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- Question 7, Exercise 10.2 @math-11-kpk:sol:unit10
- ====== Question 7, Exercise 10.2 ====== Solutions of Question 7 of Exercise 10.2 of Unit 10: Trigonometric I... \sin }^{4}}\theta =\dfrac{1}{\sec 2\theta }$. ====Solution==== \begin{align}L.H.S&={{\cos }^{4}}\theta -{{\s... }\dfrac{\theta }{2}=\dfrac{2}{\sin \theta }$. ====Solution==== \begin{align}L.H.S&=\tan \dfrac{\theta }{2}+c... eta }{1-\cos 2\theta }={{\cot }^{2}}\theta $. ====Solution==== \begin{align}L.H.S&=\dfrac{1+\cos 2\theta }{1
- Question 5, Exercise 1.3 @math-11-kpk:sol:unit01
- ====== Question 5, Exercise 1.3 ====== Solutions of Question 5 of Exercise 1.3 of Unit 01: Complex Numbers. ... awar, Pakistan. =====Question 5(i)===== Find the solutions of the equation ${{z}^{2}}+z+3=0$. ====Solution==== Given: $${{z}^{2}}+z+3=0.$$ According to the quadrati... &=\dfrac{-1\pm \sqrt{11}}{2}i\end{align} Thus the solutions of the given equation are $-\dfrac{1}{2}\pm\dfra
- Question 6, Exercise 1.3 @math-11-kpk:sol:unit01
- ====== Question 6, Exercise 1.3 ====== Solutions of Question 6 of Exercise 1.3 of Unit 01: Complex Numbers. ... awar, Pakistan. =====Question 6(i)===== Find the solutions of the equation ${{z}^{4}}+{{z}^{2}}+1=0$. ====Solution==== $$z^4+z^2+1=0$$ $$z^4+2z^2+1-z^2=0$$ $$( z^2... qrt{3}}{2}i$$ =====Question 6(ii)===== Find the solutions of the equation ${{z}^{3}}=-8$. ====Solution====
- Question 7 Exercise 6.4 @math-11-kpk:sol:unit06
- ====== Question 7 Exercise 6.4 ====== Solutions of Question 7 of Exercise 6.4 of Unit 06: Permutation, Combi... obability of getting doublet of even numbers. ====Solution==== The sample space rolling a pair of dice is \b... probability of getting a sum less than $6$. ====Solution==== The sample space rolling a pair of dice is \b... e probability of getting a sum more than $7.$ ====Solution==== The sample space rolling a pair of dice is \b
- Question 1 and 2 Exercise 4.1 @math-11-kpk:sol:unit04
- ====== Question 1 and 2 Exercise 4.1 ====== Solutions of Question 1 and 2 of Exercise 4.1 of Unit 04: Seque... nd infinite sequences\\ $2,4,6,8, \ldots ,50$ ====Solution==== It is finite sequence whose last term is $50... and infinite sequences. $1,0,1,0,1, \ldots$. ====Solution==== It is infinite sequence, the last term may be... nfinite sequences: $...,-4,0,4,8, \ldots, 60$ ====Solution==== This is infinite sequence. =====Question 1(i
- Question 1 and 2 Exercise 6.1 @math-11-kpk:sol:unit06
- ====== Question 1 and 2 Exercise 6.1 ====== Solutions of Question 1 and 2 of Exercise 6.1 of Unit 06: Permut... aluate the $\dfrac{10 !}{3 ! .3 ! \cdot 4 !}$ ====Solution==== \begin{align}\dfrac{10 !}{3 ! \cdot 3 ! \cdot... ===== Evaluate the $\dfrac{3 !+4 !}{5 !-4 !}$ ====Solution==== \begin{align}\dfrac{3 !+4 !}{5 !-4 !}&=\dfrac... ===== Evaluate the $\dfrac{(n-1) !}{(n+1) !}$ ====Solution==== \begin{align}\dfrac{(n-1) !}{(n+1) !} &= \df
- Unit 06: Permutation, Combination and Probability (Solutions) @math-11-kpk:sol
- nit 06: Permutation, Combination and Probability (Solutions) ===== This is a sixth unit of the book Mathema... awar, Pakistan. On this page we have provided the solutions of the questions. After reading this unit the s... ems. <panel type="default" title="Exercise 6.1 (Solutions)"> * [[math-11-kpk:sol:unit06:ex6-1-p1|Questio... anel> <panel type="default" title="Exercise 6.2 (Solutions)"> * [[math-11-kpk:sol:unit06:ex6-2-p1|Questio
- Solutions: Math 11 KPK
