Edexcel C4 (Core Mathematics 4) 2014 June

A curve \(C\) has the equation $$x^3 + 2xy - x - y^3 - 20 = 0$$

1. [(a)] Find \(\frac{dy}{dx}\) in terms of \(x\) and \(y\). \hfill [5]
2. [(b)] Find an equation of the tangent to \(C\) at the point \((3, -2)\), giving your answer in the form \(ax + by + c = 0\), where \(a\), \(b\) and \(c\) are integers. \hfill [2]

Given that the binomial expansion of \((1 + kx)^{-4}\), \(|kx| < 1\), is $$1 - 6x + Ax^2 + \ldots$$

1. [(a)] find the value of the constant \(k\), \hfill [2]
2. [(b)] find the value of the constant \(A\), giving your answer in its simplest form. \hfill [3]

\includegraphics{figure_1} Figure 1 shows a sketch of part of the curve with equation \(y = \frac{10}{2x + 5\sqrt{x}}\), \(x > 0\) The finite region \(R\), shown shaded in Figure 1, is bounded by the curve, the \(x\)-axis, and the lines with equations \(x = 1\) and \(x = 4\) The table below shows corresponding values of \(x\) and \(y\) for \(y = \frac{10}{2x + 5\sqrt{x}}\)

1. [(a)] Complete the table above by giving the missing value of \(y\) to 5 decimal places. \hfill [1]
2. [(b)] Use the trapezium rule, with all the values of \(y\) in the completed table, to find an estimate for the area of \(R\), giving your answer to 4 decimal places. \hfill [3]
3. [(c)] By reference to the curve in Figure 1, state, giving a reason, whether your estimate in part (b) is an overestimate or an underestimate for the area of \(R\). \hfill [1]
4. [(d)] Use the substitution \(u = \sqrt{x}\), or otherwise, to find the exact value of $$\int_1^4 \frac{10}{2x + 5\sqrt{x}} dx$$ \hfill [6]

\includegraphics{figure_2} A vase with a circular cross-section is shown in Figure 2. Water is flowing into the vase. When the depth of the water is \(h\) cm, the volume of water \(V\) cm\(^3\) is given by $$V = 4\pi h(h + 4), \quad 0 \leq h \leq 25$$ Water flows into the vase at a constant rate of \(80\pi\) cm\(^3\)s\(^{-1}\) Find the rate of change of the depth of the water, in cm s\(^{-1}\), when \(h = 6\) \hfill [5]

\includegraphics{figure_3} Figure 3 shows a sketch of the curve \(C\) with parametric equations $$x = 4\cos\left(t + \frac{\pi}{6}\right), \quad y = 2\sin t, \quad 0 \leq t < 2\pi$$

1. [(a)] Show that $$x + y = 2\sqrt{3}\cos t$$ \hfill [3]
2. [(b)] Show that a cartesian equation of \(C\) is $$(x + y)^2 + ay^2 = b$$ where \(a\) and \(b\) are integers to be determined. \hfill [2]

1. [(i)] Find $$\int xe^{4x} dx$$ \hfill [3]
2. [(ii)] Find $$\int \frac{8}{(2x - 1)^3} dx, \quad x > \frac{1}{2}$$ \hfill [2]
3. [(iii)] Given that \(y = \frac{\pi}{6}\) at \(x = 0\), solve the differential equation $$\frac{dy}{dx} = e^x \cosec 2y \cosec y$$ \hfill [7] \end{enumerate}

\includegraphics{figure_4} Figure 4 shows a sketch of part of the curve \(C\) with parametric equations $$x = 3\tan\theta, \quad y = 4\cos^2\theta, \quad 0 \leq \theta < \frac{\pi}{2}$$ The point \(P\) lies on \(C\) and has coordinates \((3, 2)\). The line \(l\) is the normal to \(C\) at \(P\). The normal cuts the \(x\)-axis at the point \(Q\).

1. [(a)] Find the \(x\) coordinate of the point \(Q\). \hfill [6]

The finite region \(S\), shown shaded in Figure 4, is bounded by the curve \(C\), the \(x\)-axis, the \(y\)-axis and the line \(l\). This shaded region is rotated \(2\pi\) radians about the \(x\)-axis to form a solid of revolution.

1. [(b)] Find the exact value of the volume of the solid of revolution, giving your answer in the form \(p\pi + q\pi^2\), where \(p\) and \(q\) are rational numbers to be determined. [You may use the formula \(V = \frac{1}{3}\pi r^2 h\) for the volume of a cone.] \hfill [9] \end{enumerate} \end{enumerate}

Relative to a fixed origin \(O\), the point \(A\) has position vector \(\begin{pmatrix} -2 \\ 4 \\ 7 \end{pmatrix}\) and the point \(B\) has position vector \(\begin{pmatrix} -1 \\ 3 \\ 8 \end{pmatrix}\) The line \(l_1\) passes through the points \(A\) and \(B\).

1. [(a)] Find the vector \(\overrightarrow{AB}\). \hfill [2]
2. [(b)] Hence find a vector equation for the line \(l_1\) \hfill [1]

The point \(P\) has position vector \(\begin{pmatrix} 0 \\ 2 \\ 3 \end{pmatrix}\) Given that angle \(PBA\) is \(\theta\),

1. [(c)] show that \(\cos\theta = \frac{1}{3}\) \hfill [3]

The line \(l_2\) passes through the point \(P\) and is parallel to the line \(l_1\)

1. [(d)] Find a vector equation for the line \(l_2\) \hfill [2]

The points \(C\) and \(D\) both lie on the line \(l_2\) Given that \(AB = PC = DP\) and the \(x\) coordinate of \(C\) is positive,

1. [(e)] find the coordinates of \(C\) and the coordinates of \(D\). \hfill [3]
2. [(f)] find the exact area of the trapezium \(ABCD\), giving your answer as a simplified surd. \hfill [4] \end{enumerate} \end{enumerate} \end{enumerate} \end{enumerate}