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- Question 2 & 3, Exercise 1.1 @math-11-kpk:sol:unit01
- ====== Question 2 & 3, Exercise 1.1 ====== Solutions of Question 2 & 3 of Exercise 1.1 of Unit 01: Complex N... +{{i}^{112}}+{{i}^{122}}+{{i}^{153}}=0$. GOOD ====Solution==== \begin{align}L.H.S.&={{i}^{107}}+{{i}^{112}}+... $3\left( 1+2i \right),-2\left( 1-3i \right)$. ====Solution==== \begin{align}& 3\left( 1+2i \right)+-2\left( ... 2}-\dfrac{2}{3}i,\dfrac{1}{4}-\dfrac{1}{3}i$. ====Solution==== \begin{align}&\left( \dfrac{1}{2}-\dfrac{2}{3
- Question 1, Exercise 1.1 @math-11-kpk:sol:unit01
- ====== Question 1, Exercise 1.1 ====== Solutions of Question 1 of Exercise 1.1 of Unit 01: Complex Numbers. ... i)===== Simplify ${{i}^{9}}+{{i}^{19}}$. GOOD ====Solution==== \begin{align}{{i}^{9}}+{{i}^{19}}&=i\cdot{{i}... = Simplify ${{\left( -i \right)}^{23}}$. GOOD ====Solution==== \begin{align}{{\left( -i \right)}^{23}}&={{\l... ${{\left( -1 \right)}^{\frac{-23}{2}}}$. GOOD ====Solution==== \begin{align}{{\left( -1 \right)}^{\frac{-23}
- Question 1, Review Exercise 10 @math-11-kpk:sol:unit10
- ====== Question 1, Review Exercise 10 ====== Solutions of Question 1 of Review Exercise 10 of Unit 10: Trig
- Question 3 & 4 Exercise 4.3 @math-11-kpk:sol:unit04
- ====== Question 3 & 4 Exercise 4.3 ====== Solutions of Question 3 & 4 of Exercise 4.3 of Unit 04: Sequence ... ers divisible by $5$ from $25$ to $350$. GOOD ====Solution==== The numbers divisible by $5$ from $25$ tò $35... the sum of their cubes is $6336$ . Find them. ====Solution==== Let us suppose the three numbers are $a-d, a,
- Question 2 Exercise 4.3 @math-11-kpk:sol:unit04
- ====== Question 2 Exercise 4.3 ====== Solutions of Question 2 of Exercise 4.3 of Unit 04: Sequence and Seri... one that is missing: $a_1=2, n=17, d=3$. GOOD ====Solution==== Given: $a_1=2, n=17, d=3$ \\ We need to find ... that are missing $a_1=-40, S_{21}=210$. GOOD ====Solution==== Given: $a_1=-40$ and $S_{21}=210$.\\ So we ha... that are missing $a_1=-7, d=8, S_n=225$. GOOD ====Solution==== Given: $a_1=-7, d=8, S_n=225$, we have to fin
- Question 1 Exercise 4.3 @math-11-kpk:sol:unit04
- ====== Question 1 Exercise 4.3 ====== Solutions of Question 1 of Exercise 4.3 of Unit 04: Sequence and Seri... $9,7,5,3, \ldots$; 20th term; 20 terms. GOOD ====Solution==== Let $a_1$ be first term and $d$ be common dif... {7}{3}, 2, \ldots$; 11th term; 11 terms. GOOD ====Solution==== Let $a_1$ be first term and $d$ be common dif
- Question 14 Exercise 4.2 @math-11-kpk:sol:unit04
- ====== Question 14 Exercise 4.2 ====== Solutions of Question 14 of Exercise 4.2 of Unit 04: Sequence and Se... three arithmetic means between 6 and 41. GOOD ====Solution==== Let $A_1, A_2, A_3$ be three arithmetic means... four arithmetic means between 17 and 32. GOOD ====Solution==== Let $A_1, A_2, A_3, A_4$ be four arithmetic m
- Question 17 Exercise 4.2 @math-11-kpk:sol:unit04
- ====== Question 17 Exercise 4.2 ====== Solutions of Question 17 of Exercise 4.2 of Unit 04: Sequence and Se... means is $7: 13$, find the value of $n$. GOOD ====Solution==== Let $A_1, A_2, A_3, \ldots, A_n$ be $n$ arith
- Question 16 Exercise 4.2 @math-11-kpk:sol:unit04
- ====== Question 16 Exercise 4.2 ====== Solutions of Question 16 of Exercise 4.2 of Unit 04: Sequence and Se... the arithmetic mean between $5$ and $8$. GOOD ====Solution==== Let $A_1, A_2, A_3, A_4, A_5$ be five arithme
- Question 15 Exercise 4.2 @math-11-kpk:sol:unit04
- ====== Question 15 Exercise 4.2 ====== Solutions of Question 15 of Exercise 4.2 of Unit 04: Sequence and Se... here $a$ and $b$ are not zero simultaneously. ====Solution==== Suppose $A$ represents the arithmetic mean be
- Question 12 & 13 Exercise 4.2 @math-11-kpk:sol:unit04
- ====== Question 12 & 13 Exercise 4.2 ====== Solutions of Question 12 & 13 of Exercise 4.2 of Unit 04: Seque... ry during his twenty first year of work? GOOD ====Solution==== Suppose $a_1$ represents salary of worker at ... e arithmetic mean between $12$ and $18$. GOOD ====Solution==== Here $a=12, b=18$.\\ Let say $A$ be arithmeti... an between $\dfrac{1}{3}$ and $\dfrac{1}{4}$. ====Solution==== Here $a=\dfrac{1}{3}, b=\dfrac{1}{4}$,\\ Let
- Question 11 Exercise 4.2 @math-11-kpk:sol:unit04
- ====== Question 11 Exercise 4.2 ====== Solutions of Question 11 of Exercise 4.2 of Unit 04: Sequence and Se... they reach the top of a 5400 feet hill? GOOD ====Solution==== Suppose $a_1$ represent the distance climb by
- Question 10 Exercise 4.2 @math-11-kpk:sol:unit04
- ====== Question 10 Exercise 4.2 ====== Solutions of Question 10 of Exercise 4.2 of Unit 04: Sequence and Se... 20135, what was its population in 1970? GOOD ====Solution==== Suppose $a_1$ represents the population in 19
- Question 9 Exercise 4.2 @math-11-kpk:sol:unit04
- ====== Question 9 Exercise 4.2 ====== Solutions of Question 9 of Exercise 4.2 of Unit 04: Sequence and Seri... y meters it covered in the eight second? GOOD ====Solution==== Suppose that $$a_1=24,$$ $$a_2=21,$$ $$a_3=1
- Question 8 Exercise 4.2 @math-11-kpk:sol:unit04
- ====== Question 8 Exercise 4.2 ====== Solutions of Question 8 of Exercise 4.2 of Unit 04: Sequence and Seri... \dfrac{1}{b}, \dfrac{1}{c}$ are in A.P. GOOD ====Solution==== Since $\dfrac{b+c-a}{a}, \dfrac{c+a-b}{b}, \d