MCQ 11 Mark
Statement-1 (A): $-5,-\frac{5}{2}, 0, \frac{5}{2}, \ldots .$. is an A.P.
Statement-2 (R): The terms of an A.P. cannot have both positive and negative rational numbers.
- A
Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1.
- B
Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1.
- ✓
Statement-1 is true, Statement-2 is false.
- D
Statement-1 is false, Statement-2 is true.
AnswerCorrect option: C. Statement-1 is true, Statement-2 is false.
View full question & answer→MCQ 21 Mark
Statement-1 (A): $a, b, c$ are in A.P. if and only if $2 b=a+c$
Statement-2 (R): The sum of first $n$ odd natural numbers is $n^2$.
- A
Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1.
- ✓
Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1.
- C
Statement-1 is true, Statement-2 is false.
- D
Statement-1 is false, Statement-2 is true.
AnswerCorrect option: B. Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1.
View full question & answer→MCQ 31 Mark
Statement-1 (A): If $a_n$ denotes the nth term of the A.P. 2, 7, 12, 17, ..., then $a_{5160}-a_{2020}=15150$.
Statement-2 (R): If $a_n$ denotes the nth term of an A.P. with common difference $d$, then $a_p-a_q=(p-q) d$.
- ✓
Statement-1 and Statement-2 are True; Statement-2 is a correct explanation for Statement-1.
- B
Statement-1 and Statement- 2 are True; Statement- 2 is not a correct explanation for Statement-1.
- C
Statement-1 is True, Statement-2 is False.
- D
Statement- 1 is False, Statement- 2 is True.
AnswerCorrect option: A. Statement-1 and Statement-2 are True; Statement-2 is a correct explanation for Statement-1.
(A)Statement-1 and Statement-2 are True; Statement-2 is a correct explanation for Statement-1.
Let $a$ be the first term of the A.P. whose common difference is $d$. Then,
$
\begin{aligned}
& a_p=a+(p-1) d \text { and } a_q=a+(q-1) d \\
\therefore \quad & a_p-a_q=|a+(p-1) d|-|a+(q-1) d|=(p-q) d
\end{aligned}
$
So, statement-2 is true. Using statement-2 for the A.P. 2, 7, 12, 17, ... we obtain
$
a_{5050}-a_{2020}=(5050-2020) \times 5=15150
$
So, statement- 1 is also true and statement-2 is a correct explanation for statement-1. Hence, option (a) is correct.
View full question & answer→MCQ 41 Mark
Statement-1 (A): $\quad a, b, c$ arc in A.P. iff $2 b=a+c$.
Statement-2 (R): In an A.P. the sum of the terms cquidistant from the beginning and the end is aluays same and is equal to the sum of first and least term.
- A
Statement-1 and Statement-2 are True; Statement-2 is a correct explanation for Statement-1.
- ✓
Statement-1 and Statement- 2 are True; Statement- 2 is not a correct explanation for Statement-1.
- C
Statement-1 is True, Statement-2 is False.
- D
Statement- 1 is False, Statement- 2 is True.
AnswerCorrect option: B. Statement-1 and Statement- 2 are True; Statement- 2 is not a correct explanation for Statement-1.
(B)Statement-1 and Statement- 2 are True; Statement- 2 is not a correct explanation for Statement-1.
$a, b, c$ are in A.P. $\Leftrightarrow b-a=c-b \Leftrightarrow 2 b=a+c$. So, statement -1 is true.
Let $a_1, a_2, \ldots, a_n$ be an A.P. with common difference $d$. Then,
$k^{\text {th }}$ term from the beginning $+k^{\text {th }}$ term from the end
$
=a_1+a_{n-k+1}=\left\{a_1+(k-1) d\right\}+\left\{a_1+(n-k+1-1) d\right\}=a_1+a_1+(n-1) d=a_1+a_n
$
So, statement -2 is correct. Clearly, statement -2 is not a correct explanation for statement-1. Hence, option (b) is correct.
View full question & answer→MCQ 51 Mark
Statement-1 (A): The sum of the $n$ terms of the A.P. $1,5,9,13, \ldots$ is $2 n^2+n$.
