SPS SPS FM (SPS FM) 2020 December

Solve \(2 \sin x = \tan x\) exactly, where \(-\frac{\pi}{2} < x < \frac{\pi}{2}\). [4]

Let \(a, b\) satisfy \(0 < a < b\).

1. Find, in terms of \(a\) and \(b\), the value of $$\int_a^b \frac{81}{x^4} dx$$ [2]
2. Explaining clearly any limiting processes used, find the value of \(a\), given that $$\int_a^{\infty} \frac{81}{x^4} dx = \frac{216}{125}$$ [2]

1. Sketch the graph of \(y = |3x - 1|\). [1]
2. Hence, solve \(5x + 3 < |3x - 1|\). [3]

The following diagram shows the curve \(y = a \sin(b(x + c)) + d\), where \(a, b, c\) and \(d\) are all positive constants and \(x\) is measured in radians. The curve has a maximum point at \((1, 3.5)\) and a minimum point at \((2, 0.5)\). \includegraphics{figure_4}

1. Write down the value of \(a\) and the value of \(d\). [2]
2. Find the value of \(b\). [2]
3. Find the smallest possible value of \(c\), given that \(c > 0\). [2]

The \(2 \times 2\) matrix A represents a rotation by \(90°\) anticlockwise about the origin. The \(2 \times 2\) matrix B represents a reflection in the line \(y = -x\). The matrix B is given by $$\mathbf{B} = \begin{pmatrix} 0 & -1 \\ -1 & 0 \end{pmatrix}$$

1. Write down the matrix representing A. [1]
2. The \(2 \times 2\) matrix C represents a rotation by \(90°\) anticlockwise about the origin, followed by a reflection in the line \(y = -x\). Compute the matrix C and describe geometrically the single transformation represented by C. [3]

Given that \(z\) is the complex number \(x + iy\) and satisfies $$|z| + z = 6 - 2i$$ find the value of \(x\) and the value of \(y\). [4]

The diagram below shows part of a curve C with equation \(y = 1 + 3x - \frac{1}{2}x^2\). \includegraphics{figure_7}

1. The curve crosses the \(y\) axis at the point A. The straight line L is normal to the curve at A and meets the curve again at B. Find the equation of L and the \(x\) coordinate of the point B. [4]
2. The region R is bounded by the curve C and the line L. Find the exact area of R. [3]

1. The \(2 \times 2\) matrix A is given by $$\mathbf{A} = \begin{pmatrix} 7 & 3 \\ 2 & 1 \end{pmatrix}.$$ The \(2 \times 2\) matrix B satisfies $$\mathbf{BA}^2 = \mathbf{A}.$$ Find the matrix B. [3]
2. The \(2 \times 2\) matrix C is given by $$\mathbf{C} = \begin{pmatrix} -2 & 4 \\ -1 & 2 \end{pmatrix}.$$ By considering \(\mathbf{C}^2\), show that the matrices \(\mathbf{I} - \mathbf{C}\) and \(\mathbf{I} + \mathbf{C}\) are inverse to each other. [2]

Sketch on an Argand diagram the locus of all points that satisfy \(|z + 4 - 4i| = 2\sqrt{2}\) and hence find \(\theta, \phi \in (-\pi, \pi]\) such that \(\theta \leq \arg z \leq \phi\). [5]

The \(2 \times 2\) matrix M is defined by $$\mathbf{M} = \begin{pmatrix} 0 & 0.25 \\ 0.36 & 0 \end{pmatrix}$$ Find, by calculation, the equations of the two lines that pass through the origin, that remain invariant under the transformation represented by M. [4]

In the triangle \(PQR\), \(PQ = 6\), \(PR = k\), \(P\hat{Q}R = 30°\).

1. For the case \(k = 4\), find the two possible values of \(QR\) exactly. [3]
2. Determine the value(s) of \(k\) for which the conditions above define a unique triangle. [3]

Consider the binomial expansion of \(\left(1 + \frac{x}{5}\right)^n\) in ascending powers of \(x\), where \(n\) is a positive integer.

1. Write down the first four terms of the expansion, giving the coefficients as polynomials in \(n\). [1]

The coefficients of the second, third and fourth terms of the expansion are consecutive terms of an arithmetic sequence.

1. Show that \(n^3 - 33n^2 + 182n = 0\). [3]
2. Hence find the possible values of \(n\) and the corresponding values of the common difference. [3]

A series is given by $$\sum_{r=1}^k 9^{r-1}$$

1. Write down a formula for the sum of this series. [1]
2. Prove by induction that \(P(n) = 9^n - 8n - 1\) is divisible by 64 if \(n\) is a positive integer greater than 1. [4]