Given below are two statements: Statement I: The sum of integers from 110 to…
2023
Given below are two statements:
Statement I: The sum of integers from 110 to 630 is over 300,000.
Statement II: The ratio of the circumference of a circle with its area is inversely proportional to its diameter.
In the light of the above statements, choose the most appropriate answer from the options given below:
Answer: D. Statement I is incorrect but Statement II is correct — CONCEPTAn inclusive arithmetic series from a to b has n = b − a + 1 terms and sum S = n(a + b)/2. For a circle, C = 2πr, A = πr2, and d = 2r; proportionality…
- A.
Both Statement I and Statement II are correct
- B.
Both Statement I and Statement II are incorrect
- C.
Statement I is correct but Statement II is incorrect
- D.
Statement I is incorrect but Statement II is correct
Show answer & explanation
Correct answer: D
CONCEPT
An inclusive arithmetic series from a to b has n = b − a + 1 terms and sum S = n(a + b)/2.
For a circle, C = 2πr, A = πr2, and d = 2r; proportionality is identified after common factors are simplified.
APPLICATION
Count the integers inclusively: n = 630 − 110 + 1 = 521.
Compute the sum: S = 521(110 + 630)/2 = 521 × 370 = 192,770. This is below 300,000, so Statement I is incorrect.
Form the circle ratio: C/A = 2πr/(πr2) = 2/r.
Use r = d/2: 2/(d/2) = 4/d. Thus the ratio varies inversely with diameter, so Statement II is correct.
CROSS-CHECK
The average of the endpoints is 370, and 370 × 521 = 192,770, confirming the series result.
If d doubles, 4/d halves, confirming inverse proportionality.
RESULT
Therefore, Statement I is incorrect but Statement II is correct.