Questions — Edexcel C3 (403 questions)

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Edexcel C3 Q3
8 marks Moderate -0.3
The function f is defined by $$f: x \mapsto |2x - a|, \quad x \in \mathbb{R}$$ where \(a\) is a positive constant.
  1. Sketch the graph of \(y = f(x)\), showing the coordinates of the points where the graph cuts the axes. [2]
  2. On a separate diagram, sketch the graph of \(y = f(2x)\), showing the coordinates of the points where the graph cuts the axes. [2]
  3. Given that a solution of the equation f(x) = \(\frac{1}{2}x\) is \(x = 4\), find the two possible values of \(a\). [4]
Edexcel C3 Q4
6 marks Moderate -0.3
Prove that $$\frac{1 - \tan^2 \theta}{1 + \tan^2 \theta} = \cos 2\theta$$ [6]
Edexcel C3 Q5
7 marks Standard +0.3
Express \(\frac{3}{x^2 + 2x} + \frac{x - 4}{x^2 - 4}\) as a single fraction in its simplest form. [7]
Edexcel C3 Q6
10 marks Standard +0.3
The function f, defined for \(x \in \mathbb{R}, x > 0\), is such that $$f'(x) = x^2 - 2 + \frac{1}{x^2}$$
  1. Find the value of f''(x) at \(x = 4\). [3]
  2. Given that f(3) = 0, find f(x). [4]
  3. Prove that f is an increasing function. [3]
Edexcel C3 Q7
10 marks Moderate -0.3
$$f(x) = \frac{2}{x - 1} - \frac{6}{(x - 1)(2x + 1)}, \quad x > 1$$
  1. Prove that f(x) = \(\frac{4}{2x + 1}\). [4]
  2. Find the range of f. [2]
  3. Find \(f^{-1}(x)\). [3]
  4. Find the range of \(f^{-1}(x)\). [1]
Edexcel C3 Q8
13 marks Standard +0.2
The function f is given by $$f: x \mapsto \ln(3x - 6), \quad x \in \mathbb{R}, \quad x > 2$$
  1. Find \(f^{-1}(x)\). [3]
  2. Write down the domain of \(f^{-1}\) and the range of \(f^{-1}\). [2]
  3. Find, to 3 significant figures, the value of \(x\) for which f(x) = 3. [2]
The function g is given by $$g: x \mapsto \ln|3x - 6|, \quad x \in \mathbb{R}, \quad x \neq 2$$
  1. Sketch the graph of \(y = g(x)\). [3]
  2. Find the exact coordinates of all the points at which the graph of \(y = g(x)\) meets the coordinate axes. [3]
Edexcel C3 Q1
9 marks Moderate -0.3
The function f is given by $$f : x \alpha \frac{x}{x^2-1} - \frac{1}{x+1}, \quad x > 1.$$
  1. Show that \(\text{f}(x) = \frac{1}{(x-1)(x+1)}\). [3]
  2. Find the range of f. [2]
The function g is given by $$g : x \alpha \frac{2}{x}, \quad x > 0.$$
  1. Solve gf(x) = 70. [4]
Edexcel C3 Q2
5 marks Moderate -0.3
Express \(\frac{y+3}{(y+1)(y+2)} - \frac{y+1}{(y+2)(y+3)}\) as a single fraction in its simplest form. [5]
Edexcel C3 Q3
9 marks Standard +0.3
The function f is even and has domain \(\mathbb{R}\). For \(x \geq 0\), f(x) = \(x^2 - 4ax\), where \(a\) is a positive constant.
  1. In the space below, sketch the curve with equation \(y = \text{f}(x)\), showing the coordinates of all the points at which the curve meets the axes. [3]
  2. Find, in terms of \(a\), the value of f(2a) and the value of f(-2a). [2]
Given that \(a = 3\),
  1. use algebra to find the values of \(x\) for which f(x) = 45. [4]
Edexcel C3 Q4
10 marks Standard +0.2
\(\text{f}(x) = x^3 + x^2 - 4x - 1\). The equation f(x) = 0 has only one positive root, \(\alpha\).
  1. Show that f(x) = 0 can be rearranged as $$x = \sqrt{\frac{4x+1}{x+1}}, \quad x \neq -1.$$ [2]
The iterative formula \(x_{n+1} = \sqrt{\frac{4x_n+1}{x_n+1}}\) is used to find an approximation to \(\alpha\).
