Questions P3 (1243 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 2018 November Q3
5 marks Moderate -0.3
  1. Find \(\int \frac{\ln x}{x^3} \, dx\). [3]
  2. Hence show that \(\int_1^2 \frac{\ln x}{x^3} \, dx = \frac{1}{16}(3 - \ln 4)\). [2]
CAIE P3 2018 November Q4
5 marks Standard +0.3
Showing all necessary working, solve the equation $$\frac{e^x + e^{-x}}{e^x + 1} = 4,$$ giving your answer correct to 3 decimal places. [5]
CAIE P3 2018 November Q5
8 marks Standard +0.3
The equation of a curve is \(y = x \ln(8 - x)\). The gradient of the curve is equal to 1 at only one point, when \(x = a\).
  1. Show that \(a\) satisfies the equation \(x = 8 - \frac{8}{\ln(8 - x)}\). [3]
  2. Verify by calculation that \(a\) lies between 2.9 and 3.1. [2]
  3. Use an iterative formula based on the equation in part (i) to determine \(a\) correct to 2 decimal places. Give the result of each iteration to 4 decimal places. [3]
CAIE P3 2018 November Q6
8 marks Standard +0.3
A certain curve is such that its gradient at a general point with coordinates \((x, y)\) is proportional to \(\frac{y^2}{x}\). The curve passes through the points with coordinates \((1, 1)\) and \((e, 2)\). By setting up and solving a differential equation, find the equation of the curve, expressing \(y\) in terms of \(x\). [8]
CAIE P3 2018 November Q7
10 marks Standard +0.3
A curve has equation \(y = \frac{3 \cos x}{2 + \sin x}\), for \(-\frac{1}{2}\pi \leqslant x \leqslant \frac{1}{2}\pi\).
  1. Find the exact coordinates of the stationary point of the curve. [6]
  2. The constant \(a\) is such that \(\int_0^a \frac{3 \cos x}{2 + \sin x} \, dx = 1\). Find the value of \(a\), giving your answer correct to 3 significant figures. [4]
CAIE P3 2018 November Q8
10 marks Standard +0.3
Let \(f(x) = \frac{7x^2 - 15x + 8}{(1 - 2x)(2 - x)^2}\).
  1. Express \(f(x)\) in partial fractions. [5]
  2. Hence obtain the expansion of \(f(x)\) in ascending powers of \(x\), up to and including the term in \(x^2\). [5]
CAIE P3 2018 November Q9
10 marks Standard +0.3
    1. Without using a calculator, express the complex number \(\frac{2 + 6i}{1 - 2i}\) in the form \(x + iy\), where \(x\) and \(y\) are real. [2]
    2. Hence, without using a calculator, express \(\frac{2 + 6i}{1 - 2i}\) in the form \(r(\cos \theta + i \sin \theta)\), where \(r > 0\) and \(-\pi < \theta \leqslant \pi\), giving the exact values of \(r\) and \(\theta\). [3]
  2. On a sketch of an Argand diagram, shade the region whose points represent complex numbers \(z\) satisfying both the inequalities \(|z - 3i| \leqslant 1\) and \(\text{Re } z \leqslant 0\), where \(\text{Re } z\) denotes the real part of \(z\). Find the greatest value of \(\arg z\) for points in this region, giving your answer in radians correct to 2 decimal places. [5]
CAIE P3 2018 November Q10
11 marks Standard +0.8
The line \(l\) has equation \(\mathbf{r} = 5\mathbf{i} - 3\mathbf{j} - \mathbf{k} + \lambda(\mathbf{i} - 2\mathbf{j} + \mathbf{k})\). The plane \(p\) has equation $$(\mathbf{r} - \mathbf{i} - 2\mathbf{j}) \cdot (3\mathbf{i} + \mathbf{j} + \mathbf{k}) = 0.$$ The line \(l\) intersects the plane \(p\) at the point \(A\).
