Edexcel C4 (Core Mathematics 4) 2013 June

1. Find the binomial expansion of $$\sqrt{(9 + 8x)}, \quad |x| < \frac{9}{8}$$ in ascending powers of \(x\), up to and including the term in \(x^2\). Give each coefficient as a simplified fraction. [5]
2. Use your expansion to estimate the value of \(\sqrt{11}\), giving your answer as a single fraction. [3]

\includegraphics{figure_1} Figure 1 shows a sketch of part of the curve with equation \(y = xe^{-\frac{1}{2}x}\), \(x > 0\). The finite region \(R\), shown shaded in Figure 1, is bounded by the curve, the \(x\)-axis, and the line \(x = 4\). The table shows corresponding values of \(x\) and \(y\) for \(y = xe^{-\frac{1}{2}x}\).

1. Complete the table with the value of \(y\) corresponding to \(x = 2\) [1]
2. Use the trapezium rule, with all the values of \(y\) in the completed table, to obtain an estimate for the area of \(R\), giving your answer to 2 decimal places. [4]
3. 1. Find \(\int xe^{-\frac{1}{2}x} \, dx\).
   2. Hence find the exact area of \(R\), giving your answer in the form \(a + be^{-2}\), where \(a\) and \(b\) are integers. [6]

A curve \(C\) has parametric equations $$x = 2t + 5, \quad y = 3 + \frac{4}{t}, \quad t \neq 0$$

1. Find the value of \(\frac{dy}{dx}\) at the point on \(C\) with coordinates \((9, 5)\). [4]
2. Find a cartesian equation of the curve in the form $$y = \frac{ax + b}{cx + d}$$ where \(a\), \(b\), \(c\) and \(d\) are integers. [3]

With respect to a fixed origin \(O\), the line \(l_1\) has vector equation $$\mathbf{r} = \begin{pmatrix} -9 \\ 8 \\ 5 \end{pmatrix} + \mu \begin{pmatrix} 5 \\ -4 \\ -3 \end{pmatrix}$$ where \(\mu\) is a scalar parameter. The point \(A\) is on \(l_1\) where \(\mu = 2\).

1. Write down the coordinates of \(A\). [1] The acute angle between \(OA\) and \(l_1\) is \(\theta\), where \(O\) is the origin.
2. Find the value of \(\cos \theta\). [3] The point \(B\) is such that \(\overrightarrow{OB} = 3\overrightarrow{OA}\). The line \(l_2\) passes through the point \(B\) and is parallel to the line \(l_1\).
3. Find a vector equation of \(l_2\). [2]
4. Find the length of \(OB\), giving your answer as a simplified surd. [1] The point \(X\) lies on \(l_2\). Given that the vector \(\overrightarrow{OX}\) is perpendicular to \(l_2\),
5. find the length of \(OX\), giving your answer to 3 significant figures. [3]

The curve \(C\) has the equation $$\sin(\pi y) - y - x^2 y = -5, \quad x > 0$$

1. Find \(\frac{dy}{dx}\) in terms of \(x\) and \(y\). [5] The point \(P\) with coordinates \((2, 1)\) lies on \(C\). The tangent to \(C\) at \(P\) meets the \(x\)-axis at the point \(A\).
2. Find the exact value of the \(x\)-coordinate of \(A\). [4]

1. 1. Express \(\frac{7x}{(x + 3)(2x - 1)}\) in partial fractions. [3]
   2. Given that \(x > \frac{1}{2}\), find $$\int \frac{7x}{(x + 3)(2x - 1)} \, dx$$ [3]
2. Using the substitution \(u^3 = x\), or otherwise, find $$\int \frac{1}{x + x^3} \, dx, \quad x > 0$$ [5]

\includegraphics{figure_2} Figure 2 shows a sketch of part of the curve \(C\) with parametric equations $$x = \tan \theta, \quad y = 1 + 2\cos 2\theta, \quad 0 \leq \theta < \frac{\pi}{2}$$ The curve \(C\) crosses the \(x\)-axis at \((\sqrt{3}, 0)\). The finite shaded region \(S\) shown in Figure 2 is bounded by \(C\), the line \(x = 1\) and the \(x\)-axis. This shaded region is rotated through \(2\pi\) radians about the \(x\)-axis to form a solid of revolution.

1. Show that the volume of the solid of revolution formed is given by the integral $$k \int_{\frac{\pi}{6}}^{\frac{\pi}{4}} (16 \cos^2 \theta - 8 + \sec^2 \theta) \, d\theta$$ where \(k\) is a constant. [5]
2. Hence, use integration to find the exact value for this volume. [5]

\includegraphics{figure_3} Figure 3 shows a large vertical cylindrical tank containing a liquid. The radius of the circular cross-section of the tank is 40 cm. At time \(t\) minutes, the depth of liquid in the tank is \(h\) centimetres. The liquid leaks from a hole \(P\) at the bottom of the tank. The liquid leaks from the tank at a rate of \(32\pi \sqrt{h}\) cm\(^3\) min\(^{-1}\).

1. Show that at time \(t\) minutes, the height \(h\) cm of liquid in the tank satisfies the differential equation $$\frac{dh}{dt} = -0.02\sqrt{h}$$ [4]
2. Find the time taken, to the nearest minute, for the depth of liquid in the tank to decrease from 100 cm to 50 cm. [5]