Statement-2 (R): Let $S_n$ denote the sum of $n$ terms of an A.P. with first term a and common difference $d$ such that $d=2 a$. Then for any natural number $m, \frac{S_{m n}}{S_m}$ is independent of $m$.
- A
Statement-1 and Statement-2 are True; Statement-2 is a correct explanation for Statement-1.
- B
Statement-1 and Statement- 2 are True; Statement- 2 is not a correct explanation for Statement-1.
- C
Statement-1 is True, Statement-2 is False.
- ✓
Statement- 1 is False, Statement- 2 is True.
AnswerCorrect option: D. Statement- 1 is False, Statement- 2 is True.
(D)Statement- 1 is False, Statement- 2 is True.
$1,5,9,13, \ldots$ is an A.P. with first term 1 and common difference 4 . Let $S_n$ denote the sum of its $n$ terms. Then,
$
S_n=\frac{n}{2}\{2 \times 1+(n-1) \times 4\}=2 n^2-n
$
So, statement-1 is not true i.e. it is false.
Now, $\quad \frac{S_{m n}}{S_m}=\frac{\frac{m m}{2}\{2 a+(m n-1) d\}}{\frac{m}{2}\{2 a+(m-1) d\}}=\frac{n\{(2 a-d)+m n d\}}{\{(2 a-d)+m d\}}=\frac{n(m n d)}{m d}=n^2$, if $d=2 a$.
Clearly, $\frac{S_{m m}}{S_m}$ is independent of $m$, if $d=2 a$. So, statement- 2 is true. Hence, option (d) is correct.
View full question & answer→MCQ 61 Mark
Statement-1 (A): The sum of 20 terms of the series $11,13,15,17, \ldots$ is 600.
Statement-2 (R) : The sum of first $n$ odd natural numbers is $n^2$.
- A
Statement-1 and Statement-2 are True; Statement-2 is a correct explanation for Statement-1.
- ✓
Statement-1 and Statement- 2 are True; Statement- 2 is not a correct explanation for Statement-1.
- C
Statement-1 is True, Statement-2 is False.
- D
Statement- 1 is False, Statement- 2 is True.
AnswerCorrect option: B. Statement-1 and Statement- 2 are True; Statement- 2 is not a correct explanation for Statement-1.
(B)Statement-1 and Statement- 2 are True; Statement- 2 is not a correct explanation for Statement-1.
Clearly, statement-2 is true (See example 26).
Let $S$ denote the sum of the series $11,13,15,17 \ldots$ upto 20 terms. Clearly, it is an A.P. with first term $a=11$ and common difference $d=2$.
$
\therefore \quad S=\frac{20}{2}\{2 \times 11+(20-1) \times 2\}=10(22+38)=600
$
So, statement- 1 is also true. But, statement- 2 is not a correct explanation for statement- 1 . Hence, option (b) is correct.
View full question & answer→MCQ 71 Mark
Statement-1 (A) : The sum of 20 terms of the A.P. $1,3,5,7, \ldots$ is 400.
Statement-2 (R) : The sum of first $n$ odd natural numbers is $n^2$.
- ✓
Statement-1 and Statement-2 are True; Statement-2 is a correct explanation for Statement-1.
- B
Statement-1 and Statement- 2 are True; Statement- 2 is not a correct explanation for Statement-1.
- C
Statement-1 is True, Statement-2 is False.
- D
Statement- 1 is False, Statement- 2 is True.
AnswerCorrect option: A. Statement-1 and Statement-2 are True; Statement-2 is a correct explanation for Statement-1.
(A)Statement-1 and Statement-2 are True; Statement-2 is a correct explanation for Statement-1.
$1,3,5,7,9, \ldots,(2 n-1)$ are first $n$ odd natural numbers. Let $S$ be their sum. Then,
$
\begin{array}{rlr}
S & =1+3+5+7+9+\ldots+(2 n-1) \\
\Rightarrow \quad S & =\frac{n}{2}(1+2 n-1)=n^2
\end{array} \quad\left[\text { Using } S_n=\frac{n}{2}\left(a_1+a_n\right)\right]
$
So, statement- 2 is true. Using this statement, we find that
$
1+3+5+7+\ldots \text { upto } 20 \text { terms }=20^2=400
$
So, statement -1 is also true and statement- 2 is a correct explanation for statement- 1 .
