OCR C4 (Core Mathematics 4)

2. (i) Find the binomial expansion of \(( 2 - 3 x ) ^ { - 3 }\) in ascending powers of \(x\) up to and including the term in \(x ^ { 3 }\), simplifying each coefficient.
(ii) State the set of values of \(x\) for which your expansion is valid.

3. (i) Express \(\frac { x + 11 } { ( x + 4 ) ( x - 3 ) }\) as a sum of partial fractions.
(ii) Evaluate $$\int _ { 0 } ^ { 2 } \frac { x + 11 } { ( x + 4 ) ( x - 3 ) } d x$$ giving your answer in the form \(\ln k\), where \(k\) is an exact simplified fraction.

4. A curve has the equation $$4 x ^ { 2 } - 2 x y - y ^ { 2 } + 11 = 0$$ Find an equation for the normal to the curve at the point with coordinates \(( - 1 , - 3 )\).

5.
\includegraphics[max width=\textwidth, alt={}, center]{5840974b-b08a-4818-9a59-97b2d3ce9890-1_469_809_1777_484} The diagram shows the curve with equation \(y = x \sqrt { 1 - x } , 0 \leq x \leq 1\).
Use the substitution \(u ^ { 2 } = 1 - x\) to show that the area of the region bounded by the curve and the \(x\)-axis is \(\frac { 4 } { 15 }\).

6. The number of people, \(n\), in a queue at a Post Office \(t\) minutes after it opens is modelled by the differential equation $$\frac { \mathrm { d } n } { \mathrm {~d} t } = \mathrm { e } ^ { 0.5 t } - 5 , \quad t \geq 0$$

1. Find, to the nearest second, the time when the model predicts that there will be the least number of people in the queue.
2. Given that there are 20 people in the queue when the Post Office opens, solve the differential equation.
3. Explain why this model would not be appropriate for large values of \(t\).

7. (i) Show that ( \(2 x + 3\) ) is a factor of ( \(\left. 2 x ^ { 3 } - x ^ { 2 } + 4 x + 15 \right)\) and hence, simplify $$\frac { 2 x ^ { 2 } + x - 3 } { 2 x ^ { 3 } - x ^ { 2 } + 4 x + 15 } .$$ (ii) Show that $$\int _ { 2 } ^ { 5 } \frac { 2 x ^ { 2 } + x - 3 } { 2 x ^ { 3 } - x ^ { 2 } + 4 x + 15 } \mathrm {~d} x = \ln k$$ where \(k\) is an integer.

8. The points \(A\) and \(B\) have coordinates \(( 3,9 , - 7 )\) and \(( 13 , - 6 , - 2 )\) respectively.

1. Find, in vector form, an equation for the line \(l\) which passes through \(A\) and \(B\).
2. Show that the point \(C\) with coordinates \(( 9,0 , - 4 )\) lies on \(l\). The point \(D\) is the point on \(l\) closest to the origin, \(O\).
3. Find the coordinates of \(D\).
4. Find the area of triangle \(O A B\) to 3 significant figures.

9. A curve has parametric equations $$x = \sec \theta + \tan \theta , \quad y = \operatorname { cosec } \theta + \cot \theta , \quad 0 < \theta < \frac { \pi } { 2 }$$

1. Show that \(x + \frac { 1 } { x } = 2 \sec \theta\). Given that \(y + \frac { 1 } { y } = 2 \operatorname { cosec } \theta\),
2. find a cartesian equation for the curve.
3. Show that \(\frac { \mathrm { d } x } { \mathrm {~d} \theta } = \frac { 1 } { 2 } \left( x ^ { 2 } + 1 \right)\).
4. Find an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\) and \(y\).