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- Unit 01: Complex Numbers (Solutions)
- ===== Unit 01: Complex Numbers (Solutions) ===== {{ :math-11-nbf:sol:math-11-nbf-sol-unit01.jpg?nolink&400x335|Unit 01: Complex Numbers (Solutions)}} This is a first unit of the book Mo... * Know the condition for equality of two complex numbers. * Revising the basic operations on complex numbers. * Find conjugate and modulus of a complex number.
- Question 18 and 19, Exercise 6.2 @math-11-nbf:sol:unit06
- ad, Pakistan. =====Question 18===== Howmany odd numbers less than $10,000$ can be formed using the digits... digits. ** Solution. ** We must make $4$ digit numbers to keep the number less that $10000$\\ and digit ... either $3$ or $5$ to make number odd.\\ Possible numbers starting with $0$ and ending with $3={ }^{3} P=6$\\ Possible numbers starting with $0$ and ending with $5=3 p_{6}=6$\\
- Exercise 6.2 (Solutions) @math-11-nbf:sol:unit06
- ion 3 ]] **Question 4.** How many 3 -digit even numbers can be formed from the digits $1,2,3,4,5,6$, if t... stion 4 & 5 ]] **Question 6.** How many 4 -digit numbers can be formed with the digits $1,2,3,4,5,6$ when ... ution: Question 6 & 7]] **Question 7.** How many numbers can be formed with the digits $1,1,2,2,3,3,4$ so ... Question 14 & 15]] **Question 16.** How many odd numbers can be formed by using the digits $1,2,3,4,5,6$ w
- Question 2, Exercise 1.2 @math-11-nbf:sol:unit01
- of Question 2 of Exercise 1.2 of Unit 01: Complex Numbers. This is unit of Model Textbook of Mathematics fo... ion 2==== Use the algebraic properties of complex numbers to prove that $$ \left(z_{1} z_{2}\right)\left(z_... quired result. **Remark:** For any three complex numbers $z_1$, $z_2$ and $z_3$, we have $$z_1 (z_2 z_3) =... that the order in which we multiply three complex numbers doesn't matter; we will always end up with the sa
- Question 23 and 24, Exercise 4.7 @math-11-nbf:sol:unit04
- es 2+3 \times 2^{2}+4 \times 2^{3}+\ldots $$ The numbers $1,2,3,4,\ldots$ are in A.P. with $a=1$ and $d=1$. The numbers $1, 2, 2^2, 2^3, \ldots$ are in G.P. with first t... es is: \[ 1 + 4y + 7y^2 + 10y^3 + \ldots \] The numbers \(1, 4, 7, 10, \ldots\) are in A.P. with \(a = 1\) and \(d = 3\). The numbers \(1, y, y^2, y^3, \ldots\) are in G.P. with first
- Question 25 and 26, Exercise 4.7 @math-11-nbf:sol:unit04
- \frac{7}{7^2} + \frac{10}{7^3} + \ldots \] The numbers \(1, 4, 7, 10, \ldots\) are in AP with \(a = 1\) and \(d = 3\). The numbers \(1, \frac{1}{7}, \frac{1}{7^2}, \frac{1}{7^3}, \... + \frac{19}{8} + \frac{25}{16} + \ldots \] The numbers \(1, 7, 13, 19, 25, \ldots\) are in AP with \(a = 1\) and \(d = 6\). The numbers \(1, \frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \frac
- Question 27 and 28, Exercise 4.7 @math-11-nbf:sol:unit04
- es\frac{1}{9}+11\times\frac{1}{27}+\ldots $$ The numbers $5,7,9,11,4,\ldots$ are in A.P. with $a=5$ and $d=7-5=2$. The numbers $1, \dfrac{1}{3}, \dfrac{1}{9}, \dfrac{1}{27}, \l... 1}{25} + 4 \times \frac{1}{125} + \ldots \] The numbers \(1, 2, 3, 4, \ldots\) are in AP with \(a = 1\) and \(d = 1\). The numbers \(1, \frac{1}{5}, \frac{1}{25}, \frac{1}{125}, \l
