SPS SPS SM Pure (SPS SM Pure) 2022 June

1. The expression \((2 + x^2)^3\) can be written in the form $$8 + px^2 + qx^4 + x^6$$ Demonstrate clearly, using the binomial expansion, that \(p = 12\) and find the value of \(q\). [3 marks]
2. Hence find \(\int \frac{(2 + x^2)^3}{x^4} dx\). [3 marks]

The trapezium \(ABCD\) is shown below. \includegraphics{figure_2} The line \(AB\) has equation \(2x + 3y = 14\) and \(DC\) is parallel to \(AB\). The point D has coordinates \((3, 7)\).

1. Find an equation of the line DC [2 marks]
2. The angle BAD is a right angle. Find an equation of the line AD, giving your answer in the form \(mx + ny + p = 0\), where \(m\), \(n\) and \(p\) are integers. [3 marks]

A circle has centre \(C(3, -8)\) and radius \(10\).

1. Express the equation of the circle in the form $$(x - a)^2 + (y - b)^2 = k$$ [2 marks]
2. Find the \(x\)-coordinates of the points where the circle crosses the \(x\)-axis. [3 marks]
3. The line with equation \(y = 2x + 1\) intersects the circle.
   1. Show that the \(x\)-coordinates of the points of intersection satisfy the equation $$x^2 + 6x - 2 = 0$$ [3 marks]
   2. Hence show that the \(x\)-coordinates of the points of intersection are of the form \(m \pm \sqrt{n}\), where \(m\) and \(n\) are integers. [2 marks]

The function \(f\) is defined by $$f(x) = \frac{5x}{7x - 5}$$

1. The domain of \(f\) is the set \(\{x \in \mathbb{R} : x \neq a\}\) State the value of \(a\) [1 mark]
2. Prove that \(f\) is a self-inverse function [3 marks]
3. Find the range of \(f\) [1 mark]

Relative to a fixed origin \(O\),

• the point \(A\) has position vector \(-2\mathbf{i} + 3\mathbf{j}\),
• the point \(B\) has position vector \(3\mathbf{i} + p\mathbf{j}\), where \(p\) is constant,

Given that \(|\overrightarrow{AB}| = 5\sqrt{2}\), find the possible values for \(p\). [3]

A small company which makes batteries for electric cars has a 10 year plan for growth. In year 1 the company will make 2600 batteries. In year 10 the company aims to make 12000 batteries. In order to calculate the number of batteries it will need to make each year from year 2 to year 9, the company considers two models. Model A assumes that the number of batteries it will make each year will increase by the same number each year.

1. According to model A, determine the number of batteries the company will make in year 2. Give your answer to the nearest whole number of batteries. [3]

Model B assumes that the numbers of batteries it will make each year will increase by the same percentage each year.

1. According to model B, determine the number of batteries the company will make in year 2. Give your answer to the nearest 10 batteries. [3]

Sam calculates the total number of batteries made from year 1 to year 10 inclusive, using each of the two models.

1. Calculate the difference between the two totals, giving your answer to the nearest 100 batteries. [3]

\includegraphics{figure_1} Figure 1 shows the plan view of a design for a stage at a concert. The stage is modelled as a sector \(BCDF\), of a circle centre \(F\), joined to two congruent triangles \(ABF\) and \(EDF\). Given that \(AFE\) is a straight line, \(AF = FE = 10.7\) m, \(BF = FD = 9.2\) m and angle \(BFD = 1.82\) radians, find the perimeter of the stage, in metres, to one decimal place. [4]

The function \(f(x)\) is such that \(f(x) = -x^3 + 2x^2 + kx - 10\) The graph of \(y = f(x)\) crosses the \(x\)-axis at the points with coordinates \((a, 0)\), \((2, 0)\) and \((b, 0)\) where \(a < b\)

