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- Exercise 6.2 (Solutions) @math-11-nbf:sol:unit06
- }^n P_{n-2}$\\ [[math-11-nbf:sol:unit06:ex6-2-p1|Solution Question 1]] **Question 2.** Find $n$, if:\\ (i)... -1}=22: 7$ \\ [[math-11-nbf:sol:unit06:ex6-2-p2|Solution: Question 2 ]] **Question 3.** Find $r$, if:\\ (... +6}=1: 30800$\\ [[math-11-nbf:sol:unit06:ex6-2-p3|Solution: Question 3 ]] **Question 4.** How many 3 -digi... not repeated?\\ [[math-11-nbf:sol:unit06:ex6-2-p4|Solution: Question 4 & 5 ]] **Question 5.** How many 7 -d
- Exercise 6.3 (Solutions) @math-11-nbf:sol:unit06
- ^{n+1} C_{r}$\\ [[math-11-nbf:sol:unit06:ex6-3-p1|Solution: Question 1(i-v)]] **Question 1(vi-x).** Prove t... ible by $k$ !\\ [[math-11-nbf:sol:unit06:ex6-3-p2|Solution: Question 1(vi-x) ]] **Question 2.** Find $n$, i... C_{2}=12: 1$\\ [[math-11-nbf:sol:unit06:ex6-3-p3|Solution: Question 2 ]] **Question 3.** Find $r$, if:\\ (... {r}-1}=11: 5$\\ [[math-11-nbf:sol:unit06:ex6-3-p4|Solution: Question 3 ]] **Question 4.** Find $n$ and $r$,
- Question 1, Exercise 2.6 @math-11-nbf:sol:unit02
- em of homogeneous linear equation for non-trivial solution if exists\\ $ 2 x_{1}-3 x_{2}+4 x_{3}=0$\\ $x_{1}... _{2}+3 x_{3}=0$\\ $4 x_{1}+x_{2}-6 x_{3}=0$\\ ** Solution. ** \begin{align*} &2 x_{1}-3 x_{2}+4 x_{3}=0\cdo... 4+36=0 \end{align*} So the system has non-trivial solution. \text{By}\quad(i)-2(ii), we have \begin{align*... es of $x_3$, there are infinite solutions. Hence solution is; \begin{align*} \left[ \begin{array}{c} x_3
- Question 1, Exercise 1.3 @math-11-nbf:sol:unit01
- polynomial into linear functions: $z^{2}+169$. **Solution.** \begin{align} & z^{2} + 169 \\ = & z^{2} - (... olynomial into linear functions: $2 z^{2}+18$. **Solution.** \begin{align} & 2z^2 + 18 \\ = &2(z^2 - (3i)^... nomial into linear functions: $3 z^{2}+363$. **Solution.** \begin{align} & 3z^2 + 363 \\ = & 3(z^2 - (1... into linear functions: $z^{2}+\dfrac{3}{25}$. **Solution.** \begin{align} & z^2 + \dfrac{3}{25} \\ = & z^
- Question 6(i-ix), Exercise 1.4 @math-11-nbf:sol:unit01
- os 315^{\circ}+i \sin 315^{\circ}\right)$ ** Solution. ** \begin{align} &\sqrt{2}\left(\cos 315^{\circ... t(\cos 210^{\circ}+i \sin 210^{\circ}\right)$ ** Solution. ** \begin{align*} &5\left(\cos 210^\circ + i \s... c{3 \pi}{2}+i \sin \dfrac{3 \pi}{2}\right)$ ** Solution. ** \begin{align*} &2\left(\cos \frac{3\pi}{2} +... {5 \pi}{6}+i \sin \dfrac{5 \pi}{6}\right)$ ** Solution. ** \begin{align*} &4\left(\cos \frac{5\pi}{6} +
- Exercise 6.1 (Solutions) @math-11-nbf:sol:unit06
- !}{(6!)^2}$ \\ [[math-11-nbf:sol:unit06:ex6-1-p1|Solution: Question 1]] **Question 2.** Write the followin... -1)}{n(n-4)}$\\ [[math-11-nbf:sol:unit06:ex6-1-p2|Solution: Question 2 ]] **Question 3.** Prove the followi... n^{2}-3 n+2$ \\ [[math-11-nbf:sol:unit06:ex6-1-p3|Solution: Question 3 & 4]] **Question 4.** Show that:\\ (... dots(2 n-1))$\\ [[math-11-nbf:sol:unit06:ex6-1-p3|Solution: Question 3 & 4]] **Question 5.** Find the value
- Question 2, Exercise 6.2 @math-11-nbf:sol:unit06
- )===== Find $n$, if: $\quad ^nP_4=20\, ^nP_2$ ** Solution. ** \begin{align*} \dfrac{m}{(n-4)!}&=20 \cdot \... = Find $n$, if: $\quad ^{2n}P_3=100 \, ^nP_2$ ** Solution. ** \begin{align*} \dfrac{(2 n)!}{(2 n-3)!}&=100... ind $n$, if: $\quad16\, ^nP_3=13\, ^{n+1}P_3$ ** Solution. ** \begin{align*}16 \dfrac{n!}{(n-3)!}&=13 \dfr... )===== Find $n$, if: $\quad ^nP_5=20, ^nP_3$ ** Solution. ** \begin{align*}{ }^{n} P_{5}&=20{ }^{n} P_{3}
- Question 2, Exercise 1.1 @math-11-nbf:sol:unit01
