AQA C3 (Core Mathematics 3) 2011 June

The diagram shows the curve with equation \(y = \ln(6x)\). \includegraphics{figure_1}

1. State the \(x\)-coordinate of the point of intersection of the curve with the \(x\)-axis. [1]
2. Find \(\frac{dy}{dx}\). [2]
3. Use Simpson's rule with 6 strips (7 ordinates) to find an estimate for \(\int_1^7 \ln(6x) \, dx\), giving your answer to three significant figures. [4]

1. 1. Find \(\frac{dy}{dx}\) when \(y = xe^{2x}\). [3]
   2. Find an equation of the tangent to the curve \(y = xe^{2x}\) at the point \((1, e^2)\). [2]
2. Given that \(y = \frac{2\sin 3x}{1 + \cos 3x}\), use the quotient rule to show that $$\frac{dy}{dx} = \frac{k}{1 + \cos 3x}$$ where \(k\) is an integer. [4]

The curve \(y = \cos^{-1}(2x - 1)\) intersects the curve \(y = e^x\) at a single point where \(x = \alpha\).

1. Show that \(\alpha\) lies between 0.4 and 0.5. [2]
2. Show that the equation \(\cos^{-1}(2x - 1) = e^x\) can be written as \(x = \frac{1}{2} + \frac{1}{2}\cos(e^x)\). [1]
3. Use the iteration \(x_{n+1} = \frac{1}{2} + \frac{1}{2}\cos(e^{x_n})\) with \(x_1 = 0.4\) to find the values of \(x_2\) and \(x_3\), giving your answers to three decimal places. [2]

1. 1. Solve the equation \(\cosec \theta = -4\) for \(0° < \theta < 360°\), giving your answers to the nearest 0.1°. [2]
   2. Solve the equation $$2\cot^2(2x + 30°) = 2 - 7\cosec(2x + 30°)$$ for \(0° < x < 180°\), giving your answers to the nearest 0.1°. [6]
2. Describe a sequence of two geometrical transformations that maps the graph of \(y = \cosec x\) onto the graph of \(y = \cosec(2x + 30°)\). [4]

The functions f and g are defined with their respective domains by $$f(x) = x^2 \quad \text{for all real values of } x$$ $$g(x) = \frac{1}{2x + 1} \quad \text{for real values of } x, \quad x \neq -0.5$$

1. Explain why f does not have an inverse. [1]
2. The inverse of g is \(g^{-1}\). Find \(g^{-1}(x)\). [3]
3. State the range of \(g^{-1}\). [1]
4. Solve the equation \(fg(x) = g(x)\). [3]

1. Given that \(3\ln x = 4\), find the exact value of \(x\). [1]
2. By forming a quadratic equation in \(\ln x\), solve \(3\ln x + \frac{20}{\ln x} = 19\), giving your answers for \(x\) in an exact form. [5]

1. On separate diagrams:
   1. sketch the curve with equation \(y = |3x + 3|\); [2]
   2. sketch the curve with equation \(y = |x^2 - 1|\). [3]
2. 1. Solve the equation \(|3x + 3| = |x^2 - 1|\). [5]
   2. Hence solve the inequality \(|3x + 3| < |x^2 - 1|\). [2]

Use the substitution \(u = 1 + 2\tan x\) to find $$\int \frac{1}{(1 + 2\tan x)^2 \cos^2 x} \, dx$$ [5]

1. Use integration by parts to find \(\int x\ln x \, dx\). [3]
2. Given that \(y = (\ln x)^2\), find \(\frac{dy}{dx}\). [2]
3. The diagram shows part of the curve with equation \(y = \sqrt{x\ln x}\). \includegraphics{figure_9} The shaded region \(R\) is bounded by the curve \(y = \sqrt{x\ln x}\), the line \(x = e\) and the \(x\)-axis from \(x = 1\) to \(x = e\). Find the volume of the solid generated when the region \(R\) is rotated through 360° about the \(x\)-axis, giving your answer in an exact form. [6]