Suppose, the positive number be x.
According to the question,
x² – 21x = 100
⇒  x² – 21x – 100 = 0
⇒  x² – 25x + 4x – 100 = 0
⇒  x (x – 25) + 4 (x – 25) = 0
⇒  (x – 25) (x + 4) = 0
⇒  x = 25 because x ≠ –4

Correct Option: A

Suppose, the positive number be x.
According to the question,
x² – 21x = 100
⇒  x² – 21x – 100 = 0
⇒  x² – 25x + 4x – 100 = 0
⇒  x (x – 25) + 4 (x – 25) = 0
⇒  (x – 25) (x + 4) = 0
⇒  x = 25 because x ≠ –4

Let the numbers be x and y where x > y.
∴  x + y = 80
x – y = 20
∴  (x + y) (x – y) = 80 × 20
⇒  x² – y² = 1600

Correct Option: B

Let the numbers be x and y where x > y.
∴  x + y = 80
x – y = 20
∴  (x + y) (x – y) = 80 × 20
⇒  x² – y² = 1600

The smallest 6-digit number = 100000

Clearly,
316 < √100000 < 317
317 × 317 = 100489
∴  Required number = 100489

Correct Option: A

The smallest 6-digit number = 100000

Clearly,
316 < √100000 < 317
317 × 317 = 100489
∴  Required number = 100489

Now, 63520 – 16 = 63504
and √63504 = 252
∴  Required number = 16

Correct Option: D

Now, 63520 – 16 = 63504
and √63504 = 252
∴  Required number = 16

Let the numbers be a and b where a > b.
According to the question,
a – b = 9 .... (i)
and a² – b² = 207
⇒  (a + b) (a – b) = 207
⇒  9 (a + b) = 207

Correct Option: B

Let the numbers be a and b where a > b.
According to the question,
a – b = 9 .... (i)
and a² – b² = 207
⇒  (a + b) (a – b) = 207
⇒  9 (a + b) = 207