of these lines is \(\frac{1}{4}\pi\), find the possible values of \(a\). [6]
  2. Given instead that the lines intersect, find the value of \(a\) and the position vector of the point of intersection. [5]
CAIE P3 2024 June Q11
9 marks Standard +0.8
Use the substitution \(2x = \tan \theta\) to find the exact value of $$\int_0^{\frac{1}{2}} \frac{12}{(1 + 4x^2)^2} \, dx .$$ Give your answer in the form \(a + b\pi\), where \(a\) and \(b\) are rational numbers. [9]
CAIE P3 2021 March Q1
3 marks Moderate -0.3
Solve the equation \(\ln(x^3 - 3) = 3 \ln x - \ln 3\). Give your answer correct to 3 significant figures. [3]
CAIE P3 2021 March Q2
5 marks Moderate -0.8
The polynomial \(ax^3 + 5x^2 - 4x + b\), where \(a\) and \(b\) are constants, is denoted by p\((x)\). It is given that \((x + 2)\) is a factor of p\((x)\) and that when p\((x)\) is divided by \((x + 1)\) the remainder is 2. Find the values of \(a\) and \(b\). [5]
CAIE P3 2021 March Q3
6 marks Standard +0.3
By first expressing the equation \(\tan(x + 45°) = 2 \cot x + 1\) as a quadratic equation in \(\tan x\), solve the equation for \(0° < x < 180°\). [6]
CAIE P3 2021 March Q4
7 marks Standard +0.3
The variables \(x\) and \(y\) satisfy the differential equation $$(1 - \cos x)\frac{dy}{dx} = y \sin x.$$ It is given that \(y = 4\) when \(x = \pi\).
  1. Solve the differential equation, obtaining an expression for \(y\) in terms of \(x\). [6]
  2. Sketch the graph of \(y\) against \(x\) for \(0 < x < 2\pi\). [1]
CAIE P3 2021 March Q5
8 marks Moderate -0.3
  1. Express \(\sqrt{7} \sin x + 2 \cos x\) in the form \(R \sin(x + \alpha)\), where \(R > 0\) and \(0° < \alpha < 90°\). State the exact value of \(R\) and give \(\alpha\) correct to 2 decimal places. [3]
  2. Hence solve the equation \(\sqrt{7} \sin 2\theta + 2 \cos 2\theta = 1\), for \(0° < \theta < 180°\). [5]
CAIE P3 2021 March Q6
7 marks Standard +0.3
Let \(\text{f}(x) = \frac{5a}{(2x - a)(3a - x)}\), where \(a\) is a positive constant.
  1. Express f\((x)\) in partial fractions. [3]
  2. Hence show that \(\int_a^{2a} \text{f}(x) \, dx = \ln 6\). [4]
CAIE P3 2021 March Q7
8 marks Standard +0.3
Two lines have equations \(\mathbf{r} = \begin{pmatrix} 1 \\ 3 \\ 2 \end{pmatrix} + s \begin{pmatrix} 2 \\ -1 \\ 3 \end{pmatrix}\) and \(\mathbf{r} = \begin{pmatrix} 2 \\ 1 \\ 4 \end{pmatrix} + t \begin{pmatrix} 1 \\ -1 \\ 4 \end{pmatrix}\).
  1. Show that the lines are skew. [5]
  2. Find the acute angle between the directions of the two lines. [3]
CAIE P3 2021 March Q8
9 marks Standard +0.3
The complex numbers \(u\) and \(v\) are defined by \(u = -4 + 2\text{i}\) and \(v = 3 + \text{i}\).
  1. Find \(\frac{u}{v}\) in the form \(x + \text{i}y\), where \(x\) and \(y\) are real. [3]
  2. Hence express \(\frac{u}{v}\) in the form \(re^{\text{i}\theta}\), where \(r\) and \(\theta\) are exact. [2]
In an Argand diagram, with origin \(O\), the points \(A\), \(B\) and \(C\) represent the complex numbers \(u\), \(v\) and \(2u + v\) respectively.
  1. State fully the geometrical relationship between \(OA\) and \(BC\). [2]
  2. Prove that angle \(AOB = \frac{3}{4}\pi\). [2]
CAIE P3 2021 March Q9
11 marks Standard +0.3
Let \(\text{f}(x) = \frac{e^{2x} + 1}{e^{2x} - 1}\), for \(x > 0\).
  1. The equation \(x = \text{f}(x)\) has one root, denoted by \(a\). Verify by calculation that \(a\) lies between 1 and 1.5. [2]
  2. Use an iterative formula based on the equation in part (a) to determine \(a\) correct to 2 decimal places. Give the result of each iteration to 4 decimal places. [3]
  3. Find f\('(x)\). Hence find the exact value of \(x\) for which f\('(x) = -8\). [6]
CAIE P3 2021 March Q10
11 marks Standard +0.8
\includegraphics{figure_10} The diagram shows the curve \(y = \sin 2x \cos^2 x\) for \(0 \leqslant x \leqslant \frac{1}{2}\pi\), and its maximum point \(M\).
