Edexcel C3 (Core Mathematics 3)

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\includegraphics{figure_2} Figure 2 shows part of the curve \(C\) with equation \(y = \text{f}(x)\), where $$\text{f}(x) = 0.5e^x - x^2.$$ The curve \(C\) cuts the \(y\)-axis at \(A\) and there is a minimum at the point \(B\).

1. Find an equation of the tangent to \(C\) at \(A\). [4]

The \(x\)-coordinate of \(B\) is approximately 2.15. A more exact estimate is to be made of this coordinate using iterations \(x_{n+1} = \ln g(x_n)\).

1. Show that a possible form for \(g(x)\) is \(g(x) = 4x\). [3]
2. Using \(x_{n+1} = \ln 4x_n\), with \(x_0 = 2.15\), calculate \(x_1\), \(x_2\) and \(x_3\). Give the value of \(x_3\) to 4 decimal places. [2]

1. Sketch the graph of \(y = |2x + a|\), \(a > 0\), showing the coordinates of the points where the graph meets the coordinate axes. [2]
2. On the same axes, sketch the graph of \(y = \frac{1}{x}\). [1]
3. Explain how your graphs show that there is only one solution of the equation $$x|2x + a| - 1 = 0.$$ [1]
4. Find, using algebra, the value of \(x\) for which \(x|2x + 1| - 1 = 0\). [3]

\includegraphics{figure_1} Figure 1 shows a sketch of the curve with equation \(y = \text{f}(x)\), \(-1 \leq x \leq 3\). The curve touches the \(x\)-axis at the origin \(O\), crosses the \(x\)-axis at the point \(A(2, 0)\) and has a maximum at the point \(B(\frac{4}{3}, 1)\). In separate diagrams, show a sketch of the curve with equation

1. \(y = \text{f}(x + 1)\), [3]
2. \(y = |\text{f}(x)|\), [3]
3. \(y = \text{f}(|x|)\), [4]

marking on each sketch the coordinates of points at which the curve

1. has a turning point,
2. meets the \(x\)-axis.

1. Given that \(\sin x = \frac{3}{5}\), use an appropriate double angle formula to find the exact value of \(\sec 2x\). [4]
2. Prove that $$\cot 2x + \cosec 2x \equiv \cot x, \quad \left(x \neq \frac{n\pi}{2}, n \in \mathbb{Z}\right).$$ [4]

The function f is defined by \(\text{f}: x \to \frac{3x - 1}{x - 3}\), \(x \in \mathbb{R}\), \(x \neq 3\).

1. Prove that \(\text{f}^{-1}(x) = \text{f}(x)\) for all \(x \in \mathbb{R}\), \(x \neq 3\). [3]
2. Hence find, in terms of \(k\), \(\text{f}f(k)\), where \(x \neq 3\). [2]

\includegraphics{figure_3} Figure 3 shows a sketch of the one-one function g, defined over the domain \(-2 \leq x \leq 2\).

1. Find the value of \(\text{f}g(-2)\). [3]
2. Sketch the graph of the inverse function \(\text{g}^{-1}\) and state its domain. [3]

The function h is defined by \(\text{h}: x \mapsto 2g(x - 1)\).

1. Sketch the graph of the function h and state its range. [3]

1. 1. Express \((12 \cos \theta - 5 \sin \theta)\) in the form \(R \cos (\theta + \alpha)\), where \(R > 0\) and \(0 < \alpha < 90°\). [4]
2. Hence solve the equation $$12 \cos \theta - 5 \sin \theta = 4,$$ for \(0 < \theta < 90°\), giving your answer to 1 decimal place. [3]
3. Solve $$8 \cot \theta - 3 \tan \theta = 2,$$ for \(0 < \theta < 90°\), giving your answer to 1 decimal place. [5]

The curve \(C\) has equation \(y = \text{f}(x)\), where $$\text{f}(x) = 3 \ln x + \frac{1}{x}, \quad x > 0.$$ The point \(P\) is a stationary point on \(C\).

1. Calculate the \(x\)-coordinate of \(P\). [4]
2. Show that the \(y\)-coordinate of \(P\) may be expressed in the form \(k - k \ln k\), where \(k\) is a constant to be found. [2]

The point \(Q\) on \(C\) has \(x\)-coordinate 1.

1. Find an equation for the normal to \(C\) at \(Q\). [4]

The normal to \(C\) at \(Q\) meets \(C\) again at the point \(R\).

1. Show that the \(x\)-coordinate of \(R\)
   1. satisfies the equation \(6 \ln x + x + \frac{2}{x} - 3 = 0\),
   2. lies between 0.13 and 0.14. [4]