Hence, option (a) is correct.
View full question & answer→MCQ 81 Mark
Statement-1 (A): The sequence whose $n^{\text {th }}$ term is given by $a_n=7 n-5$ is an A.P. with common difference 7.
Statement-2 (R): A sequence is an A.P. with common difference ' $A$ ' if and only if its $n^{\text {th }}$ terms is of the form $a_n=A n+B$.
- ✓
Statement-1 and Statement-2 are True; Statement-2 is a correct explanation for Statement-1.
- B
Statement-1 and Statement- 2 are True; Statement- 2 is not a correct explanation for Statement-1.
- C
Statement-1 is True, Statement-2 is False.
- D
Statement- 1 is False, Statement- 2 is True.
AnswerCorrect option: A. Statement-1 and Statement-2 are True; Statement-2 is a correct explanation for Statement-1.
(A)Statement-1 and Statement-2 are True; Statement-2 is a correct explanation for Statement-1.
Statement-2 is true (See Example 2 on page 5.6 in main book). Statement-1 is also true and statement-2 is a correct explanation for statement-1. Hence, option (a) is correct.
View full question & answer→MCQ 91 Mark
Statement-1 (A): If $a_1, a_2, a_3, \ldots, a_n$ is an AP such that $a_1+a_4+a_7+\ldots+a_{16}=147$, then $a_1+a_6+a_{11}+a_{16}=98$
Statement-2 (R): In an A.P., the sum of the terms equidistant from the beginning andthe end is always same and is equal to the sum of first and last term.
- ✓
Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1.
- B
Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1.
- C
Statement-1 is true, Statement-2 is false.
- D
Statement-1 is false, Statement-2 is true.
AnswerCorrect option: A. Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1.
View full question & answer→MCQ 101 Mark
Statement-1 (A): The sum of first $n$ even natural numbers is $n(n+1)$.
Statement-2 (R): The sum of first $n$ odd natural numbers is $n(n-1)$.
- A
Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1.
- B
Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1.
- ✓
Statement-1 is true, Statement-2 is false.
- D
Statement-1 is false, Statement-2 is true.
AnswerCorrect option: C. Statement-1 is true, Statement-2 is false.
View full question & answer→MCQ 111 Mark
Statement-1 (A): The sum of $n$ terms of an AP with first and last terms as $a_1$ and $a_n$ respectively, is $S_n=\frac{n}{2}\left(a_1+a_n\right)$.
Statement-2 (R): The sum of the terms equidistant from the beginning and end in the A.P. $a_1, a_2, a_3, \ldots, a_{n-2}, a_{n-1}, a_n$ is equal to $a_1+a_n$.
- ✓
Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1.
- B
Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1.
- C
Statement-1 is true, Statement-2 is false.
- D
Statement-1 is false, Statement-2 is true.
AnswerCorrect option: A. Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1.
View full question & answer→MCQ 121 Mark
Statement-1 (A): The sum of $n$ terms of the series $\sqrt{5}+\sqrt{20}+\sqrt{45}+\sqrt{80}+\ldots$ is $\frac{\sqrt{5}}{2} n(n+1)$.
Statement-2 (R): The sum of first $n$ natural numbers is $\frac{n(n+1)}{2}$.
- ✓
Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1.
- B
Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1.
- C
Statement-1 is true, Statement-2 is false.
- D
Statement-1 is false, Statement-2 is true.
AnswerCorrect option: A. Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1.
View full question & answer→MCQ 131 Mark
Statement-1 (A): The $n^{\text {th }}$ term $a_n$ of an A.P., the sum of whose $n$ terms is $S_n$, is given by $a_n=S_n-S_{n-1}, n >1$.
Statement-2 (R): The common difference ' $d$ ' of an A.P., the sum of whose $n$ terms $S_n$ is given by $d=S_n-2 S_{n-1}+S_{n-2}, n >2$.
- A
Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1.
- ✓
Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1.
- C
Statement-1 is true, Statement-2 is false.
- D
Statement-1 is false, Statement-2 is true.
AnswerCorrect option: B. Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1.
View full question & answer→