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 2013 June Q8
11 marks Standard +0.3
  1. Express \(\frac{1}{x^2(2x + 1)}\) in the form \(\frac{A}{x^2} + \frac{B}{x} + \frac{C}{2x + 1}\). [4]
  2. The variables \(x\) and \(y\) satisfy the differential equation $$y = x^2(2x + 1)\frac{dy}{dx},$$ and \(y = 1\) when \(x = 1\). Solve the differential equation and find the exact value of \(y\) when \(x = 2\). Give your value of \(y\) in a form not involving logarithms. [7]
CAIE P3 2013 June Q9
11 marks Standard +0.3
  1. The complex number \(w\) is such that \(\text{Re } w > 0\) and \(w + 3w^* = iw^2\), where \(w^*\) denotes the complex conjugate of \(w\). Find \(w\), giving your answer in the form \(x + iy\), where \(x\) and \(y\) are real. [5]
  2. On a sketch of an Argand diagram, shade the region whose points represent complex numbers \(z\) which satisfy both the inequalities \(|z - 2i| \leq 2\) and \(0 \leq \arg(z + 2) \leq \frac{1}{4}\pi\). Calculate the greatest value of \(|z|\) for points in this region, giving your answer correct to 2 decimal places. [6]
CAIE P3 2013 June Q10
11 marks Standard +0.8
The points \(A\) and \(B\) have position vectors \(\mathbf{2i - 3j + 2k}\) and \(\mathbf{5i - 2j + k}\) respectively. The plane \(p\) has equation \(x + y = 5\).
  1. Find the position vector of the point of intersection of the line through \(A\) and \(B\) and the plane \(p\). [4]
  2. A second plane \(q\) has an equation of the form \(x + by + cz = d\), where \(b\), \(c\) and \(d\) are constants. The plane \(q\) contains the line \(AB\), and the acute angle between the planes \(p\) and \(q\) is \(60°\). Find the equation of \(q\). [7]
CAIE P3 2014 June Q1
4 marks Standard +0.8
Find the set of values of \(x\) satisfying the inequality $$|x + 2a| > 3|x - a|,$$ where \(a\) is a positive constant. [4]
CAIE P3 2014 June Q2
4 marks Standard +0.3
Solve the equation $$2\ln(5 - e^{-2x}) = 1,$$ giving your answer correct to 3 significant figures. [4]
CAIE P3 2014 June Q3
5 marks Standard +0.3
Solve the equation $$\cos(x + 30°) = 2\cos x,$$ giving all solutions in the interval \(-180° < x < 180°\). [5]
CAIE P3 2014 June Q4
7 marks Standard +0.3
The parametric equations of a curve are $$x = t - \tan t, \quad y = \ln(\cos t),$$ for \(-\frac{1}{4}\pi < t < \frac{1}{4}\pi\).
  1. Show that \(\frac{dy}{dx} = \cot t\). [5]
  2. Hence find the \(x\)-coordinate of the point on the curve at which the gradient is equal to 2. Give your answer correct to 3 significant figures. [2]
CAIE P3 2014 June Q5
7 marks Standard +0.8
  1. The polynomial \(f(x)\) is of the form \((x - 2)^2g(x)\), where \(g(x)\) is another polynomial. Show that \((x - 2)\) is a factor of \(f'(x)\). [2]
  2. The polynomial \(x^5 + ax^4 + 3x^3 + bx^2 + a\), where \(a\) and \(b\) are constants, has a factor \((x - 2)^2\). Using the factor theorem and the result of part (i), or otherwise, find the values of \(a\) and \(b\). [5]
CAIE P3 2014 June Q6
8 marks Standard +0.3
\includegraphics{figure_6} In the diagram, \(A\) is a point on the circumference of a circle with centre \(O\) and radius \(r\). A circular arc with centre \(A\) meets the circumference at \(B\) and \(C\). The angle \(\angle OAB\) is equal to \(x\) radians. The shaded region is bounded by \(AB\), \(AC\) and the circular arc with centre \(A\) joining \(B\) and \(C\). The perimeter of the shaded region is equal to half the circumference of the circle.