- Question 6 and 7, Exercise 6.2 @math-11-nbf:sol:unit06
- kistan. =====Question 6===== How many $4$-digit numbers can be formed with the digits $$1,2,3,4,5,6$ when... times 6=6^4=1296$$ =====Question 7===== How many numbers can be formed with the digits $1,1,2,2,3,3,4$ so ... 7^{\text {th }}$ place.\\ We have to arrange even numbers $2,2,4$ on even places and\\ odd values $1,1,3,3$... !}{2!2!}=\dfrac{24}{4}=6$ $$\text{Total possible numbers }=3 \times 6=18$$ ====Go to ==== <text align="
- Question 16 and 17, Exercise 6.2 @math-11-nbf:sol:unit06
- d, Pakistan. =====Question 16===== How many odd numbers can be formed by using the digits $1,2,3,4,5,6$ w... ts of remaining digits.\\ Number of six digit odd numbers $=3 \times{ }^{5} P_{5}=360$ =====Question 17===== How many $4$-digit odd numbers can be formed using the digits $1,2,3,4$ and $5$ ... al, we have $24+24+24=72$ possible $4$ digits odd numbers. ====Go to ==== <text align="left"><btn type
- Question 2 and 3, Review Exercise 6 @math-11-nbf:sol:unit06
- 58800$$ =====Question 3===== How many $3$-digit numbers are there which have $0$ at unit place? ** Solution. ** In $3$ digit numbers unit, tens and $100's$ place may be filled in 10 ... digits #0# to $9$.\\ So total possible $3$ digit numbers are $$=10\times 10\times10=1000$$\\ If we fix uni... illed in $10$ ways,\\ so total possible $3-$digit numbers having $0$ at units place are $$=10\times 10=100$
- Unit 04: Sequences and Seeries
- neral term. * Know arithmetic means between two numbers. Also insert $n$ arithmetic means between them. ... Show that sum of $n$ arithmatic means between two numbers is equal to $n$ times their A.M. * Solve real ... eneral term. * Know geometric means between two numbers, Also insert $n$ geometric means between them.
- Question 7 and 8, Exercise 4.3 @math-11-nbf:sol:unit04
- D =====Question 8===== Find the sum of the even numbers from $2$ to $100$. ** Solution. ** Sum of the even numbers from $2$ to $100$ is $$2+4+6+...+100 (50 \text{ t... ]\\ &=2550. \end{align} Hence the sum of the even numbers from $2$ to $100$ is $2550$. GOOD ====Go to
- Question 9 and 10, Exercise 4.3 @math-11-nbf:sol:unit04
- n. =====Question 9===== Find the sum of the odd numbers from $1$ to $99$. ** Solution. ** ** Solution. ** Sum of the odd numbers from $1$ to $99$ is $$1+3+5+...+99 (50 \text{ ter... 8]\\ &=2500. \end{align} Hence the sum of the odd numbers from $1$ to $99$ is $2500$. GOOD =====Question
- Question 1, Review Exercise 6 @math-11-nbf:sol:unit06
- "a1" collapsed="true">%%(b)%%: $6$</collapse> ii. Numbers of ways of arrangement of the word "GARDEN"\\ ... apse> iii. The product of $r$ consective positive numbers is divisible by \\ * (a) $r!$\\ * (b)$(... " collapsed="true">%%(c)%%: $3$</collapse> x. The numbers of ways in which $r$ latters can be posted in $n$
- Question 4, 5 and 6, Review Exercise 6 @math-11-nbf:sol:unit06
- kistan. =====Question 4===== How mant six-digit numbers can be formed using the digits $0,2,3,4,5,7$ with... h permutations is $$5!=120$$ Number of $6-$digits numbers formed $$=720-120=600$$ =====Question 5===== The numbers of ways of arranging $7$ keys in a key chain? **