SPS SPS FM (SPS FM) 2022 January

1. The matrix \(\mathbf { A }\) is given by \(\mathbf { A } = \left( \begin{array} { l l } 4 & 1
5 & 2 \end{array} \right)\) and \(\mathbf { I }\) is the \(2 \times 2\) identity matrix. Find

1. \(\mathbf { A } - 3 \mathbf { I }\),
2. \(\mathbf { A } ^ { - 1 }\).
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2. The complex number \(3 + 4 \mathrm { i }\) is denoted by \(a\).

1. Find \(| a |\) and \(\arg a\).
2. Sketch on a single Argand diagram the loci given by
   (a) \(| z - a | = | a |\),
   (b) \(\quad \arg ( z - 3 ) = \arg a\).
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3. Relative to an origin \(O\), the points \(A\) and \(B\) have position vectors \(3 \mathbf { i } + 2 \mathbf { j } + 3 \mathbf { k }\) and \(\mathbf { i } + 3 \mathbf { j } + 4 \mathbf { k }\) respectively.

1. Find a vector equation of the line passing through \(A\) and \(B\).
2. Find the position vector of the point \(P\) on \(A B\) such that \(O P\) is perpendicular to \(A B\).
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4. Express \(\frac { ( x - 7 ) ( x - 2 ) } { ( x + 2 ) ( x - 1 ) ^ { 2 } }\) in partial fractions.
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5.

1. Expand \(( 1 + a x ) ^ { - 4 }\) in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\).
2. The coefficients of \(x\) and \(x ^ { 2 }\) in the expansion of \(( 1 + b x ) ( 1 + a x ) ^ { - 4 }\) are 1 and - 2 respectively. Given that \(a > 0\), find the values of \(a\) and \(b\).
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6. The figure below shows part of the curve \(y = 1 + x ^ { 2 }\), together with the line \(y = 2\).
\includegraphics[max width=\textwidth, alt={}, center]{143eb5e6-a5f5-4c7b-b357-dea3fabec794-14_572_734_258_685} The region enclosed by the curve, the \(y\)-axis, and the line \(y = 2\) is rotated through \(360 ^ { \circ }\) about the \(y\)-axis. Find the volume of the solid generated, giving your answer in terms of \(\pi\).
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7.

1. Use an algebraic method to find the square roots of the complex number \(16 + 30 \mathrm { i }\).
2. Use your answers to part (i) to solve the equation \(z ^ { 2 } - 2 z - ( 15 + 30 \mathrm { i } ) = 0\), giving your answers in the form \(x + \mathrm { i } y\).
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8.

1. A group of four different letters is chosen from the alphabet of 26 letters, regardless of order.
   1. How many different groups can be chosen?
   2. Find the probability that a randomly chosen group includes the letter P .
2. A three-digit number greater than 100 is formed using three different digits from the ten digits \(0,1,2,3,4,5,6,7,8,9\).
   1. Show that 648 different numbers can be formed. One of these 648 numbers is chosen at random.
   2. Find the probability that all three digits in the number are even. (You are reminded that 0 is an even number.)
   3. Find the probability that the number is even.
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9.
\includegraphics[max width=\textwidth, alt={}, center]{143eb5e6-a5f5-4c7b-b357-dea3fabec794-20_719_969_207_525} The diagram shows the unit square \(O A B C\) and its image \(O A ^ { \prime } B ^ { \prime } C ^ { \prime }\) under a transformation T .

1. Write down the matrix that represents T . The transformation T is equivalent to a transformation P followed by a transformation Q . The matrix that represents \(P\) is \(\left( \begin{array} { r r } 0 & - 1
   1 & 0 \end{array} \right)\).
2. Give a geometrical description of transformation P .
3. Find the matrix that represents transformation Q and give a geometrical description of transformation Q .
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