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NDA Math Determinants
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© examsnet.com
Question : 15
Total: 68
If
a
1
,
a
2
,
a
3
a
9
are in GP, then what is the value of the following determinant?
|
ln
a
1
ln
a
2
ln
a
3
ln
a
4
ln
a
5
ln
a
6
ln
a
7
ln
a
8
ln
a
9
|
[NDA-I 2021]
0
1
2
4
Validate
Solution:
👈: Video Solution
Concept:
Let us consider sequence
a
1
,
a
2
,
a
3
.
.
.
a
n
is a G.P.
So, common ratio
=
r
=
a
p
+
1
a
p
,
Where
p
∈
Positive Integer Logarithmic formula:
log
a
−
log
b
=
log
a
log
b
Calculation:
Let
r
be the common ratio of the given GP
Now,
|
ln
a
1
ln
a
2
ln
a
3
ln
a
4
ln
a
5
ln
a
6
ln
a
7
ln
a
8
ln
a
9
|
Apply,
C
2
→
C
2
−
C
1
and
C
3
→
C
3
−
C
1
=
|
ln
a
1
ln
a
2
−
ln
a
1
ln
a
3
−
ln
a
2
ln
a
4
ln
a
5
−
ln
a
4
ln
a
6
−
ln
a
5
ln
a
7
ln
a
8
−
ln
a
7
ln
a
9
−
ln
a
8
|
|
ln
a
1
lnr
lnr
ln
a
4
lnr
lnr
ln
a
7
lnr
lnr
|
[
Since
a
p
+
1
a
p
=
r
and
log
a
p
+
1
−
log
a
p
=
log
a
p
+
1
a
p
=
log
r
]
=
0
[
Since
C
3
=
C
2
]
© examsnet.com
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