OCR C4 (Core Mathematics 4)

Find \(\int xe^{3x} dx\). [4]

Find the quotient and remainder when \((x^4 + x^3 - 5x^2 - 9)\) is divided by \((x^2 + x - 6)\). [4]

Differentiate each of the following with respect to \(x\) and simplify your answers.

1. \(\cot x^2\) [2]
2. \(\frac{\sin x}{3 + 2\cos x}\) [4]

1. Expand \((1 - 3x)^{-2}, |x| < \frac{1}{3}\), in ascending powers of \(x\) up to and including the term in \(x^3\), simplifying each coefficient. [4]
2. Hence, or otherwise, show that for small \(x\), $$\left(\frac{2-x}{1-3x}\right)^2 \approx 4 + 20x + 85x^2 + 330x^3.$$ [3]

\includegraphics{figure_5} The diagram shows the curve with parametric equations $$x = a\sqrt{t}, \quad y = at(1-t), \quad t \geq 0,$$ where \(a\) is a positive constant.

1. Find \(\frac{dy}{dx}\) in terms of \(t\). [3]

The curve meets the \(x\)-axis at the origin, \(O\), and at the point \(A\). The tangent to the curve at \(A\) meets the \(y\)-axis at the point \(B\) as shown.

1. Show that the area of triangle \(OAB\) is \(a^2\). [5]

Relative to a fixed origin, two lines have the equations $$\mathbf{r} = (7\mathbf{i} - 4\mathbf{k}) + s(4\mathbf{i} - 3\mathbf{j} + \mathbf{k}),$$ and $$\mathbf{r} = (-7\mathbf{i} + \mathbf{j} + 8\mathbf{k}) + t(-3\mathbf{i} + 2\mathbf{k}),$$ where \(s\) and \(t\) are scalar parameters.

1. Show that the two lines intersect and find the position vector of the point where they meet. [5]
2. Find, in degrees to 1 decimal place, the acute angle between the lines. [4]

At time \(t = 0\), a tank of height 2 metres is completely filled with water. Water then leaks from a hole in the side of the tank such that the depth of water in the tank, \(y\) metres, after \(t\) hours satisfies the differential equation $$\frac{dy}{dt} = -ke^{-0.2t},$$ where \(k\) is a positive constant.

1. Find an expression for \(y\) in terms of \(k\) and \(t\). [4]

Given that two hours after being filled the depth of water in the tank is 1.6 metres,

1. find the value of \(k\) to 4 significant figures. [2]

Given also that the hole in the tank is \(h\) cm above the base of the tank,

1. show that \(h = 79\) to 2 significant figures. [3]

A curve has the equation $$x^2 - 4xy + 2y^2 = 1.$$

1. Find an expression for \(\frac{dy}{dx}\) in its simplest form in terms of \(x\) and \(y\). [4]
2. Show that the tangent to the curve at the point \(P(1, 2)\) has the equation $$3x - 2y + 1 = 0.$$ [3]

The tangent to the curve at the point \(Q\) is parallel to the tangent at \(P\).

1. Find the coordinates of \(Q\). [4]

1. Show that the substitution \(u = \sin x\) transforms the integral $$\int \frac{6}{\cos x(2 - \sin x)} dx$$ into the integral $$\int \frac{6}{(1-u^2)(2-u)} du.$$ [4]
2. Express \(\frac{6}{(1-u^2)(2-u)}\) in partial fractions. [4]
3. Hence, evaluate $$\int_0^{\pi/6} \frac{6}{\cos x(2 - \sin x)} dx,$$ giving your answer in the form \(a \ln 2 + b \ln 3\), where \(a\) and \(b\) are integers. [6]