OCR C4 (Core Mathematics 4) 2007 January

It is given that $$f(x) = \frac{x^2 + 2x - 24}{x^2 - 4x} \quad \text{for } x \neq 0, x \neq 4.$$ Express \(f(x)\) in its simplest form. [3]

Find the exact value of \(\int_1^2 x \ln x \, dx\). [5]

The points \(A\) and \(B\) have position vectors \(\mathbf{a}\) and \(\mathbf{b}\) relative to an origin \(O\), where \(\mathbf{a} = 4\mathbf{i} + 3\mathbf{j} - 2\mathbf{k}\) and \(\mathbf{b} = -7\mathbf{i} + 5\mathbf{j} + 4\mathbf{k}\).

1. Find the length of \(AB\). [3]
2. Use a scalar product to find angle \(OAB\). [3]

Use the substitution \(u = 2x - 5\) to show that \(\int_2^3 (4x - 8)(2x - 5)^7 \, dx = \frac{17}{72}\). [5]

1. Expand \((1 - 3x)^{-\frac{1}{2}}\) in ascending powers of \(x\), up to and including the term in \(x^3\). [4]
2. Hence find the coefficient of \(x^3\) in the expansion of \(\left(1 - 3(x + x^3)\right)^{-\frac{1}{2}}\). [3]

1. Express \(\frac{2x + 1}{(x - 3)^2}\) in the form \(\frac{A}{x - 3} + \frac{B}{(x - 3)^2}\), where \(A\) and \(B\) are constants. [3]
2. Hence find the exact value of \(\int_4^{10} \frac{2x + 1}{(x - 3)^2} \, dx\), giving your answer in the form \(a + b \ln c\), where \(a\), \(b\) and \(c\) are integers. [4]

The equation of a curve is \(2x^2 + xy + y^2 = 14\). Show that there are two stationary points on the curve and find their coordinates. [8]

The parametric equations of a curve are \(x = 2t^2\), \(y = 4t\). Two points on the curve are \(P(2p^2, 4p)\) and \(Q(2q^2, 4q)\).

1. Show that the gradient of the normal to the curve at \(P\) is \(-p\). [2]
2. Show that the gradient of the chord joining the points \(P\) and \(Q\) is \(\frac{2}{p + q}\). [2]
3. The chord \(PQ\) is the normal to the curve at \(P\). Show that \(p^2 + pq + 2 = 0\). [2]
4. The normal at the point \(R(8, 8)\) meets the curve again at \(S\). The normal at \(S\) meets the curve again at \(T\). Find the coordinates of \(T\). [4]

1. Find the general solution of the differential equation $$\frac{\sec^2 y}{\cos^2(2x)} \frac{dy}{dx} = 2.$$ [7]
2. For the particular solution in which \(y = \frac{1}{4}\pi\) when \(x = 0\), find the value of \(y\) when \(x = \frac{1}{8}\pi\). [3]

The position vectors of the points \(P\) and \(Q\) with respect to an origin \(O\) are \(5\mathbf{i} + 2\mathbf{j} - 9\mathbf{k}\) and \(4\mathbf{i} + 4\mathbf{j} - 6\mathbf{k}\) respectively.

1. Find a vector equation for the line \(PQ\). [2]

The position vector of the point \(T\) is \(\mathbf{i} + 2\mathbf{j} - \mathbf{k}\).

1. Write down a vector equation for the line \(OT\) and show that \(OT\) is perpendicular to \(PQ\). [4]

It is given that \(OT\) intersects \(PQ\).

1. Find the position vector of the point of intersection of \(OT\) and \(PQ\). [3]
2. Hence find the perpendicular distance from \(O\) to \(PQ\), giving your answer in an exact form. [2]