An A.P. with 3rd term = -8 and 9th term = 4. Assertion (A): Common difference = -2. Reason (R): If first term of the A.P. is a, then (a + 8d) - (a + 2d) = -8 - 4. 1. A is true, R is false. 2. Both A and R are false. 3. Both A and R are true and R is correct reason for A. 4. Both A and R are true and R is incorrect reason for A.

Given, in an A.P. 3rd term = -8 and 9th term = 4. Let a be the first term of the A.P. and d be the common difference. By formula : ⇒ an = a + (n - 1)d ⇒ a3 = a + (3 - 1)d ⇒ -8 = a + 2d .................(1) ⇒ a9 = a + (9 - 1)d ⇒ 4 = a + 8d .................(2) Subtracting equation (1) from (2), we get : ⇒ (a + 8d) - (a + 2d) = 4 - (-8) ⇒ a + 8d - a - 2d = 4 + 8 ⇒ 6d = 12 ⇒ d = ⇒ d = 2. According to assertion d = -2, which is incorrect. So, assertion (A) is false. From above calculation, we get : ⇒ (a + 8d) - (a + 2d) = 4 - (-8) ⇒ (a + 8d) - (a + 2d) = 12 According to reason, ⇒ (a + 8d) - (a + 2d) = -8 - 4 ⇒ (a + 8d) - (a + 2d) = -12, which is incorrect. So, reason (R) is false. Hence, option 2 is the correct option.

For the given numbers $\sqrt{3}, \sqrt{12}, \sqrt{27}, \sqrt{48}…………$
Assertion (A):
To find whether these terms form an A.P. or not. Express each term as the product of a natural number and $\sqrt{3 }$ i.e.,
$\sqrt{3} = 1 \times \sqrt{3}, \sqrt{12} = 2\sqrt{3}, \sqrt{27} = 3\sqrt{3}, \sqrt{48} = 4\sqrt{3}$ , etc.
Reason (R): Since, for the given number difference between the consecutive term is same. It is an A.P.
A is true, R is false.
A is false, R is true.
Both A and R are true and R is correct reason for A.
Both A and R are true and R is incorrect reason for A.

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5, 8, 11, 14, …………… are in AP.
Assertion (A): $\dfrac{5}{2}, 4, \dfrac{11}{2}, 7,$ …………… are also in AP.
Reason (R): If each term of a given A.P. is divided by the same non zero number, the resulting sequence is an A.P..
A is true, R is false.
A is false, R is true.
Both A and R are true and R is correct reason for A.
Both A and R are true and R is incorrect reason for A.

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The n^th term of a sequence = 5n² - 3.
Statement (1): The sequence is an A.P.
Statement (2): If the n^th term of a sequence is not linear, the sequence does not form an A.P.
Both the statements are true.
Both the statements are false.
Statement 1 is true, and statement 2 is false.
Statement 1 is false, and statement 2 is true.

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The sum of first 10 term of an A.P. = 3 and the sum of its first 15 term = 16.
Statement (1): The sum of first five terms of the given AP equals to 16 - 3 = 13.
Statement (2): The sum of last 5 terms of the given AP equals to sum of first 15 term minus sum of first 10 terms.
Both the statements are true.
Both the statements are false.
Statement 1 is true, and statement 2 is false.
Statement 1 is false, and statement 2 is true.

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Mathematics

A.P.

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