SPS SPS SM (SPS SM) 2025 February

1. Given that \(( x - 2 )\) is a factor of \(2 x ^ { 3 } + k x - 4\), find the value of the constant \(k\).
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2. (a)
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The diagram shows a model for the roof of a toy building. The roof is in the form of a solid triangular prism \(A B C D E F\). The base \(A C F D\) of the roof is a horizontal rectangle, and the cross-section \(A B C\) of the roof is an isosceles triangle with \(A B = B C\). The lengths of \(A C\) and \(C F\) are \(2 x \mathrm {~cm}\) and \(y \mathrm {~cm}\) respectively, and the height of \(B E\) above the base of the roof is \(x \mathrm {~cm}\). The total surface area of the five faces of the roof is \(600 \mathrm {~cm} ^ { 2 }\) and the volume of the roof is \(V \mathrm {~cm} ^ { 3 }\). Show that \(V = k x \left( 300 - x ^ { 2 } \right)\), where \(k = \sqrt { a } + b\) and \(\alpha\) and \(b\) are integers to be determined.
(b) Use differentiation to determine the value of \(x\) for which the volume of the roof is a maximum.
(c) Find the maximum volume of the roof. Give your answer in \(\mathrm { cm } ^ { 3 }\), correct to the nearest integer.
(d) Explain why, for this roof, \(x\) must be less than a certain value, which you should state.
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3.
\includegraphics[max width=\textwidth, alt={}, center]{9eff9a1d-7d5c-4cee-87c9-8811dad16ffb-08_469_471_130_877} The diagram shows a sector \(A O B\) of a circle with centre \(O\). The length of the \(\operatorname { arc } A B\) is 6 cm and the area of the sector \(A O B\) is \(24 \mathrm {~cm} ^ { 2 }\). Find the area of the shaded segment enclosed by the \(\operatorname { arc } A B\) and the chord \(A B\), giving your answer correct to 3 significant figures.
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4. (a) The number \(K\) is defined by \(K = n ^ { 3 } + 1\), where \(n\) is an integer greater than 2 . Given that \(n ^ { 3 } + 1 \equiv ( n + 1 ) \left( n ^ { 2 } + b n + c \right)\), find the constants \(b\) and \(c\).
(b) Prove that \(K\) has at least two distinct factors other than 1 and \(K\).
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5. A curve has the following properties:

• The gradient of the curve is given by \(\frac { \mathrm { d } y } { \mathrm {~d} x } = - 2 x\).
• The curve passes through the point \(( 4 , - 13 )\).

Determine the coordinates of the points where the curve meets the line \(y = 2 x\).
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6. For all real values of \(x\), the functions f and g are defined by \(\mathrm { f } ( x ) = x ^ { 2 } + 8 a x + 4 a ^ { 2 }\) and \(\mathrm { g } ( x ) = 6 x - 2 a\), where \(a\) is a positive constant.

1. Find \(\mathrm { fg } ( x )\). Determine the range of \(\mathrm { fg } ( x )\) in terms of \(a\).
2. If \(f g ( 2 ) = 144\), find the value of \(a\).
3. Determine whether the function fg has an inverse.
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7. (a) Show that the equation $$2 \sin x \tan x = \cos x + 5$$ can be expressed in the form $$3 \cos ^ { 2 } x + 5 \cos x - 2 = 0$$ (b) Hence solve the equation $$2 \sin 2 \theta \tan 2 \theta = \cos 2 \theta + 5$$ giving all values of \(\theta\) between \(0 ^ { \circ }\) and \(180 ^ { \circ }\), correct to 1 decimal place.
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8.
\includegraphics[max width=\textwidth, alt={}, center]{9eff9a1d-7d5c-4cee-87c9-8811dad16ffb-18_680_942_118_651} The diagram shows the curve with equation \(y = 5 x ^ { 4 } + a x ^ { 3 } + b x\), 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 \(P\).
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