Questions — CAIE P3 (1110 questions)

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CAIE P3 2013 November Q6
8 marks Challenging +1.2
\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 \(OAB\) is \(\theta\) radians. The shaded region is bounded by the circumference of the circle and the arc with centre \(A\) joining \(B\) and \(C\). The area of the shaded region is equal to half the area of the circle.
  1. Show that \(\cos 2\theta = \frac{2\sin 2\theta - \pi}{4\theta}\). [5]
  2. Use the iterative formula $$\theta_{n+1} = \frac{1}{2}\cos^{-1}\left(\frac{2\sin 2\theta_n - \pi}{4\theta_n}\right),$$ with initial value \(\theta_1 = 1\), to determine \(\theta\) correct to 2 decimal places, showing the result of each iteration to 4 decimal places. [3]
CAIE P3 2013 November Q7
10 marks Standard +0.3
Let \(f(x) = \frac{2x^2 - 7x - 1}{(x-2)(x^2+3)}\).
  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 2013 November Q8
10 marks Standard +0.3
Throughout this question the use of a calculator is not permitted.
  1. The complex numbers \(u\) and \(v\) satisfy the equations $$u + 2v = 2i \quad \text{and} \quad iu + v = 3.$$ Solve the equations for \(u\) and \(v\), giving both answers in the form \(x + iy\), where \(x\) and \(y\) are real. [5]
  2. On an Argand diagram, sketch the locus representing complex numbers \(z\) satisfying \(|z + i| = 1\) and the locus representing complex numbers \(w\) satisfying \(\arg(w - 2) = \frac{\pi}{4}\). Find the least value of \(|z - w|\) for points on these loci. [5]
CAIE P3 2013 November Q9
11 marks Standard +0.3
\includegraphics{figure_9} The diagram shows three points \(A\), \(B\) and \(C\) whose position vectors with respect to the origin \(O\) are given by \(\overrightarrow{OA} = \begin{pmatrix} 2 \\ -1 \\ 1 \end{pmatrix}\), \(\overrightarrow{OB} = \begin{pmatrix} 0 \\ 3 \\ 1 \end{pmatrix}\) and \(\overrightarrow{OC} = \begin{pmatrix} 3 \\ 0 \\ 4 \end{pmatrix}\). The point \(D\) lies on \(BC\), between \(B\) and \(C\), and is such that \(CD = 2DB\).
  1. Find the equation of the plane \(ABC\), giving your answer in the form \(ax + by + cz = d\). [6]
  2. Find the position vector of \(D\). [1]
  3. Show that the length of the perpendicular from \(A\) to \(OD\) is \(\frac{1}{3}\sqrt{(65)}\). [4]
CAIE P3 2013 November Q10
11 marks Standard +0.3
\includegraphics{figure_10} A tank containing water is in the form of a cone with vertex \(C\). The axis is vertical and the semi-vertical angle is \(60°\), as shown in the diagram. At time \(t = 0\), the tank is full and the depth of water is \(H\). At this instant, a tap at \(C\) is opened and water begins to flow out. The volume of water in the tank decreases at a rate proportional to \(\sqrt{h}\), where \(h\) is the depth of water at time \(t\). The tank becomes empty when \(t = 60\).
  1. Show that \(h\) and \(t\) satisfy a differential equation of the form $$\frac{dh}{dt} = -Ah^{-\frac{1}{2}},$$ where \(A\) is a positive constant. [4]
  2. Solve the differential equation given in part (i) and obtain an expression for \(t\) in terms of \(h\) and \(H\). [6]
  3. Find the time at which the depth reaches \(\frac{1}{2}H\). [1]
[The volume \(V\) of a cone of vertical height \(h\) and base radius \(r\) is given by \(V = \frac{1}{3}\pi r^2 h\).]
CAIE P3 2017 November Q1
3 marks Moderate -0.8
Find the quotient and remainder when \(x^4\) is divided by \(x^2 + 2x - 1\). [3]
CAIE P3 2017 November Q2
5 marks Moderate -0.8
Two variable quantities \(x\) and \(y\) are believed to satisfy an equation of the form \(y = C(a^x)\), where \(C\) and \(a\) are constants. An experiment produced four pairs of values of \(x\) and \(y\). The table below gives the corresponding values of \(x\) and \(\ln y\).
\(x\)0.91.62.43.2
\(\ln y\)1.71.92.32.6
By plotting \(\ln y\) against \(x\) for these four pairs of values and drawing a suitable straight line, estimate the values of \(C\) and \(a\). Give your answers correct to 2 significant figures. [5] \includegraphics{figure_2}
CAIE P3 2017 November Q3
6 marks Standard +0.3
The equation \(x^3 = 3x + 7\) has one real root, denoted by \(\alpha\).
