UPSEE 2018 Solved Paper

Given, a, b and c are pth, qth and rth terms of GP, respectively. Let a1 and k be the first term and common ratio of a GP.
Then,
Now,
=|

log‌a	p	1
log‌b	q	1
log‌c	r	1

| =|

log(a1.kp−1)	p	1
log(a1.kq−1)	q	1
log(a1.kr−1)	r	1

| [from Eq. (i)]
=|

log‌a1+log‌kp−1	p	1
log‌a1+log‌kq−1	q	1
log‌a1+log‌kr−1	r	1

| [∵log‌x‌y=log‌x+log‌y]
=|

log‌a1	p	1
log‌a1	q	1
log‌a1	r	1

| +|

p−1‌log‌k	p	1
q−1‌log‌k	q	1
r−1‌log‌k	r	1

| [by property of determinant ]
=log‌a1|

1	p	1
1	q	1
1	r	1

| +log‌k|

p−1	p	1
q−1	q	1
r−1	r	1

|
[∵ taking log a ‌1 common from C in I determinant and log k common from C1 in II determinant ]
On applying C1→C1+C3 in second determinant,
we get
=log‌a1(0)+log‌k|

p	p	1
q	q	1
r	r	1

|
[∵C1 and C3 are identical in first determinant ]
=0+log‌k(0)=0
[∵C1 and C2 are identical ]