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- Question 1, Exercise 2.5 @math-11-nbf:sol:unit02
- Matrices and Determinants. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. =====Question 1... & 8 & 3 \\ -4 & 6 & 5\end{array}\right]\\ \sim & \text{R} \left[\begin{array}{ccc} 1 & 3 & 5 \\ 0 & 26 &... right]\quad R_2 + 6R_1 \quad R_3 + 4R_1\\ \sim & \text{R} \left[\begin{array}{ccc} 1 & 3 & 5 \\ 0 & 1 &
- Question 2, Exercise 2.5 @math-11-nbf:sol:unit02
- Matrices and Determinants. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. =====Question 2... & 6 \\ 2 & 10 & 6 \end{array} \right]\\ \sim & \text{R}\left[ \begin{array}{ccc} 1 & \frac{9}{5} & \fr... \end{array} \right]\quad \frac{1}{5} R1\\ \sim & \text{R}\left[ \begin{array}{ccc} 1 & \frac{9}{5} & \fr
- Question 4, Exercise 2.6 @math-11-nbf:sol:unit02
- Matrices and Determinants. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. =====Question 4... & 7 \\ 4 & 2 & -5 & : & 10 \end{bmatrix}\\ &\sim \text{R}\begin{bmatrix} 1 & -\frac{1}{2} & -\frac{1}{2}... ert & 10 \end{bmatrix}\quad \dfrac{1}{2}\\ &\sim \text{R}\begin{bmatrix} 1 & -\frac{1}{2} & -\frac{1}{2}
- Question 3, Exercise 9.1 @math-11-nbf:sol:unit09
- 9: Trigonometric Functions. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. =====Question 3... Domain of $y=\left\{\theta: \theta\in \mathbb{R} \text{ and } \theta \neq n\pi, n\text{ is integer} \right\}$ Range of $y=\mathbb{R}$ As \begin{align*} &
- Question 7 and 8, Exercise 2.6 @math-11-nbf:sol:unit02
- Matrices and Determinants. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. =====Question 7... 9 \\ 9 & -16 & 5 \\ 5 & -2 & -11 \end{bmatrix}\\ \text{adj}(A) &= \begin{bmatrix} -3 & 9 & 5 \\ 26 & -16... & 3 & 4 \\ 1 & 2 & -3 & 0 \end{bmatrix}\\ &\sim \text{R} = \begin{bmatrix} 1 & 2 & -3 & 0\\ 2 & -1 & 3
- Question 9, Exercise 1.2 @math-11-nbf:sol:unit01
- f Unit 01: Complex Numbers. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. ====Question 9(i... =3-2i$. We will use the following formulas: \[\text{Re}(z^{-2}) = \frac{(\text{Re}(z))^2 - (\text{Im}(z))^2}{|z|^4},\] \[\text{Im}(z^{-2}) = \frac{-2 \text
- Question 3, Exercise 2.5 @math-11-nbf:sol:unit02
- Matrices and Determinants. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. =====Question 3... -1 & 4 & 0 & 0 & 1 \end{array} \right]\\ \sim &{\text{R}} \left[ \begin{array}{ccc|ccc} 1 & -1 & 4 & 0 ... & -1 & -1 & 1 & 0 & 0 \end{array} \right] \quad \text{by swapping } R1 \text{ and } R3\\ \sim&{\text{R}
- Question 5 and 6, Exercise 4.8 @math-11-nbf:sol:unit04
- it 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. =====Question ... _{n-1})-T_{n}. \\ \implies 0=&3+(1+2+4+8+\ldots \text { up to } (n-1) \text { terms })-T_{n} \end{align*} Then \begin{align*} T_{n} & =3+(1+2+4+8+\ldots \te
- Question 2, Exercise 1.2 @math-11-nbf:sol:unit01
- f Unit 01: Complex Numbers. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. ====Question 2==... ign} &(z_1 z_2)(z_3 z_4) \\ =&(z_1 z_2)z_5 \quad \text {Let }z_5=z_3 z_4 \\ =&z_1 (z_2 z_5) \quad \text{Multiplicative assocative law}\\ =&z_1\left(z_2 (z_3 z
- Question 2, Exercise 2.6 @math-11-nbf:sol:unit02
- Matrices and Determinants. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. =====Question 2... ** \begin{align*} &x_{1}-4 x_{2}+3 x_{3}=0 \quad \text{(i)}\\ &2 x_{1}+\lambda x_{2}+x_{3}=0 \quad \text{(ii)}\\ &x_{1}-2 x_{2}+\lambda x_{3}=0 \quad \text{(i
- Question 3, Exercise 2.6 @math-11-nbf:sol:unit02
- Matrices and Determinants. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. =====Question 3... 5 \\ 3 & -2 & 1 & -3 \end{array}\right]\\ & \sim \text{R}\left[\begin{array}{cccc} 2 & 3 & 4 & 2 \\ 0 & ... }{2} & -5 & -6 \end{array}\right]\quad R_2 - R_1 \text{and}\quad R_3 - \frac{3}{2}R_1\\ & \sim \text{R}
- Question 1 and 2, Exercise 4.8 @math-11-nbf:sol:unit04
- it 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. =====Question ... ight)-T_{n}. \\ \implies 0=&3+(4+6+8+10+\ldots \text { up to } (n-1) \text { terms })-T_{n} \end{align*} Then \begin{align*} T_{n} & =3+(4+6+8+10+\ldots \t
- Question 3 and 4, Exercise 4.8 @math-11-nbf:sol:unit04
- it 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. =====Question 3... -1})-T_{n}. \\ \implies 0=&1+(3+9+27+81+\ldots \text { up to } (n-1) \text { terms })-T_{n} \end{align*} Then \begin{align*} T_{n} & =1+(3+9+27+81+\ldots
- Definitions: Mathematics 11 NBF
- ==== Definitions: Mathematics 11 NBF ====== Model Textbook of Mathematics for Class XI is published by N... mabad, Pakistan. NBF can be considered as Federal Textbook Board Islamabad. The book has total of nine (... $ and $Im(z^{-1}) = -\dfrac{Im(z)}{|z|^2}$. * $\text{Re}(z^{-2}) = \frac{(\text{Re}(z))^2 - (\text{Im}(z))^2}{|z|^4}$ and $\text{Im}(z^{-2}) = \frac{-2 \tex
- Question 3, Exercise 1.2 @math-11-nbf:sol:unit01
- f Unit 01: Complex Numbers. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. ====Question 3(i... f $z=\bar{z}$. **Solution.** Let $$z=a+ib\quad \text{where}\quad a,b\in \mathbb{R}\, ... (1)$$ First s... $z$ is real or pure imaginary, then $$z=x \quad \text{ or } \quad z=iy.$$ This gives $$\bar{z}=x \quad