  1. Find the position vector of \(A\). [3]
  2. Calculate the acute angle between \(l\) and \(p\). [4]
  3. Find the equation of the line which lies in \(p\) and intersects \(l\) at right angles. [4]
CAIE P3 2018 November Q1
4 marks Standard +0.8
Find the set of values of \(x\) satisfying the inequality \(2|2x - a| < |x + 3a|\), where \(a\) is a positive constant. [4]
CAIE P3 2018 November Q2
4 marks Moderate -0.3
Showing all necessary working, solve the equation \(\frac{2e^x + e^{-x}}{e^x - e^{-x}} = 4\), giving your answer correct to 2 decimal places. [4]
CAIE P3 2018 November Q3
7 marks Standard +0.3
  1. By sketching a suitable pair of graphs, show that the equation \(x^3 = 3 - x\) has exactly one real root. [2]
  2. Show that if a sequence of real values given by the iterative formula $$x_{n+1} = \frac{2x_n^3 + 3}{3x_n^2 + 1}$$ converges, then it converges to the root of the equation in part (i). [2]
  3. Use this iterative formula to determine the root correct to 3 decimal places. Give the result of each iteration to 5 decimal places. [3]
CAIE P3 2018 November Q4
7 marks Standard +0.3
The parametric equations of a curve are $$x = 2\sin\theta + \sin 2\theta, \quad y = 2\cos\theta + \cos 2\theta,$$ where \(0 < \theta < \pi\).
  1. Obtain an expression for \(\frac{dy}{dx}\) in terms of \(\theta\). [3]
  2. Hence find the exact coordinates of the point on the curve at which the tangent is parallel to the \(y\)-axis. [4]
CAIE P3 2018 November Q5
7 marks Standard +0.3
The coordinates \((x, y)\) of a general point on a curve satisfy the differential equation $$x\frac{dy}{dx} = (2 - x^2)y.$$ The curve passes through the point \((1, 1)\). Find the equation of the curve, obtaining an expression for \(y\) in terms of \(x\). [7]
CAIE P3 2018 November Q6
8 marks Standard +0.8
  1. Show that the equation \((\sqrt{2})\cos ec x + \cot x = \sqrt{3}\) can be expressed in the form \(R\sin(x - \alpha) = \sqrt{2}\), where \(R > 0\) and \(0° < \alpha < 90°\). [4]
  2. Hence solve the equation \((\sqrt{2})\cos ec x + \cot x = \sqrt{3}\), for \(0° < x < 180°\). [4]
CAIE P3 2018 November Q7
9 marks Standard +0.3
\includegraphics{figure_7} The diagram shows the curve \(y = 5\sin^2 x \cos^3 x\) for \(0 \leqslant x \leqslant \frac{1}{2}\pi\), and its maximum point \(M\). The shaded region \(R\) is bounded by the curve and the \(x\)-axis.
  1. Find the \(x\)-coordinate of \(M\), giving your answer correct to 3 decimal places. [5]
  2. Using the substitution \(u = \sin x\) and showing all necessary working, find the exact area of \(R\). [4]
CAIE P3 2018 November Q8
9 marks Standard +0.3
  1. Showing all necessary working, express the complex number \(\frac{2 + 3i}{1 - 2i}\) in the form \(re^{i\theta}\), where \(r > 0\) and \(-\pi < \theta \leqslant \pi\). Give the values of \(r\) and \(\theta\) correct to 3 significant figures. [5]
  2. On an Argand diagram sketch the locus of points representing complex numbers \(z\) satisfying the equation \(|z - 3 + 2i| = 1\). Find the least value of \(|z|\) for points on this locus, giving your answer in an exact form. [4]
CAIE P3 2018 November Q9
10 marks Standard +0.3
Let \(f(x) = \frac{6x^2 + 8x + 9}{(2 - x)(3 + 2x)^2}\).
  1. Express \(f(x)\) in partial fractions. [5]
  2. Hence, showing all necessary working, show that \(\int_{-1}^0 f(x) dx = 1 + \frac{1}{2}\ln\left(\frac{4}{3}\right)\). [5]
CAIE P3 2018 November Q10
10 marks Standard +0.3
The planes \(m\) and \(n\) have equations \(3x + y - 2z = 10\) and \(x - 2y + 2z = 5\) respectively. The line \(l\) has equation \(\mathbf{r} = 4\mathbf{i} + 2\mathbf{j} + \mathbf{k} + \lambda(\mathbf{i} + \mathbf{j} + 2\mathbf{k})\).
  1. Show that \(l\) is parallel to \(m\). [3]
  2. Calculate the acute angle between the planes \(m\) and \(n\). [3]
  3. A point \(P\) lies on the line \(l\). The perpendicular distance of \(P\) from the plane \(n\) is equal to 2. Find the position vectors of the two possible positions of \(P\). [4]