Questions P3 (1243 questions)

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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]
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]