MCQ 11 Mark
Directions: In the following questions, the Assertions $(A)$ and Reason(s) $(R)$ have been put forward. Read both the statements carefully and choose the correct alternative from the following:
Assertion $(A)$ The fourth term of a $GP$ is the square of its second term and the first term is $-3,$ then its $7^{th}$ term is equal to $2187.$
Reason $(R)$ Sum of first $10$ terms of the $AP\ 6, 8, 10, .....$ is equal to $150.$
- A
$A$ is true, $R$ is true; $R$ is acorrect explanation of $A.$
- B
$A$ is true, $R$ is true; $R$ is not a correct explanation of $A.$
- C
$A$ is true; $R$ is false
- ✓
$A$ is false; $R$ is true.
AnswerCorrect option: D. $A$ is false; $R$ is true.
Assertion Let a be the first term and $r$ be the common ratio of the given $GP.$ According to the question,
$T_4 = (T_2)^2 $ and $a = -3$
$\because T_4 = (T_2)^2$
$\therefore ar^3 = (ar)^2$
$\Rightarrow -3r^3 = (-3)^2 r^2 [\because a = -3]$
$\Rightarrow r = -3$
Now, $T_7 = ar^6 = -3(-3)^6 = -3 \times 729 = -2187$
Reason Given $AP$ is $6, 8, 10, ...$
$\because\text{a} = \text{6}, \text{d} = 8 - 6 = 2$
$\therefore\text{s}_{10}=\frac{10}{2}[2\times6+(10-1)\times2]$
$=5[12+18]$
$=5\times30=150$
View full question & answer→MCQ 21 Mark
Directions: In the following questions, the Assertions $(A)$ and Reason $(s)(R)$ have been put forward. Read both the statements carefully and choose the correct alternative from the following:
Assertion $;(A)$ The sum of first $n$ terms of the series $0.6 + 0.66 + 0.666 +......$ is $\frac{3}{2}\Big[\text{n}-\frac{1}{9}\Big(1-\frac{1}{10}\Big)\Big].$
Reason $;(R)$ General term of a $GP$ is $T_n = ar^{n-1}$, where $a =$ first term and $r =$common ratio.
- A
$A$ is true, $R$ is true; $R$ is acorrect explanation of $A.$
- ✓
$A$ is true, $R$ is true; $R$ is not a correct explanation of $A.$
- C
$A$ is true; $R$ is false
- D
$A$ is false; $R$ is true.
AnswerCorrect option: B. $A$ is true, $R$ is true; $R$ is not a correct explanation of $A.$
$A$ is true, $R$ is true; $R$ is not a correct explanation of $A.$
View full question & answer→MCQ 31 Mark
Directions: In the following questions, the Assertions $(A)$ and Reason $(s)(R)$ have been put forward. Read both the statements carefully and choose the correct alternative from the following: If $n^{th} $ term of a sequence is $a_n = 2n^2 - n + 1.$
Assertion $(A)$ First and second terms of same sequence are $2$ and $7$ respectively.
Reason $(R)$ Third and fourth terms of same sequence are $16$ and $29,$
- ✓
$A$ is true, $R$ is true; $R$ is acorrect explanation of $A.$
- B
$A$ is true, $R$ is true; $R$ is not a correct explanation of $A.$
- C
$A$ is true; $R$ is false
- D
$A$ is false; $R$ is true.
AnswerCorrect option: A. $A$ is true, $R$ is true; $R$ is acorrect explanation of $A.$
We have, $a_n = 2n^2 - n + 1$
Assertion Putting $n = 1,$ we get
$a_1 = 2 (1)^2 - 1 + 1 = 2 - 1 + 1 = 2$
Putting $n = 2,$ we get
$a_2 = 2(2)^2 - 2 + 1 = 7$
Reason Putting $n = 3,$ we get
$a_3 = 2(3)^2 - 3 + 1 = 18 - 3 + 1 = 16$
Putting $n= 4,$ we get
$a_4= 2(4)^2 - 4 + 1 = 32 - 4 + 1 = 29$
View full question & answer→MCQ 41 Mark
Directions: In the following questions, the Assertions $(A)$ and Reason$(s) (R)$ have been put forward. Read both the statements carefully and choose the correct alternative from the following:
Assertion $(A)$ The first three terms of the sequence are $\frac{3}{2},\text{x},\frac{21}{2}$ whose $n^{th}$ term is $\text{a}_\text{n}=\frac{\text{n}(\text{n}^2+5)}{4}.$Then $\text{x}=\frac{9}{2}$
Reason $(R)$ The third term of the sequence whose nth term is $\text{a}_\text{n}=(-1)^\text{n-1}5^\text{n+1}$ is $620.$
- A
$A$ is true, $R$ is true; $R$ is acorrect explanation of $A.$
- B
$A$ is true, $R$ is true; $R$ is not a correct explanation of $A.$
- ✓
$A$ is true; $R$ is false
- D
$A$ is false; $R$ is true.
