Mathematics Times – July 2019

To find M^1 we use (1). As already noted,

   2. Also adj M is given to us. Hence

1

1 1 1

1 1

8 6 2

2 2

5 3 1

M adjM

 

      

 

 

1 / 2 1 / 2 1 / 2

4 3 1

5 / 2 3 / 2 1 / 2

  

   

 

  

(5)

Putting this into (4), we get

1 / 2 1 / 2 1 / 2 1

4 3 1 1

5 / 2 3 / 2 1 / 2 1







      

        

     

          

(6)

which gives   1, 1 and 1 .Hence

     3. Therefore (D) is correct.

For (B), we already know M adjM I  23.

So,adjM M  2 ^1.

Taking inverses of both the sides, we get

 

1 1

2

adjM M



  (7)

As for the second term, we apply (1) with M

replaced by M^1. Since detM 2, det(M^1 )

1

2

 .

Therefore

   

1 1 1 1 1

2 2

adjM M M

  

    (8)

Adding (7) and (8), we see that (B) is correct
too.
Finally, for (C), we apply (1) again with M
replaced by M^2. Since

2 2

detM 2, det(M ) ( 2)   4. Therfore

adj M M I^2 ^2  4 3 (9)

We take determinants of both the sides. We

already know detM^2  4. The determinant

of 4 I 3 is 43  64 .Therefore

detadjM^2 equals 16 and not 81. So, (C) is

false.

SECTION-

13.Sol: Given

1 1 1 1 1

(1, 0, 0), , , 0 and , ,

2 2 3 3 3

A B  C 

   

Hence,

(^1) ˆ (^1) ˆ (^2) ˆ (^1) ˆ (^1) ˆ
and
2 2 3 3 3
AB i j AC i j k      
 
So, area
1 1 1 2 1
2 2 2 3 4
  AB AC   
 
1
2 2 3


now
SECTION-
13.Sol:
14.Sol:
(6 )^23 0.
4
  
14.Sol: By Bayes theorem
1 2
1 2
2
( )
( / )
( )
P E E
P E E
P E

 (1)
The sample space S has 29 elements in all
and they are equally likely. Therefore, the
probabilities of events are proportional to the
numbers of elements in the respective subsets.
Hence
1 2
1 2
2
( / )
E E
P E E
E

 (2)
It is easier to find E 2. As each of the 9 entries
of a matrix A S has only 0 and 1 as possible
values, their sum will equal 7 if and only if
exactly 7 of the entries are 1 and the remaining
2 are 0. This gives
2
(^9) 9.
36
2 2
E
 
   
 
(3)
For a matrix A E 2 , the determinant will