OCR C4 (Core Mathematics 4)

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

1. \(\ln(\cos x)\) [2]
2. \(x^2 \sin 3x\) [2]

A curve has the equation $$x^2 + 3xy - 2y^2 + 17 = 0.$$

1. Find an expression for \(\frac{dy}{dx}\) in terms of \(x\) and \(y\). [4]
2. Find an equation for the normal to the curve at the point \((3, -2)\). [3]

$$f(x) = 3 - \frac{x-1}{x-3} + \frac{x+11}{2x^2-5x-3}, \quad |x| < \frac{1}{2}.$$

1. Show that $$f(x) = \frac{4x-1}{2x+1}.$$ [4]
2. Find the series expansion of \(f(x)\) in ascending powers of \(x\) up to and including the term in \(x^3\), simplifying each coefficient. [5]

A curve has parametric equations $$x = t^3 + 1, \quad y = \frac{2}{t}, \quad t \neq 0.$$

1. Find an equation for the normal to the curve at the point where \(t = 1\), giving your answer in the form \(y = mx + c\). [6]
2. Find a cartesian equation for the curve in the form \(y = f(x)\). [3]

$$f(x) = \frac{15-17x}{(2+x)(1-3x)^2}, \quad x \neq -2, \quad x \neq \frac{1}{3}.$$

1. Find the values of the constants \(A\), \(B\) and \(C\) such that $$f(x) = \frac{A}{2+x} + \frac{B}{1-3x} + \frac{C}{(1-3x)^2}.$$ [5]
2. Find the value of $$\int_{-1}^{0} f(x) \, dx,$$ giving your answer in the form \(p + \ln q\), where \(p\) and \(q\) are integers. [5]

Relative to a fixed origin, \(O\), the line \(l\) has the equation $$\mathbf{r} = \begin{pmatrix} 1 \\ p \\ -5 \end{pmatrix} + \lambda \begin{pmatrix} 3 \\ -1 \\ q \end{pmatrix},$$ where \(p\) and \(q\) are constants and \(\lambda\) is a scalar parameter. Given that the point \(A\) with coordinates \((-5, 9, -9)\) lies on \(l\),

1. find the values of \(p\) and \(q\), [3]
2. show that the point \(B\) with coordinates \((25, -1, 11)\) also lies on \(l\). [2]

The point \(C\) lies on \(l\) and is such that \(OC\) is perpendicular to \(l\).

1. Find the coordinates of \(C\). [3]
2. Find the ratio \(AC : CB\) [2]

1. Use the substitution \(x = 2 \sin u\) to evaluate $$\int_0^{\sqrt{3}} \frac{1}{\sqrt{4-x^2}} \, dx.$$ [6]
2. Evaluate $$\int_0^{\frac{\pi}{2}} x \cos x \, dx.$$ [5]

The rate of increase in the number of bacteria in a culture, \(N\), at time \(t\) hours is proportional to \(N\).

1. Write down a differential equation connecting \(N\) and \(t\). [1]

Given that initially there are \(N_0\) bacteria present in a culture,

1. Show that \(N = N_0 e^{kt}\), where \(k\) is a positive constant. [6]

Given also that the number of bacteria present doubles every six hours,

1. find the value of \(k\), [3]
2. Find how long it takes for the number of bacteria to increase by a factor of ten, giving your answer to the nearest minute. [2]