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CAIE P2 2015 June Q4
7 marks Moderate -0.3
\includegraphics{figure_4} The diagram shows the curve \(y = e^x + 4e^{-2x}\) and its minimum point \(M\).
  1. Show that the \(x\)-coordinate of \(M\) is \(\ln 2\). [3]
  2. The region shaded in the diagram is enclosed by the curve and the lines \(x = 0\), \(x = \ln 2\) and \(y = 0\). Use integration to show that the area of the shaded region is \(\frac{5}{2}\). [4]
CAIE P2 2015 June Q5
12 marks Standard +0.3
  1. By sketching a suitable pair of graphs, show that the equation $$|3x| = 16 - x^4$$ has two real roots. [3]
  2. Use the iterative formula \(x_{n+1} = \sqrt[4]{16 - 3x_n}\) to find one of the real roots correct to 3 decimal places. Give the result of each iteration to 5 decimal places. [3]
  3. Hence find the coordinates of each of the points of intersection of the graphs \(y = |3x|\) and \(y = 16 - x^4\), giving your answers correct to 3 decimal places. [2]
CAIE P2 2015 June Q6
9 marks Standard +0.3
\includegraphics{figure_6} The diagram shows part of the curve with equation $$y = 4\sin^2 x + 8\sin x + 3$$ and its point of intersection \(P\) with the \(x\)-axis.
  1. Find the exact \(x\)-coordinate of \(P\). [3]
  2. Show that the equation of the curve can be written $$y = 5 + 8\sin x - 2\cos 2x,$$ and use integration to find the exact area of the shaded region enclosed by the curve and the axes. [6]
CAIE P2 2015 June Q7
10 marks Standard +0.3
  1. Find the gradient of the curve $$3\ln x + 4\ln y + 6xy = 6$$ at the point \((1, 1)\). [4]
  2. The parametric equations of a curve are $$x = \frac{10}{t} - t, \quad y = \sqrt{2t - 1}.$$ Find the gradient of the curve at the point \((-3, 3)\). [6]
CAIE P2 2003 November Q1
3 marks Moderate -0.8
Find the set of values of \(x\) satisfying the inequality \(|8 - 3x| < 2\). [3]
CAIE P2 2003 November Q2
5 marks Moderate -0.3
\includegraphics{figure_2} Two variable quantities \(x\) and \(y\) are related by the equation $$y = k(a^{-x}),$$ where \(a\) and \(k\) are constants. Four pairs of values of \(x\) and \(y\) are measured experimentally. The result of plotting \(\ln y\) against \(x\) is shown in the diagram. Use the diagram to estimate the values of \(a\) and \(k\). [5]
CAIE P2 2003 November Q3
6 marks Moderate -0.8
The polynomial \(x^4 - 6x^2 + x + a\) is denoted by \(f(x)\).
  1. It is given that \((x + 1)\) is a factor of \(f(x)\). Find the value of \(a\). [2]
  2. When \(a\) has this value, verify that \((x - 2)\) is also a factor of \(f(x)\) and hence factorise \(f(x)\) completely. [4]
CAIE P2 2003 November Q4
7 marks Moderate -0.3
  1. Express \(\cos \theta + (\sqrt{3}) \sin \theta\) in the form \(R \cos(\theta - \alpha)\), where \(R > 0\) and \(0 < \alpha < \frac{1}{2}\pi\), giving the exact value of \(\alpha\). [3]
  2. Hence show that one solution of the equation $$\cos \theta + (\sqrt{3}) \sin \theta = \sqrt{2}$$ is \(\theta = \frac{7}{12}\pi\), and find the other solution in the interval \(0 < \theta < 2\pi\). [4]
CAIE P2 2003 November Q5
7 marks Standard +0.3
  1. By sketching a suitable pair of graphs, for \(x < 0\), show that exactly one root of the equation \(x^2 = 2^x\) is negative. [2]
  2. Verify by calculation that this root lies between \(-1.0\) and \(-0.5\). [2]
  3. Use the iterative formula $$x_{n+1} = -\sqrt{(2^{x_n})}$$ to determine this root correct to 2 significant figures, showing the result of each iteration. [3]
CAIE P2 2003 November Q6
11 marks Standard +0.3
\includegraphics{figure_6} The diagram shows the curve \(y = (4 - x)e^x\) and its maximum point \(M\). The curve cuts the \(x\)-axis at \(A\) and the \(y\)-axis at \(B\).
