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Byju's Answer
Standard XII
Mathematics
Change of Variables
Prove that:1a...
Question
Prove that :
1
a
b
c
1
b
c
a
1
c
a
b
=
1
a
a
2
1
b
b
2
1
c
c
2
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Solution
Let
LHS
=
∆
=
1
a
b
c
1
b
c
a
1
c
a
b
=
1
abc
a
a
2
a
b
c
b
b
2
b
c
a
c
c
2
a
b
c
Applying
R
1
→
a
R
1
,
R
2
→
b
R
2
and
R
3
→
c
R
3
and
then
dividing
it
by
a
b
c
=
a
b
c
a
b
c
a
a
2
1
b
b
2
1
c
c
2
1
Taking
out
a
b
c
common
from
C
3
=
-
1
1
a
2
a
1
b
2
b
1
c
2
c
Interchanging
C
3
and
C
1
to
get
-
ve
value
of
original
determinant
=
-
1
-
1
1
a
a
2
1
b
b
2
1
c
c
2
Applying
C
2
↔
C
3
=
1
a
a
2
1
b
b
2
1
c
c
=
RHS
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0
Similar questions
Q.
(i) Solve the equation
∣
∣ ∣
∣
x
−
1
2
3
0
x
−
2
4
0
0
x
−
3
∣
∣ ∣
∣
(ii) Show that
∣
∣ ∣
∣
1
a
b
c
1
b
c
a
1
c
a
b
∣
∣ ∣
∣
=
∣
∣ ∣ ∣
∣
1
a
a
2
1
b
b
2
1
c
c
2
∣
∣ ∣ ∣
∣
Q.
The determinants
∣
∣ ∣
∣
1
a
b
c
1
b
c
a
1
c
a
b
∣
∣ ∣
∣
and
∣
∣ ∣ ∣
∣
1
a
a
2
1
b
b
2
1
c
c
2
∣
∣ ∣ ∣
∣
are not identically equal.
Q.
Prove that
∣
∣ ∣
∣
1
a
b
c
1
b
c
a
1
c
a
b
∣
∣ ∣
∣
=
(
a
−
b
)
(
b
−
c
)
(
c
−
a
)
.
Q.
show that
∣
∣ ∣
∣
1
a
b
c
1
b
c
a
1
c
a
b
∣
∣ ∣
∣
=
(
a
−
b
)
(
b
−
c
)
(
c
−
a
)
Q.
Evaluate :
1
a
b
c
1
b
c
a
1
c
a
b
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