SPS SPS SM (SPS SM) 2022 February

1.

1. Evaluate \(27 ^ { - \frac { 2 } { 3 } }\).
2. Express \(5 \sqrt { 5 }\) in the form \(5 ^ { n }\).
3. Express \(\frac { 1 - \sqrt { 5 } } { 3 + \sqrt { 5 } }\) in the form \(a + b \sqrt { 5 }\).

2.

1. Solve the equation \(x ^ { 4 } - 10 x ^ { 2 } + 25 = 0\).
2. Given that \(y = \frac { 2 } { 5 } x ^ { 5 } - \frac { 20 } { 3 } x ^ { 3 } + 50 x + 3\), find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
3. Hence find the number of stationary points on the curve \(y = \frac { 2 } { 5 } x ^ { 5 } - \frac { 20 } { 3 } x ^ { 3 } + 50 x + 3\).

3. Solve each of the following equations, for \(0 ^ { \circ } \leqslant x \leqslant 180 ^ { \circ }\).

1. \(2 \sin ^ { 2 } x = 1 + \cos x\).
2. \(\sin 2 x = - \cos 2 x\).

4.

1. By expanding the brackets, show that $$( x - 4 ) ( x - 3 ) ( x + 1 ) = x ^ { 3 } - 6 x ^ { 2 } + 5 x + 12 .$$
2. Sketch the curve $$y = x ^ { 3 } - 6 x ^ { 2 } + 5 x + 12$$ giving the coordinates of the points where the curve meets the axes. Label the curve \(C _ { 1 }\).
3. On the same diagram as in part (ii), sketch the curve $$y = - x ^ { 3 } + 6 x ^ { 2 } - 5 x - 12$$ Label this curve \(C _ { 2 }\).

5. The gradient of a curve is given by \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 2 x ^ { - \frac { 1 } { 2 } }\), and the curve passes through the point \(( 4,5 )\). Find the equation of the curve. \section*{6.} The diagram shows the curve \(y = 4 - x ^ { 2 }\) and the line \(y = x + 2\).
\includegraphics[max width=\textwidth, alt={}, center]{6831b037-2ec7-499a-8bf6-62728a5681b7-2_408_435_27_1560}

1. Find the \(x\)-coordinates of the points of intersection of the curve and the line.
2. Use integration to find the area of the shaded region bounded by the line and the curve.

7. The diagram shows a triangle \(A B C\), and a sector \(A C D\) of a circle with centre \(A\). It is given that \(A B = 11 \mathrm {~cm} , B C = 8 \mathrm {~cm}\), angle \(A B C = 0.8\) radians and angle \(D A C = 1.7\) radians. The shaded segment is bounded by the line \(D C\) and the \(\operatorname { arc } D C\).

1. Show that the length of \(A C\) is 7.90 cm , correct to 3 significant figures.
2. Find the area of the shaded segment.
3. Find the perimeter of the shaded segment.
   \includegraphics[max width=\textwidth, alt={}, center]{6831b037-2ec7-499a-8bf6-62728a5681b7-2_641_1360_1219_310}

8. The diagram shows the graph of \(y = \mathrm { f } ( x )\), where

1. Evaluate \(\mathrm { ff } ( - 3 )\).
2. Find an expression for \(\mathrm { f } ^ { - 1 } ( x )\). $$\mathrm { f } ( x ) = 2 - x ^ { 2 } , \quad x \leqslant 0 .$$
3. Sketch the graph of \(y = \mathrm { f } ^ { - 1 } ( x )\). Indicate the coordinates of the points where the graph meets the axes.
   \includegraphics[max width=\textwidth, alt={}, center]{6831b037-2ec7-499a-8bf6-62728a5681b7-2_476_490_2014_1352}