Question 11 Mark
Find the coefficient $x^6$ in the expression of $e^{2 x}$ using series expansion.
View full question & answer→Question 21 Mark
Find $k$ so that $k-1, k, k+2$ are consecutive terms of a G.P.
View full question & answer→Question 31 Mark
Find $\sum_{r=1}^n \frac{1^3+2^3+3^3+\ldots r^3}{(r+1)^2}$
View full question & answer→Question 41 Mark
Find $\sum_{r=1}^n\left(5 r^2+4 r-3\right)$
View full question & answer→Question 51 Mark
For a G.P. $a=\frac{4}{3}$ and $t_7=\frac{243}{1024}$, find the value of $r$.
View full question & answer→Question 61 Mark
Find the sum of the first 5 terms of the G.P. whose first term is 1 and the common ratio is$\frac{2}{3}$.
View full question & answer→Question 71 Mark
Find the sum $2^2+4^2+6^2+8^2+\ldots .$. upto $n$ terms.
View full question & answer→Question 81 Mark
Find G.M. of two positive numbers whose A.M. and H.M. are 75 and 48.
Question 91 Mark
Find H.M. of two positive numbers whose A.M. and G.M. are $\frac{15}{2}$ and 6 .
View full question & answer→Question 101 Mark
Find A.M. of two positive numbers whose G.M. and H.M. are 4 and $\frac{16}{5}$ respectively.
View full question & answer→Question 111 Mark
Find the nth term and hence find the 8th term of the following HPs.$\frac{1}{5}, \frac{1}{10}, \frac{1}{15}, \frac{1}{20}, \ldots$
View full question & answer→Question 121 Mark
Find the nth term and hence find the 8th term of the following HPs.$\frac{1}{4}, \frac{1}{6}, \frac{1}{8}, \frac{1}{10}, \ldots$
View full question & answer→Question 131 Mark
Find the nth term and hence find the 8th term of the following HPs.$\frac{1}{2}, \frac{1}{5}, \frac{1}{8}, \frac{1}{11}, \ldots$
View full question & answer→Question 141 Mark
Verify whether the following sequences are H.P.$5, \frac{10}{17}, \frac{10}{32}, \frac{10}{47}, \ldots$
View full question & answer→Question 151 Mark
Verify whether the following sequences are H.P.$\frac{1}{3}, \frac{1}{6}, \frac{1}{12}, \frac{1}{24}, \ldots$
View full question & answer→Question 161 Mark
Verify whether the following sequences are H.P.$\frac{1}{3}, \frac{1}{5}, \frac{1}{7}, \frac{1}{9}, \ldots$
View full question & answer→Question 171 Mark
If the first term of the G.P. is 6 and its sum to infinity is $\frac{96}{17}$, find the common ratio.
Answer$a=6$, sum to infinity $=\frac{96}{17} \ldots . .[$ Given $]$Sum to infinity $=\frac{ a }{1- r }$
$\begin{array}{ll}\therefore & \frac{96}{17}=\frac{6}{1-r} \\ \therefore & \frac{16}{17}=\frac{1}{1-r} \\ \therefore & 16-16 r=17 \\ \therefore & 16 r=16-17 \\ \therefore & r=\frac{-1}{16}\end{array}$
View full question & answer→Question 181 Mark
If the common ratio of a G.P. is $\frac{2}{3}$ and the sum to infinity is $12,$ find the first term.
Answer$r=\frac{2}{3}$, sum to infinity $=12 \ldots .$. [Given]
Sum to infinity $=\frac{ a }{1- r }$
$12=\frac{a}{1-\frac{2}{3}}$
$a=12 \times \frac{1}{3}$
$\therefore a=4$
View full question & answer→Question 191 Mark
Determine whether the sum to infinity of the following G.P.s exist, if exists find them.
9, 8.1, 7.29, ……
Question 201 Mark
Determine whether the sum to infinity of the following G.P.s exist, if exists find them.$\frac{1}{5}, \frac{-2}{5}, \frac{4}{5}, \frac{-8}{5}, \frac{16}{5}, \ldots$
View full question & answer→Question 211 Mark
Determine whether the sum to infinity of the following G.P.s exist, if exists find them.$-3,1, \frac{-1}{3}, \frac{1}{9}, \ldots$
View full question & answer→Question 221 Mark
Determine whether the sum to infinity of the following G.P.s exist, if exists find them.$2, \frac{4}{3}, \frac{8}{9}, \frac{16}{27}, \ldots$
View full question & answer→Question 231 Mark
Determine whether the sum to infinity of the following G.P.s exist, if exists find them.$\frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \frac{1}{16}, \ldots$
View full question & answer→Question 241 Mark
The value of a house appreciates 5% per year. How much is the house worth after 6 years if its current worth is Rs. 15 Lac. [Given: $(1.05)^5=1.28,(1.05)^6=1.34$ ]
View full question & answer→Question 251 Mark
Find$\sum_{r=1}^{10} 5 \times 3^r$
View full question & answer→Question 261 Mark
Find$\sum_{r=1}^{10}\left(3 \times 2^r\right)$
View full question & answer→Question 271 Mark
For a G.P.
If $a=2, r=3, S_n=242$, find $n$.
Answer$a=2, r=3, S_n=242$
$S_n=a\left(\frac{r^2-1}{r-1}\right), \text { for } r>1$
$\therefore 242=2\left(\frac{3^n-1}{3-1}\right)$
$\therefore 242=3^n-1$
$\therefore 3^n=243$
$\therefore 3^n=3^5$
$\therefore \quad n=5$
View full question & answer→Question 281 Mark
For a G.P.If $S_5=1023, r=4$, find $a$.