AnswerCorrect option: C. $A$ is true; $R$ is false
Assertion We have, $\text{a}_\text{n}=\frac{\text{n}^2+5}{4}$
Now, we need to find $x$ which is second term of the sequence, so put $m = 2$ in $a_n.$
$\therefore \text{a}_2=\frac{2(4+5)}{4}=\frac{18}{4}=\frac{9}{2}$
Reason We have, $\text{a}_\text{n} =(-1)^{\text{n}-1}\ 5^{\text{n}+1}$
$\therefore\text{a}_3=(-1)^{3-1}5^4=625$
View full question & answer→MCQ 51 Mark
Directions: In the following questions, the Assertions $(A)$ and Reason$(s) (R)$ have been put forward. Read both the statements carefully and choose the correct alternative from the following:
Assertion $(A)$ first and second terms of same sequence are $2$ and $7$ repectively.
Reason $(R)$ Third and fourt terms of same sequence are $16$ and $29.$ respectively.
- A
Both assertion and reason are true and reason is the correct explanation of assertion.
- ✓
Both assertion and reason are true but reason is not the correct explanation of assertion.
- C
Assertion is true but reason is false.
- D
Assertion is false but reason is true
AnswerCorrect option: B. Both assertion and reason are true but reason is not the correct explanation of assertion.
View full question & answer→MCQ 61 Mark
Directions: In the following questions, the Assertions $(A)$ and Reason$(s) (R)$ have been put forward. Read both the statements carefully and choose the correct alternative from the following:
Assertion $(A)$ The sum of the series $\frac{3}{\sqrt{5}}+\frac{4}{\sqrt{5}}+\sqrt{5}....25$ terms is $75\sqrt{5}.$
Reason $(R)$ If $27, x, 3$ are in $GP,$ then $\text{x}=\pm\ 4.$
- A
$A$ is true, $R$ is true; $R$ is a correct explanation of $A.$
- B
$A$ is true, $R$ is true; $R$ is not a correct explanation of $A.$
- ✓
$A$ is true; $R$ is false.
- D
$A$ is false; $R$ is true.
AnswerCorrect option: C. $A$ is true; $R$ is false.
Assertion
Let $\text{s}_\text{n}=\frac{3}{\sqrt{5}}+\frac{4}{\sqrt{5}}+\sqrt{5}...25^\text{th}$ terms
$\Rightarrow\text{s}_\text{n}=\frac{3}{\sqrt{5}}+\frac{4}{\sqrt{5}}+\frac{5}{\sqrt{5}}...25^\text{th}$ terms
Clearly, the successive difference of the terms is same.
So, $\text{RHS}$ of the above series forms an $AP,$ with first term,
$\text{a}=\frac{3}{\sqrt{5}}$ and common $V5$ difference,
$\text{d}=\frac{4}{\sqrt{5}}-\frac{3}{\sqrt{5}}=\frac{1}{\sqrt{5}}.$
$\therefore\text{x}_{25}=\frac{25}{2}\Big[2\times\frac{3}{\sqrt{5}}+(25-1)\frac{1}{\sqrt{5}}\Big]$
$=25\Big[\frac{3}{\sqrt{5}}+\frac{12}{\sqrt{5}}\Big]$
$=25\times\frac{15}{5}\times\sqrt{5}$
$=75\sqrt{5}$
Reason Given, $27, x , 3$ are in $GP.$
$\therefore\frac{\text{x}}{27}=\frac{3}{\text{x}}$
$\Rightarrow\text{x}^2=81$
$\Rightarrow\text{x}=\pm9$
View full question & answer→MCQ 71 Mark
Directions: In the following questions, the Assertions $(A)$ and Reason$(s) (R)$ have been put forward. Read both the statements carefully and choose the correct alternative from the following:
Assertion $(A)$ If the sequence of even natural number is $2, 4, 6, 8, ...,$ then $m^{th}$ term of the sequence is $a_n$ given by $a_n = 2n,$ where $n\ € N.$
Reason $(R)$ If the sequence of odd natural numbers is $1, 3, 5, 7, ...,$ then $m^{th}$ term of the sequence is given by $a_n = 2n - 1,$ wherene $n\ € N.$
- A
$A$ is true, $R$ is true; $R$ is acorrect explanation of $A.$
- ✓
$A$ is true,$ R$ is true; $R$ is not a correct explanation of $A.$
- C
$A$ is true; $R $ is false
- D
$A$ is false; $R$ is true.
