Let A denotes the sum of the roots of the equation 
+ 
= 3.
B denotes the value of the product of m and n, if 2 m = 3 and 3 n = 4.
C denotes the sum of the integral roots of the equation log 3x 
+ (log 3 x) 2 = 1.
(i) The value of A + B equals
Text Solution
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(i)c (ii)a (iii)b
Sol. Let log 4 x = t ⇒ 1 + t + 4(5 – 4t) = 3(5 – 4t)(1 + t)
21 – 15t = 15 + 3t – 12t 2
⇒ 12t 2 – 18t + 6 = 0 ⇒ 2t 2 – 3t + 1 = 0
t = 1, t = 1/2 ⇒ x = 4' or x = 4 1/2 = 2
A = sum of roots = 4 + 2 = 6
for B , 2 m = 3, 3 n = 4 ⇒ m = log 2 3, ⇒ n = log 3 4
⇒ Now mn = log 2 3.(log 3 2 2 ) = 2, B = 2
for C : 
= 1 ⇒ 1 – t + t 2 (1 + t) = 1 + t
⇒ t 2 + t 3 – t = t
t 3 + t 2 – 2t = 0 ⇒ t(t 2 + t – 2) = 0 ⇒ t = 0, t = –2, t = 1
x = 3 t ⇒ x = 3° = 1, x = 3' = 3
x = 3 –2 = 1/9
sum of integral root = C = 1 + 3 = 4
(i) A + B = 6 + 2 = 8
(ii) B + C = 2 + 4 = 6
(iii) A + C ÷ B = 6 + 4 ÷ 2 = 6 + 2 = 8
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