Questions P3 (1243 questions)

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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]
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]