Questions — CAIE P3 (1110 questions)

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CAIE P3 2006 June Q3
5 marks Moderate -0.3
The parametric equations of a curve are $$x = 2\theta + \sin 2\theta, \quad y = 1 - \cos 2\theta.$$ Show that \(\frac{dy}{dx} = \tan \theta\). [5]
CAIE P3 2006 June Q4
7 marks Moderate -0.3
  1. Express \(7\cos \theta + 24\sin \theta\) in the form \(R\cos(\theta - \alpha)\), where \(R > 0\) and \(0° < \alpha < 90°\), giving the exact value of \(R\) and the value of \(\alpha\) correct to 2 decimal places. [3]
  2. Hence solve the equation $$7\cos \theta + 24\sin \theta = 15,$$ giving all solutions in the interval \(0° \leqslant \theta \leqslant 360°\). [4]
CAIE P3 2006 June Q5
8 marks Standard +0.3
In a certain industrial process, a substance is being produced in a container. The mass of the substance in the container \(t\) minutes after the start of the process is \(x\) grams. At any time, the rate of formation of the substance is proportional to its mass. Also, throughout the process, the substance is removed from the container at a constant rate of 25 grams per minute. When \(t = 0\), \(x = 1000\) and \(\frac{dx}{dt} = 75\).
  1. Show that \(x\) and \(t\) satisfy the differential equation $$\frac{dx}{dt} = 0.1(x - 250).$$ [2]
  2. Solve this differential equation, obtaining an expression for \(x\) in terms of \(t\). [6]
CAIE P3 2006 June Q6
8 marks Standard +0.3
  1. By sketching a suitable pair of graphs, show that the equation $$2\cot x = 1 + e^x,$$ where \(x\) is in radians, has only one root in the interval \(0 < x < \frac{1}{2}\pi\). [2]
  2. Verify by calculation that this root lies between 0.5 and 1.0. [2]
  3. Show that this root also satisfies the equation $$x = \tan^{-1}\left(\frac{2}{1 + e^x}\right).$$ [1]
  4. Use the iterative formula $$x_{n+1} = \tan^{-1}\left(\frac{2}{1 + e^{x_n}}\right),$$ with initial value \(x_1 = 0.7\), to determine this root correct to 2 decimal places. Give the result of each iteration to 4 decimal places. [3]
CAIE P3 2006 June Q7
9 marks Standard +0.3
The complex number \(2 + \mathrm{i}\) is denoted by \(u\). Its complex conjugate is denoted by \(u^*\).
  1. Show, on a sketch of an Argand diagram with origin \(O\), the points \(A\), \(B\) and \(C\) representing the complex numbers \(u\), \(u^*\) and \(u + u^*\) respectively. Describe in geometrical terms the relationship between the four points \(O\), \(A\), \(B\) and \(C\). [4]
  2. Express \(\frac{u}{u^*}\) in the form \(x + \mathrm{i}y\), where \(x\) and \(y\) are real. [3]
  3. By considering the argument of \(\frac{u}{u^*}\), or otherwise, prove that $$\tan^{-1}\left(\frac{4}{3}\right) = 2\tan^{-1}\left(\frac{1}{2}\right).$$ [2]
CAIE P3 2006 June Q8
9 marks Standard +0.3
\includegraphics{figure_8} The diagram shows a sketch of the curve \(y = x^2\ln x\) and its minimum point \(M\). The curve cuts the \(x\)-axis at the point \((1, 0)\).
  1. Find the exact value of the \(x\)-coordinate of \(M\). [4]
  2. Use integration by parts to find the area of the shaded region enclosed by the curve, the \(x\)-axis and the line \(x = 4\). Give your answer correct to 2 decimal places. [5]
CAIE P3 2006 June Q9
10 marks Standard +0.3
  1. Express \(\frac{10}{(2-x)(1+x^2)}\) in partial fractions. [5]
  2. Hence, given that \(|x| < 1\), obtain the expansion of \(\frac{10}{(2-x)(1+x^2)}\) in ascending powers of \(x\), up to and including the term in \(x^3\), simplifying the coefficients. [5]
CAIE P3 2006 June Q10
12 marks Standard +0.3
The points \(A\) and \(B\) have position vectors, relative to the origin \(O\), given by $$\overrightarrow{OA} = \begin{pmatrix} -1 \\ 3 \\ 5 \end{pmatrix} \quad \text{and} \quad \overrightarrow{OB} = \begin{pmatrix} 3 \\ -1 \\ -4 \end{pmatrix}.$$ The line \(l\) passes through \(A\) and is parallel to \(OB\). The point \(N\) is the foot of the perpendicular from \(B\) to \(l\).
