KC Sinha Mathematics Solution Class 12 Chapter 5 आव्यूह ( Matrices ) Exercise 5.1 (Q11-Q20)

Ex-5.1

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Exercise 5.1

Question 11

सुधा तथा उसकी दो सहेलियाँ सयिदा तथा सिमरन के पास पुस्तिकाओं तथा कलमों की संख्या के बारे में निम्नलिखित सूचनाओं पर विचार करें।

[Consider the following information regarding number of note books and pens possessed by Sudha and her two friends Syeeda and Simran]

	पुस्तिकाओं की संख्या (Number of note nooks)	कलमों की संख्या (Number of pens)
सुधा (Sudha)	15	6
सयिदा (Syeeda)	10	2
सिमरन(Simran)	13	5

ऊपर दिए गए सूचनाओं को 3×2 आव्यूह के रूप में लिखें। दूसरी पंक्ति और पहले स्तम्भ की प्रविष्टि क्या प्रकट करती है ?

[Represent the above informations in the form of a 3×2 matrix. What does the entry in the second row and first column represent ?]

Sol :

Question 12
(i) यदि (if) $\left[\begin{array}{cc}x+y & 2 \\ 1 & x-y\end{array}\right]=\left[\begin{array}{ll}3 & 2 \\ 1 & 7\end{array}\right]$ , तो (then) x=__ , y=__
Sol :
$\begin{aligned} x+y &=3..(i) \\ x-y &=7..(ii) \\ \hline 2 x &=10\\x=5 \end{aligned}$

Putting the value of x in equation (i)
⇒x+y=3
⇒5+y=3
⇒y=3-5=-2


(ii) यदि (if) $\left[\begin{array}{cc}x-y & 2 x-x_{1} \\ 2 x-y & 3 x+y_{1}\end{array}\right]=\left[\begin{array}{cc}-1 & 5 \\ 0 & 13\end{array}\right]$ ,
तथा P और Q के नियामक क्रमश
(x,y) तथा (x₁y₁) है तो PQ=__
[and co-ordinates of points P and Q be (x,y) and (x₁y₁) respectively] then PQ=__
Sol :
$\begin{aligned} x-y &=-1 ..(i)\\ 2 x-y &=0 ..(ii)\\ \hline x&=1 \end{aligned}$

⇒1-y=-1
⇒y=2

$\begin{aligned} 2 x-x_{1} &=5..(iii) \\ 3 x+y_{1} &=13...(iv) \end{aligned}$

From equation (iii)
⇒$2 x-x_{1}=5$
⇒$2 (1)-x_{1}=5$
⇒x₁=-3

From equation (iv)

$3 x+y_{1}=13$

$3(1)+y_{1}=13$
$y_{1}=10$

Diagram

$PQ=\sqrt{(-3-1)^2+(10-2)^2}$
=$\sqrt{-4^2+8^2}$
=$\sqrt{16+64}=\sqrt{80}$
=$4\sqrt{5}$

Question 13
(i) यदि (if) $\left[\begin{array}{cc}x-y & 2 x+z \\ 2 x-y & 3 z+\omega\end{array}\right]=\left[\begin{array}{rr}-1 & 5 \\ 0 & 13\end{array}\right]$  , (find) x, y,z,ω , निकाले
Sol :
$\begin{aligned} x-y &=-1..(i) \\2 x+y &=0..(ii) \\\hline x &=1 \end{aligned}$

Putting x=1 in equation (1)
⇒1-y=-1
⇒y=2

2x+z=5..(iii)

3z+ω=13...(iv)

Putting x=1 in equation (iii)
⇒2x+z=5
⇒2(1)+z=5
⇒z=3

putting z=3 in equation (iv)
⇒3(3)+ω=13
⇒ω=4

(ii) यदि (If) $\left[\begin{array}{cc}x-y & z \\ 2 x-y & \omega\end{array}\right]=\left[\begin{array}{rr}-1 & 4 \\ 0 & 5\end{array}\right]$, (find) $x, y, z, \omega$ निकालें।

Sol :