AnswerCorrect option: B. $A$ is true,$ R$ is true; $R$ is not a correct explanation of $A.$
Assertion It is given that
$a_1= 2 = 2 \times 1$
$a_2 = 4 = 2 \times 2$
$a_3 = 6 = 2 \times 3$
$a_4 = 8 = 2 \times 4$
Reason It is given that
$a_1 = 1 = 2 - 1$
$a_2 = 3 = 2 \times 2 - 1$
$a_3 = 5 = 2 \times 3 - 1$
$a_4 = 7 = 2 \times 4 - 1$
View full question & answer→MCQ 81 Mark
Directions: In the following questions, the Assertions $(A)$ and Reason$(s) (R)$ have been put forward. Read both the statements carefully and choose the correct alternative from the following:
Assertion $(A)$ The sum of first $20$ terms of an $AP, 4, 8, 12, ...$ is equal to $840.$
Reason $(R)$ Sum of first $m$ terms of an $AP$ is given by $\text{s}_\text{n}=\frac{\text{n}}{2}[2\text{a}+(\text{n-1)}\text{d}],$ where $a =$ first term and $d =$ common difference.
- ✓
$A$ is true, $R$ is true; $R$ is a correct explanation of $A.$
- B
$A$ is true, $R$ is true; $R$ is not a correct explanation of $A.$
- C
$A$ is true; $R$ is false.
- D
$A$ is false; $R$ is true.
AnswerCorrect option: A. $A$ is true, $R$ is true; $R$ is a correct explanation of $A.$
Assertion Given $AP$ is $4, 8, 12, ...$
$\therefore \text{a}=4,\text{d}=8-4=4 $
Now, $\text{s}_{20}=\frac{20}{2}[2\times4+(20-1)\times4]$
$=10[8+76]$
$=840$
View full question & answer→MCQ 91 Mark
Directions: In the following questions, the Assertions $(A)$ and Reason$(s) (R)$ have been put forward. Read both the statements carefully and choose the correct alternative from the following:
Assertion $(A)$ The sum of first $6$ terms of the $GP\ 4, 16, 64, ...$ is equal to $5460.$
Reason $(R)$ Sum of first $n$ terms of the $G. P$ is given by $\text{s}_\text{n}=\frac{\text{a}(\text{r}^\text{n}-1)}{\text{r}-1},$ where $a =$ first term $r =$ common ratio and $|r| >1$
- ✓
$A$ is true, $R$ is true; $R$ is a correct explanation of $A.$
- B
$A$ is true, $R$ is true; $R$ is not a correct explanation of $A.$
- C
$A$ is true; $R$ is false.
- D
$A$ is false; $R$ is true.
AnswerCorrect option: A. $A$ is true, $R$ is true; $R$ is a correct explanation of $A.$
Assertion Given $GP\ 4, 16, 64, ...$
$\therefore\text{a}=4,\text{r}=\frac{16}{4}=4>1$
$\therefore\text{s}_6=\frac{4(4)^6-1)}{4-1}=\frac{4(4095)}{3}$
$=5460$
View full question & answer→MCQ 101 Mark
Directions: In the following questions, the Assertions $(A)$ and Reason$(s) (R)$ have been put forward. Read both the statements carefully and choose the correct alternative from the following:
Assertion $(A)$ The sum of first m terms of the series $5 + 55 + 555 +...$ is $\frac{5}{9}\Big[\frac{10(10^\text{n}-1)}{9}-\text{n}\Big].$
Reason $(R)$ General term of an $AP$ is $T_n = a + (n - 1) d,$ where $a =$ first term and $d =$ common difference.
- A
$A$ is true, $R$ is true; $R$ is acorrect explanation of $A$.
- ✓
$A$ is true, $R$ is true; $R$ is not a correct explanation of $A.$
- C
$A$ is true; $R$ is false
- D
$A$ is false; $R$ is true.