Answer$ r=4, S_5=1023$
$ S_n=a\left(\frac{r^n-1}{r-1}\right), \text { for } r>1$
$\therefore S_5=a\left(\frac{4^5-1}{4-1}\right)$
$\therefore 1023=a\left(\frac{1024-1}{3}\right)$
$\therefore 1023=\frac{a}{3}(1023)$
$\therefore a=3$
View full question & answer→Question 291 Mark
For the following G.P.s, find $S_n$ : √5, -5, 5√5, -25, …….
Answer$a=\sqrt{5}, r=\frac{-5}{\sqrt{5}}=-\sqrt{5}<1$
$S_n=\frac{a\left(1-r^n\right)}{1-r}, \text { for } r<1$
$=\frac{\sqrt{5}\left[1-(-\sqrt{5})^{ n }\right]}{1-(-\sqrt{5})}$
$=\frac{\sqrt{5}}{1+\sqrt{5}}\left[1-(-\sqrt{5})^{ n }\right]$
$=\frac{-\sqrt{5}}{(\sqrt{5}+1)}\left[(-\sqrt{5})^{ n }-1\right]$
View full question & answer→Question 301 Mark
For the following G.P.s, find $S_n.: 0.7, 0.07, 0.007, …….$
Answer$a=0.7=\frac{7}{10}, r=\frac{0.07}{0.7}=\frac{1}{10}<1$
$S_n=\frac{a\left(1-r^n\right)}{1-r}, \text { for } r<1$
$=\frac{\frac{7}{10}\left[1-\left(\frac{1}{10}\right)^n\right]}{1-\frac{1}{10}}=\frac{7}{9}\left(1-\frac{1}{10^n}\right)$
View full question & answer→Question 311 Mark
For the following G.P.s, find $S_n$. 3, 6, 12, 24, ……..
Answer$3,6,12,24, \ldots$Here, $a=3, r=\frac{6}{3}=2>1$
$\begin{aligned} & S_n=\frac{a\left(r^n-1\right)}{r-1}, \text { for } r>1 \\ \therefore & S_n=\frac{3\left(2^n-1\right)}{2-1} \\ \therefore & S_n=3\left(2^n-1\right)\end{aligned}$
View full question & answer→Question 321 Mark
The numbers 3, x and x + 6 are in G. P. Find : nth term.
Answer$t_n=3(2)^{n-1}$ or $t_n=3(-1)^{n-1}$
View full question & answer→Question 331 Mark
The numbers $3, x$ and $x + 6$ are in G. P. Find : $20$th term
Answer$r =\frac{6}{3}=2 \quad$ or $\quad r =\frac{-3}{3}=-1$$t_n=a r^{n-1}$
$\therefore \quad t_{20}=3\left(2^{19}\right) \text { or } t_{20}=3(-1)^{19}$
$\therefore \quad t _{20}=3\left(2^{19}\right) \text { or } t _{20}=-3$
View full question & answer→Question 341 Mark
The numbers $3, x$ and $x + 6$ are in G. P. Find $: x$
Answer$3, x$ and $x + 6$ are in G. P.
$\frac{x}{3}=\frac{x+6}{x}$
$x^2=3 x+18$
$x^2-3 x-18=0$
$(x-6)(x+3)=0$
$x=6,-3$
View full question & answer→Question 351 Mark
The fifth term of a G. P. is x, the eighth term of a G.P. is y and the eleventh term of a G.P. is$z$, verify whether $y^2=x z$.
View full question & answer→Question 361 Mark
For the G.P.If $a=\frac{2}{3}, t_6=162$, find $r$.
AnswerGiven, $a=\frac{2}{3}, t_6=162$$\begin{array}{ll}\therefore & t_n=a r^{n-1} \\ \therefore & t_6=\left(\frac{2}{3}\right)\left(r^{6-1}\right) \\ \therefore & 162=\frac{2}{3} r^5 \\ \therefore & r^5=162 \times \frac{3}{2} \\ \therefore & r^5=3^5 \\ \therefore & r=3\end{array}$
View full question & answer→Question 371 Mark
For the G.P.If $r=-3$ and $t_6=1701$, find $a$.
AnswerGiven, $r=-3, t_6=1701$$\begin{array}{ll} & t _{ n }=a r^{n-1} \\ \therefore & t _6=a(-3)^{6-1} \\ \therefore & 1701=a(-3)^5 \\ \therefore & a =\frac{1701}{-243} \\ \therefore & a =-7\end{array}$
View full question & answer→Question 381 Mark
For the G.P.If $a=\frac{7}{243}, r=3$, find $t_6$
AnswerGiven, $a=\frac{7}{243}, r=3$$\begin{aligned} t _{ n } & = ar ^{ n -1} \\ \therefore \quad t _6 & =\frac{7}{243} \times(3)^{6-1} \\ & =\frac{7}{243} \times 3^5 \\ & =7\end{aligned}$
View full question & answer→Question 391 Mark
For the G.P.If $r=\frac{1}{3}, a=9$, find $t_7$.
AnswerGiven, $r=\frac{1}{3}, a=9$$t_n=a r^{n-1}$
$\therefore \quad t_7=9 \times\left(\frac{1}{3}\right)^{7-1}$
$=\frac{9}{3^6}$
$=\frac{1}{81}$
View full question & answer→Question 401 Mark
Check whether the following sequences are G.P. If so, write $t_n$.
7, 14, 21, 28, ……
View full question & answer→Question 411 Mark
Check whether the following sequences are G.P. If so, write $t_n$.
3, 4, 5, 6, ……
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