Questions C3 (1301 questions)

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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]
Edexcel C3 Q5
11 marks Standard +0.2
\includegraphics{figure_1} Figure 1 shows a sketch of the curve with equation \(y = e^{-x} - 1\).
  1. Copy Fig. 1 and on the same axes sketch the graph of \(y = \frac{1}{2}|x - 1|\). Show the coordinates of the points where the graph meets the axes. [2]
The \(x\)-coordinate of the point of intersection of the graph is \(\alpha\).
  1. Show that \(x = \alpha\) is a root of the equation \(x + 2e^{-x} - 3 = 0\). [3]
  2. Show that \(-1 < \alpha < 0\). [2]
The iterative formula \(x_{n+1} = -\ln[\frac{1}{2}(3 - x_n)]\) is used to solve the equation \(x + 2e^{-x} - 3 = 0\).
  1. Starting with \(x_0 = -1\), find the values of \(x_1\) and \(x_2\). [2]
  2. Show that, to 2 decimal places, \(\alpha = -0.58\). [2]
Edexcel C3 Q6
14 marks Standard +0.3
f(x) = \(x^2 - 2x - 3\), \(x \in \mathbb{R}\), \(x \geq 1\).
  1. Find the range of f. [1]
  2. Write down the domain and range of \(f^{-1}\). [2]
  3. Sketch the graph of \(f^{-1}\), indicating clearly the coordinates of any point at which the graph intersects the coordinate axes. [4]
Given that g(x) = \(|x - 4|\), \(x \in \mathbb{R}\),
  1. find an expression for gf(x). [2]
  2. Solve gf(x) = 8. [5]
Edexcel C3 Q7
7 marks Moderate -0.3
f(x) = \(x + \frac{e^x}{5}\), \(x \in \mathbb{R}\).
  1. Find f'(x). [2]
The curve \(C\), with equation \(y = \)f(x), crosses the \(y\)-axis at the point \(A\).
  1. Find an equation for the tangent to \(C\) at \(A\). [3]
  2. Complete the table, giving the values of \(\sqrt{x + \frac{e^x}{5}}\) to 2 decimal places.
\(x\)00.511.52
\(\sqrt{x + \frac{e^x}{5}}\)0.450.91
[2]
Edexcel C3 Q8
15 marks Standard +0.3
  1. Express \(2\cos\theta + 5\sin\theta\) in the form \(R\cos(\theta - \alpha)\), where \(R > 0\) and \(0 < \alpha < \frac{\pi}{2}\). Give the values of \(R\) and \(\alpha\) to 3 significant figures. [3]
  2. Find the maximum and minimum values of \(2\cos\theta + 5\sin\theta\) and the smallest possible value of \(\theta\) for which the maximum occurs. [2]
The temperature \(T\) °C, of an unheated building is modelled using the equation $$T = 15 + 2\cos\left(\frac{\pi t}{12}\right) + 5\sin\left(\frac{\pi t}{12}\right), \quad 0 \leq t < 24,$$ where \(t\) hours is the number of hours after 1200.
  1. Calculate the maximum temperature predicted by this model and the value of \(t\) when this maximum occurs. [4]
  2. Calculate, to the nearest half hour, the times when the temperature is predicted to be 12 °C. [6]
Edexcel C3 Q1
4 marks Moderate -0.5
Use the derivatives of \(\sin x\) and \(\cos x\) to prove that the derivative of \(\tan x\) is \(\sec^2 x\). [4]
Edexcel C3 Q2
5 marks Moderate -0.3
The function f is given by \(f: x \mapsto 2 + \frac{3}{x + 2}\), \(x \in \mathbb{R}\), \(x \neq -2\).
  1. Express \(2 + \frac{3}{x + 2}\) as a single fraction. [1]
  2. Find an expression for \(f^{-1}(x)\). [3]
  3. Write down the domain of \(f^{-1}\). [1]
Edexcel C3 Q3
6 marks Moderate -0.3
  1. Express as a fraction in its simplest form $$\frac{2}{x - 3} + \frac{13}{x^2 + 4x - 21}.$$ [3]
  2. Hence solve $$\frac{2}{x - 3} + \frac{13}{x^2 + 4x - 21} = 1.$$ [3]
Edexcel C3 Q4
6 marks Moderate -0.3
  1. Simplify \(\frac{x^2 + 4x + 3}{x^2 + x}\). [2]
  2. Find the value of \(x\) for which \(\log_2(x^2 + 4x + 3) - \log_2(x^2 + x) = 4\). [4]
Edexcel C3 Q5
7 marks Standard +0.3
  1. Prove, by counter-example, that the statement "\(\sec(A + B) \equiv \sec A + \sec B\), for all \(A\) and \(B\)" is false [2]
  2. Prove that $$\tan \theta + \cot \theta = 2\cosec 2\theta, \quad \theta \neq \frac{n\pi}{2}, n \in \mathbb{Z}.$$ [5]
Edexcel C3 Q6
9 marks Standard +0.3
  1. Prove that $$\frac{1 - \cos 2\theta}{\sin 2\theta} \equiv \tan \theta, \quad \theta \neq \frac{n\pi}{2}, n \in \mathbb{Z}.$$ [3]
  2. Solve, giving exact answers in terms of \(\pi\), $$2(1 - \cos 2\theta) = \tan \theta, \quad 0 < \theta < \pi.$$ [6]
Edexcel C3 Q7
10 marks Moderate -0.3
Given that \(y = \log_a x\), \(x > 0\), where \(a\) is a positive constant,
    1. express \(x\) in terms of \(a\) and \(y\), [1]
    2. deduce that \(\ln x = y \ln a\). [1]
  2. Show that \(\frac{\mathrm{d}y}{\mathrm{d}x} = \frac{1}{x \ln a}\). [2]
The curve \(C\) has equation \(y = \log_{10} x\), \(x > 0\). The point \(A\) on \(C\) has \(x\)-coordinate 10. Using the result in part (b),
  1. find an equation for the tangent to \(C\) at \(A\). [4]
The tangent to \(C\) at \(A\) crosses the \(x\)-axis at the point \(B\).