AnswerCorrect option: B. $A$ is true, $R$ is true; $R$ is not a correct explanation of $A.$
Let $S = 5 + 55 + 555 +...$ up to $m$ terms
$= 5(1 + 11 + 111 +...$ upton terms$)$
$=\frac{5}{9}(9+99+999+... $up to $n$ terms
$=\frac{5}{9}[(10-1)+(100-1)+(1000-1)+...$ up to $n$ terms $] $
$=\frac{5}{9}[(10+100+1000+...$ up to $n$ terms $-(1+1+1+...$ up to $n$ term $)]$
$=\frac{5}{9}\Big[\frac{10(10^\text{n}-1)}{10-1}-\text{n}\Big]$
$\Big[\because\text{sum of GP }=\frac{\text{a}(\text{r}^\text{n}-1)}{\text{r}-1},\text{r}>1\text{ and}\sum1=\text{n}\Big]$
$=\frac{5}{9}\Big[\frac{10(10^\text{n}-1)}{9}-\text{n}\Big]$
View full question & answer→MCQ 111 Mark
Directions: In the following questions, the Assertions $(A)$ and Reason$(s) (R)$ have been put forward. Read both the statements carefully and choose the correct alternative from the following:
Assertion $(A)$ If $n^{th}$ term of a sequence is $\text{a}_\text{n}=\frac{\text{n}^2}{2^\text{n}},$ then its $7^{th}$ term is $\frac{49}{128}.$
Reason $(R)$ If $n^{th}$ term of a sequence is $\text{a}_\text{n}=\frac{\text{n}\text{(n}-2)}{\text{n}+3},$ then its $20^{th}$ term is $\frac{323}{22}.$
- A
$A$ is true, $R$ is true; $R$ is a correct explanation of $A.$
- B
$A$ is true, $R$ is true; $R$ is not a correct explanation of $A.$
- ✓
$A$ is true; $R$ is false.
- D
$A$ is false; $R$ is true.
AnswerCorrect option: C. $A$ is true; $R$ is false.
Assertion We have, $\text{a}_\text{n} \frac{\text{n}^2}{2^\text{n}}$
Putting $n = 7, \text{a}_7=\frac{7^2}{2^7}=\frac{49}{128}$
Reason We have, $\text{a}_\text{n} \frac{\text{n}\text{(n}-2)}{\text{n}+3}$
Puttingn $=20, \text{a}_{20}=\frac{20(20-2)}{20+3}$
$\Rightarrow\text{a}_{20}=\frac{20\times18}{23}$
$=\frac{360}{23}$
View full question & answer→MCQ 121 Mark
Directions: In the following questions, the Assertions $(A)$ and Reason$(s) (R)$ have been put forward. Read both the statements carefully and choose the correct alternative from the following:
Assertion $(A)$ If the numbers $\frac{-2}{7},\text{k},\frac{-7}{2}$ are in $GP,$ then $\text{k}=\pm=1.$
Reason $(R)$ If $a_1, a_2, a_3$ are in $GP,$ then $\frac{\text{a}^2}{\text{a}_1}=\frac{\text{a}_3}{\text{a}_2}.$
- ✓
$A$ is true, $R$ is true; $R$ is acorrect explanation of $A.$
- B
$A$ is true, $R$ is true; $R$ is not a correct explanation of $A.$
- C
$A$ is true; $R$ is false
- D
$A$ is false; $R$ is true.
AnswerCorrect option: A. $A$ is true, $R$ is true; $R$ is acorrect explanation of $A.$
Assertion If $-\frac{2}{7},\text{k},-\frac{7}{2}$ are in $GP.$
Then, $\frac{\text{a}_2}{\text{a}_1}=\frac{\text{a}_3}{\text{a}_2}$
$\Big[\because\text{ common ratio }\text{(r)}=\frac{\text{a}_2}{\text{a}_1}=\frac{\text{a}_3}{\text{a}_2}=\frac{\text{a}_4}{\text{a}_3}= ...\Big]$
$\therefore\frac{\text{k}}{\frac{-2}{7}}=\frac{\frac{-7}{2}}{\text{k}}$
$\Rightarrow\frac{7}{-2}\text{k}=\frac{-7}{2}\times\frac{1}{\text{k}}$
$\Rightarrow7\text{k}\times2\text{k}=-7\times(-2)$
$\Rightarrow14\text{k}^2=14$
$\text{k}^2=1\Rightarrow\text{k}=\pm1$
View full question & answer→MCQ 131 Mark
Directions: In the following questions, the Assertions $(A)$ and Reason(s) $(R)$ have been put forward. Read both the statements carefully and choose the correct alternative from the following:
Assertion $(A)$ If the $n^{th} $ term ofa sequence is $a_n = 4n - 3.$ Here, $a_{17}$ and $a_{24}$ are $65$ and $93$ respectively.