  1. Write down the coordinates of \(A\) and \(B\). [2]
  2. Find the \(x\)-coordinate of \(M\). [4]
  3. The point \(P\) on the curve has \(x\)-coordinate \(p\). The tangent to the curve at \(P\) passes through the origin \(O\). Calculate the value of \(p\). [5]
CAIE P2 2003 November Q7
11 marks Moderate -0.3
  1. By differentiating \(\frac{\cos x}{\sin x}\), show that if \(y = \cot x\) then \(\frac{dy}{dx} = -\cosec^2 x\). [3]
  2. Hence show that \(\int_{\frac{\pi}{6}}^{\frac{\pi}{2}} \cosec^2 x \, dx = \sqrt{3}\). [2]
By using appropriate trigonometrical identities, find the exact value of
  1. \(\int_{\frac{\pi}{6}}^{\frac{\pi}{2}} \cot^2 x \, dx\), [3]
  2. \(\int_{\frac{\pi}{6}}^{\frac{\pi}{2}} \frac{1}{1 - \cos 2x} \, dx\). [3]
CAIE P2 2016 November Q1
5 marks Moderate -0.3
  1. It is given that \(x\) satisfies the equation \(3^{2x} = 5(3^x) + 14\). Find the value of \(3^x\) and, using logarithms, find the value of \(x\) correct to 3 significant figures. [4]
  2. Hence state the values of \(x\) satisfying the equation \(3^{2|x|} = 5(3^{|x|}) + 14\). [1]
CAIE P2 2016 November Q2
5 marks Moderate -0.8
\includegraphics{figure_2} The variables \(x\) and \(y\) satisfy the equation \(y = Ae^{px}\), where \(A\) and \(p\) are constants. The graph of \(\ln y\) against \(x\) is a straight line passing through the points \((5, 3.17)\) and \((10, 4.77)\), as shown in the diagram. Find the values of \(A\) and \(p\) correct to 2 decimal places. [5]
CAIE P2 2016 November Q3
6 marks Standard +0.3
A curve has equation \(y = 2\sin 2x - 5\cos 2x + 6\) and is defined for \(0 \leq x \leq \pi\). Find the \(x\)-coordinates of the stationary points of the curve, giving your answers correct to 3 significant figures. [6]
CAIE P2 2016 November Q4
7 marks Standard +0.3
It is given that the positive constant \(a\) is such that $$\int_{-a}^a (4e^{2x} + 5) dx = 100.$$
  1. Show that \(a = \frac{1}{4}\ln(50 + e^{-2a} - 5a)\). [4]
  2. Use the iterative formula \(a_{n+1} = \frac{1}{4}\ln(50 + e^{-2a_n} - 5a_n)\) to find \(a\) correct to 3 decimal places. Give the result of each iteration to 5 decimal places. [3]
CAIE P2 2016 November Q5
7 marks Standard +0.3
  1. Show that \(\frac{\cos 2x + 9\cos x + 5}{\cos x + 4} \equiv 2\cos x + 1\). [3]
  2. Hence find the exact value of \(\int_{-\pi}^{\pi} \frac{\cos 4x + 9\cos 2x + 5}{\cos 2x + 4} dx\). [4]
CAIE P2 2016 November Q6
8 marks Standard +0.3
The equation of a curve is \(3x^2 + 4xy + y^2 = 24\). Find the equation of the normal to the curve at the point \((1, 3)\), giving your answer in the form \(ax + by + c = 0\) where \(a\), \(b\) and \(c\) are integers. [8]
CAIE P2 2016 November Q7
12 marks Moderate -0.3
The polynomial \(p(x)\) is defined by $$p(x) = ax^3 + 3x^2 + bx + 12,$$ where \(a\) and \(b\) are constants. It is given that \((x + 3)\) is a factor of \(p(x)\). It is also given that the remainder is 18 when \(p(x)\) is divided by \((x + 2)\).