- ex number in the form $x+iy$: $(3+2i)+(2+4i)$ ** Solution. ** \begin{align}&(3+i2)+(2+i4)\\ =&(3+2)+(i2+i... lex number in the form $x+iy$: $(4+3i)-(2+5i)$ **Solution.** \begin{align}&(4+3i)-(2+5i)\\ =&(4-2)+(3i-5i... lex number in the form $x+iy$: $(4+7i)+(4-7i)$ **Solution.** \begin{align} &(4+7i)+(4-7i)\\ =&(4+4)+(7i-7... lex number in the form $x+iy$: $(2+5i)-(2-5i)$ **Solution.** \begin{align} &(2+5i)-(2-5i)\\ =&(2-2)+(5i+5i
- Question 3, Exercise 1.3 @math-11-nbf:sol:unit01
- ratic equation: $\dfrac{1}{3} z^{2}+2 z-16=0$. **Solution.** Given \begin{align}&\dfrac{1}{3}z^{2}+2 z-16=0... 57}}}{2} \\ &= -3 \pm \sqrt{57} \end{align} Hence Solution set $=\{ -3 \pm \sqrt{57} \}$. ====Question 3(i... uadratic equation: $z^{2}-\frac{1}{2} z+17=0$. **Solution.** Given $$ z^{2} - \frac{1}{2}z + 17 = 0 $$ Usi... 1 \pm \sqrt{271}i}{4} \end{align} Therefore, the solution set is: $\left\{\dfrac{1 \pm \sqrt{271}i}{4}\righ
- Question 6(x-xvii), Exercise 1.4 @math-11-nbf:sol:unit01
- rac{5 \pi}{4}+i \sin \dfrac{5 \pi}{4}\right)$ ** Solution. ** //Do yourself as previous parts.// =====Ques... rac{7 \pi}{4}+i \sin \dfrac{7 \pi}{4}\right)$ ** Solution. ** //Do yourself as previous parts.// =====Q... \dfrac{5\pi}{2}+i \sin \dfrac{5\pi}{2}\right)$ ** Solution. ** //Do yourself as previous parts.// =====Ques... \dfrac{\pi}{4}+i \sin \dfrac{\pi}{4}\right)$ ** Solution. ** //Do yourself as previous parts.// =====Que
- Question 2, Exercise 2.6 @math-11-nbf:sol:unit02
- homogeneous linear equation may have non-trivial solution. Also solve the system for value of $\lambda$.\\ ... _{2}-x_{3}=0$\\ $3 x_{1}-2 x_{2}+4 x_{3}=0$\\ ** Solution. ** \begin{align*} &2 x_{1}-\lambda x_{2}+x_{3}=0... \\ \end{align*} Homogenous system has non-trivial solution, if \begin{align*} &\left| \begin{array}{ccc} 2 &... homogeneous linear equation may have non-trivial solution. Also solve the system for value of $\lambda$.\\
- Question 10, Exercise 8.1 @math-11-nbf:sol:unit08
- ft(\dfrac{\pi}{2}-\alpha\right)=\cos \alpha$ ** Solution. ** \begin{align*} L.H.S & = \sin \left(\frac{\p... === Verify: $\cos (\pi-\alpha)=-\cos \alpha$ ** Solution. ** \begin{align*} L.H.S & = \cos(\pi - \alpha) ... frac{1}{\sqrt{2}}(\cos \alpha-\sin \alpha)$ ** Solution. ** \begin{align*} L.H.S &= \cos \left(\alpha+\f... =\frac{\sqrt{2}}{2}(\cos \beta+\sin \beta)$ ** Solution. ** \begin{align*} L.H.S & = \cos \left(\alpha +
- Question 11, Exercise 8.1 @math-11-nbf:sol:unit08
- ht) \cos \left(270^{\circ}-\lambda\right)}=1$ ** Solution. ** \begin{align*} L.H.S & = \dfrac{\sin \left(1... ght)+\cos \left(90^{\circ}+\alpha\right)}=-1$ ** Solution. ** \begin{align*} L.H.S & = \frac{\sin \left(90... \sin (\alpha+\beta)}{\cos \alpha \cos \beta}$ ** Solution. ** \begin{align*} L.H.S & = \tan \alpha+\tan \b... ^{2} \alpha=\sin ^{2} \alpha-\sin ^{2} \beta$ ** Solution. ** \begin{align*} L.H.S & = \sin (\alpha + \bet
- Question 2, Exercise 1.3 @math-11-nbf:sol:unit01
- quation by completing square: $z^{2}-6 z+2=0$. **Solution.** \begin{align} & z^2 - 6z + 2 = 0 \\ \implies ... pleting square: $-\dfrac{1}{2} z^{2}-5 z+2=0$. **Solution.** \begin{align} -\dfrac{1}{2} z^{2} - 5z + 2& =... \implies z &= -5 \pm \sqrt{29} \end{align} Hence solution set$=\{-5 \pm \sqrt{29}\}$ ====Question 2(iii)=... uation by completing square: $4 z^{2}+5 z=14$. **Solution.** \begin{align} 4z^{2} + 5z &= 14\\ z^{2} + \d
- Question 3, Exercise 2.6 @math-11-nbf:sol:unit02
- x+3 y+4 z=2$\\ $2 x+y+z=5$\\ $3 x-2 y+z=-3$\\ ** Solution. ** Given the system of equations: \begin{align*}... x &= \frac{46}{19} \end{align*} Therefore, the solution to the system is: $$x = \frac{46}{19}, \quad y ... y+z=2$\\ $2 x+2 y+6 z=1$\\ $3 x-4 y-5 z=3$\\ ** Solution. ** Given the system of equations: \begin{align*}... h is inconsistent, the system of equations has no solution. =====Question 3(iii)===== Solve the system o