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- Question 7, Exercise 1.4 @math-11-nbf:sol:unit01
- . =====Question 7(i)===== Convert the following equation in Cartesian form: $\arg (z-1)=-\dfrac{\pi}{4}$ ... . =====Question 7(ii)===== Convert the following equations and inequations in Cartesian form: $z \bar{z}=4\left|e^{i \theta}\right|$ ** Solution. ** Suppose $z=x+i... } =====Question 7(iii)===== Convert the following equations and inequations in Cartesian form: $-\dfrac{\pi
- Question 3, Exercise 2.6 @math-11-nbf:sol:unit02
- ====Question 3(i)===== Solve the system of linear equation by Gauss elimination method.\\ $2 x+3 y+4 z=2$\\ ... -2 y+z=-3$\\ ** Solution. ** Given the system of equations: \begin{align*} \begin{aligned} 2x + 3y + 4z &= ... ===Question 3(ii)===== Solve the system of linear equation by Gauss elimination method.\\ $5 x-2 y+z=2$\\ $2... 4 y-5 z=3$\\ ** Solution. ** Given the system of equations: \begin{align*} 5x - 2y + z &= 2 \quad \cdots (i
- Question 6, Exercise 2.6 @math-11-nbf:sol:unit02
- ===Question 6(i)===== Solve the system of linear equation by matrix inversion method.FIXME\\ $5 x+3 y+z=6$\... 2 y+4 z=25$\\ ** Solution. ** For this system of equations; we have \begin{align*} A &= \begin{bmatrix} 5 &... lign*} Therefore, the solution to the system of equations is: $$x = \frac{1}{11}, \quad y =\frac{119}{11},... ===Question 6(ii)===== Solve the system of linear equation by matrix inversion method.\\ $x+2 y-3 z=5$\\ $2
- Question 4, Exercise 1.3 @math-11-nbf:sol:unit01
- 4(i)===== Solve the simultaneous system of linear equation with complex coefficients: $(1-i) z+(1+i) \omega=... (ii)===== Solve the simultaneous system of linear equation with complex coefficients: $2 i z+(3-2 i) \omega=... ga = 3-i \quad \cdots(3) \end{align} Multiplying equation (2) by $2i$, we get: \begin{align} &2i(1-2i) z +... \end{align} Now, substituting $\omega$ back into equation $(1)$ to find $z$: \begin{align} &2iz + (3-2i)\le
- Question 1, Exercise 2.6 @math-11-nbf:sol:unit02
- 1(i)===== Solve the system of homogeneous linear equation for non-trivial solution if exists\\ $ 2 x_{1}-3 ... x_{3}=0\cdots (iii)\\ \end{align*} For system of equation, \begin{align*} A &= \left[ \begin{array}{ccc} 2... 1(ii)===== Solve the system of homogeneous linear equation for non-trivial solution if exists\\ $2 x_{1}-3 x... \quad \text{(iii)} \end{align*} For the system of equations, we have: \begin{align*} A &= \left[ \begin{arra
- Question 4, Exercise 2.6 @math-11-nbf:sol:unit02
- ====Question 4(i)===== Solve the system of linear equation by Gauss-Jordan method.\\ $2 x_{1}-x_{2}-x_{3}=2$... 2\end{align*} Thus, the solution to the system of equations is: $$\boxed{x_1 = \frac{13}{3}, \quad x_2 = \fr... ==Question 4(ii)===== Solve the system of linear equation by Gauss-Jordan method.\\ $2 x_{1}-3 x_{2}+7 x_{3... =Question 4(iii)===== Solve the system of linear equation by Gauss-Jordan method.\\ $x_{1}+x_{2}+x_{3}=3$\\
- Question 5, Exercise 2.6 @math-11-nbf:sol:unit02
- ====Question 5(i)===== Solve the system of linear equation by using Cramer's rule.\\ $x_{1}+x_{2}+2 x_{3}=8$... ===Question 5(ii)===== Solve the system of linear equation by using Cramer's rule.\\ $2 x_{1}+2 x_{2}+x_{3}=... {align*} The solution set for the given system of equations using Cramer's rule is: $$( -\frac{1}{7}, \frac{... ==Question 5(iii)===== Solve the system of linear equation by using Cramer's rule.\\ $-2 x_{2}+3 x_{3}=1$\\
- Unit 01: Complex Numbers (Solutions) @math-11-nbf:sol
- complex number. * Solve the simultaneous linear equations with complex coefficients. * Factorize the giv... e $z^2+a^2$ or $z^3-3z^2+z=5$ * Solve quadratic equation of the form $pz^2+qz+r=0$, by completing squares,... inary operations in polar form. * Solve complex equations and inequations in polar form. * Using the complex numbers in real world problems. <panel type="default
- Question 2, Exercise 1.3 @math-11-nbf:sol:unit01
- abad, Pakistan. ====Question 2(i)==== Solve the equation by completing square: $z^{2}-6 z+2=0$. **Solutio... m \sqrt{7}\}$. ====Question 2(ii)==== Solve the equation by completing square: $-\dfrac{1}{2} z^{2}-5 z+2=... \sqrt{29}\}$ ====Question 2(iii)==== Solve the equation by completing square: $4 z^{2}+5 z=14$. **Soluti... 9}}{8}\right\}$ ====Question 2(iv)==== Solve the equation by completing square: $z^{2}=5 z-3$. **Solution.
