Edexcel C3 (Core Mathematics 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]

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]

The root of the equation \(f(x) = 0\), where $$f(x) = x + \ln 2x - 4$$ is to be estimated using the iterative formula \(x_{n+1} = 4 - \ln 2x_n\), with \(x_0 = 2.4\).

1. Showing your values of \(x_1, x_2, x_3, \ldots\), obtain the value, to 3 decimal places, of the root. [4]
2. By considering the change of sign of \(f(x)\) in a suitable interval, justify the accuracy of your answer to part (a). [2]

1. Prove, by counter-example, that the statement "\(\sec(A + B) = \sec A + \sec B\), 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]

The function \(f\) is given by $$f : x \mapsto \frac{x}{x^2 - 1} - \frac{1}{x + 1}, \quad x > 1.$$

1. Show that \(f(x) = \frac{1}{(x-1)(x+1)}\). [3]
2. Find the range of \(f\). [2]

The function \(g\) is given by $$g : x \mapsto \frac{2}{x}, \quad x > 0.$$

1. Solve \(gf(x) = 70\). [4]

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]

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]

1. Prove that $$\frac{1 - \cos 2\theta}{\sin 2\theta} = \tan \theta, \quad \theta \neq \frac{n\pi}{2}, \quad n \in \mathbb{Z}.$$ [3]
2. Solve, giving exact answers in terms of \(\pi\), $$2(1 - \cos 2\theta) = \tan \theta, \quad 0 < \theta < \pi.$$ [6]

\includegraphics{figure_2} Figure 2 shows part of the curve \(C\) with equation \(y = f(x)\), where $$f(x) = 0.5e^x - x^2.$$ 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]

$$f(x) = \frac{2}{x-1} - \frac{6}{(x-1)(2x+1)}, \quad x > 1.$$

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]

Use the derivatives of \(\sin x\) and \(\cos x\) to prove that the derivative of \(\tan x\) is \(\sec^2 x\). [4]

Express \(\frac{3}{x^2 + 2x} + \frac{x - 4}{x^2 - 4}\) as a single fraction in its simplest form. [7]

\includegraphics{figure_1} Figure 1 shows a sketch of the curve with equation \(y = f(x)\), where $$f(x) = 10 + \ln(3x) - \frac{1}{2}e^x, \quad 0.1 \leq x \leq 3.3.$$ 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. 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]

$$f(x) = x^2 - 2x - 3, \quad x \in \mathbb{R}, x \geq 1.$$

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]

Express \(\frac{y + 3}{(y + 1)(y + 2)} - \frac{y + 1}{(y + 2)(y + 3)}\) as a single fraction in its simplest form. [5]

1. Express \(1.5 \sin 2x + 2 \cos 2x\) in the form \(R \sin (2x + \alpha)\), where \(R > 0\) and \(0 < \alpha < \frac{1}{2}\pi\), giving your values of \(R\) and \(\alpha\) to 3 decimal places where appropriate. [4]
2. Express \(3 \sin x \cos x + 4 \cos^2 x\) in the form \(a \cos 2x + b \sin 2x + c\), where \(a\), \(b\) and \(c\) are constants to be found. [2]
3. Hence, using your answer to part (a), deduce the maximum value of \(3 \sin x \cos x + 4 \cos^2 x\). [2]

The curve \(C\) with equation \(y = p + qe^x\), where \(p\) and \(q\) are constants, passes through the point \((0, 2)\). At the point \(P(\ln 2, p + 2q)\) on \(C\), the gradient is \(5\).

1. Find the value of \(p\) and the value of \(q\). [5]

The normal to \(C\) at \(P\) crosses the \(x\)-axis at \(L\) and the \(y\)-axis at \(M\).