  1. State a vector equation for the line \(l\). [1]
  2. Find the position vector of \(N\) and show that \(BN = 3\). [6]
  3. Find the equation of the plane containing \(A\), \(B\) and \(N\), giving your answer in the form \(ax + by + cz = d\). [5]
CAIE P3 2010 June Q1
4 marks Standard +0.3
Solve the inequality \(|x - 3| > 2|x + 1|\). [4]
CAIE P3 2010 June Q2
4 marks Moderate -0.8
The variables \(x\) and \(y\) satisfy the equation \(y^3 = Ae^{2x}\), where \(A\) is a constant. The graph of \(\ln y\) against \(x\) is a straight line.
  1. Find the gradient of this line. [2]
  2. Given that the line intersects the axis of \(\ln y\) at the point where \(\ln y = 0.5\), find the value of \(A\) correct to 2 decimal places. [2]
CAIE P3 2010 June Q3
5 marks Standard +0.3
Solve the equation $$\tan(45° - x) = 2\tan x,$$ giving all solutions in the interval \(0° < x < 180°\). [5]
CAIE P3 2010 June Q4
7 marks Standard +0.3
Given that \(x = 1\) when \(t = 0\), solve the differential equation $$\frac{dx}{dt} = \frac{1}{x} - \frac{x}{4},$$ obtaining an expression for \(x^2\) in terms of \(t\). [7]
CAIE P3 2010 June Q5
8 marks Standard +0.3
\includegraphics{figure_5} The diagram shows the curve \(y = e^{-x} - e^{-2x}\) and its maximum point \(M\). The \(x\)-coordinate of \(M\) is denoted by \(p\).
  1. Find the exact value of \(p\). [4]
  2. Show that the area of the shaded region bounded by the curve, the \(x\)-axis and the line \(x = p\) is equal to \(\frac{1}{4}\). [4]
CAIE P3 2010 June Q6
8 marks Standard +0.3
The curve \(y = \frac{\ln x}{x + 1}\) has one stationary point.
  1. Show that the \(x\)-coordinate of this point satisfies the equation $$x = \frac{x + 1}{\ln x},$$ and that this \(x\)-coordinate lies between 3 and 4. [5]
  2. Use the iterative formula $$x_{n+1} = \frac{x_n + 1}{\ln x_n}$$ to determine the \(x\)-coordinate correct to 2 decimal places. Give the result of each iteration to 4 decimal places. [3]
CAIE P3 2010 June Q7
8 marks Standard +0.3
  1. Prove the identity \(\cos 3\theta \equiv 4\cos^3 \theta - 3\cos \theta\). [4]
  2. Using this result, find the exact value of $$\int_{\frac{\pi}{4}}^{\frac{\pi}{2}} \cos^3 \theta \, d\theta.$$ [4]
CAIE P3 2010 June Q8
9 marks Standard +0.3
  1. The equation \(2x^3 - x^2 + 2x + 12 = 0\) has one real root and two complex roots. Showing your working, verify that \(1 + i\sqrt{3}\) is one of the complex roots. State the other complex root. [4]
  2. On a sketch of an Argand diagram, show the point representing the complex number \(1 + i\sqrt{3}\). On the same diagram, shade the region whose points represent the complex numbers \(z\) which satisfy both the inequalities \(|z - 1 - i\sqrt{3}| \leq 1\) and \(\arg z \leq \frac{1}{4}\pi\). [5]
CAIE P3 2010 June Q9
10 marks Standard +0.3
  1. Express \(\frac{4 + 5x - x^2}{(1 - 2x)(2 + x)^2}\) in partial fractions. [5]
  2. Hence obtain the expansion of \(\frac{4 + 5x - x^2}{(1 - 2x)(2 + x)^2}\) in ascending powers of \(x\), up to and including the term in \(x^2\). [5]
CAIE P3 2010 June Q10
12 marks Standard +0.3
The straight line \(l\) has equation \(\mathbf{r} = 2\mathbf{i} - \mathbf{j} - 4\mathbf{k} + \lambda(\mathbf{i} + 2\mathbf{j} + 2\mathbf{k})\). The plane \(p\) has equation \(3x - y + 2z = 9\). The line \(l\) intersects the plane \(p\) at the point \(A\).
