KC Sinha Mathematics Solution Class 12 Chapter 5 आव्यूह ( Matrices ) Exercise 5.2 (Q22-Q28)

Exercise 5.2

Ex-5.2

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Question 22

(i) x निकाले यदि (find x if) $\left[\begin{array}{ll}x & 1\end{array}\right]\left[\begin{array}{rr}1 & 0 \\ -2 & -3\end{array}\right]\left[\begin{array}{l}x \\ 3\end{array}\right]=0$

Sol :

(ii) $\left[\begin{array}{ll}2 x & 3\end{array}\right]\left[\begin{array}{rr}1 & 2 \\ -3 & 0\end{array}\right]\left[\begin{array}{l}x \\ 3\end{array}\right]=0$

Sol :

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

$\left[x^{2}-2 x-9\right]=[0]$

$x^{2}-2 x-9=0$

a=1 , b=-2 , c=-9

$x=\frac{-(-2) \pm \sqrt{(-2)^{2}-4 \times1\times(-9)}}{2 \times 1}$

$x=\frac{2\pm \sqrt{4+36}}{2}$

$x=\frac{2 \pm 2 \sqrt{10}}{2}=$\frac{2(1 \pm \sqrt{10})}{2}$$

$x=1 \pm \sqrt{10}$

(iii) a और b का मान निकालें जिसके लिए निम्नलिखित सत्य है:

(Find the values of a and b for which the following holds)

$\begin{bmatrix}4&2\\3&-1\end{bmatrix}\begin{bmatrix}a\\b\end{bmatrix}=\begin{bmatrix}-4\\2\end{bmatrix}$

Sol :

(iv) माना कि (Let) $A=\left[\begin{array}{rr}2 & -1 \\ 3 & 4\end{array}\right], B=\left[\begin{array}{ll}5 & 2 \\ 7 & 4\end{array}\right], C=\left[\begin{array}{ll}2 & 5 \\ 3 & 8\end{array}\right]$

एक आव्यूह D निकालें ताकि (find a matrix D such that) CD-AB=0

Sol :

C D-A B=D

$\left[\begin{array}{ll}2 & 5 \\ 3 & 8\end{array}\right] D-\left[\begin{array}{ll}2 & -1 \\ 3 & \phantom{-}4\end{array}\right]\left[\begin{array}{ll}5 & 2 \\ 7 & 4\end{array}\right]=0$

Let $D=\left[\begin{array}{ll}x & a \\ y & b\end{array}\right]$

$\left[\begin{array}{cc}2 & 5 \\ 3 & 8\end{array}\right]\left[\begin{array}{cc}x & a \\ y & b\end{array}\right]=\left[\begin{array}{cc}2 & -1 \\ 3 & 4\end{array}\right]\left[\begin{array}{cc}5 & 2 \\ 7 & 4\end{array}\right]$

$\left[\begin{array}{cc}2 x+5 y & 2 a+5 b \\ 3 x+8 y & 3 a+8 b\end{array}\right]=\left[\begin{array}{cc}10-7 & 4-4 \\ 15+28 & 6+16\end{array}\right]$

$\left[\begin{array}{cc}2 x+5 y & 2 a+5 b \\ 3 x+8 y & 3 a+8 b\end{array}\right]=\left[\begin{array}{cc}3 & 0 \\ 43 & 22\end{array}\right]$

$2 x+5 y=3..(i)\times 3$
$3 x+8 y=43..(ii)\times 2$
2a+5 b=0..(iii)
3a+8b=22...(iv)

$\begin{array}{rl}6 x+15 y^{2}=9 \\ -6 x+16 y=86 \\\hline y=77\end{array}$

Putting the value of y in equation (i)

2x+5(77)=3

2x+385=3

2x=-382

$x=\dfrac{-382}{2}=-191$

$\begin{aligned} 6 a +15 b &=0 \\6 a+16 b &=44 \\\hline b &=44 \end{aligned}$

Putting the value of b in equation (ii)

2a+5(44)=0

2a+220=0
$a=\dfrac{-220}{2}$

a=-110

$\therefore \quad D=\left[\begin{array}{cc}-191 & -110 \\ 77 & 44\end{array}\right]$

(v) आव्यूह X निकाले ताकि (Find the matrix X so that) $x\left[\begin{array}{lll}1 & 2 & 3 \\ 4 & 5 & 6\end{array}\right]=\left[\begin{array}{rrr}-7 & -8 & -9 \\ 2 & 4 & 6\end{array}\right]$

Sol :

