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- Unit 04: Sequences and Seeries
- find its general term. * Know arithmetic means between two numbers. Also insert $n$ arithmetic means between them. * Define an arithmetic series and establish ... series. * Show that sum of $n$ arithmatic means between two numbers is equal to $n$ times their A.M. *... ce and its general term. * Know geometric means between two numbers, Also insert $n$ geometric means betw
- Question 14 and 15, Exercise 4.2 @math-11-nbf:sol:unit04
- =====Question 14===== Find '$b$' if $10$ is A.M between $b$ and $20$. ** Solution. ** Let $a= b$ and $... can be as follows: * Find '$b$' if $25$ is A.M between $b$ and $20$. * Find '$b$' if $10$ is A.M between $b$ and $-10$. </callout> =====Question 15===== Find ... and $y$ if $2$ and $13$ are two arithmetic means between $x$ and $y$. ** Solution. ** Given: $2$ and $13
- Question 6(vi-ix), Exercise 6.1 @math-11-nbf:sol:unit06
- $ (n!+1)$ is not divisible by any natural number between $2$ and $n$. ** Solution. ** We know $$n!=n(n-1... s 3.2.1$$ Hence $n!$ is divisible by every number between $1$ and $n$.\\ $n!$ can also divides by any natural number between $2$ and $n$.\\ For $(n!+1)$, $1$ is not divisible by any natural number between $2$ and $n$.\\ So $ (n!+1)$ is not divisible by a
- Question 13, Exercise 4.2 @math-11-nbf:sol:unit04
- , Pakistan. =====Question 13(i)===== Find A.M. between $7$ and $17$ ** Solution. ** Here $a=7$ and $b=... = $12$. GOOD =====Question 13(ii)===== Find A.M. between $3+3 \sqrt{2}$ and $7-3 \sqrt{2}$ ** Solution. *... = $5$. GOOD =====Question 13(iii)===== Find A.M. between $7 \sqrt{5}$ and $\sqrt{5}$ ** Solution. ** Here... rt{5}$. GOOD =====Question 13(iv)===== Find A.M. between $2y+5$ and $5y+3$ ** Solution. ** Here $a=2y+5$
- Question 16 and 17, Exercise 4.2 @math-11-nbf:sol:unit04
- ===Question 16===== Find the two arithmetic means between $5$ and $17$. ** Solution. ** Let $A_1$ and $A_2$ be two arithmetic means between $5$ and $17$.\\ Then $5$, $A_1$, $A_2$, $17$ are ... ===Question 16===== Find the two arithmetic means between $5$ and $17$. ** Solution. ** Let $A_1$, $A_2$ and $A_3$ be thre arithmetic means between $2$ and $-18$.\\ Then $2$, $A_1$, $A_2$, $A_3$, $
- Question 14, Exercise 8.1 @math-11-nbf:sol:unit08
- os \theta$.\\ **c.** Find the cosine of the angle between the wires where they meet at the ground. **d.** F... , to the nearest degree, the measure of the angle between the wires. ** Solution. ** {{ :math-11-nbf:sol:... m the figure, we see $\theta-\alpha$ is and angle between the wires where they meet at the ground. Thus \b... \circ \end{align*} Hence the measure of the angle between the wires is $22^\circ$ approximately. GOOD ====G
- Question 9 and 10, Exercise 4.3 @math-11-nbf:sol:unit04
- ===== Find the sum of all multiples of 4 that are between $14$ and $523$. ** Solution. ** Sum of all multiples of 4 that are between $14$ and $523$. $$16+20+24+...+520.$$ This is ar
- Question 11, Exercise 4.6 @math-11-nbf:sol:unit04
- abad, Pakistan. =====Question 11===== Find H.M. between $\dfrac{2}{3}$ and $\dfrac{4}{7}$. ** Solution. ... {13} \\ \end{align*} Hence $\dfrac{8}{13}$ is H.M between $\dfrac{2}{3}$ and $\dfrac{4}{7}$. GOOD ===
- Question 12, Exercise 4.6 @math-11-nbf:sol:unit04
- Pakistan. =====Question 12===== Find four H.Ms. between $\dfrac{1}{3}$ and $\dfrac{1}{11}$. ** Solution. ** Let $H_1, H_2, H_3, H_4$ be four $H.Ms$ between $\dfrac{1}{3}$ and $\dfrac{1}{11}$.\\ Then $$\dfr
- Question 9 & 10, Exercise 4.6 @math-11-nbf:sol:unit04
- {0} $ is in A.P. =====Question 10===== Find H.M. between 9 and 11 . Also find $A, H, G$ and show that $A H
- Exercise 6.1 (Solutions) @math-11-nbf:sol:unit06
- ) $(n!+1)$ is not divisible by any natural number between 2 and $n$. (ix) $\quad(n!)^{2} \leq n^{n} . n!<(2
- Question 8 and 9, Exercise 6.2 @math-11-nbf:sol:unit06
- -1)! arrangements\\ while $4$ men can be adjusted between seats of women in $41$ ways,\\ so total possible
- Question 20 and 21, Exercise 6.2 @math-11-nbf:sol:unit06
- ssible arrangements\\ and $6$ men may be adjusted between in $6!$ possible arrangements\\ and hence total p
- Question 9 and 10, Exercise 6.3 @math-11-nbf:sol:unit06
- etermined by $n$-points \\ and all possible lines between $n$ points is $C_{2}$\\ but $n$ lines are sides o