  1. Find the position vector of \(A\). [3]
  2. Find the acute angle between \(l\) and \(p\). [4]
  3. Find an equation for the plane which contains \(l\) and is perpendicular to \(p\), giving your answer in the form \(ax + by + cz = d\). [5]
CAIE P3 2013 June Q1
3 marks Moderate -0.8
Solve the equation \(|x - 2| = |\frac{1}{3}x|\). [3]
CAIE P3 2013 June Q2
5 marks Moderate -0.3
The sequence of values given by the iterative formula $$x_{n+1} = \frac{x_n(x_n^2 + 100)}{2(x_n^2 + 25)},$$ with initial value \(x_1 = 3.5\), converges to \(\alpha\).
  1. Use this formula to calculate \(\alpha\) correct to 4 decimal places, showing the result of each iteration to 6 decimal places. [3]
  2. State an equation satisfied by \(\alpha\) and hence find the exact value of \(\alpha\). [2]
CAIE P3 2013 June Q3
5 marks Moderate -0.3
\includegraphics{figure_3} The variables \(x\) and \(y\) satisfy the equation \(y = Ae^{-kx^2}\), where \(A\) and \(k\) are constants. The graph of \(\ln y\) against \(x^2\) is a straight line passing through the points \((0.64, 0.76)\) and \((1.69, 0.32)\), as shown in the diagram. Find the values of \(A\) and \(k\) correct to 2 decimal places. [5]
CAIE P3 2013 June Q4
6 marks Moderate -0.8
The polynomial \(ax^3 - 20x^2 + x + 3\), where \(a\) is a constant, is denoted by \(\text{p}(x)\). It is given that \((3x + 1)\) is a factor of \(\text{p}(x)\).
  1. Find the value of \(a\). [3]
  2. When \(a\) has this value, factorise \(\text{p}(x)\) completely. [3]
CAIE P3 2013 June Q5
6 marks Challenging +1.2
\includegraphics{figure_5} The diagram shows the curve with equation $$x^3 + xy^2 + ay^2 - 3ax^2 = 0,$$ where \(a\) is a positive constant. The maximum point on the curve is \(M\). Find the \(x\)-coordinate of \(M\) in terms of \(a\). [6]
CAIE P3 2013 June Q6
8 marks Standard +0.3
  1. By differentiating \(\frac{1}{\cos x}\), show that the derivative of \(\sec x\) is \(\sec x \tan x\). Hence show that if \(y = \ln(\sec x + \tan x)\) then \(\frac{dy}{dx} = \sec x\). [4]
  2. Using the substitution \(x = (\sqrt{3}) \tan \theta\), find the exact value of $$\int_1^3 \frac{1}{\sqrt{(3 + x^2)}} dx,$$ expressing your answer as a single logarithm. [4]
CAIE P3 2013 June Q7
9 marks Standard +0.3
  1. By first expanding \(\cos(x + 45°)\), express \(\cos(x + 45°) - (\sqrt{2}) \sin x\) in the form \(R \cos(x + \alpha)\), where \(R > 0\) and \(0° < \alpha < 90°\). Give the value of \(R\) correct to 4 significant figures and the value of \(\alpha\) correct to 2 decimal places. [5]
  2. Hence solve the equation $$\cos(x + 45°) - (\sqrt{2}) \sin x = 2,$$ for \(0° < x < 360°\). [4]