Reason $(R)$ If the $n^{th}$ term of a sequence is $a_n =(-1)^{n-1}n^3.$ Here, $9^{th}$ term is $729.$
- A
$A$ is true, $R$ is true; $R$ is acorrect explanation of $A.$
- ✓
$A$ is true, $R$ is true; $R$ is not a correct explanation of $A.$
- C
$A$ is true; $R$ is false
- D
$A$ is false; $R$ is true.
AnswerCorrect option: B. $A$ is true, $R$ is true; $R$ is not a correct explanation of $A.$
Assertion $a_n = 4n - 3$
Then, $a_{17} = 4 (17) - 3 = 65$
and $a_{24}= 4 (24) - 3 = 93$
Reason $a_9 = (-1)^{9 - 1} \times (9)^3$
$= (-1)^8\times 729$
$= 729$
View full question & answer→MCQ 141 Mark
Directions: In the following questions, the Assertions $(A)$ and Reason$(s) (R)$ have been put forward. Read both the statements carefully and choose the correct alternative from the following:
Assertion $(A)$ If the sum of first two terms of an infinite $GP$ is $5$ and each term is three times the sum of the succeeding terms, then the common ratio is $\frac{1}{4}.$
Reason $(R)$ In an $AP\ 3, 6, 9, 12.....$ the $10^{th}$ term is equal to $30.$
- A
$A$ is true, $R$ is true; $R$ is a correct explanation of $A.$
- B
$A$ is true, $R$ is true; $R$ is not a correct explanation of $A.$
- C
$A$ is true; $R$ is false.
- ✓
$A$ is false; $R$ is true.
AnswerCorrect option: D. $A$ is false; $R$ is true.
View full question & answer→MCQ 151 Mark
Directions: In the following questions, the Assertions $(A)$ and Reason$(s) (R)$ have been put forward. Read both the statements carefully and choose the correct alternative from the following:
Assertion $(A)$ The sum of first $23$ terms of the $AP\ 16, 11, 6, ...... is - 897.$
Reason $(R)$ The sum of first $22$ terms of the $AP\ x + y, x - y, x - 3y, .....$ is $22 [x - 20y].$
- A
$A$ is true, $R$ is true; $R$ is acorrect explanation of $A.$
- ✓
$A$ is true, $R$ is true; $R$ is not a correct explanation of $A.$
- C
$A$ is true; $R$ is false
- D
$A$ is false; $R$ is true.
AnswerCorrect option: B. $A$ is true, $R$ is true; $R$ is not a correct explanation of $A.$
Assertion Given $AP$ is $16, 11, 6, ...$ Here,
$\text{a} =16,\text{d} =11-16=-5 $
$\because\text{s}_\text{n}=\frac{\text{n}}{2}[2\text{a}+(\text{n}-1)\text{d}]$
$\therefore\text{s}_{23}=\frac{23}{2}[2\times16+(23-1)(-5)]$
$=\frac{23}{2}[32+(22)(-5)]$
$=\frac{23}{2}[32-110]$
$=\frac{23}{2}[-78]$
$=-897$
Reason Given $AP$ is $x + y, x - y, x - 3y, ...$ Here, $a = x + y$
$\text{d} = (\text{x} - \text{y}) - (\text{x} + \text{y}) = -2\text{y}$
$\because\text{s}_\text{n}=\frac{\text{n}}{2}[2\text{a}+(\text{n}-1)\text{d}]$
$\therefore\text{s}_{22}\frac{22}{2}[2\times(\text{x}+\text{y})+(22-1)(-2)]$
$=11[2\text{x}+2\text{y}+(21)(-2\text{y})]$
$=11[2\text{x}+2\text{y}-42\text{y}]$
$=11[2\text{x}-40\text{y}]$
$=22[\text{x}-20\text{y}]$
View full question & answer→MCQ 161 Mark
Directions: In the following questions, the Assertions $(A)$ and Reason $(s)(R)$ have been put forward. Read both the statements carefully and choose the correct alternative from the following:
Assertion $(A)$ if $5^{th}$ and $8^{th}$ terms of a $GP$ be $48$ and $384$ respectively, then the common ratio of $GP$ is $2$.
Reason $(R)$ if $18, x,14$ are in $AP, $ then $x = 16.$
- A
Both assertion and reason are true and reason is the correct explanation of assertion.
- ✓
Both assertion and reason are true but reason is not the correct explanation of assertion.
- C
Assertion is true but reason is false.
- D
Assertion is false but reason is true
AnswerCorrect option: B. Both assertion and reason are true but reason is not the correct explanation of assertion.
Both assertion and reason are true but reason is not the correct explanation of assertion.
View full question & answer→