Let $X=\left[\begin{array}{ll}x & a\\ y & b\end{array}\right]$

$\left[\begin{array}{ll}x & a \\ y & b\end{array}\right]\left[\begin{array}{lll}1 & 2 & 3 \\ 4 & 5 & 6\end{array}\right]=\left[\begin{array}{rrr}-7 & -8 & -9 \\ 2 & 4 & 6\end{array}\right]$

$\left[\begin{array}{ccc}x+4 a & 2 x+5 a & 3 x+64 \\ y+4 b & 2 y+5 b & 3 y+6 b\end{array}\right]=\left[\begin{array}{ccc}-7 & -8 & -9 \\ 2 & 4 & 6\end{array}\right]$

$x+4 a=-7..(i) \times 2$
$2x+5 a=-8..(ii) \times 1$
$y+4b=2..(iii)\times 2$
$2y+5b=4..(iv)\times 1$

Question 23

(i) x का मान निकाले ताकि (Find the value of x , such that)

$\left[\begin{array}{lll}1 & x & 1\end{array}\right]\left[\begin{array}{ccc}1 & 3 & 2 \\ 2 & 5 & 1 \\ 15 & 3 & 2\end{array}\right]\left[\begin{array}{l}1 \\ 2 \\ x\end{array}\right]=0$

Sol :

$\left[\begin{array}{lll}1 & x & 1\end{array}\right]\left[\begin{array}{c}1+6+2 x \\ 2+10+x \\ 15+6+2 x\end{array}\right]=0$

$\left[\begin{array}{lll}1 & x & 1\end{array}\right]\left[\begin{array}{c}7-2 x \\ 12+x \\ 21+2 x\end{array}\right]=0$

$\left[7+2 x+12 x+x^{2}+21+2 x\right]=[0]$

$2^{2}+16 x+28=0$

$x^{2}+14 x+2 x+28=0$

$x(x+14)+2(x+14)=0$

$(x+14)(x+2)=0$

⇒x=-14 , -2

(ii) x का मान निकालें यदि (Find the value of x , if)

$\begin{bmatrix}1&x&1\end{bmatrix}\begin{bmatrix}1&2&3\\4&5&6\\3&2&5\end{bmatrix}\begin{bmatrix}1\\-2\\3\end{bmatrix}=0$

Sol :

(iii) x निकालें यदि (Find x,  if) $\left[\begin{array}{lll}x&-5 & -1\end{array}\right]\left[\begin{array}{lll}1 & 0 & 2 \\ 0 & 2 & 1 \\ 2 & 0 & 3\end{array}\right]\left[\begin{array}{l}x \\ 4 \\ 1\end{array}\right]=0$

Sol :

(iv) x के किस मान के लिए 

(For what value of x)

$\left[\begin{array}{lll}1 & 2 & 1\end{array}\right]\left[\begin{array}{lll}1 & 2 & 0 \\ 2 & 0 & 1 \\ 1 & 0 & 2\end{array}\right]\left[\begin{array}{l}0 \\ 2 \\ x\end{array}\right]=0$?

Sol :

Question 24

(i) यदि (If) $\left[\begin{array}{lll}x & 4 & 1\end{array}\right]\left[\begin{array}{ccc}2 & 1 & 2 \\ 1 & 0 & 2 \\ 0 & 2 & -4\end{array}\right]\left[\begin{array}{r}x \\ 4 \\ -1\end{array}\right]=0$, x निकालें (find x)

Sol :

(ii) यदि (If) $\left[\begin{array}{lll}1 & 1 & x\end{array}\right]\left[\begin{array}{lll}1 & 0 & 2 \\ 0 & 2 & 1 \\ 2 & 1 & 0\end{array}\right]\left[\begin{array}{l}1 \\ 1 \\ 1\end{array}\right]=0$, x निकाले (find x)

Sol :

Question 25

एक 2×2 आवयूह B निकाले ताकि (Find the 2×2 matrix B such that)

$\left[\begin{array}{ll}6 & 5 \\ 5 & 6\end{array}\right] \mathrm{B}=\left[\begin{array}{ll}11 & 0 \\ 0 & 11\end{array}\right]$

Sol :

Let $B=\left[\begin{array}{ll}a & c \\ b & d\end{array}\right]$

$\left[\begin{array}{ll}6 & 5 \\ 5 & 6\end{array}\right]\left[\begin{array}{ll}a & c \\ b & d\end{array}\right]=\left[\begin{array}{ll}11 & 0 \\ 0 & 11\end{array}\right]$

$\left[\begin{array}{ll}6 a+5 b & 6 c+5 d \\ 5 a+6 b & 5 c+6 d\end{array}\right]=\left[\begin{array}{cc}11 & 0 \\ 0 & 1\end{array}\right]$

6a+5b=11..(i)
5a+6b=0..(ii)
6c+5d=0...(iii)
5c+6d=11...(iv)