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- Question 9, Exercise 1.2 @math-11-nbf:sol:unit01
- n. ====Question 9(i)==== Find real and imaginary parts of $(2+4 i)^{-1}$. **Solution.** Suppose $z=2+4i$. \... OD ====Question 9(ii)==== Find real and imaginary parts of $(3-\sqrt{-4})^{-2}$. **Solution.** Suppose $z=3 - \sqrt{-4}=3-2i... = \frac{12}{169}. \end{align} Therefore, the real part is \(\dfrac{5}{169}\) and the imaginary part is \(\dfrac{12}{169}\). GOOD ====Question 9(iii)==== Find real and imaginary parts of $\left(\dfrac{7+2 i}{3-i}\right)^{-1}$. **Solution.** We use the
- Definitions: Mathematics 11 NBF
- the form $x+iy$, where $x,y\in\mathbb{R}$ and $i^2=1$. Set of all complex numbers is usually denoted... \mathbb{C}$. Every complex number $x+i y$ has two parts $x$ and $y$. $x$ is called the real part and $y$ is called the imaginary part i.e., $Re(z)=x$ and $Im(z)=y$. **Conjugate of a Complex Number:**
- Question 8, Exercise 2.2 @math-11-nbf:sol:unit02
- akistan. =====Question 8===== Consider any two particular matrices $A$ and $B$ of your choice of order $2 \times 3$ and $3 \times 2$ respectively and show that $(A B)^{t}=B^{t} A^{t}$. ** Solution. ** Le... onsider matrices \( A \) and \( B \) of orders \( 2 \times 3 \) and \( 3 \times 2 \). Let \begin{align*} A &= \begin{bmatrix} a_{11} & a_{12} & a_{1
- Question 4, Exercise 1.1 @math-11-nbf:sol:unit01
- al number $x$ and $y$ in each of the following: $(2+3i)x+(1+3i)y+2=0$ **Solution.** \begin{align}&(2+3i)x+(1+3i)y+2=0\\ \implies &(2x+y+2)+(3x+3y)i=0.\end{align} Comparing real and imaginary parts \beg
- Question 3, Exercise 1.4 @math-11-nbf:sol:unit01
- 3(i)===== If $\left(x_{1}+i y_{1}\right)\left(x_{2}+i y_{2}\right)\left(x_{3}+i y_{3}\right) \ldots\left(x_{n}+i y_{n}\right)=a+i b$, show that: (i) $\left(x_{1}^{2}+y_{1}^{2}\right)\left(x_{2}^{2}+y_{2}^{2}\right)\left(x_{3}^{2}+y_{3}^{2}\right) \ldots\left(x_{n}^
- Exercise 6.2 (Solutions) @math-11-nbf:sol:unit06
- y ways they be seated at a round table\\ if three particular secretaries want to sit together?\\ [[math-11-nbf:sol:unit06:ex6-2-p11|Solution: Question 18 & 19]] **Question 20.** Find the number of ways that 6 men and 6 women se... alternative seats.\\ [[math-11-nbf:sol:unit06:ex6-2-p12|Solution: Question 20 & 21]] **Question 21.** Make all the permutations of the following word
- Question 9, Exercise 2.2 @math-11-nbf:sol:unit02
- ====== Question 9, Exercise 2.2 ====== Solutions of Question 9 of Exercise 2.2 of Unit 02: Matrices and Determinants. This is unit of Model Textbook of Mathematics for Class XI
- Question 11 and 12, Exercise 4.8 @math-11-nbf:sol:unit04
