SPS SPS SM (SPS SM) 2021 January

1. $$f ( x ) = 6 x + 9 \sqrt { x } - \frac { 4 } { x ^ { 2 } } , x > 0 .$$ Find a fully simplified expression for $$\int f ( x ) d x$$

2. The figure above shows the graph of a curve with equation \(y = f ( x )\). The curve meets the \(x\) axis at \(( - 3,0 )\) and the \(y\) axis at \(( 0,2 )\). The curve has a maximum at \(( 3,4 )\) and a minimum at \(( - 3,0 )\). The line with equation \(y = 2\) is a horizontal asymptote to the curve. Sketch on separate diagrams the graph of ...
a) \(\ldots \quad y = f ( x + 3 )\).
b) \(. . \quad y = f ( x ) - 2\).
c) \(\ldots \quad y = \frac { 1 } { 2 } f ( x )\). Each of the sketches must include

• the coordinates of any points where the graph meets the coordinate axes.
• the coordinates of any minimum or maximum points of the curve.
• any asymptotes to the curve, clearly labelled.

3. a) Solve the linear inequality $$6 - 2 ( x + 2 ) < 10 .$$ b) Solve the quadratic inequality $$( x + 1 ) ^ { 2 } \geq 4 x + 9$$ c) Hence determine the range of values of \(x\) that satisfy both the inequalities of part (a) and part (b).

4. \includegraphics[max width=\textwidth, alt={}, center]{fbe229f8-d390-487d-87fc-90edc50c3325-3_554_988_1153_486} The figure above shows the graph of the curve with equation \(y = f ( x )\). The curve crosses the \(x\) axis at the points \(( - 4,0 ) , ( 2,0 )\) and \(( 4,0 )\), and the \(y\) axis at the point \(( 0,16 )\). Determine the equation of \(f ( x )\) in the form $$f ( x ) \equiv a x ^ { 3 } + b x ^ { 2 } + c x + d$$ where \(a , b , c\) and \(d\) are constants.

5. A curve \(C\) and a straight line \(L\) have respective equations $$y = x ^ { 2 } - 4 x - 5 \text { and } y = 2 x - 14$$ a) Find the coordinates of any points of intersection between \(C\) and \(L\).
b) Sketch in the same diagram the graph of \(C\) and the graph of \(L\). The sketch must include of any points of intersection between the graph of \(C\) and the coordinate axes, and any points of intersection between the graph of \(L\) and the coordinate axes.

6. Solve the following trigonometric equation in the range given. $$\tan ( 5 y - 35 ) ^ { \circ } = - 2 - \sqrt { 3 } , \quad 0 \leq y < 90$$

7. A circle whose centre is at \(( 3 , - 5 )\) has equation $$x ^ { 2 } + y ^ { 2 } - 6 x + a y = 15$$ where \(a\) is a constant.
a) Find the value of \(a\).
b) Determine the radius of the circle.

8. Solve each of the following equations, giving the final answers correct to three significant figures, where appropriate.
a) \(\quad 7 ^ { x } = 10\).
b) \(\quad \log _ { 2 } y = \frac { 9 } { \log _ { 2 } y }\).

9. The total cost \(C\), in \(\pounds\), for a certain car journey, is modelled by $$C = \frac { 200 } { V } + \frac { 2 V } { 25 } , V > 30 ,$$ where \(V\) is the average speed in miles per hour.
a) Find the value of \(V\) for which \(C\) is stationary.