OCR C4 (Core Mathematics 4)

1. \(\mathrm { f } ( x ) = \frac { x ^ { 4 } + x ^ { 3 } - 13 x ^ { 2 } + 26 x - 17 } { x ^ { 2 } - 3 x + 3 }\).

Find the values of the constants \(A , B , C\) and \(D\) such that $$f ( x ) = x ^ { 2 } + A x + B + \frac { C x + D } { x ^ { 2 } - 3 x + 3 }$$

1. Use the substitution \(u = 1 - x ^ { \frac { 1 } { 2 } }\) to find

$$\int \frac { 1 } { 1 - x ^ { \frac { 1 } { 2 } } } \mathrm {~d} x$$

1. A curve has the equation

$$4 \cos x + 2 \sin y = 3$$

1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 2 \sin x \sec y\).
2. Find an equation for the tangent to the curve at the point ( \(\frac { \pi } { 3 } , \frac { \pi } { 6 }\) ), giving your answer in the form \(a x + b y = c\), where \(a\) and \(b\) are integers.

4. (i) Express \(\frac { 3 x + 6 } { 3 x - x ^ { 2 } }\) in partial fractions.
(ii) Evaluate \(\int _ { 1 } ^ { 2 } \frac { 3 x + 6 } { 3 x - x ^ { 2 } } \mathrm {~d} x\).

5.
\includegraphics[max width=\textwidth, alt={}, center]{825f6c7d-5399-4e7f-bacd-b7c0831aab06-1_408_858_1893_488} The diagram shows the curve with equation \(y = 4 x ^ { \frac { 1 } { 2 } } \mathrm { e } ^ { - x }\).
The shaded region bounded by the curve, the \(x\)-axis and the line \(x = 2\) is rotated through four right angles about the \(x\)-axis. Find, in terms of \(\pi\) and e, the exact volume of the solid formed.

6. $$f ( x ) = \frac { 3 } { \sqrt { 1 - x } } , | x | < 1$$

1. Show that \(\mathrm { f } \left( \frac { 1 } { 10 } \right) = \sqrt { 10 }\).
2. Expand \(\mathrm { f } ( x )\) in ascending powers of \(x\) up to and including the term in \(x ^ { 3 }\), simplifying each coefficient.
3. Use your expansion to find an approximate value for \(\sqrt { 10 }\), giving your answer to 8 significant figures.
4. Find, to 1 significant figure, the percentage error in your answer to part (c).

7. Relative to a fixed origin, two lines have the equations
and $$\begin{aligned} & \mathbf { r } = \left( \begin{array} { c } 7
0
- 3 \end{array} \right) + s \left( \begin{array} { c } 5
4
- 2 \end{array} \right)
& \mathbf { r } = \left( \begin{array} { l } a
6
3 \end{array} \right) + t \left( \begin{array} { c } - 5
14
2 \end{array} \right) , \end{aligned}$$ where \(a\) is a constant and \(s\) and \(t\) are scalar parameters.
Given that the two lines intersect,

1. find the position vector of their point of intersection,
2. find the value of \(a\). Given also that \(\theta\) is the acute angle between the lines,
3. find the value of \(\cos \theta\) in the form \(k \sqrt { 5 }\) where \(k\) is rational.

8. A small town had a population of 9000 in the year 2001. In a model, it is assumed that the population of the town, \(P\), at time \(t\) years after 2001 satisfies the differential equation $$\frac { \mathrm { d } P } { \mathrm {~d} t } = 0.05 P \mathrm { e } ^ { - 0.05 t }$$

1. Show that, according to the model, the population of the town in 2011 will be 13300 to 3 significant figures.
2. Find the value which the population of the town will approach in the long term, according to the model.

14
2 \end{array} \right) , \end{aligned}$$ where \(a\) is a constant and \(s\) and \(t\) are scalar parameters.
Given that the two lines intersect,

1. find the position vector of their point of intersection,
2. find the value of \(a\). Given also that \(\theta\) is the acute angle between the lines,
3. find the value of \(\cos \theta\) in the form \(k \sqrt { 5 }\) where \(k\) is rational.
   8. A small town had a population of 9000 in the year 2001. In a model, it is assumed that the population of the town, \(P\), at time \(t\) years after 2001 satisfies the differential equation $$\frac { \mathrm { d } P } { \mathrm {~d} t } = 0.05 P \mathrm { e } ^ { - 0.05 t }$$
4. Show that, according to the model, the population of the town in 2011 will be 13300 to 3 significant figures.
5. Find the value which the population of the town will approach in the long term, according to the model.
   9. A curve has parametric equations $$x = t ( t - 1 ) , \quad y = \frac { 4 t } { 1 - t } , \quad t \neq 1$$
6. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(t\). The point \(P\) on the curve has parameter \(t = - 1\).
7. Show that the tangent to the curve at \(P\) has the equation $$x + 3 y + 4 = 0$$ The tangent to the curve at \(P\) meets the curve again at the point \(Q\).
8. Find the coordinates of \(Q\).