(iii) यदि (If) $\left[\begin{array}{cc}x & 3 x-y \\ 2 x+z & 3 y-\omega\end{array}\right]=\left[\begin{array}{ll}3 & 2 \\ 4 & 7\end{array}\right]$, (find) $x, y, z, \omega$ निकाले।

Sol :

Question 14
निम्नलिखित समिकरणों x,y,z निकाले ।
[Find x,y,z from the following equations]
(i) $\left[\begin{array}{cc}x+y & 2 \\ 5+z & x y\end{array}\right]=\left[\begin{array}{cc}6 & 2 \\ 5 & 8\end{array}\right]$
Sol :
x+y=6⇒y=6-x

⇒xy=8
⇒x(6-x)=8
⇒6x-x²=8
⇒x²-6x+8=0
⇒x²-4x-2x+8=0
⇒x(x-4)-2(x-4)=0
⇒(x-4)(x-2)=0

⇒x-4=0 , x-2=0
⇒x=4 , x=2

If x=4 , then y=6-4=2

If x=2 , then y=6-2=4

5+z=5
z=0

x+4 , y=2 , z=0
x=2 , y=4 , z=0



(ii) $\left[\begin{array}{ll}4 & 3 \\ x & 5\end{array}\right]=\left[\begin{array}{ll}y & z \\ 1 & 5\end{array}\right]$
Sol :

y=4

z=3

x=1


(iii) $\left[\begin{array}{c}x+y+z \\ x+z \\ y+z\end{array}\right]=\left[\begin{array}{l}9 \\ 5 \\ 7\end{array}\right]$
Sol :
⇒x+y+z=9..(i)
⇒x+z=5..(ii)
⇒y+z=7..(iii)

⇒x+y+z=9
⇒5+y=9
⇒y=4

Putting y=4 in equation (iii)
⇒4+z=7
⇒z=3

Putting z=3 in equation (ii)
⇒x+3=5
⇒x=2


Question 15
a, b,c,d निकाले यदि (Find a,b,c,d if) $\left[\begin{array}{cc}2 a+b & a-2 b \\ 5 c-d & 4 c+3 d\end{array}\right]=\left[\begin{array}{cc}4 & -3 \\ 11 & 24\end{array}\right]$
Sol :
2a+b=4..(i)
a-2b=-3..(ii)
5c-d=11..(iii)
4c+3d=24...(iv)


Question 16
माना कि (let) $A=\left[\begin{array}{ll}2 & 4 \\ 3 & 2\end{array}\right]$,$B=\left[\begin{array}{rr}1 & 3 \\ -2 & 5\end{array}\right]$, $C=\left[\begin{array}{rr}-2 & 5 \\ 3 & 4\end{array}\right]$ निकाले (find)
(i) A+B
Sol :
$A+B=\left[\begin{array}{cc}2 & 4 \\ 3 & 2\end{array}\right]+\left[\begin{array}{cc}1 & 3 \\ -2 & 5\end{array}\right]$

$=\left[\begin{array}{ll}3 & 7 \\ 1 & 7\end{array}\right]$

(ii) A-B
Sol :
$A-B=\left[\begin{array}{ll}2 & 4 \\ 3 & 2\end{array}\right]-\left[\begin{array}{cc}1 & 3 \\ -2 & 5\end{array}\right]$

$=\left[\begin{array}{cc}1 & 1 \\ 5 & -3\end{array}\right]$

(iii) 3A-C
Sol :
$3 A-C=3\left[\begin{array}{cc}2 & 4 \\ 3 & 2\end{array}\right]-\left[\begin{array}{rr}-2 & 5 \\ 3 & 4\end{array}\right]$

$=\left[\begin{array}{cc}6 & 12 \\ 9 & 6\end{array}\right]-\left[\begin{array}{cc}-2 & 5 \\ 3 & 4\end{array}\right]$

