Questions — Edexcel C3 (403 questions)

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Edexcel C3 Q6
10 marks Standard +0.8
  1. Sketch on the same diagram the graphs of \(y = |x| - a\) and \(y = |3x + 5a|\), where \(a\) is a positive constant. Show on your diagram the coordinates of any points where each graph meets the coordinate axes. [6]
  2. Solve the equation $$|x| - a = |3x + 5a|.$$ [4]
Edexcel C3 Q7
12 marks Standard +0.3
  1. Use the identity $$\cos (A + B) = \cos A \cos B - \sin A \sin B$$ to prove that $$\cos x \equiv 1 - 2 \sin^2 \frac{x}{2}.$$ [3]
  2. Prove that, for \(\sin x \neq 0\), $$\frac{1 - \cos x}{\sin x} \equiv \tan \frac{x}{2}.$$ [3]
  3. Find the values of \(x\) in the interval \(0 \leq x \leq 360^{\circ}\) for which $$\frac{1 - \cos x}{\sin x} = 2 \sec^2 \frac{x}{2} - 5,$$ giving your answers to 1 decimal place where appropriate. [6]
Edexcel C3 Q8
14 marks Standard +0.3
A curve has the equation \(y = (2x + 3)e^{-x}\).
  1. Find the exact coordinates of the stationary point of the curve. [4]
The curve crosses the \(y\)-axis at the point \(P\).
  1. Find an equation for the normal to the curve at \(P\). [2]
The normal to the curve at \(P\) meets the curve again at \(Q\).
  1. Show that the \(x\)-coordinate of \(Q\) lies in the interval \([-2, -1]\). [3]
  2. Use the iterative formula $$x_{n+1} = \frac{3 - 3e^{x_n}}{e^{x_n} - 2}$$ with \(x_0 = -1\), to find \(x_1\), \(x_2\), \(x_3\) and \(x_4\). Give the value of \(x_4\) to 2 decimal places. [3]
  3. Show that your value for \(x_4\) is the \(x\)-coordinate of \(Q\) correct to 2 decimal places. [2]