Questions — CAIE P2 (709 questions)

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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]
CAIE P2 2018 November Q1
5 marks Moderate -0.3
  1. Solve the equation \(|9x - 2| = |3x + 2|\). [3]
  2. Hence, using logarithms, solve the equation \(|3^{x+2} - 2| = |3^{x+1} + 2|\), giving your answer correct to 3 significant figures. [2]
CAIE P2 2018 November Q2
5 marks Moderate -0.5
Show that \(\int_1^7 \frac{6}{2x + 1} \, dx = \ln 125\). [5]
CAIE P2 2018 November Q3
5 marks Standard +0.3
Solve the equation \(\sec^2 \theta = 3 \cosec \theta\) for \(0° < \theta < 180°\). [5]
CAIE P2 2018 November Q4
11 marks Moderate -0.3
\includegraphics{figure_4} The diagram shows the curve with equation $$y = x^4 + 2x^3 + 2x^2 - 12x - 32.$$ The curve crosses the \(x\)-axis at points with coordinates \((\alpha, 0)\) and \((\beta, 0)\).
  1. Use the factor theorem to show that \((x + 2)\) is a factor of $$x^4 + 2x^3 + 2x^2 - 12x - 32.$$ [2]
  2. Show that \(\beta\) satisfies an equation of the form \(x = \sqrt[3]{p + qx}\), and state the values of \(p\) and \(q\). [3]
  3. Use an iterative formula based on the equation in part (ii) to find the value of \(\beta\) correct to 4 significant figures. Give the result of each iteration to 6 significant figures. [3]
CAIE P2 2018 November Q5
9 marks Standard +0.3
A curve has parametric equations $$x = t + \ln(t + 1), \quad y = 3te^{2t}.$$
  1. Find the equation of the tangent to the curve at the origin. [5]
  2. Find the coordinates of the stationary point, giving each coordinate correct to 2 decimal places. [4]
CAIE P2 2018 November Q6
8 marks Standard +0.3
\includegraphics{figure_6} The diagram shows the curve with equation \(y = \sqrt{1 + 3\cos^2(\frac{1}{2}x)}\) for \(0 \leqslant x \leqslant \pi\). The region \(R\) is bounded by the curve, the axes and the line \(x = \pi\).
  1. Use the trapezium rule with two intervals to find an approximation to the area of \(R\), giving your answer correct to 3 significant figures. [3]
  2. The region \(R\) is rotated completely about the \(x\)-axis. Without using a calculator, find the exact volume of the solid produced. [5]
CAIE P2 2018 November Q7
10 marks Standard +0.3
\includegraphics{figure_7} The diagram shows the curve with equation \(y = \sin 2x + 3\cos 2x\) for \(0 \leqslant x \leqslant \pi\). At the points \(P\) and \(Q\) on the curve, the gradient of the curve is 3.
  1. Find an expression for \(\frac{dy}{dx}\). [2]
  2. By first expressing \(\frac{dy}{dx}\) in the form \(R\cos(2x + \alpha)\), where \(R > 0\) and \(0 < \alpha < \frac{1}{2}\pi\), find the \(x\)-coordinates of \(P\) and \(Q\), giving your answers correct to 4 significant figures. [8]