Correct Answer: A. the beauty of nature

Correct Answer: D. pawpaw

Explanation

(a)(i) The matrix of rotation of point anticlockwise through angle 60° about the origin is given by:

\(M = \begin{pmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{pmatrix} = \begin{pmatrix} \cos 60° & -\sin 60° \\ \sin 60° & \cos 60° \end{pmatrix}\)

= \(\begin{pmatrix} \frac{1}{2} & \frac{-\sqrt{3}}{2} \\ \frac{\sqrt{3}}{2} & \frac{1}{2} \end{pmatrix}\)

(ii) Using the transformation,

\(T :P \to MP\) or \(T : \begin{pmatrix} x \\ y \end{pmatrix} \to \begin{pmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix}\)

\(T : \begin{pmatrix} 3 \\ 5 \end{pmatrix} \to \begin{pmatrix} \frac{1}{2} & \frac{-\sqrt{3}}{2} \\ \frac{\sqrt{3}}{2} & \frac{1}{2} \end{pmatrix} \begin{pmatrix} 3 \\ 5 \end{pmatrix}\)

Image of P, P' = \(\begin{pmatrix} \frac{3 + 5\sqrt{3}}{2} \\ \frac{3\sqrt{3} - 5}{2} \end{pmatrix}\)

(b) \(T : \begin{pmatrix} x \\ y \end{pmatrix} \to \begin{pmatrix} 2x + 3y \\ 3x - y \end{pmatrix} = \begin{pmatrix} 2 & 3 \\ 3 & -1 \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix}\)

(i) \(N = \begin{pmatrix} 2 & 3 \\ 3 & -1 \end{pmatrix}\)

(ii) \(N^{2} + aN + bI = 0\)

\(\begin{pmatrix} 2 & 3 \\ 3 & -1 \end{pmatrix} ^{2} + a \begin{pmatrix} 2 & 3 \\ 3 & -1 \end{pmatrix} + b \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} = \begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix}\)

\(\begin{pmatrix} 2 & 3 \\ 3 & -1 \end{pmatrix} \begin{pmatrix} 2 & 3 \\ 3 & -1 \end{pmatrix} + a \begin{pmatrix} 2 & 3 \\ 3 & -1 \end{pmatrix} + b \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} = \begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix}\)

\(\begin{pmatrix} 4 + 9 & 6 - 3 \\ 6 - 3 & 9 + 1 \end{pmatrix} + \begin{pmatrix} 2a & 3a \\ 3a & -a \end{pmatrix} + \begin{pmatrix} b & 0 \\ 0 & b \end{pmatrix} = \begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix}\)

Adding corresponding elements of each matrix,

\(13 + 2a + b = 0 \implies 2a + b = -13 ... (1)\)

\(3 + 3a = 0 \implies 3a = -3 ; a = -1\)

\(2(-1) + b = -13 \implies -2 + b = -13\)

\(b = -13 + 2 = -11\)

\(\therefore a = -1 ; b = -11\)

Correct Answer: B) Testing arithmetical accuracies of the ledger account balance

Correct Answer: D. Phloem

Explanation

The main supporting tissues in plants are parenchyma, collenchyma, sclerenchyma (fibre) and wood (xylem).

Correct Answer: C. Gas molecules colliding in a flask

Correct Answer: B. a symbol

Correct Answer: A. N1,000

Correct Answer: D. caveat emptor

Explanation

Caveat emptor is a neo-Latin phrase meaning let the buyer beware. the principle that the seller of a product cannot be held responsible for its quality unless it is guaranteed in a warranty. The buyer is required to properly examine goods before making payment because the seller wont be held responsible for any defects in the goods unless it comes with a warranty

Correct Answer: D. sent Babylon and other nations against it