Edexcel C4 (Core Mathematics 4)

The curve \(C\) has equation \(5x^2 + 2xy - 3y^2 + 3 = 0\). The point \(P\) on the curve \(C\) has coordinates \((1, 2)\).

1. Find the gradient of the curve at \(P\). [5]
2. Find the equation of the normal to the curve \(C\) at \(P\), in the form \(y = ax + b\), where \(a\) and \(b\) are constants. [3]

\includegraphics{figure_1} In Fig. 1, the curve \(C\) has equation \(y = f(x)\), where $$f(x) = x + \frac{2}{x^2}, \quad x > 0.$$ The shaded region is bounded by \(C\), the \(x\)-axis and the lines with equations \(x = 1\) and \(x = 2\). The shaded region is rotated through \(2\pi\) radians about the \(x\)-axis. Using calculus, calculate the volume of the solid generated. Give your answer in the form \(\pi(a + \ln b)\), where \(a\) and \(b\) are constants. [8]

\includegraphics{figure_2} Figure 2 shows part of the curve with equation $$y = e^x \cos x, \quad 0 \leq x \leq \frac{\pi}{2}.$$ The finite region \(R\) is bounded by the curve and the coordinate axes.

1. Calculate, to 2 decimal places, the \(y\)-coordinates of the points on the curve where \(x = 0\), \(\frac{\pi}{6}\), \(\frac{\pi}{3}\) and \(\frac{\pi}{2}\). [3]
2. Using the trapezium rule and all the values calculated in part (a), find an approximation for the area of \(R\). [4]
3. State, with a reason, whether your approximation underestimates or overestimates the area of \(R\). [2]

A curve is given parametrically by the equations $$x = 5 \cos t, \quad y = -2 + 4 \sin t, \quad 0 \leq t < 2\pi.$$

1. Find the coordinates of all the points at which \(C\) intersects the coordinate axes, giving your answers in surd form where appropriate. [4]
2. Sketch the graph at \(C\). [2]

\(P\) is the point on \(C\) where \(t = \frac{1}{6}\pi\).

1. Show that the normal to \(C\) at \(P\) has equation $$8\sqrt{3}y = 10x - 25\sqrt{3}.$$ [4]

\includegraphics{figure_1} The curve \(C\) has equation \(y = f(x)\), \(x \in \mathbb{R}\). Figure 1 shows the part of \(C\) for which \(0 \leq x \leq 2\). Given that $$\frac{dy}{dx} = e^x - 2x^2,$$ and that \(C\) has a single maximum, at \(x = k\),

1. show that \(1.48 < k < 1.49\). [3]

Given also that the point \((0, 5)\) lies on \(C\),

1. find \(f(x)\). [4]

The finite region \(R\) is bounded by \(C\), the coordinate axes and the line \(x = 2\).

1. Use integration to find the exact area of \(R\). [4]

When \((1 + ax)^n\) is expanded as a series in ascending powers of \(x\), the coefficients of \(x\) and \(x^2\) are \(-6\) and \(27\) respectively.

1. Find the value of \(a\) and the value of \(n\). [5]
2. Find the coefficient of \(x^3\). [2]
3. State the set of values of \(x\) for which the expansion is valid. [1]

Two submarines are travelling in straight lines through the ocean. Relative to a fixed origin, the vector equations of the two lines, \(l_1\) and \(l_2\), along which they travel are \begin{align} \mathbf{r} &= 3\mathbf{i} + 4\mathbf{j} - 5\mathbf{k} + \lambda(\mathbf{i} - 2\mathbf{j} + 2\mathbf{k})
\text{and} \quad \mathbf{r} &= 9\mathbf{i} + \mathbf{j} - 2\mathbf{k} + \mu (4\mathbf{i} + \mathbf{j} - \mathbf{k}), \end{align} where \(\lambda\) and \(\mu\) are scalars.

1. Show that the submarines are moving in perpendicular directions. [2]
2. Given that \(l_1\) and \(l_2\) intersect at the point \(A\), find the position vector of \(A\). [5]

The point \(B\) has position vector \(10\mathbf{j} - 11\mathbf{k}\).

1. Show that only one of the submarines passes through the point \(B\). [3]
2. Given that 1 unit on each coordinate axis represents 100 m, find, in km, the distance \(AB\). [2]

In a chemical reaction two substances combine to form a third substance. At time \(t\), \(t \geq 0\), the concentration of this third substance is \(x\) and the reaction is modelled by the differential equation $$\frac{dx}{dt} = k(1 - 2x)(1 - 4x), \text{ where } k \text{ is a positive constant.}$$

1. Solve this differential equation and hence show that $$\ln \left| \frac{1 - 2x}{1 - 4x} \right| = 2kt + c, \text{ where } c \text{ is an arbitrary constant.}$$ [7]
2. Given that \(x = 0\) when \(t = 0\), find an expression for \(x\) in terms of \(k\) and \(t\). [4]
3. Find the limiting value of the concentration \(x\) as \(t\) becomes very large. [2]