OCR C2 (Core Mathematics 2)

1. \(f ( x ) = 3 x ^ { 3 } - 2 x ^ { 2 } + k x + 9\).

Given that when \(\mathrm { f } ( x )\) is divided by \(( x + 2 )\) there is a remainder of - 35 ,

1. find the value of the constant \(k\),
2. find the remainder when \(\mathrm { f } ( x )\) is divided by \(( 3 x - 2 )\).

2.
\includegraphics[max width=\textwidth, alt={}, center]{e4afa57d-5be3-42a6-ab35-39b0fdcc1681-1_554_848_685_461} The diagram shows the curve with equation \(y = 4 x + \frac { 1 } { x } , x > 0\).
Use the trapezium rule with three intervals, each of width 1 , to estimate the area of the shaded region bounded by the curve, the \(x\)-axis and the lines \(x = 1\) and \(x = 4\).

3. The sides of a triangle have lengths of \(7 \mathrm {~cm} , 8 \mathrm {~cm}\) and 10 cm . Find the area of the triangle correct to 3 significant figures.

4. Find all values of \(x\) in the interval \(0 \leq x < 360 ^ { \circ }\) for which $$2 \sin ^ { 2 } x - 2 \cos x - \cos ^ { 2 } x = 1$$ giving non-exact answers to 1 decimal place.

5. (i) Describe fully a single transformation that maps the graph of \(y = 3 ^ { x }\) onto the graph of \(y = \left( \frac { 1 } { 3 } \right) ^ { x }\).
(ii) Sketch on the same diagram the curves \(y = \left( \frac { 1 } { 3 } \right) ^ { x }\) and \(y = 2 \left( 3 ^ { x } \right)\), showing the coordinates of any points where each curve crosses the coordinate axes. The curves \(y = \left( \frac { 1 } { 3 } \right) ^ { x }\) and \(y = 2 \left( 3 ^ { x } \right)\) intersect at the point \(P\).
(iii) Find the \(x\)-coordinate of \(P\) to 2 decimal places and show that the \(y\)-coordinate of \(P\) is \(\sqrt { 2 }\).

6. Evaluate

1. \(\quad \int _ { 1 } ^ { 4 } \left( x ^ { 2 } - 5 x + 4 \right) \mathrm { d } x\),
2. \(\int _ { - \infty } ^ { - 1 } \frac { 1 } { x ^ { 4 } } \mathrm {~d} x\).

7.
\includegraphics[max width=\textwidth, alt={}, center]{e4afa57d-5be3-42a6-ab35-39b0fdcc1681-2_364_666_1338_568} The diagram 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.

8. Given that for small values of \(x\) $$( 1 + a x ) ^ { n } \approx 1 - 24 x + 270 x ^ { 2 }$$ where \(n\) is an integer and \(n > 1\),

1. show that \(n = 16\) and find the value of \(a\),
2. use your value of \(a\) and a suitable value of \(x\) to estimate the value of (0.9985) \({ } ^ { 16 }\), giving your answer to 5 decimal places.

9.
\includegraphics[max width=\textwidth, alt={}, center]{e4afa57d-5be3-42a6-ab35-39b0fdcc1681-3_559_732_824_388} The diagram shows the curve with equation \(y = \mathrm { f } ( x )\) which crosses the \(x\)-axis at the origin and at the points \(A\) and \(B\). Given that $$f ^ { \prime } ( x ) = 4 - 6 x - 3 x ^ { 2 }$$

1. find an expression for \(y\) in terms of \(x\),
2. show that \(A\) has coordinates ( \(- 4,0\) ) and find the coordinates of \(B\),
3. find the total area of the two regions bounded by the curve and the \(x\)-axis.