[4 marks sum]

Question 14 Marks

Find $X$ and $Y,$ if $\left|\begin{array}{cc}1 & 2 \\ 2 & -3\end{array}\right|\left|\begin{array}{l}x \\ y\end{array}\right|=\left|\begin{array}{c}-1 \\ 12\end{array}\right|$

Answer

$\begin{aligned} & \left|\begin{array}{cc}1 & 2 \\ 2 & -3\end{array}\right|_{2 \times 2}\left|\begin{array}{c}x \\ y\end{array}\right|_{2 \times 1}=\left|\begin{array}{c}-1 \\ 12\end{array}\right|_{2 \times 1}\end{aligned} $
$ \left|\begin{array}{c}x+2 y \\ 2 x-3 y\end{array}\right|$
$=\left|\begin{array}{c}-1 \\ 12\end{array}\right|$
$ x+2 y=-1 \ldots(1)$
$ \Rightarrow x=-1-2 y$
$ 2 x-3 y=12 \ldots(2)$
Putting the value of $x$ in $(2)$
$2(-1-2 y)-3 y=12$
$ \Rightarrow-7 y=14$
$ \Rightarrow y=-2$
from $(1)$
$x+2(-2)=-1$
$ \Rightarrow x=-1+4$
$ \Rightarrow x=3$

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Question 24 Marks

If $P=\left|\begin{array}{ll}1 & 2 \\ 3 & 4\end{array}\right|, Q=\left|\begin{array}{ll}5 & 1 \\ 7 & 4\end{array}\right|$ and $R=\left|\begin{array}{ll}2 & 1 \\ 4 & 2\end{array}\right|$ find the value of $(R+Q) P$

Answer

$\begin{array}{l}R+Q=\left|\begin{array}{ll}2 & 1 \\ 4 & 2\end{array}\right|+\left|\begin{array}{ll}5 & 1 \\ 7 & 4\end{array}\right| \end{array} $
$ =\left|\begin{array}{cc}7 & 2 \\ 11 & 6\end{array}\right|_{2 \times 2} $
$ (R+Q) P=\left|\begin{array}{cc}7 & 2 \\ 11 & 6\end{array}\right|\left|\begin{array}{ll}1 & 2 \\ 3 & 4\end{array}\right| $
$ =\left|\begin{array}{cc}7+6 & 14+8 \\ 11+18 & 22+24\end{array}\right| $
$ =\left|\begin{array}{cc}13 & 22 \\ 29 & 46\end{array}\right|_{2 \times 2}$

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Question 34 Marks

Answer

$\begin{array}{l}P=\left|\begin{array}{ll}1 & 2 \\ 3 & 4\end{array}\right|_{2 \times 2}, Q=\left|\begin{array}{ll}5 & 1 \\ 7 & 4\end{array}\right|_{2 \times 2} R=\left|\begin{array}{ll}2 & 1 \\ 4 & 2\end{array}\right|_{2 \times 2} \end{array} $
$ Q+R=\left|\begin{array}{ll}5 & 1 \\ 7 & 4\end{array}\right|+\left|\begin{array}{ll}2 & 1 \\ 4 & 2\end{array}\right| $
$ =\left|\begin{array}{ll}7 & 2 \\ 2 & 6\end{array}\right| $
$ P(Q+R)=\left|\begin{array}{ll}1 & 2 \\ 3 & 4\end{array}\right|\left|\begin{array}{ll}7 & 2 \\ 2 & 6\end{array}\right| $
$ =\left|\begin{array}{ll}7+22 & 2+12 \\ 21+44 & 6+24\end{array}\right| $
$ =\left|\begin{array}{ll}29 & 14 \\ 65 & 30\end{array}\right|_{2 \times 2}$

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Question 44 Marks

If $P=\left|\begin{array}{ll}1 & 2 \\ 2 & 1\end{array}\right|$ and $Q=\left|\begin{array}{ll}2 & 1 \\ 1 & 2\end{array}\right|$ find $P(Q P)$.

Answer

$\begin{array}{l}P=\left|\begin{array}{ll}1 & 2 \\ 2 & 1\end{array}\right|_{2 \times 2}, Q=\left|\begin{array}{ll}2 & 1 \\ 1 & 2\end{array}\right|_{2 \times 2} \end{array} $
$ QP =\left|\begin{array}{ll}2 & 1 \\ 1 & 2\end{array}\right|\left|\begin{array}{ll}1 & 2 \\ 2 & 1\end{array}\right| $
$=\left|\begin{array}{ll}2+2 & 4+1 \\ 1+4 & 2+2\end{array}\right| $
$ =\left|\begin{array}{ll}4 & 5 \\ 5 & 4\end{array}\right|_{2 \times 2} $
$ P(Q P)=\left|\begin{array}{ll}1 & 2 \\ 2 & 1\end{array}\right|\left|\begin{array}{cc}4 & 5 \\ 5 & 4\end{array}\right| $
$ =\left|\begin{array}{cc}4+10 & 5+8 \\ 8+5 & 10+4\end{array}\right| $
$ =\left|\begin{array}{ll}14 & 13 \\ 13 & 14\end{array}\right|_{2 \times 2} $

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Question 54 Marks

Find the values of $a , b , c$ and $d$, if $\left|\begin{array}{ll} a +3 b & 3 c + d \\ 2 a - b & c -2 d \end{array}\right|=\left|\begin{array}{ll}5 & 8 \\ 3 & 5\end{array}\right|$

Answer

$\left|\begin{array}{ll}a+3 b & 3 c+d \\ 2 a-b & c-2 d\end{array}\right|=\left|\begin{array}{ll}5 & 8 \\ 3 & 5\end{array}\right|$
$a+3 b=5....(1)$
$ 2 a-b=3....(2)$
$ \Rightarrow b=2 a-3$
putting this value of $b$ in $(1)$
$a+6 a-9=5$
$ \Rightarrow 7 a=14$
$ \Rightarrow a=2$
from $(i)$
$2+3 b=5$
$ \Rightarrow 3 b=3$
$ \Rightarrow b=1$
$3 c+d=8....(3)$
$ c-2 d=5....(4)$
$ c=5+2 d$
putting the value of $c$ in $(3)$
$15+6 d+d=8$
$ \Rightarrow 7 d=-7$
$ \Rightarrow d=-1$
from $(4)$
$ c+2=5$
$ \Rightarrow c=3$

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Question 64 Marks

If $\left|\begin{array}{cc}2 a + b & c \\ d & 3 a - b \end{array}\right|=\left|\begin{array}{cc}4 & 3 a \\ 7 & 6\end{array}\right|$, find the values of $a , b , c$ and $d$.

Answer

Given,
$\left|\begin{array}{cc} 2 a+b & c \\ d & 3 a-b \end{array}\right|_{2 \times 2}=\left|\begin{array}{cc} 4 & 3 a \\ 7 & 6 \end{array}\right|_{2 \times 2}$
$2 a+b=4......(1)$
$3 a-b=6......(2)$
Adding $(1)$ and $(2),$ we get
$5 a=10$
$\Rightarrow a=2$
from $(1)$
$2(2)+b=4$
$\Rightarrow b=0$
$C=3 a$
$\Rightarrow C=3 \times 2$
$\Rightarrow c=6$
$\Rightarrow d=7$

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Mathematics [4 marks sum]

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