  1. Find the exact \(x\)-coordinate of \(B\). [2]
Edexcel C3 Q8
11 marks Standard +0.3
The curve with equation \(y = \ln 3x\) crosses the \(x\)-axis at the point \(P(p, 0)\).
  1. Sketch the graph of \(y = \ln 3x\), showing the exact value of \(p\). [2]
The normal to the curve at the point \(Q\), with \(x\)-coordinate \(q\), passes through the origin.
  1. Show that \(x = q\) is a solution of the equation \(x^2 + \ln 3x = 0\). [4]
  2. Show that the equation in part (b) can be rearranged in the form \(x = \frac{1}{3}e^{-x^2}\). [2]
  3. Use the iteration formula \(x_{n + 1} = \frac{1}{3}e^{-x_n^2}\), with \(x_0 = \frac{1}{3}\), to find \(x_1, x_2, x_3\) and \(x_4\). Hence write down, to 3 decimal places, an approximation for \(q\). [3]
Edexcel C3 Q9
14 marks Standard +0.3
\includegraphics{figure_3} Figure 3 shows a sketch of the curve with equation \(y = \text{f}(x)\), \(x \geq 0\). The curve meets the coordinate axes at the points \((0, c)\) and \((d, 0)\). In separate diagrams sketch the curve with equation
  1. \(y = \text{f}^{-1}(x)\), [2]
  2. \(y = 3\text{f}(2x)\). [3]
Indicate clearly on each sketch the coordinates, in terms of \(c\) or \(d\), of any point where the curve meets the coordinate axes. Given that f is defined by $$\text{f}: x \mapsto 3(2^{-x}) - 1, \quad x \in \mathbb{R}, x \geq 0,$$
  1. state
    1. the value of \(c\),
    2. the range of f.
    [3]
  2. Find the value of \(d\), giving your answer to 3 decimal places. [3]
The function g is defined by $$\text{g}: x \mapsto \log_2 x, \quad x \in \mathbb{R}, x \geq 1.$$
  1. Find fg(x), giving your answer in its simplest form. [3]
OCR C3 Q1
4 marks Moderate -0.8
The function f is defined for all real values of \(x\) by $$f(x) = 10 - (x + 3)^2.$$
  1. State the range of f. [1]
  2. Find the value of ff(-1). [3]
OCR C3 Q2
4 marks Moderate -0.3
Find the exact solutions of the equation \(|6x - 1| = |x - 1|\). [4]
OCR C3 Q3
6 marks Moderate -0.3
The mass, \(m\) grams, of a substance at time \(t\) years is given by the formula $$m = 180e^{-0.017t}.$$
  1. Find the value of \(t\) for which the mass is 25 grams. [3]
  2. Find the rate at which the mass is decreasing when \(t = 55\). [3]
OCR C3 Q4
8 marks Standard +0.2
  1. \includegraphics{figure_4a} The diagram shows the curve \(y = \frac{2}{\sqrt{x}}\). The region \(R\), shaded in the diagram, is bounded by the curve and by the lines \(x = 1\), \(x = 5\) and \(y = 0\). The region \(R\) is rotated completely about the \(x\)-axis. Find the exact volume of the solid formed. [4]
  2. Use Simpson's rule, with 4 strips, to find an approximate value for $$\int_1^5 \sqrt{(x^2 + 1)} \, dx,$$ giving your answer correct to 3 decimal places. [4]
OCR C3 Q5
8 marks Standard +0.3
  1. Express \(3 \sin \theta + 2 \cos \theta\) in the form \(R \sin(\theta + \alpha)\), where \(R > 0\) and \(0° < \alpha < 90°\). [3]
  2. Hence solve the equation \(3 \sin \theta + 2 \cos \theta = \frac{5}{2}\), giving all solutions for which \(0° < \theta < 360°\). [5]