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(i) 3 3

3 3

3 3

3 3

c d

b c

b c

a b









(^) (ii) (a^2 b^2 c^2 )(b^2 c^2 d^2 ) (abbccd)^2

11. If
    a b

ab

x



4

, show that, 2 ,.

2

2

2

2

a b

x b

x b

x a

x a

z











12. If 3 3

3 3

1 1

1 1

  

  

m m

m m

x , prove that, x^3  3 mx^2  3 xm 0

13. If
    a b a b

a b a b

x

2 3 2 3

2 3 2 3

  

  

, show that,^3430.

bx^2  ax b

14. If 2

2

2 2

2 2

( )

( )

a c

a b

b c

a b









, prove that,a,b,c are ordered proportional.

15. If
    a b

z

c a

y

b c

x







, prove that,.

x y z

c

z x y

b

y z x

a

 

 

 

16. If
    c

ay bx

b

cx az

a

bz cy 





, prove that,.

c

z

b

y

a

x

17. If
    c a

c a b

b c

b c a

a b

a b c



 



 



 

and abcz 0 , prove that,

a b c.

18. If
    za xb yc

z

ya zb xc

y

xa yb zc

x

 

 

 

and xyzz 0 , show

that, each of the ratio is.

1

abc

19. If(abc)p (bca)q (cab)r (abc)s,

prove that,.

1 1 1 1

q r s p

 

20. If lx my nz, show that, 2 2 2.

2 2 2

n

lm

m

nl

l

mn

xy

z

zx

y

yz

x

   

21. If 2

2

b

a

q

p

and

a q

a q

b

a





, show that,.

q

p q

a

p q 



11 ⋅4. Successive Ratio
Let Roni’s earning be Tk. 1000, Soni’s earning be Tk. 1500 and Somi’s earning be
Tk. 2500. Here, Roni’s earning : Soni’s earning = 1000 : 1500 = 2 : 3 ; Soni’s