  1. Find the values of \(a\) and \(b\). [5]
  2. When \(a\) and \(b\) have these values,
    1. show that the equation \(p(x) = 0\) has exactly one real root, [4]
    2. solve the equation \(p(\sec y) = 0\) for \(-180° < y < 180°\). [3]
CAIE P2 2016 November Q1
5 marks Moderate -0.3
The sequence of values given by the iterative formula $$x_{n+1} = \frac{4}{x_n^2} + \frac{2x_n}{3},$$ with initial value \(x_1 = 2\), converges to \(\alpha\).
  1. Use this iterative formula to find \(\alpha\) correct to 3 decimal places. Give the result of each iteration to 5 decimal places. [3]
  2. State an equation that is satisfied by \(\alpha\), and hence find the exact value of \(\alpha\). [2]
CAIE P2 2016 November Q2
5 marks Moderate -0.8
\includegraphics{figure_2} The variables \(x\) and \(y\) satisfy the equation \(y = Kx^p\), where \(K\) and \(p\) are constants. The graph of \(\ln y\) against \(\ln x\) is a straight line passing through the points \((1.28, 3.69)\) and \((2.11, 4.81)\), as shown in the diagram. Find the values of \(K\) and \(p\) correct to 2 decimal places. [5]
CAIE P2 2016 November Q3
5 marks Standard +0.3
  1. Find \(\int \tan^2 4x \, dx\). [2]
  2. Without using a calculator, find the exact value of \(\int_0^{\frac{\pi}{2}} (4 \cos 2x + 6 \sin 3x) \, dx\). [3]
CAIE P2 2016 November Q4
8 marks Moderate -0.3
The polynomial \(\mathrm{p}(x)\) is defined by $$\mathrm{p}(x) = ax^3 + 3x^2 + 4ax - 5,$$ where \(a\) is a constant. It is given that \((2x - 1)\) is a factor of \(\mathrm{p}(x)\).
  1. Use the factor theorem to find the value of \(a\). [2]
  2. Factorise \(\mathrm{p}(x)\) and hence show that the equation \(\mathrm{p}(x) = 0\) has only one real root. [4]
  3. Use logarithms to solve the equation \(\mathrm{p}(6^x) = 0\) correct to 3 significant figures. [2]
CAIE P2 2016 November Q5
8 marks Moderate -0.3
\includegraphics{figure_5} The diagram shows the curve \(y = \sqrt{1 + e^{4x}}\) for \(0 \leq x \leq 6\). The region bounded by the curve and the lines \(x = 0\), \(x = 6\) and \(y = 0\) is denoted by \(R\).
  1. Use the trapezium rule with 2 strips to find an estimate of the area of \(R\), giving your answer correct to 2 decimal places. [3]
  2. With reference to the diagram, explain why this estimate is greater than the exact area of \(R\). [1]
  3. The region \(R\) is rotated completely about the \(x\)-axis. Find the exact volume of the solid produced. [4]
CAIE P2 2016 November Q6
9 marks Standard +0.3
A curve has parametric equations $$x = \ln(t + 1), \quad y = t^2 \ln t.$$
  1. Find an expression for \(\frac{dy}{dx}\) in terms of \(t\). [5]
  2. Find the exact value of \(t\) at the stationary point. [2]
  3. Find the gradient of the curve at the point where it crosses the \(x\)-axis. [2]
CAIE P2 2016 November Q7
10 marks Standard +0.3
  1. Express \(\sin 2\theta (3 \sec \theta + 4 \cosec \theta)\) in the form \(a \sin \theta + b \cos \theta\), where \(a\) and \(b\) are integers. [3]
  2. Hence express \(\sin 2\theta (3 \sec \theta + 4 \cosec \theta)\) in the form \(R \sin(\theta + \alpha)\) where \(R > 0\) and \(0° < \alpha < 90°\). [3]
  3. Using the result of part (ii), solve the equation \(\sin 2\theta (3 \sec \theta + 4 \cosec \theta) = 7\) for \(0° \leq \theta \leq 360°\). [4]