  1. Show by calculation that \(\alpha\) lies between 2 and 3. [2]
Two iterative formulae, \(A\) and \(B\), derived from this equation are as follows: $$x_{n+1} = (3x_n + 7)^{\frac{1}{3}}, \quad (A)$$ $$x_{n+1} = \frac{x_n^3 - 7}{3}. \quad (B)$$ Each formula is used with initial value \(x_1 = 2.5\).
  1. Show that one of these formulae produces a sequence which fails to converge, and use the other formula to calculate \(\alpha\) correct to 2 decimal places. Give the result of each iteration to 4 decimal places. [4]
CAIE P3 2017 November Q4
7 marks Standard +0.3
  1. Prove the identity \(\tan(45° + x) + \tan(45° - x) = 2 \sec 2x\). [4]
  2. Hence sketch the graph of \(y = \tan(45° + x) + \tan(45° - x)\) for \(0° \leqslant x \leqslant 90°\). [3]
CAIE P3 2017 November Q5
8 marks Standard +0.3
The equation of a curve is \(2x^4 + xy^3 + y^4 = 10\).
  1. Show that \(\frac{dy}{dx} = -\frac{8x^3 + y^3}{3xy^2 + 4y^3}\). [4]
  2. Hence show that there are two points on the curve at which the tangent is parallel to the \(x\)-axis and find the coordinates of these points. [4]
CAIE P3 2017 November Q6
8 marks Standard +0.3
The variables \(x\) and \(y\) satisfy the differential equation $$\frac{dy}{dx} = 4 \cos^2 y \tan x,$$ for \(0 \leqslant x < \frac{1}{2}\pi\), and \(x = 0\) when \(y = \frac{1}{4}\pi\). Solve this differential equation and find the value of \(x\) when \(y = \frac{1}{8}\pi\). [8]
CAIE P3 2017 November Q7
9 marks Standard +0.3
  1. The complex number \(u\) is given by \(u = 8 - 15\text{i}\). Showing all necessary working, find the two square roots of \(u\). Give answers in the form \(a + ib\), where the numbers \(a\) and \(b\) are real and exact. [5]
  2. On an Argand diagram, shade the region whose points represent complex numbers satisfying both the inequalities \(|z - 2 - \text{i}| \leqslant 2\) and \(0 \leqslant \arg(z - \text{i}) \leqslant \frac{1}{4}\pi\). [4]
CAIE P3 2017 November Q8
9 marks Standard +0.3
Let \(\text{f}(x) = \frac{4x^2 + 9x - 8}{(x + 2)(2x - 1)}\).
  1. Express \(\text{f}(x)\) in the form \(A + \frac{B}{x + 2} + \frac{C}{2x - 1}\). [4]
  2. Hence show that \(\int_1^4 \text{f}(x) \, dx = 6 + \frac{1}{2} \ln\left(\frac{16}{7}\right)\). [5]
CAIE P3 2017 November Q9
9 marks Standard +0.8
\includegraphics{figure_9} The diagram shows the curve \(y = (1 + x^2)\text{e}^{-\frac{3x}{4}}\) for \(x \geqslant 0\). The shaded region \(R\) is enclosed by the curve, the \(x\)-axis and the lines \(x = 0\) and \(x = 2\).
  1. Find the exact values of the \(x\)-coordinates of the stationary points of the curve. [4]
  2. Show that the exact value of the area of \(R\) is \(18 - \frac{42}{\text{e}}\). [5]
CAIE P3 2017 November Q10
11 marks Standard +0.3
The equations of two lines \(l\) and \(m\) are \(\mathbf{r} = 3\mathbf{i} - \mathbf{j} - 2\mathbf{k} + \lambda(-\mathbf{i} + \mathbf{j} + 4\mathbf{k})\) and \(\mathbf{r} = 4\mathbf{i} + 4\mathbf{j} - 3\mathbf{k} + \mu(2\mathbf{i} + \mathbf{j} - 2\mathbf{k})\) respectively.
  1. Show that the lines do not intersect. [3]
  2. Calculate the acute angle between the directions of the lines. [3]
  3. Find the equation of the plane which passes through the point \((3, -2, -1)\) and which is parallel to both \(l\) and \(m\). Give your answer in the form \(ax + by + cz = d\). [5]
CAIE P3 2018 November Q1
4 marks Moderate -0.3
Solve the inequality \(3|2x - 1| > |x + 4|\). [4]
CAIE P3 2018 November Q2
4 marks Moderate -0.3
Showing all necessary working, solve the equation \(\sin(\theta - 30°) + \cos \theta = 2 \sin \theta\), for \(0° < \theta < 180°\). [4]
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]