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- Question 6(i-ix), Exercise 1.4 @math-11-nbf:sol:unit01
- bad, Pakistan. =====Question 6(i)===== Write a given complex number in the algebraic form: $\sqrt{2}\l... 1-i. \end{align} =====Question 6(ii)===== Write a given complex number in the algebraic form: $5\left(\co... }i \end{align*} =====Question 6(iii)===== Write a given complex number in the algebraic form: $2\left(\co... -2i \end{align*} =====Question 6(iv)===== Write a given complex number in the algebraic form: $4\left(\co
- Question 6(x-xvii), Exercise 1.4 @math-11-nbf:sol:unit01
- abad, Pakistan. =====Question 6(x)===== Write a given complex number in the algebraic form: $7 \sqrt{2}... revious parts.// =====Question 6(xi)===== Write a given complex number in the algebraic form: $10 \sqrt{2... vious parts.// =====Question 6(xii)===== Write a given complex number in the algebraic form: $2\left(\co... vious parts.// =====Question 6(xiii)===== Write a given complex number in the algebraic form: $\dfrac{1}{
- Question 5 Exercise 8.2 @math-11-nbf:sol:unit08
- s \theta$ and $\tan \theta$ using the information given: $\sin 2 \theta=\frac{24}{25}, 2 \theta$ in QII ** Solution. ** Given: $\sin 2\theta=\dfrac{24}{25}$, $2\theta$ in QII.... s \theta$ and $\tan \theta$ using the information given: $\cos 2 \theta=-\frac{7}{25}, 2 \theta$ in QIII ** Solution. ** Given: \(\cos 2\theta = -\dfrac{7}{25}\) and \(2\theta\
- Question 8, Exercise 1.2 @math-11-nbf:sol:unit01
- $x$ and $y$ by taking $z=x+i y$. **Solution.** Given: $$|2z-i|=4.$$ Put $z=x+i y$, we have \begin{alig... $x$ and $y$ by taking $z=x+i y$. **Solution.** Given: $$|z-1|=|\bar{z}+i|.$$ Put $z=x+iy$, we have \be... $x$ and $y$ by taking $z=x+i y$. **Solution.** Given: $$|z-4i| + |z+4i| = 10.$$ Put $z = x + iy$, we h... $x$ and $y$ by taking $z=x+i y$. **Solution.** Given: $$\dfrac{1}{2} Re(i \bar{z}) = 4.$$ Put $z = x
- Question 1, Exercise 4.2 @math-11-nbf:sol:unit04
- ic sequence with $a_{1}=4, d=3$ ** Solution. ** Given: $a_1= 4$, $d=3$.\\ The general term of an arithm... ic sequence with $a_1=7$, $d=5$ ** Solution. ** Given: $a_1= 7$, $d=5$.\\ The general term of an arithm... c sequence. $a_{1}=16$, $d=-2$. ** Solution. ** Given: $a_1= 16$, $d=-2$.\\ We have $$a_n = a_1 + (n - ... tic sequence. $a_1=38$, $d=-4$. ** Solution. ** Given: $a_1= 38$, $d=-4$.\\ We have $$a_n = a_1 + (n -
- Question 17, 18 and 19, Exercise 4.3 @math-11-nbf:sol:unit04
- ic series. $6+12+18+\ldots+96$. ** Solution. ** Given arithmetic series: $$6+12+18+\ldots+96.$$ So, $a... \times 102\\ &=1224. \end{align} Hence the sum of given series is $1224$. =====Question 18===== Find sum ... etic series. $34+30+26+\ldots+2$ ** Solution. ** Given arithmetic series: $$34+30+26+\ldots+2.$$ So, $a... imes 36\\ &=162. \end{align} Hence the sum of the given series is $162$. =====Question 19===== Find
- Question 1 and 2, Exercise 4.4 @math-11-nbf:sol:unit04
- n ratio. $5,20,100,500, \ldots$ ** Solution. ** Given sequence is $5, 20, 100, 500, \ldots $.\\ A seque... {5} = 4\neq \frac{100}{20} = 5.\end{align*} Hence given sequence is not geometric. GOOD **Alternative Method** Given sequence is $5, 20, 100, 500, \ldots $.\\ Suppose... two consective terms has different ratio.\\ Hence given sequence is not geometric. GOOD =====Question
- Question 1, Exercise 8.1 @math-11-nbf:sol:unit08