  1. Taking \(x_1 = 1\), find, to 2 decimal places, the values of \(x_2\), \(x_3\) and \(x_4\). [3]
  2. By choosing values of \(x\) in a suitable interval, prove that \(\alpha = 1.70\), correct to 2 decimal places. [3]
  3. Write down a value of \(x_1\) for which the iteration formula \(x_{n+1} = \sqrt{\frac{4x_n+1}{x_n+1}}\) does not produce a valid value for \(x_2\). Justify your answer. [2]
Edexcel C3 Q5
13 marks Standard +0.3
The functions f and g are defined by $$\text{f}: x \alpha |x - a| + a, \quad x \in \mathbb{R},$$ $$\text{g}: x \alpha 4x + a, \quad x \in \mathbb{R}.$$ where \(a\) is a positive constant.
  1. On the same diagram, sketch the graphs of f and g, showing clearly the coordinates of any points at which your graphs meet the axes. [5]
  2. Use algebra to find, in terms of \(a\), the coordinates of the point at which the graphs of f and g intersect. [3]
  3. Find an expression for fg(x). [2]
  4. Solve, for \(x\) in terms of \(a\), the equation $$\text{fg}(x) = 3a.$$ [3]
Edexcel C3 Q6
10 marks Standard +0.2
\includegraphics{figure_1} Figure 1 shows a sketch of the curve with equation \(y = \text{f}(x)\), where $$\text{f}(x) = 10 + \ln(3x) - \frac{1}{2}e^x, \quad 0.1 \leq x \leq 3.3.$$ Given that f(k) = 0,
  1. show, by calculation, that \(3.1 < k < 3.2\). [2]
  2. Find f'(x). [3]
The tangent to the graph at \(x = 1\) intersects the \(y\)-axis at the point \(P\).
    1. Find an equation of this tangent.
    2. Find the exact \(y\)-coordinate of \(P\), giving your answer in the form \(a + \ln b\). [5]
Edexcel C3 Q7
8 marks Standard +0.3
  1. Express \(\sin x + \sqrt{3} \cos x\) in the form \(R \sin (x + \alpha)\), where \(R > 0\) and \(0 < \alpha < 90°\). [4]
  2. Show that the equation \(\sec x + \sqrt{3} \cosec x = 4\) can be written in the form $$\sin x + \sqrt{3} \cos x = 2 \sin 2x.$$ [3]
  3. Deduce from parts (a) and (b) that \(\sec x + \sqrt{3} \cosec x = 4\) can be written in the form $$\sin 2x - \sin (x + 60°) = 0.$$ [1]
Edexcel C3 Q1
Standard +0.0
[No content visible for question 1]
Edexcel C3 Q2
9 marks Standard +0.2
\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]
Edexcel C3 Q3
7 marks Moderate -0.3
  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]
Edexcel C3 Q4
10 marks Standard +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.
Edexcel C3 Q5
8 marks Moderate -0.3
  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]
Edexcel C3 Q6
14 marks Standard +0.3
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]
Edexcel C3 Q7
12 marks Standard +0.3
    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]
Edexcel C3 Q8
14 marks Standard +0.3
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]
Edexcel C3 Q1
5 marks Moderate -0.8
The curve \(C\) has equation \(y = 2e^x + 3x^2 + 2\). The point \(A\) with coordinates \((0, 4)\) lies on \(C\). Find the equation of the tangent to \(C\) at \(A\). [5]
Edexcel C3 Q2
6 marks Standard +0.3
Express \(\frac{x}{(x+1)(x+3)} + \frac{x+12}{x^2-9}\) as a single fraction in its simplest form. [6]
Edexcel C3 Q3
6 marks Moderate -0.3
The functions f and g are defined by \(\text{f: } x \mapsto x^2 - 2x + 3, x \in \mathbb{R}, 0 \leq x \leq 4,\) \(\text{g: } x \mapsto \lambda x^2 + 1, \text{ where } \lambda \text{ is a constant, } x \in \mathbb{R}.\)
  1. Find the range of f. [3]
  2. Given that gf(2) = 16, find the value of \(\lambda\). [3]
Edexcel C3 Q4
10 marks Standard +0.2
  1. Sketch, on the same set of axes, the graphs of $$y = 2 - e^{-x} \text{ and } y = \sqrt{x}.$$ [3] [It is not necessary to find the coordinates of any points of intersection with the axes.] Given that f(x) = \(e^{-x} + \sqrt{x} - 2\), \(x \geq 0\),
  2. explain how your graphs show that the equation f(x) = 0 has only one solution, [1]
  3. show that the solution of f(x) = 0 lies between \(x = 3\) and \(x = 4\). [2]
The iterative formula \(x_{n+1} = (2 - e^{-x_n})^2\) is used to solve the equation f(x) = 0.
  1. Taking \(x_0 = 4\), write down the values of \(x_1\), \(x_2\), \(x_3\) and \(x_4\), and hence find an approximation to the solution of f(x) = 0, giving your answer to 3 decimal places. [4]