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- Definitions: FSc Part1 KPK
- llout> ===== Chapters 10 and 11 ===== * **Trigonometry:** The word trigonometry has been derived from three Greek words Trei (three) Goni (angles) and Metron (measurement). its mean mea... em.\\ e.g. $16^\circ 13' 9''$ * **Period of Trigonometric Function:** The smallest +ve number which
- Important Questions
- alculator find $tan(1110^{\circ})$ --- // BISE Sargodha(2015), BISE Gujrawala(2017)// * Prove that $... irc}-\alpha)=-sin\alpha cos\alpha$ --- // BISE Sargodha(2015), BISE Lahore(2017)// * Prove that (wit... 3}sin\frac{4\pi}{9}=\frac{3}{16}$ --- // BISE Sargodha(2015)// * Prove that $cot\alpha-tan\alpha=2cot2\alpha$ --- // BISE Sargodha(2015)// * Find the value of $cos15^{circ}$ u
- Question 12 & 13 Exercise 4.2 @math-11-kpk:sol:unit04
- his salary during his twenty first year of work? GOOD ====Solution==== Suppose $a_1$ represents salar... n during his 21st year of work is dollars 18,500. GOOD =====Question 13(i)===== Find the arithmetic mean between $12$ and $18$. GOOD ====Solution==== Here $a=12, b=18$.\\ Let say $... 15.\end{align} Hence 15 is A.M between 12 and 18. GOOD =====Question 13(ii)===== Find the arithmetic
- Question 1, Exercise 1.1 @math-11-kpk:sol:unit01
- estion 1(i)===== Simplify ${{i}^{9}}+{{i}^{19}}$. GOOD ====Solution==== \begin{align}{{i}^{9}}+{{i}^{1... i\cdot\left( -1 \right)\\ &=i-i\\ &=0.\end{align} GOOD =====Question 1(ii)===== Simplify ${{\left( -i \right)}^{23}}$. GOOD ====Solution==== \begin{align}{{\left( -i \righ... 1}}\\ &=-i\cdot\left( -1 \right)\\ &=i\end{align} GOOD =====Question 1(iii)===== Simplify ${{\left( -1
- Question 9 Exercise 3.4 @math-11-kpk:sol:unit03
- 9(i)===== Find the area of parallelogram whose diagonals are $\vec{a}=4 \hat{i}+\hat{j}-2 \hat{k}\quad... }+4 \hat{k}$. ====Solution==== We are give the diagonal as shown in figure, instead of two adjacent si... arallelogram.\\ $E$ is the intersection of two diagonals, so $E$ is the midpoint of both diagonals. Thus\\ \begin{align}\overrightarrow{A E}&=\overrightar
- Question 2 Exercise 4.3 @math-11-kpk:sol:unit04
- Find the one that is missing: $a_1=2, n=17, d=3$. GOOD ====Solution==== Given: $a_1=2, n=17, d=3$ \\ W... 2.\end{align} Hence $a_{17}=50$ and $S_{17}=442$. GOOD =====Question 2(ii)===== Some of the component... d the one that are missing $a_1=-40, S_{21}=210$. GOOD ====Solution==== Given: $a_1=-40$ and $S_{21}=2... \end{align} Hence $a_{21}=60$, $d=5$ and $n=21$. GOOD =====Question 2(iii)===== Some of the component
- Question 1, Exercise 10.1 @math-11-kpk:sol:unit10
- ons of Question 1 of Exercise 10.1 of Unit 10: Trigonometric Identities of Sum and Difference of Angle... stion 1. ===== Question 1(i) ===== Write as a trigonometric function of a single angle. $\sin {{37}^{... align} ===== Question 1(ii)===== Write as a trigonometric function of a single angle. $\cos {{83}^{... ign}. ===== Question 1(iii)===== Write as a trigonometric function of a single angle. $\cos {{19}^{
- Question 10 Exercise 6.5 @math-11-kpk:sol:unit06
- obability that either both are apples or both are good? ====Solution===== Total number of Apples $=20$... es $=5$ number of defective oranges $=3$. Totál good apples $=15$ Defective apples $=5$ Total good oranges $=10$ number of defective oranges $=3$ number of good fruits $=22$ Now two fruits are chosen at rand
