SPS SPS SM (SPS SM) 2025 February

Given that \((x - 2)\) is a factor of \(2x^3 + kx - 4\), find the value of the constant \(k\). [2]

1. \includegraphics{figure_2} The diagram shows a model for the roof of a toy building. The roof is in the form of a solid triangular prism \(ABCDEF\). The base \(ACFD\) of the roof is a horizontal rectangle, and the cross-section \(ABC\) of the roof is an isosceles triangle with \(AB = BC\). The lengths of \(AC\) and \(CF\) are \(2x\) cm and \(y\) cm respectively, and the height of \(BE\) above the base of the roof is \(x\) cm. The total surface area of the five faces of the roof is \(600\) cm\(^2\) and the volume of the roof is \(V\) cm\(^3\). Show that \(V = kx (300 - x^2)\), where \(k = \sqrt{a + b}\) and \(a\) and \(b\) are integers to be determined. [6]
2. Use differentiation to determine the value of \(x\) for which the volume of the roof is a maximum. [4]
3. Find the maximum volume of the roof. Give your answer in cm\(^3\), correct to the nearest integer. [1]
4. Explain why, for this roof, \(x\) must be less than a certain value, which you should state. [2]

\includegraphics{figure_3} The diagram shows a sector \(AOB\) of a circle with centre \(O\). The length of the arc \(AB\) is \(6\) cm and the area of the sector \(AOB\) is \(24\) cm\(^2\). Find the area of the shaded segment enclosed by the arc \(AB\) and the chord \(AB\), giving your answer correct to \(3\) significant figures. [6]

1. The number \(K\) is defined by \(K = n^3 + 1\), where \(n\) is an integer greater than \(2\). Given that \(n^3 + 1 = (n + 1) (n^2 + bn + c)\), find the constants \(b\) and \(c\). [1]
2. Prove that \(K\) has at least two distinct factors other than \(1\) and \(K\). [5]

A curve has the following properties: • The gradient of the curve is given by \(\frac{dy}{dx} = -2x\). • The curve passes through the point \((4, -13)\). Determine the coordinates of the points where the curve meets the line \(y = 2x\). [7]

For all real values of \(x\), the functions \(f\) and \(g\) are defined by \(f (x) = x^2 + 8ax + 4a^2\) and \(g(x) = 6x - 2a\), where \(a\) is a positive constant.

1. Find \(fg(x)\). Determine the range of \(fg(x)\) in terms of \(a\). [4]
2. If \(fg(2) = 144\), find the value of \(a\). [3]
3. Determine whether the function \(fg\) has an inverse. [2]

1. Show that the equation $$2 \sin x \tan x = \cos x + 5$$ can be expressed in the form $$3 \cos^2 x + 5 \cos x - 2 = 0.$$ [3]
2. Hence solve the equation $$2 \sin 2\theta \tan 2\theta = \cos 2\theta + 5,$$ giving all values of \(\theta\) between \(0°\) and \(180°\), correct to \(1\) decimal place. [5]

\includegraphics{figure_8} The diagram shows the curve with equation \(y = 5x^4 + ax^3 + bx\), where \(a\) and \(b\) are integers. The curve has a minimum at the point \(P\) where \(x = 2\). The shaded region is enclosed by the curve, the \(x\)-axis and the line \(x = 2\). Given that the area of the shaded region is \(48\) units\(^2\), determine the \(y\)-coordinate of \(P\). [7]