OCR C4 (Core Mathematics 4)

1. Show that

$$\int _ { 2 } ^ { 4 } x \left( x ^ { 2 } - 4 \right) ^ { \frac { 1 } { 2 } } \mathrm {~d} x = 8 \sqrt { 3 }$$

1. (i) Simplify

$$\frac { 2 x ^ { 2 } + 3 x - 9 } { 2 x ^ { 2 } - 7 x + 6 }$$ (ii) Find the quotient and remainder when ( \(2 x ^ { 4 } - 1\) ) is divided by ( \(x ^ { 2 } - 2\) ).

3. A curve has the equation $$2 \sin 2 x - \tan y = 0$$

1. Find an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in its simplest form in terms of \(x\) and \(y\).
2. Show that the tangent to the curve at the point \(\left( \frac { \pi } { 6 } , \frac { \pi } { 3 } \right)\) has the equation $$y = \frac { 1 } { 2 } x + \frac { \pi } { 4 } .$$

1. The gradient at any point \(( x , y )\) on a curve is proportional to \(\sqrt { y }\).

Given that the curve passes through the point with coordinates \(( 0,4 )\),

1. show that the equation of the curve can be written in the form $$2 \sqrt { y } = k x + 4$$ where \(k\) is a positive constant. Given also that the curve passes through the point with coordinates ( 2,9 ),
2. find the equation of the curve in the form \(y = \mathrm { f } ( x )\).

5.
\includegraphics[max width=\textwidth, alt={}, center]{00ad2596-cd76-425d-a373-a0deda11e3c0-2_444_702_246_516} The diagram shows the curve with parametric equations $$x = 2 - t ^ { 2 } , \quad y = t ( t + 1 ) , \quad t \geq 0$$

1. Find the coordinates of the points where the curve meets the coordinate axes.
2. Find an equation for the tangent to the curve at the point where \(t = 2\), giving your answer in the form \(a x + b y + c = 0\).

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

1. Find the values of the constants \(A\) and \(B\) such that $$\mathrm { f } ( x ) = \frac { A } { 1 - x } + \frac { B } { 1 - 3 x }$$
2. Evaluate $$\int _ { 0 } ^ { \frac { 1 } { 4 } } f ( x ) d x$$ giving your answer as a single logarithm.
3. 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.

7. Relative to a fixed origin, two lines have the equations
and $$\begin{aligned} & \mathbf { r } = \left( \begin{array} { c } 4
1
1 \end{array} \right) + s \left( \begin{array} { l } 1
4
5 \end{array} \right)
& \mathbf { r } = \left( \begin{array} { c } - 3
1
- 6 \end{array} \right) + t \left( \begin{array} { l } 3
a
b \end{array} \right) , \end{aligned}$$ where \(a\) and \(b\) are constants and \(s\) and \(t\) are scalar parameters.
Given that the two lines are perpendicular,

1. find a linear relationship between \(a\) and \(b\). Given also that the two lines intersect,
2. find the values of \(a\) and \(b\),
3. find the coordinates of the point where they intersect.

8. (i) Find $$\int x ^ { 2 } \mathrm { e } ^ { \frac { 1 } { 2 } x } \mathrm {~d} x$$ (ii) Using the substitution \(u = \sin t\), evaluate $$\int _ { 0 } ^ { \frac { \pi } { 2 } } \sin ^ { 2 } 2 t \cos t \mathrm {~d} t$$