- ====== Question 11 and 12, Exercise 4.8 ====== Solutions of Question 11 and 12 of Exercise 4.8 of Unit 04: Sequence and Series. ... e sum of the series: $\sum_{k=1}^{n} \frac{1}{k(k+2)}$ ** Solution. ** Let $T_k$ represent the $k$t... e series. Then \begin{align*} T_k &= \frac{1}{k(k+2)}. \end{align*} Resolving it into partial fractio
- Question 13, 14 and 15, Exercise 4.8 @math-11-nbf:sol:unit04
- the series. Then \begin{align*} T_k &= \frac{1}{(2k+3)(2k+9)}. \end{align*} Resolving it into partial fractions: \begin{align*} \frac{1}{(2k+3)(2k+9)} = \frac{A}{2k+3} + \frac{B}{2k+9} \ldots (1) \en
- Question 3, Exercise 1.2 @math-11-nbf:sol:unit01
- ====== Question 3, Exercise 1.2 ====== Solutions of Question 3 of Exercise 1.2 of Unit 01: Complex Numbers. This is unit of Model Te... e $\overline{z}=z$. Since $z$ is real, imaginary part of $z$ is zero. i.e. $b=0$. Then \begin{align} ... ightarrow \quad & a+ib=a-ib\\ \Rightarrow \quad & 2ib=0\\ \Rightarrow \quad & b=0\quad \because \quad
- Question 9 and 10, Exercise 4.8 @math-11-nbf:sol:unit04
- um of the series: $$\frac{1}{1 \cdot 3}+\frac{1}{2 \cdot 5}+\frac{1}{3 \cdot 7}+\ldots \ldots \text{... of the series: $\sum_{k=3}^{n} \dfrac{1}{(k+1)(k+2)}$ ** Solution. ** Consider \begin{align*} T_k &= \frac{1}{(k+1)(k+2)}. \end{align*} Resolving it into partial fractions: \begin{align*} \frac{1}{(k+1)(k+2)} = \frac{A}{k+1
- Exercise 6.3 (Solutions) @math-11-nbf:sol:unit06
- he following for $n \in \mathbb{N}$.\\ (vi) ${ }^{2 n} \mathrm{C}_{\mathrm{n}}=\frac{2^{n} \cdot[1.3 .5 \ldots(2 n-1)]}{n!}$ (vii) ${ }^{n} C_{p}={ }^{n} C_{q} \Rightarrow p=q$ or $p+q=n$ \\ (viii) ${ }^{n} C_{r}+2{ }^{n} C_{r-1}+{ }^{n} C_{r-2}={ }^{n+2} C_{r}$ (
- Question 6(x-xvii), Exercise 1.4 @math-11-nbf:sol:unit01
- en complex number in the algebraic form: $7 \sqrt{2}\left(\cos \dfrac{5 \pi}{4}+i \sin \dfrac{5 \pi}{... ght)$ ** Solution. ** //Do yourself as previous parts.// =====Question 6(xi)===== Write a given complex number in the algebraic form: $10 \sqrt{2}\left(\cos \dfrac{7 \pi}{4}+i \sin \dfrac{7 \pi}{... t)$ ** Solution. ** //Do yourself as previous parts.// =====Question 6(xii)===== Write a given comp
- Unit 01: Complex Numbers (Solutions) @math-11-nbf:sol
- x number $z$ and recognize its real and imaginary part. * Know the condition for equality of two compl... ents. * Factorize the given polynomials like $z^2+a^2$ or $z^3-3z^2+z=5$ * Solve quadratic equation of the form $pz^2+qz+r=0$, by completing squares, wh
- Question 8, Exercise 1.4 @math-11-nbf:sol:unit01
- }}{500}(1 +i). \end{align} Hence mean position of particle is $\frac{\sqrt{2}}{500}(1 +i)$. =====Question 8(ii)===== Calculate the position of a particle from mean position when amplitude is $0.004 \... (\dfrac{\pi}{3})) \\ &= \dfrac{4}{1000}(\dfrac{1}{2} + i \dfrac{\sqrt{3}}{2}) \\ &= \dfrac{1}{250} (\dfrac{1}{2} + i \dfrac{\sqrt{3}}{2}) \\ &= \dfrac{1