- =180^{\circ}, \beta=60^{\circ}$ ** Solution. ** Given: $\alpha=180^{\circ}$, $\beta=60^{\circ}$. \begi... a=60^{\circ}, \beta=90^{\circ}$ ** Solution. ** Given: \(\alpha = 60^\circ\), \(\beta = 90^\circ\) \beg... =180^{\circ}, \beta=30^{\circ}$ ** Solution. ** Given: $\alpha = 180^\circ$, $\beta = 30^\circ$. \begin... pha=\pi, \beta=\frac{2 \pi}{3}$ ** Solution. ** Given: $\alpha = \pi$, $\beta = \frac{2\pi}{3}$. \begin
- Question 9, Exercise 8.1 @math-11-nbf:sol:unit08
- , Islamabad, Pakistan. ===== Question 9(i)===== Given $\alpha$ and $\beta$ are obtuse angles with $\sin... ind: $\sin (\alpha \pm \beta)$ ** Solution. ** Given: $\sin \alpha=\dfrac{1}{\sqrt{2}}$, $\alpha$ is o... sqrt{2}}. \end{align*} ===== Question 9(ii)===== Given $\alpha$ and $\beta$ are obtuse angles with $\sin... find: $\cos (\alpha \pm \beta)$ ** Solution. ** Given: $\sin \alpha=\dfrac{1}{\sqrt{2}}$, $\alpha$ is o
- Question 4 Exercise 8.2 @math-11-nbf:sol:unit08
- where $0<\theta<\frac{\pi}{2}$ ** Solution. ** Given: $\cos\theta=\dfrac{3}{5}$ where $0<\theta<\dfrac... re $\pi<\theta<\frac{3 \pi}{2}$ ** Solution. ** Given: \(\tan \theta = \frac{12}{5}\) where \(\pi < \th... $\frac{3 \pi}{2}<\theta<2 \pi$ ** Solution. ** Given: \(\sin \theta = -\frac{7}{25}\) where \(\frac{3\... $\frac{3 \pi}{2}<\theta<2 \pi$ ** Solution. ** Given: \(\sec \theta = \sqrt{5}\) where \(\frac{3\pi}{2
- Question 2, Review Exercise @math-11-nbf:sol:unit08
- d, Islamabad, Pakistan. =====Question 2(i)===== Given that $\sin \theta=\dfrac{3}{5}, \sin \phi=\dfrac{... values of $\sin (\theta-\phi)$. ** Solution. ** Given: $\sin \theta=\dfrac{3}{5}$ and $\sin \phi=\dfrac... ac{56}{65} \end{align*} =====Question 2(ii)===== Given that $\sin \theta=\dfrac{3}{5}, \sin \phi=\dfrac{... values of $\tan (\theta-\phi)$. ** Solution. ** Given $\sin \theta=\dfrac{3}{5}$ and $\sin \phi=\dfrac{
- Question 10, Exercise 1.2 @math-11-nbf:sol:unit01
- line{z_{1}}}{\overline{z_{2}}}$. **Solution.** Given \[z_1 = -3 + 2i, \quad z_2 = 1 - 3i\] Then \begin... ne{z_{1}}\,\, \overline{z_{2}}$. **Solution.** Given \[ z_1 = -3 + 2i, \quad z_2 = 1 - 3i. \] First w... erline{z_{1}}+\overline{z_{2}}$. **Solution.** Given \[ z_1 = -3 + 2i, \quad z_2 = 1 - 3i. \] First w... z_{1}\right|\left|z_{2}\right|$. **Solution.** Given: \begin{align} z_1 = -3 + 2i, \quad z_2 = 1 - 3i
- Question 5 and 6, Exercise 4.2 @math-11-nbf:sol:unit04
- n. ** The nth term of the arithmetic sequence is given as $$a_n=a_1+(n-1)d$$ Given \begin{align*} & a_{17} = -40 \\ \implies &a_1 + 16d = -40 \quad \cdots (1) ... the first term and 87 th term. ** Solution. ** Given: $a_5 = 19$ and $a_{11} = 43$. The nth term of the arithmetic sequence is given as $$a_n = a_1 + (n-1)d$$ Given \begin{align*} &
- Question 3, Exercise 1.3 @math-11-nbf:sol:unit01
- on: $\dfrac{1}{3} z^{2}+2 z-16=0$. **Solution.** Given \begin{align}&\dfrac{1}{3}z^{2}+2 z-16=0\\ \impli... tion: $z^{2}-\frac{1}{2} z+17=0$. **Solution.** Given $$ z^{2} - \frac{1}{2}z + 17 = 0 $$ Using the qua... ratic equation: $z^{2}-6 z+25=0$. **Solution.** Given $$ z^{2} - 6z + 25 = 0 $$ Using the quadratic for... ratic equation: $z^{2}-9 z+11=0$. **Solution.** Given $$z^{2} - 9z + 11 = 0 $$ Using the quadratic form
- Question 1, Exercise 2.2 @math-11-nbf:sol:unit02
- h is $a_{i j}=\dfrac{i+3 j}{2}$ ** Solution. ** Given \( a_{ij} = \dfrac{i + 3j}{2} \). For \( i = ... 4 \end{array}\right] \] **Alternative Method:** Given \( a_{ij}=\dfrac{i+3j}{2} \). So we have \begin... $a_{i j}=\dfrac{i \times j}{2}$ ** Solution. ** Given \( a_{ij}=\dfrac{i \times j}{2} \). So we have \b... s $a_{i j}=\dfrac{2 i-3 j}{3}$ ** Solution. ** Given \( a_{ij} = \frac{2i - 3j}{3} \), we need to find