Edexcel C2 (Core Mathematics 2)

1. Evaluate

$$\int _ { 1 } ^ { 4 } \left( x ^ { 2 } - 5 x + 4 \right) d x .$$

2. \begin{figure}[h] \end{figure} Figure 1 shows the curve with equation \(y = \sqrt { 4 x - 1 }\). Use the trapezium rule with five equally-spaced ordinates to estimate the area of the shaded region bounded by the curve, the \(x\)-axis and the lines \(x = 1\) and \(x = 3\).

3. (a) Given that \(y = \log _ { 2 } x\), find expressions in terms of \(y\) for

1. \(\quad \log _ { 2 } \left( \frac { x } { 2 } \right)\),
2. \(\log _ { 2 } ( \sqrt { x } )\).
   (b) Hence, or otherwise, solve the equation $$2 \log _ { 2 } \left( \frac { x } { 2 } \right) + \log _ { 2 } ( \sqrt { x } ) = 8$$

4. $$f ( x ) = 2 - x - x ^ { 3 }$$

1. Show that \(\mathrm { f } ( x )\) is decreasing for all values of \(x\).
2. Verify that the point \(( 1,0 )\) lies on the curve \(y = \mathrm { f } ( x )\).
3. Find the area of the region bounded by the curve \(y = \mathrm { f } ( x )\) and the coordinate axes.

5. Figure 2 Figure 2 shows triangle \(P Q R\) in which \(P Q = 7\) and \(P R = 3 \sqrt { 5 }\).
Given that \(\sin ( \angle Q P R ) = \frac { 2 } { 3 }\) and that \(\angle Q P R\) is acute,

1. find the exact value of \(\cos ( \angle Q P R )\) in its simplest form,
2. show that \(Q R = 2 \sqrt { 6 }\),
3. find \(\angle P Q R\) in degrees to 1 decimal place.

6. The polynomial \(\mathrm { p } ( x )\) is defined by $$\mathrm { p } ( x ) = 2 x ^ { 3 } + x ^ { 2 } + a x + b ,$$ where \(a\) and \(b\) are constants.
Given that when \(\mathrm { p } ( x )\) is divided by \(( x + 2 )\) there is a remainder of 20 ,

1. find an expression for \(b\) in terms of \(a\). Given also that \(( x + 3 )\) is a factor of \(\mathrm { p } ( x )\),
2. find the values of \(a\) and \(b\),
3. fully factorise \(\mathrm { p } ( x )\).

7. (a) Find, to 2 decimal places, the values of \(x\) in the interval \(0 \leq x < 2 \pi\) for which $$\tan \left( x + \frac { \pi } { 4 } \right) = 3 .$$ (b) Find, in terms of \(\pi\), the values of \(y\) in the interval \(0 \leq y < 2 \pi\) for which $$2 \sin y = \tan y .$$

1. The point \(A\) has coordinates ( 4,6 ).

Given that \(O A\), where \(O\) is the origin, is a diameter of circle \(C\),

1. find an equation for \(C\). Circle \(C\) crosses the \(x\)-axis at \(O\) and at the point \(B\).
2. Find the coordinates of \(B\).
3. Find an equation for the tangent to \(C\) at \(B\), giving your answer in the form \(a x + b y = c\), where \(a , b\) and \(c\) are integers.

9. \begin{figure}[h] \end{figure} Figure 3 shows part of a design being produced by a computer program.
The program draws a series of circles with each one touching the previous one and such that their centres lie on a horizontal straight line. The radii of the circles form a geometric sequence with first term 1 mm and second term 1.5 mm . The width of the design is \(w\) as shown.

1. Find the radius of the fourth circle to be drawn.
2. Show that when eight circles have been drawn, \(w = 98.5 \mathrm {~mm}\) to 3 significant figures.
3. Find the total area of the design in square centimetres when ten circles have been drawn.