- Question 3, Exercise 1.3 @math-11-nbf:sol:unit01
- stan. ====Question 3(i)==== Solve the quadratic equation: $\dfrac{1}{3} z^{2}+2 z-16=0$. **Solution.** Gi... \}$. ====Question 3(ii)==== Solve the quadratic equation: $z^{2}-\frac{1}{2} z+17=0$. **Solution.** Give... \} $ ====Question 3(iii)==== Solve the quadratic equation: $z^{2}-6 z+25=0$. **Solution.** Given $$ z^{2}... 4i\}$ ====Question 3(iv)==== Solve the quadratic equation: $z^{2}-9 z+11=0$. **Solution.** Given $$z^{2}
- Question 7 and 8, Exercise 2.6 @math-11-nbf:sol:unit02
- ht]$; find $A^{-1}$ and hence solve the system of equations.\\ $3 x+4 y+7 z=14 ; 2 x-y+3 z=4 ; \quad x+2 y-3... {-11}{62} \end{bmatrix}$$ Now given the system of equations: \begin{align*} 3x + 4y + 7z &= 14 \\ 2x - y + 3... d R_2-8R_3\quad R_1-13R_3 \end{align*} From above equation we get, \begin{align*} x_1&=1\\ x_2&=1\\ x_3&=1 \end{align*} Now solutions of above equations are; $$ \begin{bmatrix} \dfrac{-3}{62} & \dfrac{
- Question 1, Review Exercise @math-11-nbf:sol:unit02
- $3$</collapse> vii. System of homogeneous linear equations has non-trivial solution if: * (a) $|A|>0$ ... $</collapse> viii. For non-homogeneous system of equations; the system is inconsistent if: * (a) $\ope... s</collapse> ix. For a system of non-homogeneous equations with three variables system will have unique sol... }=3$</collapse> x. A system of non- homogeneous equation having infinite many solutions can be solved by u
- Question 5 and 6, Exercise 4.2 @math-11-nbf:sol:unit04
- = -73 \quad \cdots (2) \end{align*} Now, subtract equation (1) from equation (2): \begin{align*} \begin{array}{ccc} a_1& + 27d &= -73\\ \mathop{}\limits_{-}a_1 &\ma... = 43 \quad \cdots (2) \end{align*} Now, subtract equation (1) from equation (2): \begin{align*} \begin{array}{ccc} a_1 & + 10d &= 43\\ \mathop{}\limits_{-}a_1 &\math
- Question 2, Exercise 2.6 @math-11-nbf:sol:unit02
- ambda$ for which the system of homogeneous linear equation may have non-trivial solution. Also solve the sys... ambda$ for which the system of homogeneous linear equation may have non-trivial solution. Also solve the sys... - 7x_{3} &= 0 \quad \text{(3)} \end{align*} From equation (1), we have \begin{align*} x_{1} &= 4x_{2} - 3x_
- Question 9 and 10, Exercise 4.8 @math-11-nbf:sol:unit04
- 2) \end{align*} Now, put $k+1=0 \implies k=-1$ in equation (2): \begin{align*} 1 &= (-1+2)A + 0 \\ \implies ... . \end{align*} Next, put $k+2=0 \implies k=-2$ in equation (2): \begin{align*} 1 &= 0 + (-2+1)B \\ \implies ... . \end{align*} Using the values of $A$ and $B$ in equation (1), we get \begin{align*} \frac{1}{(k+1)(k+2)} &