Question
(a) Find the magnitude of the vector \(\binom{-1}{7}\).
(b) The determinant of the matrix \(\begin{pmatrix}6 &2m \\ 5 & m\end{pmatrix}\)Find the value of m
(c) \(L=\begin{pmatrix}2 &5 \\ 3 & 9\end{pmatrix}\) \(M= \binom{-4}{2}\) N=(\(1\) \(7)\)
Work out the following.
(i) NM
…………………………………………
(ii) LM
………………………………………..
(iii) \(L^{2}\)
…………………………………………
(iv) \(L^{-1}\)
…………………………………………
Answer/Explanation
(a) 7.07 or 7.071…
(b)−6
(c)(i) (10) final answer
(ii)\(\binom{2}{6}\) final answer
(iii)\(\begin{pmatrix}19 & 55\\ 33 & 96\end{pmatrix}\) final answer
(iv)\( \frac{1}{3}\begin{pmatrix}9 & -5 \\ -3 & 2\end{pmatrix}\)
Question
(a) \(\vec{OA}=\begin{pmatrix}4\\ 3\end{pmatrix} \) \(\vec{AB}=\begin{pmatrix}8\\ -7\end{pmatrix} \) \(\vec{AC}=\begin{pmatrix}-3\\ 6\end{pmatrix}\)
Find
(i)\(\left | \vec{OB} \right |,\)
\(\left | \vec{OB} \right |=……………………….\)
(ii)\( \left | \vec{BC} \right |,\)
\(\left | \vec{BC} \right |…………………..\)
(b)
PQRS is a parallelogram with diagonals PR and SQ intersecting at X.
\(\vec{PQ}= a\) and \(\vec{PS}=b.\)
Find \vec{QX} in terms of a and b.
Give your answer in its simplest form.
\(\vec{QX} = ………………………………………..\)
(c)\( M=\begin{pmatrix}2 & 5 \\ 1 & 8\end{pmatrix}\)
Calculate
(i)\( M^{2},\)
\(M^{2}=\left ( \right )\)
(ii)\(M^{-1}\)
\(M^{-1}=\left ( \right )\)
Answer/Explanation
(a)(i) 12.6 or 12.64 to 12.65
(ii) \(\begin{pmatrix}-11\\ 13\end{pmatrix}\)
(b)\(\frac{1}{2}(b-a)\)
(c)(i) \(\begin{pmatrix}9 & -50\\ 10 & 69\end{pmatrix}\)
(ii)\(\frac{1}{11}\begin{pmatrix}8 & -5\\ -1 & 2\end{pmatrix}\)
Question
O is the origin and OPQRST is a regular hexagon.
\(\overheadarrow{OP} = x\) and \(\overheadarrow{OT} = y\)(a) Write down, in terms of x and/or y, in its simplest form,
(i) \(\overheadarrow{QR},
\(\overheadarrow{QR}\)=…………………….
(ii) \(\overheadarrow{PQ}\),
\(\overheadarrow{PQ}\) = ……………………..
(ii) the position vector of S.
(b) The line SR is extended to G so that SR : RG = 2 : 1.
\(\overheadarrow{GQ}\) = ………………..
(c) M is the midpoint of OP.
(i) Find \(\overheadarrow{MG}\), in terms of x and y, in its simplest form.
\(\overheadarrow{MG}\) = ………………………
(ii) H is a point on TQ such that TH : HQ = 3 : 1.
Use vectors to show that H lies on MG.
Answer/Explanation
Ans:
(a) (i) y
(ii) x + y
(iii) x + 2y
(b) -(1/2 x+ y ) oe
(c) (i) \(\overheadarrow{MG}\) = 2x + 2y
(ii) \(\overheadarrow{MH}\) = x + y or \(\overheadarrow{HG} \) = x + y
\(\overheadarrow{MG} = 2 \overheadarrow{MH}\) oe