Questions — Edexcel C3 (403 questions)

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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]
Edexcel C3 Q1
8 marks Standard +0.3
  1. Find the exact value of \(x\) such that $$3 \arctan (x - 2) + \pi = 0.$$ [3]
  2. Solve, for \(-\pi < \theta < \pi\), the equation $$\cos 2\theta - \sin \theta - 1 = 0,$$ giving your answers in terms of \(\pi\). [5]
Edexcel C3 Q2
9 marks Moderate -0.8
  1. Express $$\frac{4x}{x^2 - 9} - \frac{2}{x + 3}$$ as a single fraction in its simplest form. [4]
  2. Simplify $$\frac{x^3 - 8}{3x^2 - 8x + 4}.$$ [5]
Edexcel C3 Q3
9 marks Moderate -0.3
Differentiate each of the following with respect to \(x\) and simplify your answers.
  1. \(\cot x^2\) [2]
  2. \(x^2 e^{-x}\) [3]
  3. \(\frac{\sin x}{3 + 2\cos x}\) [4]
Edexcel C3 Q4
10 marks Standard +0.3
  1. Find, as natural logarithms, the solutions of the equation $$e^{2x} - 8e^x + 15 = 0.$$ [4]
  2. Use proof by contradiction to prove that \(\log_5 3\) is irrational. [6]
Edexcel C3 Q5
12 marks Standard +0.2
The function f is defined by $$f : x \to 3e^{x-1}, \quad x \in \mathbb{R}.$$
  1. State the range of f. [1]
  2. Find an expression for \(f^{-1}(x)\) and state its domain. [4]
The function g is defined by $$g : x \to 5x - 2, \quad x \in \mathbb{R}.$$ Find, in terms of e,
  1. the value of gf(ln 2), [3]
  2. the solution of the equation $$f^{-1}g(x) = 4.$$ [4]
Edexcel C3 Q6
13 marks Standard +0.3
$$f(x) = 2x^2 + 3 \ln (2 - x), \quad x \in \mathbb{R}, \quad x < 2.$$
  1. Show that the equation \(f(x) = 0\) can be written in the form $$x = 2 - e^{kx^2},$$ where \(k\) is a constant to be found. [3]
The root, \(\alpha\), of the equation \(f(x) = 0\) is \(1.9\) correct to \(1\) decimal place.
  1. Use the iteration formula $$x_{n+1} = 2 - e^{kx_n^2},$$ with \(x_0 = 1.9\) and your value of \(k\), to find \(\alpha\) to \(3\) decimal places and justify the accuracy of your answer. [5]
  2. Solve the equation \(f'(x) = 0\). [5]
Edexcel C3 Q7
14 marks Standard +0.3
\includegraphics{figure_1} Figure 1 shows the curve \(y = f(x)\) which has a maximum point at \((-45, 7)\) and a minimum point at \((135, -1)\).
  1. Showing the coordinates of any stationary points, sketch on separate diagrams the graphs of
    1. \(y = f(|x|)\),
    2. \(y = 1 + 2f(x)\). [6]
Given that $$f(x) = A + 2\sqrt{2} \cos x^{\circ} - 2\sqrt{2} \sin x^{\circ}, \quad x \in \mathbb{R}, \quad -180 \leq x \leq 180,$$ where \(A\) is a constant,
  1. show that f(x) can be expressed in the form $$f(x) = A + R \cos (x + \alpha)^{\circ},$$ where \(R > 0\) and \(0 < \alpha < 90\), [3]
  2. state the value of \(A\), [1]
  3. find, to \(1\) decimal place, the \(x\)-coordinates of the points where the curve \(y = f(x)\) crosses the \(x\)-axis. [4]
Edexcel C3 Q1
6 marks Moderate -0.3
\(f(x) \equiv \frac{2x-3}{x-2}\), \(x \in \mathbb{R}\), \(x > 2\).
  1. Find the range of \(f\). [2]
  2. Show that \(f(f(x) = x\) for all \(x > 2\). [3]
  3. Hence, write down an expression for \(f^{-1}(x)\). [1]
Edexcel C3 Q2
7 marks Moderate -0.8
Solve each equation, giving your answers in exact form.
  1. \(e^{4x-3} = 2\) [3]
  2. \(\ln (2y - 1) = 1 + \ln (3 - y)\) [4]
Edexcel C3 Q3
8 marks Standard +0.3
The curve \(C\) has the equation \(y = 2e^x - 6 \ln x\) and passes through the point \(P\) with \(x\)-coordinate \(1\).
  1. Find an equation for the tangent to \(C\) at \(P\). [4]
The tangent to \(C\) at \(P\) meets the coordinate axes at the points \(Q\) and \(R\).
  1. Show that the area of triangle \(OQR\), where \(O\) is the origin, is \(\frac{9}{3-e}\). [4]
Edexcel C3 Q4
9 marks Standard +0.3
  1. Express $$\frac{x-10}{(x-3)(x+4)} - \frac{x-8}{(x-3)(2x-1)}$$ as a single fraction in its simplest form. [5]
  2. Hence, show that the equation $$\frac{x-10}{(x-3)(x+4)} - \frac{x-8}{(x-3)(2x-1)} = 1$$ has no real roots. [4]
Edexcel C3 Q5
9 marks Challenging +1.2
Find the values of \(x\) in the interval \(-180 < x < 180\) for which $$\tan (x + 45)^{\circ} - \tan x^{\circ} = 4,$$ giving your answers to 1 decimal place. [9]