  1. Show that \(x = \cos^{-1}\left(\frac{\pi}{4 + 4x}\right)\). [3]
  2. Verify by calculation that \(x\) lies between 1 and 1.5. [2]
  3. Use the iterative formula $$x_{n+1} = \cos^{-1}\left(\frac{\pi}{4 + 4x_n}\right)$$ to determine the value of \(x\) correct to 2 decimal places. Give the result of each iteration to 4 decimal places. [3]
CAIE P3 2014 June Q7
8 marks Standard +0.3
  1. It is given that \(-1 + (\sqrt{5})i\) is a root of the equation \(z^3 + 2z + a = 0\), where \(a\) is real. Showing your working, find the value of \(a\), and write down the other complex root of this equation. [4]
  2. The complex number \(w\) has modulus 1 and argument \(2\theta\) radians. Show that \(\frac{w - 1}{w + 1} = i\tan\theta\). [4]
CAIE P3 2014 June Q8
10 marks Standard +0.3
\includegraphics{figure_8} The diagram shows the curve \(y = x\cos\frac{1}{2}x\) for \(0 \leqslant x \leqslant \pi\).
  1. Find \(\frac{dy}{dx}\) and show that \(4\frac{d^2y}{dx^2} + y + 4\sin\frac{1}{2}x = 0\). [5]
  2. Find the exact value of the area of the region enclosed by this part of the curve and the \(x\)-axis. [5]
CAIE P3 2014 June Q9
10 marks Standard +0.8
The population of a country at time \(t\) years is \(N\) millions. At any time, \(N\) is assumed to increase at a rate proportional to the product of \(N\) and \((1 - 0.01N)\). When \(t = 0\), \(N = 20\) and \(\frac{dN}{dt} = 0.32\).
  1. Treating \(N\) and \(t\) as continuous variables, show that they satisfy the differential equation $$\frac{dN}{dt} = 0.02N(1 - 0.01N).$$ [1]
  2. Solve the differential equation, obtaining an expression for \(t\) in terms of \(N\). [8]
  3. Find the time at which the population will be double its value at \(t = 0\). [1]
CAIE P3 2014 June Q10
12 marks Standard +0.3
Referred to the origin \(O\), the points \(A\), \(B\) and \(C\) have position vectors given by $$\overrightarrow{OA} = \mathbf{i} + 2\mathbf{j} + 3\mathbf{k}, \quad \overrightarrow{OB} = 2\mathbf{i} + 4\mathbf{j} + \mathbf{k} \quad \text{and} \quad \overrightarrow{OC} = 3\mathbf{i} + 5\mathbf{j} - 3\mathbf{k}.$$
  1. Find the exact value of the cosine of angle \(BAC\). [4]
  2. Hence find the exact value of the area of triangle \(ABC\). [3]
  3. Find the equation of the plane which is parallel to the \(y\)-axis and contains the line through \(B\) and \(C\). Give your answer in the form \(ax + by + cz = d\). [5]
CAIE P3 2017 June Q1
3 marks Standard +0.3
Solve the equation \(\ln(x^2 + 1) = 1 + 2 \ln x\), giving your answer correct to 3 significant figures. [3]
CAIE P3 2017 June Q2
4 marks Standard +0.3
Solve the inequality \(|x - 3| < 3x - 4\). [4]
CAIE P3 2017 June Q3
5 marks Standard +0.8
  1. Express the equation \(\cot \theta - 2 \tan \theta = \sin 2\theta\) in the form \(a \cos^4 \theta + b \cos^2 \theta + c = 0\), where \(a\), \(b\) and \(c\) are constants to be determined. [3]
  2. Hence solve the equation \(\cot \theta - 2 \tan \theta = \sin 2\theta\) for \(90° < \theta < 180°\). [2]
CAIE P3 2017 June Q4
6 marks Moderate -0.3
The parametric equations of a curve are $$x = t^2 + 1, \quad y = 4t + \ln(2t - 1).$$
  1. Express \(\frac{dy}{dx}\) in terms of \(t\). [3]
  2. Find the equation of the normal to the curve at the point where \(t = 1\). Give your answer in the form \(ax + by + c = 0\). [3]
CAIE P3 2017 June Q5
8 marks Standard +0.3
In a certain chemical process a substance \(A\) reacts with and reduces a substance \(B\). The masses of \(A\) and \(B\) at time \(t\) after the start of the process are \(x\) and \(y\) respectively. It is given that \(\frac{dy}{dt} = -0.2xy\) and \(x = \frac{10}{(1 + t)^2}\). At the beginning of the process \(y = 100\).
