OCR C4 (Core Mathematics 4)

1. $$f ( x ) = 1 + \frac { 4 x } { 2 x - 5 } - \frac { 15 } { 2 x ^ { 2 } - 7 x + 5 }$$ Show that $$f ( x ) = \frac { 3 x + 2 } { x - 1 }$$

1. A curve has the equation

$$x ^ { 2 } - 3 x y - y ^ { 2 } = 12$$

1. Find an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\) and \(y\).
2. Find an equation for the tangent to the curve at the point \(( 2 , - 2 )\).

3. Find

1. \(\int \frac { x } { 2 - x ^ { 2 } } \mathrm {~d} x\),
2. \(\int x ^ { 2 } \mathrm { e } ^ { - x } \mathrm {~d} x\).

4.
\includegraphics[max width=\textwidth, alt={}, center]{c7b867af-0730-459e-9c76-15eb07b9e476-1_465_976_1539_388} The diagram shows the curve with parametric equations $$x = \tan \theta , \quad y = \cos ^ { 2 } \theta , \quad - \frac { \pi } { 2 } < \theta < \frac { \pi } { 2 }$$

1. Find a cartesian equation for the curve. The shaded region is bounded by the curve, the \(x\)-axis and the lines \(x = - 1\) and \(x = 1\).
2. Using integration, with the substitution \(x = \tan u\), find the area of the shaded region.

5. (i) Expand \(( 4 - x ) ^ { \frac { 1 } { 2 } }\) in ascending powers of \(x\) up to and including the term in \(x ^ { 2 }\), simplifying each coefficient.
(ii) State the set of values of \(x\) for which your expansion is valid.
(iii) Use your expansion with \(x = 0.01\) to find the value of \(\sqrt { 399 }\), giving your answer to 9 significant figures.

6. (i) Use the derivative of \(\cos x\) to prove that $$\frac { \mathrm { d } } { \mathrm {~d} x } ( \sec x ) = \sec x \tan x$$ The curve \(C\) has the equation \(y = \mathrm { e } ^ { 2 x } \sec x , - \frac { \pi } { 2 } < x < \frac { \pi } { 2 }\).
(ii) Find an equation for the tangent to \(C\) at the point where it crosses the \(y\)-axis.
(iii) Find, to 2 decimal places, the \(x\)-coordinate of the stationary point of \(C\).

7. The line \(l _ { 1 }\) passes through the points \(A\) and \(B\) with position vectors ( \(3 \mathbf { i } + 6 \mathbf { j } - 8 \mathbf { k }\) ) and ( \(8 \mathbf { j } - 6 \mathbf { k }\) ) respectively, relative to a fixed origin.

1. Find a vector equation for \(l _ { 1 }\). The line \(l _ { 2 }\) has vector equation $$\mathbf { r } = ( - 2 \mathbf { i } + 10 \mathbf { j } + 6 \mathbf { k } ) + \mu ( 7 \mathbf { i } - 4 \mathbf { j } + 6 \mathbf { k } ) ,$$ where \(\mu\) is a scalar parameter.
2. Show that lines \(l _ { 1 }\) and \(l _ { 2 }\) intersect.
3. Find the coordinates of the point where \(l _ { 1 }\) and \(l _ { 2 }\) intersect. The point \(C\) lies on \(l _ { 2 }\) and is such that \(A C\) is perpendicular to \(A B\).
4. Find the position vector of \(C\).

8. When a plague of locusts attacks a wheat crop, the proportion of the crop destroyed after \(t\) hours is denoted by \(x\). In a model, it is assumed that the rate at which the crop is destroyed is proportional to \(x ( 1 - x )\). A plague of locusts is discovered in a wheat crop when one quarter of the crop has been destroyed. Given that the rate of destruction at this instant is such that if it remained constant, the crop would be completely destroyed in a further six hours,

1. show that \(\frac { \mathrm { d } x } { \mathrm {~d} t } = \frac { 2 } { 3 } x ( 1 - x )\),
2. find the percentage of the crop destroyed three hours after the plague of locusts is first discovered.