25 point for this question:

for which values of a and b the following is true for any real number x?

The values of a and b are and respectively.

Step-by-step explanation:

Given that

We solve this by partial fraction method.

Taking L.C.M. in the RHS, we get

To find the value of 'b', substitute x = 2. This would make 'a' vanish leaving an equation with 'b'.

Therefore, 2 - 4 = a(2 - 2) + b(2 + 9)

⇒ -2 = 0 + b(11)

Similarly, substitute x = -9 to find the value of 'a'.

⇒ -9 - 4 = a(7) + b(0)

.

Therefore, the values of 'a' and 'b'b are: =-13/7 and -2/11 respectively.

C and D are irrational.

Generally, any square root that doesn't reduce to a rational number is irrational. Some examples:

• √2 is irrational

• √4 = 2 is rational

• √(1/2) = 1/√2 = √2/2 is irrational

• √(1/9) = 1/√9 = 1/3 is rational

Numbers with non-terminating decimal expansions are rational if there is a repeating pattern after the decimal. The pattern in 4.737737773… is pretty obvious (one 7, one 3, two 7s, one 3, three 7s, one 3, etc) but it does not repeat, the number of 7s just keeps going up. Compare this to 4.737373… or 4.773773773… - these two numbers are rational in contrast.