SPS SPS SM (SPS SM) 2022 February

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

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

Solve each of the following equations, for \(0° \leqslant x \leqslant 180°\).

1. \(2\sin^2 x = 1 + \cos x\). [4]
2. \(\sin 2x = -\cos 2x\). [4]

1. By expanding the brackets, show that \((x - 4)(x - 3)(x + 1) = x^3 - 6x^2 + 5x + 12\). [3]
2. Sketch the curve \(y = x^3 - 6x^2 + 5x + 12\), giving the coordinates of the points where the curve meets the axes. Label the curve \(C_1\). [3]
3. On the same diagram as in part (ii), sketch the curve \(y = -x^3 + 6x^2 - 5x - 12\). Label this curve \(C_2\). [2]

The gradient of a curve is given by \(\frac{dy}{dx} = 2x^{-\frac{1}{2}}\), and the curve passes through the point \((4, 5)\). Find the equation of the curve. [6]

The diagram shows the curve \(y = 4 - x^2\) and the line \(y = x + 2\). \includegraphics{figure_6}

1. Find the \(x\)-coordinates of the points of intersection of the curve and the line. [2]
2. Use integration to find the area of the shaded region bounded by the line and the curve. [6]

The diagram shows a triangle \(ABC\), and a sector \(ACD\) of a circle with centre \(A\). It is given that \(AB = 11\) cm, \(BC = 8\) cm, angle \(ABC = 0.8\) radians and angle \(DAC = 1.7\) radians. The shaded segment is bounded by the line \(DC\) and the arc \(DC\). \includegraphics{figure_7}

1. Show that the length of \(AC\) is \(7.90\) cm, correct to 3 significant figures. [3]
2. Find the area of the shaded segment. [3]
3. Find the perimeter of the shaded segment. [4]

The diagram shows the graph of \(y = f(x)\), where \(f(x) = 2 - x^2, \quad x \leqslant 0\). \includegraphics{figure_8}

1. Evaluate \(f(-3)\). [3]
2. Find an expression for \(f^{-1}(x)\). [3]
3. Sketch the graph of \(y = f^{-1}(x)\). Indicate the coordinates of the points where the graph meets the axes. [3]