1. Show that \(k = 5\) [1 mark]
2. Find the exact value of \(a\) and the exact value of \(b\) [3 marks]
3. The functions \(g(x)\) and \(h(x)\) are such that $$g(x) = x^3 + 2x^2 - 5x - 10$$ $$h(x) = -8x^3 + 8x^2 + 10x - 10$$
   1. Explain how the graph of \(y = f(x)\) can be transformed into the graph of \(y = g(x)\) Fully justify your answer. [2 marks]
   2. Explain how the graph of \(y = f(x)\) can be transformed into the graph of \(y = h(x)\) Fully justify your answer. [2 marks]

A geometric series has second term 16 and fourth term 8 All the terms of the series are positive. The \(n\)th term of the series is \(u_n\) Find the exact value of \(\sum_{n=5}^{\infty} u_n\) [5 marks]

1. Sketch the graph of \(y = \cos x\) in the interval \(0 \leq x \leq 2\pi\). State the values of the intercepts with the coordinate axes. [2 marks]
2. 1. Given that $$\sin^2 \theta = \cos \theta(2 - \cos \theta)$$ prove that \(\cos \theta = \frac{1}{2}\). [2 marks]
   2. Hence solve the equation $$\sin^2 2x = \cos 2x(2 - \cos 2x)$$ in the interval \(0 \leq x \leq \pi\) [2 marks]

A sequence is defined by $$u_1 = 600$$ $$u_{n+1} = pu_n + q$$ where \(p\) and \(q\) are constants. It is given that \(u_2 = 500\) and \(u_4 = 356\)

1. Find the two possible values of \(u_3\) [5 marks]
2. When \(u_n\) is a decreasing sequence, the limit of \(u_n\) as \(n\) tends to infinity is \(L\). Write down an equation for \(L\) and hence find the value of \(L\). [2 marks]

A curve is defined for \(x \geq 0\) by the equation $$y = 6x - 2x^{\frac{1}{2}}$$

1. Find \(\frac{dy}{dx}\). [2 marks]
2. The curve has one stationary point. Find the coordinates of the stationary point and determine whether it is a maximum or minimum point. Fully justify your answer. [3 marks]

$$\frac{1 + 11x - 6x^2}{(x - 3)(1 - 2x)} \equiv A + \frac{B}{(x - 3)} + \frac{C}{(1 - 2x)}$$ Find the values of the constants \(A\), \(B\) and \(C\). [4]

A region, R, is defined by \(x^2 - 8x + 12 \leq y \leq 12 - 2x\)

1. Sketch a graph to show the region R. Shade the region R.
2. Find the area of R [6 marks]

1. Prove that $$n - 1 \text{ is divisible by } 3 \Rightarrow n^3 - 1 \text{ is divisible by } 9$$ [3 marks]
2. Show that if \(\log_2 3 = \frac{p}{q}\), then $$2^p = 3^q.$$ Use proof by contradiction to prove that \(\log_2 3\) is irrational. [3 marks]

\includegraphics{figure_6} In this question you must show all stages of your working. Solutions relying on calculator technology are not acceptable. Figure 6 shows a sketch of part of the curve with equation $$y = 3x \cdot 2^{2x}.$$ The point \(P(a, 96\sqrt{2})\) lies on the curve.

1. Find the exact value of \(a\). [3]

The curve with equation \(y = 3x \cdot 2^{2x}\) meets the curve with equation \(y = 6^{3-x}\) at the point \(Q\).

1. Show that the \(x\) coordinate of \(Q\) is \(\frac{3 + 2\log_2 3}{3 + \log_2 3}\). [4]

\includegraphics{figure_3} Figure 3 shows a sketch of the curve with equation \(y = 2x^2 - x\). The finite region \(R\), shown shaded in Figure 3, is bounded by the curve, the line with equation \(x = -0.5\), the \(x\)-axis and the line with equation \(x = 1.5\).

1. The trapezium rule with four strips is used to find an estimate for the area of \(R\). Explain whether the estimate for R is an underestimate or overestimate to the true value for the area of \(R\). [1]

The estimate for R is found to be 2.58. Using this value, and showing your working,

1. estimate the value of \(\int_{-0.5}^{1.5} (2x^2 + 1 + 2x) \, dx\). [3]