Solution:

Let k divide both m + 2n and 3m + 4n. Then, we can write:

k divides m + 2n ⇒ m + 2n = a × k … (1)

k divides 3m + 4n ⇒ 3m + 4n = b × k … (2)

Multiply equation (1) by 3:

3m + 6n = 3a × k

Subtract equation (2) from this:

(3m + 6n) – (3m + 4n) = 3a × k – b × k

2n = (3a – b) × k

Thus, 2n is divisible by k.

Now subtract equation (1) multiplied by 2 from equation (2):

(3m + 4n) – 2(m + 2n) = b × k – 2a × k

m = (b – 2a) × k

Thus, m is divisible by k.

Therefore, k is a common divisor of 2n and m.

Answer: B

Solution:

We are given that (x/y) < ((x+3)/(y-3)). Rearranging the inequality:

(x/y) – ((x+3)/(y-3)) < 0

Multiply both sides by y(y-3):

x(y-3) – y(x+3) < 0

xy – 3x – yx – 3y < 0

-3(x + y) < 0

This implies that x + y > 0, or -x < y.

Thus, option D is correct: if y < 0, then -x < y.

Answer: D

Solution:

We are given a^m × bⁿ = 144. We can express 144 as 2⁴ × 3².

Maximizing n – m requires maximizing n and minimizing m. Let b = 3 and a = 2.

Thus, a^m × bⁿ = 2⁴ × 3² gives n = 580 and m = 1, so n – m = 579.

Answer: A

Solution:

The given figure forms a quadrilateral where we need to calculate the area. Using the formula for the area of a trapezium, we get:

Area = 1/2 × (sum of parallel sides) × height

The height is the distance along the Y-axis, and the parallel sides are the distances along the X-axis. After calculation, the area of the quadrilateral is found to be 45 square units.

Answer: 45