Given: 28c2r : 24 c2r –4 = 225 : 11

To find: The value of r

Solution:

We can simplify the given ratio as follows:

28c2r : 24 c2r –4 = 225 : 11

⇒ (28c2r)/(24 c2r –4) = 225/11

⇒ (28c2r)/[24(r2) – 4] = 225/11

⇒ (28c2r)/[4(6r2 – 1)] = 225/11 [factorizing 24(r2) – 4 as 4(6r2 – 1)]

⇒ (7c2r)/(6r2 – 1) = 225/44 [dividing both sides by 4]

⇒ 7c2r = (225/44)(6r2 – 1) [cross-multiplying]

⇒ 7c2r = (225/44)(6r2) – (225/44) [expanding the bracket]

⇒ 7c2r = (675/44)r2 – (225/44)

⇒ 7c2r – (675/44)r2 = –(225/44)

⇒ (1/44)[308c2r – 675r2] = –(225/44)

⇒ 308c2r – 675r2 = –225

⇒ 308c2r = 675r2 – 225

⇒ 4c2r = 9r2 – 3 [dividing both sides by 77]

⇒ 4c2r = 3(3r2 – 1)

⇒ c2r = (3/4)(3r2 – 1)

Now, for the ratio to be valid, both the terms in the ratio must be positive. Therefore, we can conclude that:

3r2 – 1 > 0

⇒ 3r2 > 1

⇒ r2 > 1/3

⇒ r > 1/√3

Since r > 0, we can say that:

1/√3 < r="" />< />

Hence, the value of r cannot be determined uniquely from the given information.

Therefore, the correct answer is (d) none of these.