  1. Form a differential equation in \(y\) and \(t\), and solve this differential equation. [6]
  2. Find the exact value approached by the mass of \(B\) as \(t\) becomes large. State what happens to the mass of \(A\) as \(t\) becomes large. [2]
CAIE P3 2017 June Q6
8 marks Standard +0.3
Throughout this question the use of a calculator is not permitted. The complex number \(2 - \mathrm{i}\) is denoted by \(u\).
  1. It is given that \(u\) is a root of the equation \(x^3 + ax^2 - 3x + b = 0\), where the constants \(a\) and \(b\) are real. Find the values of \(a\) and \(b\). [4]
  2. On a sketch of an Argand diagram, shade the region whose points represent complex numbers \(z\) satisfying both the inequalities \(|z - u| < 1\) and \(|z| < |z + \mathrm{i}|\). [4]
CAIE P3 2017 June Q7
9 marks Standard +0.3
  1. Prove that if \(y = \frac{1}{\cos \theta}\) then \(\frac{dy}{d\theta} = \sec \theta \tan \theta\). [2]
  2. Prove the identity \(\frac{1 + \sin \theta}{1 - \sin \theta} = 2 \sec^2 \theta + 2 \sec \theta \tan \theta - 1\). [3]
  3. Hence find the exact value of \(\int_0^{\frac{\pi}{4}} \frac{1 + \sin \theta}{1 - \sin \theta} d\theta\). [4]
CAIE P3 2017 June Q8
10 marks Standard +0.3
Let \(\mathrm{f}(x) = \frac{5x^2 - 7x + 4}{(3x + 2)(x^2 + 5)}\).
  1. Express \(\mathrm{f}(x)\) in partial fractions. [5]
  2. Hence obtain the expansion of \(\mathrm{f}(x)\) in ascending powers of \(x\), up to and including the term in \(x^2\). [5]
CAIE P3 2017 June Q9
11 marks Standard +0.3
Relative to the origin \(O\), the point \(A\) has position vector given by \(\overrightarrow{OA} = \mathbf{i} + 2\mathbf{j} + 4\mathbf{k}\). The line \(l\) has equation \(\mathbf{r} = 9\mathbf{i} - \mathbf{j} + 8\mathbf{k} + \mu(3\mathbf{i} - \mathbf{j} + 2\mathbf{k})\).
  1. Find the position vector of the foot of the perpendicular from \(A\) to \(l\). Hence find the position vector of the reflection of \(A\) in \(l\). [5]
  2. Find the equation of the plane through the origin which contains \(l\). Give your answer in the form \(ax + by + cz = d\). [3]
  3. Find the exact value of the perpendicular distance of \(A\) from this plane. [3]
CAIE P3 2017 June Q10
11 marks Standard +0.3
\includegraphics{figure_10} The diagram shows the curve \(y = x^2 \cos 2x\) for \(0 \leq x \leq \frac{1}{4}\pi\). The curve has a maximum point at \(M\) where \(x = p\).
  1. Show that \(p\) satisfies the equation \(p = \frac{1}{2} \tan^{-1} \left(\frac{1}{p}\right)\). [3]
  2. Use the iterative formula \(p_{n+1} = \frac{1}{2} \tan^{-1} \left(\frac{1}{p_n}\right)\) to determine the value of \(p\) correct to 2 decimal places. Give the result of each iteration to 4 decimal places. [3]
  3. Find, showing all necessary working, the exact area of the region bounded by the curve and the \(x\)-axis. [5]
CAIE P3 2018 June Q1
5 marks Standard +0.3
The coefficient of \(x^2\) in the expansion of \(\left(2 + \frac{x}{2}\right)^6 + (a + x)^5\) is 330. Find the value of the constant \(a\). [5]
CAIE P3 2018 June Q2
5 marks Moderate -0.8
The equation of a curve is \(y = x^2 - 6x + k\), where \(k\) is a constant.
  1. Find the set of values of \(k\) for which the whole of the curve lies above the \(x\)-axis. [2]
  2. Find the value of \(k\) for which the line \(y + 2x = 7\) is a tangent to the curve. [3]