SPS SPS SM Pure (SPS SM Pure) 2023 September

In all questions you must show all stages of your working, justifying solutions and not relying solely on calculator technology.

1. Find the first four terms, in ascending powers of \(x\), of the binomial expansion of \(\left(1+\frac{x}{2}\right)^7\), giving each coefficient in exact simplified form. [3]
2. Hence determine the coefficient of \(x\) in the expansion of $$\left(1+\frac{2}{x}\right)^2\left(1+\frac{x}{2}\right)^7.$$ [3]

\includegraphics{figure_2} The figure above shows a triangle with vertices at \(A(2,6)\), \(B(11,6)\) and \(C(p,q)\).

1. Given that the point \(D(6,2)\) is the midpoint of \(AC\), determine the value of \(p\) and the value of \(q\). [2]

The straight line \(l\) passes through \(D\) and is perpendicular to \(AC\). The point \(E\) is the intersection of \(l\) and \(AB\).

1. Find the coordinates of \(E\). [4]

$$x^2 + y^2 - 2x - 2y = 8$$ The circle with the above equation has radius \(r\) and has its centre at the point \(C\).

1. Determine the value of \(r\) and the coordinates of \(C\). [3]

The point \(P(4,2)\) lies on the circle.

1. Show that an equation of the tangent to the circle at \(P\) is [4] $$y = 14 - 3x.$$

$$f(x) = e^x, x \in \mathbb{R}, x > 0.$$ $$g(x) = 2x^3 + 11, x \in \mathbb{R}.$$

1. Find and simplify an expression for the composite function \(gf(x)\). [2]
2. State the domain and range of \(gf(x)\). [2]
3. Solve the equation $$gf(x) = 27.$$ [3]

The equation \(gf(x) = k\), where \(k\) is a constant, has solutions.

1. State the range of the possible values of \(k\). [1]

Relative to the origin \(O\), the points \(A\), \(B\) and \(C\) have position vectors \(4\mathbf{i} + 2\mathbf{j}\), \(3\mathbf{i} + 4\mathbf{j}\) and \(-\mathbf{i} + 12\mathbf{j}\), respectively.

1. Find the magnitude of the vector \(\overrightarrow{OC}\) [2]
2. Find the angle that the vector \(\overrightarrow{OB}\) makes with the vector \(\mathbf{j}\) to the nearest degree [2]
3. Show that the points \(A\), \(B\) and \(C\) are collinear [3]

Liquid is kept in containers, which due to evaporation and ongoing chemical reactions, at the end of each month the volume of the liquid in these containers reduces by 10% compared with the volume at the start of the same month. One such container is filled up with 250 litres of liquid.

1. Show that the volume of the liquid in the container at the end of the second month is 202.5 litres. [1]
2. Find the volume of the liquid in the container at the end of the twelfth month. [2]

At the start of each month a new container is filled up with 250 litres of liquid, so that at the end of twelve months there are 12 containers with liquid.

1. Use an algebraic method to calculate the total amount of liquid in the 12 containers at the end of 12 months. [5]

\includegraphics{figure_7} The figure above shows a circular sector \(OAB\) whose centre is at \(O\). The radius of the sector is 60 cm. The points \(C\) and \(D\) lie on \(OA\) and \(OB\) respectively, so that \(|OC| = |OD| = 24\) cm. Given that the length of the arc \(AB\) is 48 cm, find the area of the shaded region \(ABDC\), correct to the nearest cm\(^2\). [5 marks]

A cubic curve \(C\) has equation $$y = (3-x)(4+x)^2.$$

1. Sketch the graph of \(C\). [3] The sketch must include any points where the graph meets the coordinate axes.
2. Sketch in separate diagrams the graph of \(\ldots\)
   1. \(\ldots y = (3-2x)(4+2x)^2\). [2]
   2. \(\ldots y = (3+x)(4-x)^2\). [2]
   3. \(\ldots y = (2-x)(5+x)^2\). [2]
   Each of the sketches must include any points where the graph meets the coordinate axes.

Solve the following trigonometric equation in the range given. $$4\tan^2\theta\cos\theta = 15, \quad 0 \leq \theta < 360°.$$ [6 marks]

\includegraphics{figure_10} The figure above shows solid right prism of height \(h\) cm. The cross section of the prism is a circular sector of radius \(r\) cm, subtending an angle of 2 radians at the centre.

1. Given that the volume of the prism is 1000 cm\(^3\), show clearly that $$S = 2r^2 + \frac{4000}{r},$$ where \(S\) cm\(^2\) is the total surface area of the prism. [5]
2. Hence determine the value of \(r\) and the value of \(h\) which make \(S\) least, fully justifying your answer. [7]

It is given that $$f(x) = x^2 - kx + (k+3),$$ where \(k\) is a constant. If the equation \(f(x) = 0\) has real roots find the range of the possible values of \(k\). [4]

\includegraphics{figure_12} The figure above shows the curve \(C\) with equation $$f(x) = \frac{x+4}{\sqrt{x}}, \quad x > 0.$$

1. Determine the coordinates of the minimum point of \(C\), labelled as \(M\). [5]

The point \(N\) lies on the \(x\) axis so that \(MN\) is parallel to the \(y\) axis. The finite region \(R\) is bounded by \(C\), the \(x\) axis, the straight line segment \(MN\) and the straight line with equation \(x = 1\).

1. Use the trapezium rule with 4 strips of equal width to estimate the area of \(R\). [3]

Prove or disprove each of the following statements:

1. If \(n\) is an integer, then \(3n^2 - 11n + 13\) is a prime number. [2]
2. If \(x\) is a real number, then \(x^2 - 8x + 17\) is positive. [2]
3. If \(p\) and \(q\) are irrational numbers, then \(pq\) is irrational. [2]

\includegraphics{figure_14} The diagram above shows the curve with equation $$y = (x-4)^2, \quad x \in \mathbb{R},$$ intersected by the straight line with equation \(y = 4\), at the points \(A\) and \(B\). The curve meets the \(y\) axis at the point \(C\). Calculate the exact area of the shaded region, bounded by the curve and the straight line segments \(AB\) and \(BC\). [8]