  1. Using the substitution \(u = \sin x\), find the exact area of the region bounded by the curve and the \(x\)-axis. [5]
  2. Find the exact \(x\)-coordinate of \(M\). [6]
CAIE P3 2024 November Q1
4 marks Moderate -0.8
Expand \((9 - 3x)^{\frac{1}{2}}\) in ascending powers of \(x\), up to and including the term in \(x^2\), simplifying the coefficients. [4]
CAIE P3 2024 November Q2
3 marks Standard +0.8
  1. By sketching a suitable pair of graphs, show that the equation \(\cot 2x = \sec x\) has exactly one root in the interval \(0 < x < \frac{1}{2}\pi\). [2]
  2. Show that if a sequence of real values given by the iterative formula $$x_{n+1} = \frac{1}{2}\tan^{-1}(\cos x_n)$$ converges, then it converges to the root in part (a). [1]
CAIE P3 2024 November Q3
5 marks Standard +0.3
The square roots of \(6 - 8i\) can be expressed in the Cartesian form \(x + iy\), where \(x\) and \(y\) are real and exact. By first forming a quartic equation in \(x\) or \(y\), find the square roots of \(6 - 8i\) in exact Cartesian form. [5]
CAIE P3 2024 November Q4
3 marks Moderate -0.3
Solve the equation \(5^x = 5^{x+2} - 10\). Give your answer correct to 3 decimal places. [3]
CAIE P3 2024 November Q5
4 marks Moderate -0.8
  1. The complex number \(u\) is given by $$u = \frac{(\cos \frac{1}{4}\pi + i \sin \frac{1}{4}\pi)^4}{\cos \frac{1}{2}\pi - i \sin \frac{1}{2}\pi}$$ Find the exact value of \(\arg u\). [2]
  2. The complex numbers \(u\) and \(u^*\) are plotted on an Argand diagram. Describe the single geometrical transformation that maps \(u\) onto \(u^*\) and state the exact value of \(\arg u^*\). [2]
CAIE P3 2024 November Q6
4 marks Moderate -0.3
\includegraphics{figure_6} The variables \(x\) and \(y\) satisfy the equation \(ay = b^x\), where \(a\) and \(b\) are constants. The graph of \(\ln y\) against \(x\) is a straight line passing through the points \((0.50, 2.24)\) and \((3.40, 8.27)\), as shown in the diagram. Find the values of \(a\) and \(b\). Give each value correct to 1 significant figure. [4]
CAIE P3 2024 November Q7
6 marks Standard +0.3
  1. Show that the equation \(\tan^3 x + 2 \tan 2x - \tan x = 0\) may be expressed as $$\tan^3 x - 2 \tan^2 x - 3 = 0$$ for \(\tan x \neq 0\). [3]
  2. Hence solve the equation \(\tan^3 2\theta + 2 \tan 4\theta - \tan 2\theta = 0\) for \(0 < \theta < \pi\). Give your answers in exact form. [3]
CAIE P3 2024 November Q8
8 marks Standard +0.3
The parametric equations of a curve are $$x = \tan^2 2t, \quad y = \cos 2t,$$ for \(0 < t < \frac{1}{4}\pi\).
  1. Show that \(\frac{dy}{dx} = -\frac{1}{2}\cos^3 2t\). [4]
  2. Hence find the equation of the normal to the curve at the point where \(t = \frac{1}{8}\pi\). Give your answer in the form \(y = mx + c\). [4]
CAIE P3 2024 November Q9
11 marks Standard +0.3
With respect to the origin \(O\), the points \(A\), \(B\) and \(C\) have position vectors given by $$\overrightarrow{OA} = \begin{pmatrix} 2 \\ 1 \\ -3 \end{pmatrix}, \quad \overrightarrow{OB} = \begin{pmatrix} 0 \\ 4 \\ 1 \end{pmatrix} \quad \text{and} \quad \overrightarrow{OC} = \begin{pmatrix} -3 \\ -2 \\ 2 \end{pmatrix}.$$
  1. The point \(D\) is such that \(ABCD\) is a trapezium with \(\overrightarrow{DC} = 3\overrightarrow{AB}\). Find the position vector of \(D\). [2]
  2. The diagonals of the trapezium intersect at the point \(P\). Find the position vector of \(P\). [5]
  3. Using a scalar product, calculate angle \(ABC\). [4]
CAIE P3 2024 November Q10
13 marks Challenging +1.2
A balloon in the shape of a sphere has volume \(V\) and radius \(r\). Air is pumped into the balloon at a constant rate of \(40\pi\) starting when time \(t = 0\) and \(r = 0\). At the same time, air begins to flow out of the balloon at a rate of \(0.8\pi r\). The balloon remains a sphere at all times.
  1. Show that \(r\) and \(t\) satisfy the differential equation $$\frac{dr}{dt} = \frac{50 - r}{5r^2}.$$ [3]
  2. Find the quotient and remainder when \(5r^2\) is divided by \(50 - r\). [3]
  3. Solve the differential equation in part (a), obtaining an expression for \(t\) in terms of \(r\). [6]
  4. Find the value of \(t\) when the radius of the balloon is 12. [1]
CAIE P3 2024 November Q11
14 marks Standard +0.8
Let \(f(x) = \frac{2e^{2x}}{e^{2x} - 3e^x + 2}\).
  1. Find \(f'(x)\) and hence find the exact coordinates of the stationary point of the curve with equation \(y = f(x)\). [5]
  2. Use the substitution \(u = e^x\) and partial fractions to find the exact value of \(\int_{\ln 5} f(x) dx\). Give your answer in the form \(\ln a\), where \(a\) is a rational number in its simplest form. [9]
CAIE P3 2006 June Q1
3 marks Moderate -0.8
Given that \(x = 4(3^{-y})\), express \(y\) in terms of \(x\). [3]
CAIE P3 2006 June Q2
4 marks Moderate -0.3
Solve the inequality \(2x > |x - 1|\). [4]