- ======Solutions: Math 11 KPK====== {{ :fsc-part1-kpk:fsc-part1-kpk-sol.jpg?nolink&600x503|Solutions of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa}} <lead>Solutions of A Textbook of Mathematics for Grade XI is pub... SC-I.php|here}}. Our primary goal is to offer the solutions in accordance with this plan (SLOs based). But
- Unit 03: Vectors (Solutions) @math-11-kpk:sol
- ===== Unit 03: Vectors (Solutions) ===== This is a third unit of the book Mathematics 11 published by Khybe... awar, Pakistan. On this page we have provided the solutions of the questions. After reading this unit the s... tors. <panel type="default" title="Exercise 3.2 (Solutions)"> * [[math-11-kpk:sol:unit03:ex3-2-p1|Questio... anel> <panel type="default" title="Exercise 3.3 (Solutions)"> * [[math-11-kpk:sol:unit03:ex3-3-p1|Questio
- Unit 04: Sequence and Series (Solutions) @math-11-kpk:sol
- ===== Unit 04: Sequence and Series (Solutions) ===== This is a forth unit of the book Mathematics 11 publis... awar, Pakistan. On this page we have provided the solutions of the questions. After reading this unit the s... ence. <panel type="default" title="Exercise 4.1 (Solutions)"> * [[math-11-kpk:sol:unit04:ex4-1-p1|Questio... nel> <panel type="default" title="Exercise 4.2 (Solutions)"> * [[math-11-kpk:sol:unit04:ex4-2-p1|Questio
- Unit 05: Miscellaneous Series (Solutions) @math-11-kpk:sol
- ===== Unit 05: Miscellaneous Series (Solutions) ===== This is a fifth unit of the book Mathematics 11 publi... awar, Pakistan. On this page we have provided the solutions of the questions. After reading this unit the s... ots $ <panel type="default" title="Exercise 5.1 (Solutions)"> * [[math-11-kpk:sol:unit05:ex5-1-p1|Questio... anel> <panel type="default" title="Exercise 5.2 (Solutions)"> * [[math-11-kpk:sol:unit05:ex5-2-p1|Questio
- Question 7, Exercise 1.2 @math-11-kpk:sol:unit01
- ====== Question 7, Exercise 1.2 ====== Solutions of Question 7 of Exercise 1.2 of Unit 01: Complex Numbers. ... eal and imaginary parts $\dfrac{2+3i}{5-2i}$. ====Solution==== \begin{align}&\dfrac{2+3i}{5-2i} \\ =&\dfrac... $\dfrac{{{\left( 1+2i \right)}^{2}}}{1-3i}$. ====Solution==== \begin{align}&\dfrac{(1+2i)^2}{1-3i}\\ =&\df... ts $\dfrac{1-i}{{{\left( 1+i \right)}^{2}}}$. ====Solution==== \begin{align}&\dfrac{1-i}{{{\left( 1+i \righ
- Question 2, Exercise 2.2 @math-11-kpk:sol:unit02
- ====== Question 2, Exercise 2.2 ====== Solutions of Question 2 of Exercise 2.2 of Unit 02: Matrices and Det... & 1 & 0 \\-1 & 2 & 0 \end{matrix}\right|=0$. ====Solution==== Given $$\left| \begin{matrix} 1 & 2 & 0 \... & -12 \\2 & -1 & 3 \end{matrix} \right|=0$. ====Solution==== Given $$\left| \begin{matrix} 1 & 2 & 3 \... & -1 & 1 \\-2 & 1 & 4 \end{matrix} \right|$. ====Solution==== Given $$\left| \begin{matrix} 1 & 3 & -2
- Question 1, Exercise 3.2 @math-11-kpk:sol:unit03
- ====== Question 1, Exercise 3.2 ====== Solutions of Question 1 of Exercise 3.2 of Unit 03: Matrices and Det... i}+3\hat{j}$, then find $\vec{a}+2\vec{b}$. ====Solution==== \begin{align}\vec{a}+2\vec{b}&=3\hat{i}-5\hat... }+3\hat{j}$, then find $3\vec{a}-2\vec{b}$. ====Solution==== \begin{align}3\vec{a}-2\vec{b}&=3(3\hat{i}-5\... +3\hat{j}$, then find $2(\vec{a}-\vec{b})$. ====Solution==== First we have, \begin{align}\vec{a}-\vec{b}&=
- Question 1 Exercise 4.5 @math-11-kpk:sol:unit04
- ====== Question 1 Exercise 4.5 ====== Solutions of Question 1 of Exercise 4.5 of Unit 04: Sequence and Seri... i)===== Compute the sum $3+6+12+\ldots+3.2^9$ ====Solution==== In the given geometric series: $a_1=3, \quad ... Compute the sum $8+4+2+\ldots+\dfrac{1}{16}$ ====Solution==== In the give geometric series $$a_1=8, \quad r... = Compute the sum $2^4+2^5+2^6+\ldots+2^{10}$ ====Solution==== In the give geometric series.\\ $$a_1=2^4, \q