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- Question 1, Exercise 2.6 @math-11-nbf:sol:unit02
- ====== Question 1, Exercise 2.6 ====== Solutions of Question 1 of Exercise 2.6 of Unit 02: Matrices and Dete... 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*
- Question 1, Exercise 1.3 @math-11-nbf:sol:unit01
- ====== Question 1, Exercise 1.3 ====== Solutions of Question 1 of Exercise 1.3 of Unit 01: Complex Numbers. ... 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
- Question 6(i-ix), Exercise 1.4 @math-11-nbf:sol:unit01
- ====== Question 6(i-ix), Exercise 1.4 ====== Solutions of Question 6(i-ix) of Exercise 1.4 of Unit 01: Compl... 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} +
- Unit 01: Complex Numbers (Solutions) @math-11-nbf:sol
- ===== Unit 01: Complex Numbers (Solutions) ===== {{ :math-11-nbf:sol:math-11-nbf-sol-unit01.jpg?nolink&400x335|Unit 01: Complex Numbers (Solutions)}} This is a first unit of the book Model Textbo... abad, Pakistan. On this page we have provided the solutions of the questions. After reading this unit the s... lems. <panel type="default" title="Exercise 1.1 (Solutions)"> * [[math-11-nbf:sol:unit01:ex1-1-p1|Questio
- Unit 04: Sequences and Seeries @math-11-nbf:sol
- abad, Pakistan. On this page we have provided the solutions of the questions. After reading this unit the s... ries. <panel type="default" title="Exercise 4.1 (Solutions)"> * [[math-11-nbf:sol:unit04:ex4-1-p1|Questio... nel> <panel type="default" title="Exercise 4.2 (Solutions)"> * [[math-11-nbf:sol:unit04:ex4-2-p1|Questio... anel> <panel type="default" title="Exercise 4.3 (Solutions)"> * [[math-11-nbf:sol:unit04:ex4-3-p1|Questio
- Question 2, Exercise 1.1 @math-11-nbf:sol:unit01
- ====== Question 2, Exercise 1.1 ====== Solutions of Question 2 of Exercise 1.1 of Unit 01: Complex Numbers. ... 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
- Question 3, Exercise 1.3 @math-11-nbf:sol:unit01
- ====== Question 3, Exercise 1.3 ====== Solutions of Question 3 of Exercise 1.3 of Unit 01: Complex Numbers. ... 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
- Question 6(x-xvii), Exercise 1.4 @math-11-nbf:sol:unit01
- ====== Question 6(x-xvii), Exercise 1.4 ====== Solutions of Question 6(x-xvii) of Exercise 1.4 of Unit 01: C... 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
- Question 2, Exercise 2.6 @math-11-nbf:sol:unit02
- ====== Question 2, Exercise 2.6 ====== Solutions of Question 2 of Exercise 2.6 of Unit 02: Matrices and Dete... 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 &
- Question 10, Exercise 8.1 @math-11-nbf:sol:unit08
- ====== Question 10, Exercise 8.1 ====== Solutions of Question 10 of Exercise 8.1 of Unit 08: Fundamental of ... 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
- Question 11, Exercise 8.1 @math-11-nbf:sol:unit08
- ====== Question 11, Exercise 8.1 ====== Solutions of Question 11 of Exercise 8.1 of Unit 08: Fundamental of ... 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
- Unit 02: Matrices and Determinants (Solutions) @math-11-nbf:sol
- ===== Unit 02: Matrices and Determinants (Solutions) ===== This is a second unit of the book Model Textbook... abad, Pakistan. On this page we have provided the solutions of the questions. After reading this unit the s... to <panel type="default" title="Exercise 2.1 (Solutions)"> * [[math-11-nbf:sol:unit02:ex2-1-p1|Questio... panel> <panel type="default" title="Exercise 2.2 (Solutions)"> * [[math-11-nbf:sol:unit02:ex2-2-p1|Questio
- Question 2, Exercise 1.3 @math-11-nbf:sol:unit01
- ====== Question 2, Exercise 1.3 ====== Solutions of Question 2 of Exercise 1.3 of Unit 01: Complex Numbers. ... 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)=
- Question 3, Exercise 2.6 @math-11-nbf:sol:unit02
- ====== Question 3, Exercise 2.6 ====== Solutions of Question 3 of Exercise 2.6 of Unit 02: Matrices and Dete... 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*}
- Question 5, Exercise 2.6 @math-11-nbf:sol:unit02
- ====== Question 5, Exercise 2.6 ====== Solutions of Question 5 of Exercise 2.6 of Unit 02: Matrices and Dete... }+3 x_{3}=1$\\ $3 x_{1}-7 x_{2}+4 x_{3}=10$\\ ** Solution. ** The above system may be written as $A X=B$; w... 2}\\ &= \frac{104}{52} = 2 \end{align*} Thus, the solution set is $(3, 1, 2)$. =====Question 5(ii)===== S... {2}+2 x_{3}=1$\\ $8 x_{1}+x_{2}+4 x_{3}=-1$\\ ** Solution. ** The above system maybe written as $AX = B $,