Q. 1: Suppose, log(base3)x = log(base12)y = a, where x, y are positive numbers. If G is the geometric mean of x and y, and log(base6)G is equal to
A) √a
B) 2a
C) a/2
D) a
x=3^a and y=12^a
G = √(3^a * 12^a) = 6^a
Log (base 6) 6^a = a
Option (D)
x+1=x^2
Find out the roots of x = [1+/- root(5)]/2
X2 = [3 +/- √5]/2
X4 = [7 +/-3√5]/2
2×4 = 7 +/- 3√5
As the only option is 7 + 3√5 So, we go with that.
Option (D)
Q. 3:
A) 1/3
B) 2/3
C) 5/6
D) 7/6
Correct Answer:- C
Explanation:
Q. 4: If 9^(2x – 1) – 81^(x-1) = 1944, then x is
A) 3
B) 9/4
C) 4/9
D) 1/3
9^(2x-1) – 9^(2x-2) = 1944
It can be written as 3^(4x)/9 – 3^(4x)/81 = 1944
8(3^(4x)/81) = 1944
x =9/4
Option (B)
Q. 5: The number of solutions (x, y, z) to the equation x – y – z = 25, where x, y, and z are positive integers such that x ≤ 40, y ≤ 12, and z ≤ 12 is
A) 101
B) 99
C) 87
D) 105
x = 25 + y + z. The possible values of x, y, z and the corresponding number of values of y, z are tabulated below (x, y, z are positive integers).
We see that 27 ≤ x ≤ 40
The number of solutions is 1 + 2 + …… + 12 + 11 + 10 = 78 + 21 = 99
Option (B)
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