Directions:
In the following questions, a statement of assertion (A) is followed by a statement of reason (R). Mark the correct choice as:
- (A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).
- (B) Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).
- (C) Assertion (A) is true but Reason (R) is false.
- (D) Assertion (A) is false but Reason (R) is true.
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Question 1:
Assertion (A): The sequence 2, 4, 8, 16, ... is not an Arithmetic Progression.
Reason (R): In an AP, the difference between any two consecutive terms is constant.Solution: (A)
Step 1: Check differences: $4-2=2$, $8-4=4$. Since differences are not constant, it is not an AP. A is true.
Step 2: R states the definition of AP, which explains why the sequence is not an AP. -
Question 2:
Assertion (A): The 10th term of the AP 5, 8, 11, 14, ... is 32.
Reason (R): The $n^{th}$ term of an AP is given by $a_n = a + (n-1)d$.Solution: (A)
Step 1: $a=5, d=3$. $a_{10} = 5 + (10-1)3 = 5 + 27 = 32$. A is true.
Step 2: R is the correct formula used to find the term. -
Question 3:
Assertion (A): The sum of the first $n$ odd natural numbers is $n^2$.
Reason (R): The sum of the first $n$ even natural numbers is $n(n+1)$.Solution: (B)
Step 1: Odd numbers: 1, 3, 5... Sum $= \frac{n}{2}[2 + (n-1)2] = n^2$. A is true.
Step 2: Even numbers: 2, 4, 6... Sum $= \frac{n}{2}[4 + (n-1)2] = n(n+1)$. R is true.
Step 3: R is a correct statement but does not explain A. -
Question 4:
Assertion (A): If $S_n$ is the sum of the first $n$ terms of an AP, then its $n^{th}$ term $a_n$ is given by $a_n = S_n - S_{n-1}$.
Reason (R): The sum of first $n$ terms of an AP is given by $S_n = \frac{n}{2}[2a + (n-1)d]$.Solution: (B)
Step 1: A is a standard property relating sum and term. A is true.
Step 2: R is the formula for sum. R is true.
Step 3: While related, R is the formula for $S_n$, not the derivation of the relation in A. So B is correct. -
Question 5:
Assertion (A): The common difference of the AP whose $n^{th}$ term is $a_n = 3n + 7$ is 3.
Reason (R): The $n^{th}$ term of an AP is always a linear expression in $n$, and the coefficient of $n$ represents the common difference.Solution: (A)
Step 1: $a_1 = 10, a_2 = 13$. $d = 13 - 10 = 3$. A is true.
Step 2: R correctly explains that for linear $a_n = An + B$, $d = A$. -
Question 6:
Assertion (A): The sequence 1, 1, 1, 1, ... is an AP.
Reason (R): The common difference of an AP can be zero.Solution: (A)
Step 1: Differences are $1-1=0$. Constant difference means it is an AP. A is true.
Step 2: R explains that $d=0$ is allowed. -
Question 7:
Assertion (A): If $k, 2k-1$ and $2k+1$ are three consecutive terms of an AP, then $k = 3$.
Reason (R): For three numbers $a, b, c$ to be in AP, $2b = a + c$.Solution: (A)
Step 1: Using R: $2(2k-1) = k + (2k+1) \Rightarrow 4k - 2 = 3k + 1 \Rightarrow k = 3$. A is true.
Step 2: R is the condition used to solve A. -
Question 8:
Assertion (A): The sum of the series $1 + 2 + 3 + \dots + 100$ is 5050.
Reason (R): The sum of first $n$ natural numbers is given by $\frac{n(n+1)}{2}$.Solution: (A)
Step 1: Using R with $n=100$: Sum $= \frac{100(101)}{2} = 50(101) = 5050$. A is true.
Step 2: R is the correct formula for this specific AP. -
Question 9:
Assertion (A): The common difference of the AP 5, 2, -1, -4, ... is 3.
Reason (R): Common difference $d = a_2 - a_1$.Solution: (D)
Step 1: $d = 2 - 5 = -3$. Assertion says 3, which is incorrect. A is false.
Step 2: R is the correct definition. -
Question 10:
Assertion (A): The value of $n$ for which the $n^{th}$ terms of the APs 63, 65, 67... and 3, 10, 17... are equal is 13.
Reason (R): If two APs have the same $n^{th}$ term, their first terms must be equal.Solution: (C)
Step 1: $63 + (n-1)2 = 3 + (n-1)7 \Rightarrow 60 = 5(n-1) \Rightarrow 12 = n-1 \Rightarrow n=13$. A is true.
Step 2: R is false; APs can intersect at the $n^{th}$ term without having the same first term.