1. Show that the area of \(\triangle OLM\), where \(O\) is the origin, is approximately \(53.8\). [5]

\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 graphs 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]

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

1. Prove that \(f^{-1}(x) = f(x)\) for all \(x \in \mathbb{R}, x \neq 3\). [3]
2. Hence find, in terms of \(k\), \(ff(k)\), 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 \(fg(-2)\). [3]
2. Sketch the graph of the inverse function \(g^{-1}\) and state its domain. [3]

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

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

Express \(\frac{x}{(x+1)(x+3)} + \frac{x+12}{x^2-9}\) as a single fraction in its simplest form. [6]

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]

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 \(x_0 = \frac{1}{4}\), to find \(x_1, x_2, x_3\) and \(x_4\). Hence write down, to 3 decimal places, an approximation for \(q\). [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]

\includegraphics{figure_3} Figure 3 shows a sketch of the curve with equation \(y = f(x), x \geq 0\). The curve meets the coordinate axes at the points \((0, c)\) and \((d, 0)\). In separate diagrams sketch the curve with equation

1. \(y = f^{-1}(x)\), [2]
2. \(y = 3f(2x)\). [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 $$f : x \mapsto 3(2^{-x}) - 1, \quad x \in \mathbb{R}, x \geq 0,$$

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 $$g : x \to \log_2 x, \quad x \in \mathbb{R}, x \geq 1.$$

1. Find \(fg(x)\), giving your answer in its simplest form. [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]

The functions \(f\) and \(g\) are defined by $$f: x \mapsto x^2 - 2x + 3, x \in \mathbb{R}, \quad 0 \leq x \leq 4,$$ $$g: x \mapsto \lambda x^2 + 1, \text{ where } \lambda \text{ is a constant, } x \in \mathbb{R}.$$

1. Find the range of \(f\). [3]
2. Given that \(gf(2) = 16\), find the value of \(\lambda\). [3]

\includegraphics{figure_1} Figure 1 shows a sketch of the curve with equation \(y = f(x), -1 \leq x \leq 3\). 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. \(y = f(x + 1)\), [3]
2. \(y = |f(x)|\), [3]
3. \(y = f(|x|)\), [4]

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

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

1. 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 \(f(x) = e^{-x} + \sqrt{x} - 2, x \geq 0\),

1. explain how your graphs show that the equation \(f(x) = 0\) has only one solution, [1]
2. 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]

28a.

1. Given that \(\cos(x + 30)° = 3 \cos(x - 30)°\), prove that \(\tan x° = -\frac{\sqrt{3}}{2}\). [5]
2. 1. Prove that \(\frac{1 - \cos 2\theta}{\sin 2\theta} = \tan \theta\). [3]
   2. Verify that \(\theta = 180°\) is a solution of the equation \(\sin 2\theta = 2 - 2 \cos 2\theta\). [1]
   3. Using the result in part (a), or otherwise, find the other two solutions, \(0 < \theta < 360°\), of the equation using \(\sin 2\theta = 2 - 2 \cos 2\theta\). [4]

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]

Prove that $$\frac{1 - \tan^2 \theta}{1 + \tan^2 \theta} = \cos 2\theta.$$ [4]

The functions \(f\) and \(g\) are defined by $$f: x \mapsto |x - a| + a, \quad x \in \mathbb{R},$$ $$g: x \mapsto 4x + a, \quad x \in \mathbb{R},$$ 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 $$fg(x) = 3a.$$ [3]

The curve \(C\) has equation \(y = f(x)\), where $$f(x) = 3 \ln x + \frac{1}{x}, \quad x > 0.$$ 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]

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]

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 \(y = f(x)\), 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]

Given that \(y = \log_a x, x > 0\), where \(a\) is a positive constant,

1. 1. express \(x\) in terms of \(a\) and \(y\), [1]
   2. deduce that \(\ln x = y \ln a\). [1]
2. Show that \(\frac{dy}{dx} = \frac{1}{x \ln a}\). [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]

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]
2. Solve $$8 \cot \theta - 3 \tan \theta = 2,$$ for \(0 < \theta < 90°\), giving your answer to 1 decimal place. [5]

Express as a single fraction in its simplest form $$\frac{x^2 - 8x + 15}{x^2 - 9} \times \frac{2x^2 + 6x}{(x - 5)^2}.$$ [4]