OCR C4 (Core Mathematics 4)

1. Differentiate each of the following with respect to \(x\) and simplify your answers.
   1. \(\quad \ln ( \cos x )\)
   2. \(x ^ { 2 } \sin 3 x\)
   3. A curve has the equation
   $$x ^ { 2 } + 3 x y - 2 y ^ { 2 } + 17 = 0$$
2. Find an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\) and \(y\).
3. Find an equation for the normal to the curve at the point ( \(3 , - 2\) ).

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

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

4. 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 \(y = m x + c\).
2. Find a cartesian equation for the curve in the form \(y = \mathrm { f } ( x )\).

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

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

6. Relative to a fixed origin, \(O\), the line \(l\) has the equation $$\mathbf { r } = \left( \begin{array} { c } 1
p
- 5 \end{array} \right) + \lambda \left( \begin{array} { c } 3
- 1
q \end{array} \right)$$ 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 \(q\),
2. show that the point \(B\) with coordinates \(( 25 , - 1,11 )\) also lies on \(l\). The point \(C\) lies on \(l\) and is such that \(O C\) is perpendicular to \(l\).
3. Find the coordinates of \(C\).
4. Find the ratio \(A C : C B\)

7. (i) Use the substitution \(x = 2 \sin u\) to evaluate $$\int _ { 0 } ^ { \sqrt { 3 } } \frac { 1 } { \sqrt { 4 - x ^ { 2 } } } \mathrm {~d} x$$ (ii) Evaluate $$\int _ { 0 } ^ { \frac { \pi } { 2 } } x \cos x \mathrm {~d} x$$

1. 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 \(t\).
   Given that initially there are \(N _ { 0 }\) bacteria present in a culture,
2. Show that \(N = N _ { 0 } \mathrm { e } ^ { k t }\), where \(k\) is a positive constant. Given also that the number of bacteria present doubles every six hours,
3. find the value of \(k\),
4. find how long it takes for the number of bacteria to increase by a factor of ten, giving your answer to the nearest minute.