- Question 6 Exercise 3.3 @math-11-kpk:sol:unit03
- value of $m$ so that $\vec{a}+m \vec{b}$ is orthogonal to $\vec{a}$ ====Solution==== We know that \be... hat{k}\end{align}. If $\vec{a}+m \vec{b}$ is orthogonal to $\vec{a}$, then \begin{align}(\vec{a}+m \ve... value of $m$ so that $\vec{a}+m \vec{b}$ is orthogonal to $\vec{b}$. ====Solution==== If $\vec{a}+n \vec{b}$ is orthogonal to $\vec{b}$, then \begin{align}(\vec{a}+m \ve
- Question 3 Exercise 3.4 @math-11-kpk:sol:unit03
- uestion 3(i)===== Find a unit vector that is orthogonal to the given vector $\vec{a}=\hat{i}- 2 \hat{j... ===Solution==== Let $\hat{n}$ be unit vector orthogonal to both $\vec{a}$ and $\vec{b}$. then by cross... estion 3(ii)===== Find a unit vector that is orthogonal to the given vector $\vec{a}=3 \hat{i}-\hat{j}... ===Solution==== Let $\hat{n}$ be unit vector orthogonal to buth $\vec{a}$ and $\vec{b}$. then by cross
- Question 5(i) & 5(ii) Exercise 3.5 @math-11-kpk:sol:unit03
- $ and prove that $\vec{a} \times \vec{b}$ is orthogonal to both $\vec{a}$ and $\vec{b}$ ====Solution==== To show that $\vec{a} \times \vec{b}$ is orthogonal to both $\vec{a}$ and $\vec{b}$. We check the... and $\vec{b}$. For $\vec{a} \times \vec{b}$ orthogonal to $\vec{a}$ \begin{align}\vec{a} \cdot \vec{... perp \vec{a}$. For $\vec{a} \times \vec{b}$ orthogonal to $\vec{b}$ \begin{align}\vec{b} \cdot \vec{
- Question 14 Exercise 4.2 @math-11-kpk:sol:unit04
- = Insert three arithmetic means between 6 and 41. GOOD ====Solution==== Let $A_1, A_2, A_3$ be three a... $$14\dfrac{3}{4},23\dfrac{1}{2},32\dfrac{1}{4}.$$ GOOD =====Question 14(ii)===== Insert four arithmetic means between 17 and 32. GOOD ====Solution==== Let $A_1, A_2, A_3, A_4$ be fo... etic means between 17 and 32 are $$20,23,26,29.$$ GOOD ====Go To==== <text align="left"><btn type=
- Question 1 Exercise 4.3 @math-11-kpk:sol:unit04
- sequence: $9,7,5,3, \ldots$; 20th term; 20 terms. GOOD ====Solution==== Let $a_1$ be first term and $d... th term is -29 and sum of first 20 terms is -200. GOOD =====Question 1(ii)===== Find indicated term a... }, \dfrac{7}{3}, 2, \ldots$; 11th term; 11 terms. GOOD ====Solution==== Let $a_1$ be first term and $d... 3}$ and sum of first 11 terms is $\dfrac{44}{3}$. GOOD ====Go To==== <text align="right"><btn type="
- Question 5 and 6 Exercise 6.3 @math-11-kpk:sol:unit06
- =====Question 6===== Find the total number of diagonal of hexagon. ====Solution==== First we find the total number lines. We know one line can be drawn b... }{(6-2) ! 2 !}=15 $$ Now $6$ are sides of the hexagon so, total number of diagonal are $\quad 15-6=9$. ====Go To==== <text align="left"><btn type="pri
- Question 5 and 6 Exercise 4.2 @math-11-kpk:sol:unit04
- dent of $n$. Thus, the given sequence is in A.P. GOOD =====Question 6===== Find the value of $k$, if... k-2$, $8 k-4$ are in A.P. Also find the sequence. GOOD ====Solution==== Since the given terms are in A... ac{7}{2}$ and the sequence is $14,19,24, \ldots$. GOOD ====Go To==== <text align="left"><btn type="primary">[[math-11-kpk:sol:unit04:ex4-2-p2 |< Question