$=\left[\begin{array}{ll}8 & 7 \\ 6 & 2\end{array}\right]$


Question 18
निमनलिखित का परिकलन करे (Compute the following)
(i) $\left[\begin{array}{rrr}0 & 1 & 5 \\ -3 & 2 & 1\end{array}\right]+\left[\begin{array}{rrr}6 & 2 & -3 \\ -1 & 4 & 2\end{array}\right]$
Sol :
$\left[\begin{array}{ccc}0 & 1 & 5 \\ -3 & 2 & 1\end{array}\right]+\left[\begin{array}{ccc}6 & 2 & -3 \\ -1 & 4 & 2\end{array}\right]$


(ii) $\left[\begin{array}{rr}2 & -1 \\ 3 & 5\end{array}\right]+\left[\begin{array}{rr}4 & 3 \\ 1 & -2\end{array}\right]$
Sol :


(iii) $\left[\begin{array}{rr}a & b \\ -b & a\end{array}\right]+\left[\begin{array}{ll}a & b \\ b & a\end{array}\right]$
Sol :
$=\left[\begin{array}{cc}2 a&2b \\ 0 & 2 a\end{array}\right]$


(iv) $\left[\begin{array}{cc}\cos ^{2} x & \sin ^{2} x \\ \sin ^{2} x & \cos ^{2} x\end{array}\right]+\left[\begin{array}{cc}\sin ^{2} x & \cos ^{2} x \\ \cos ^{2} x & \sin ^{2} x\end{array}\right]$
Sol :
$=\left[\begin{array}{ll}1 & 1 \\ 1 & 1\end{array}\right]$


(v) $\left[\begin{array}{cc}a^{2}+b^{2} & b^{2}+c^{2} \\ a^{2}+c^{2} & a^{2}+b^{2}\end{array}\right]+\left[\begin{array}{cc}2 a b & 2 b c \\ -2 a c & -2 a b\end{array}\right]$
Sol :
$=\left[\begin{array}{ll}a^{2}+b^{2}+2 a b & b^{2}+c^{2}+2 b c \\ a^{2}+c^{2}-2 a c & a^{2}+b^{2}-2 ab\end{array}\right]$

$=\left[\begin{array}{ll}(a+b)^{2} & (b+c)^{2} \\ (a-c)^{2} & (a-b)^{2}\end{array}\right]$


Question 20
यदि (if) $A=\left[\begin{array}{cc}2 & -1 \\ 4 & 2\end{array}\right]$,$B=\left[\begin{array}{rr}4 & 3 \\ -2 & 1\end{array}\right]$,$C=\left[\begin{array}{rr}-2 & -3 \\ -1 & 2\end{array}\right]$
निम्नलिखित का परिकलन करे (compute the following)
(i) A+(B+C)
Sol :
$=\left[\begin{array}{cc}2 & -1 \\ 4 & 2\end{array}\right]+\left(\left[\begin{array}{cc}4 & 3 \\ -2 & 1\end{array}\right]+\left[\begin{array}{cc}-2 & -3 \\ -1 & 2\end{array}\right]\right)$

$=\left[\begin{array}{cc}2 & -1 \\ 4 & 2\end{array}\right]+\left[\begin{array}{cc}2 & 0 \\ -3 & 3\end{array}\right]$

$=\left[\begin{array}{cc}4 & -1 \\ 1 & 5\end{array}\right]$


(ii) (A+B)+C
Sol :


(iii)-2A+(B+C)


(iv)A+(2B-C)
Sol :
$=\left[\begin{array}{cc}2 & -1 \\ 4 & 2\end{array}\right]+\left(2\left[\begin{array}{cc}4 & 3 \\ -2 & 1\end{array}\right]-\left[\begin{array}{cc}-2 & -3 \\ -1 & 2\end{array}\right]\right)$

$=\left[\begin{array}{cc}2 & -1 \\ 4 & 2\end{array}\right]+\left(\left[\begin{array}{cc}8 & 6 \\ -4 & 2\end{array}\right]-\left[\begin{array}{cc}-2 & -3 \\ -1 & 2\end{array}\right]\right)$

$=\left[\begin{array}{cc}2 & -1 \\ 4 & 2\end{array}\right]+\left[\begin{array}{cc}10 & 9 \\ -3 & 0\end{array}\right]$

$=\left[\begin{array}{cc}12 & 8 \\ 1 & 2\end{array}\right]$