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- Question 14 Exercise 4.2 @math-11-kpk:sol:unit04
- Question 14(i)===== Insert three arithmetic means between 6 and 41. GOOD ====Solution==== Let $A_1, A_2, A_3$ be three arithmetic means between 6 and 41. Then $6, A_1, A_2, A_3, 41$ are in A.P.... ac{1}{4}.\end{align} Hence three arithmetic means between 6 and 41 are $$14\dfrac{3}{4},23\dfrac{1}{2},32\d... Question 14(ii)===== Insert four arithmetic means between 17 and 32. GOOD ====Solution==== Let $A_1, A_2, A
- Question 16 Exercise 4.2 @math-11-kpk:sol:unit04
- ====Question 16===== Insert five arithmetic means between $5$ and $8$ and show that their sum is five times the arithmetic mean between $5$ and $8$. GOOD ====Solution==== Let $A_1, A_2, A_3, A_4, A_5$ be five arithmetic means between $5$ and $8$. Then $5, A_1, A_2, A_3, A_4, A_5, 8$... },6,\dfrac{13}{2},7, \dfrac{15}{2}$ are five A.Ms between $5$ & $8$. Now \begin{align}A_1&+A_2+A_3+A_4+A_5
- Question 2 and 3 Exercise 3.3 @math-11-kpk:sol:unit03
- d{align} =====Question 3(i)===== Find the angles between the pairs of vectors: $\hat{i}-\hat{j}+\hat{k}, \... ime}$$. =====Question 3(ii)===== Find the angles between the pairs of vectors: $3 \hat{i}+4 \hat{j}, \quad... }=2 \hat{j}-5 \hat{k}$. Let $\theta$ be the angle between $\vec{a}$ and $\vec{b}$ \begin{align}\text { then... align} =====Question 3(iii)===== Find the angles between the pairs of vectors: $2 \hat{i}-3 \hat{k}, \quad
- Question 12 & 13 Exercise 4.2 @math-11-kpk:sol:unit04
- =====Question 13(i)===== Find the arithmetic mean between $12$ and $18$. GOOD ====Solution==== Here $a=12, ... }\\&=\dfrac{30}{2}=15.\end{align} Hence 15 is A.M between 12 and 18. GOOD =====Question 13(ii)===== Find the arithmetic mean between $\dfrac{1}{3}$ and $\dfrac{1}{4}$. ====Solution==... ===Question 13(iii)===== Find the arithmetic mean between $-6,-216$. GOOD ====Solution==== Here $a=-6, b=-2
- Unit 04: Sequence and Series (Solutions)
- ing arithmetic sequence. * Know arithmetic mean between two * Insert n arithmetic means tEtween two num... series. * Show that sum of $n$ arithmetic means between two numbers is equal to n times their arithmetic ... lving geometric sequence. * Know geometric mean between two numbers. * Insert $n$ geometric means between two numbers. * Define a geometric series. * Find t
- Question 9 Exercise 4.4 @math-11-kpk:sol:unit04
- ===Question 9(i)===== Insert five geometric means between $3 \dfrac{5}{9}=\dfrac{32}{9}\quad$ and $\quad40 ... , G_3, G_4$ and $G_5$ be the five geometric means between $\dfrac{32}{9}$ and $\dfrac{81}{2}$,\\ then $\dfr... ===Question 9(ii)===== Insert $6$ geometric means between $14$ and $-\dfrac{7}{64}$. ====Solution==== Let $... 3, G_4, G_5$ and $G_6$ be the six geometric means between $14$ and $-\dfrac{7}{64}$,\\ then $14, G_1, G_2,
- Question 10 Exercise 4.4 @math-11-kpk:sol:unit04
- estion 10===== Find two numbers if the difference between them is $48$ and their A.M exceeds their G.M by $... be $a$ and $b$ \\ Condition-$1$\\ The difference between them is $48$\\ Therefore, $$\quad a-b=48....(i)$$. The geometric mean between $a$ and $b$ is $$G=\sqrt{a b}$$ The arithmetic mean between $a$ and $b$ is $$A=\dfrac{a+b}{2}$$ Condition-$2$
- Question 3, Exercise 10.1 @math-11-kpk:sol:unit10
- and $\sin v=\dfrac{4}{5}$ where$u$ and $v$ are between $0$ and $\dfrac{\pi }{2}$, evaluate each of the f... $ and $\sin v=\dfrac{4}{5}$ and $u$ and $v$ are between $0$ and $\dfrac{\pi }{2}$, evaluate each of the ... $ and $\sin v=\dfrac{4}{5}$ and $u$ and $v$ are between $0$ and $\dfrac{\pi }{2}$, evaluate each of the f... and $\sin v=\dfrac{4}{5}$ where$u$ and $v$ are between $0$ and $\dfrac{\pi }{2}$, evaluate each of the f
- Question 9 & 10 Exercise 4.3 @math-11-kpk:sol:unit04
- estion 9===== Find the sum 'of all multiples of 9 between 300 and 700. ====Solution==== All the multiples of 9 between 300 and 700 are:\\ $$306,315,324,333, \ldots, 693... d{align} Hence, sum of all multiples of $9$ lying between $300$ and $700$ is equal to $21,978$. =====Quest
- Question 11 Exercise 4.4 @math-11-kpk:sol:unit04
- that the prodect of $\mathrm{n}$ geometric means between $a$ and $b$ is equal to the $nth$ power for the single geometric mean between them. ====Solution==== Let $G_1, G_2, G_9, \ldots, G_n$ be the $n$ geometric means between $a$ and $b$,\\ then $a, G_1, G_2, G_3, \ldots, G_
- Unit 03: Vectors (Solutions)
- s is unity. * Use dot product to find the angle between two vectors. * Find the projection of a vector ... product. * Use cross product to find the angle between two vectors. * Find the vector moment of a give
- Question 7 & 8 Exercise 3.3 @math-11-kpk:sol:unit03
- l vector along $y-a x i s$.\\ The cosine of angie between the given vector and $y-a x i s$ is now actually cosine of angle between $\vec{d}$ and $\vec{b}$.\\ Now $\vec{a} \cdot \ve
- Question 6 & 7 Review Exercise 3 @math-11-kpk:sol:unit03
- \times \vec{b}$ is a unit vector. Write the angle between $\vec{a}$ and $\vec{b}$. ====Solution==== Let $\theta$ be the angle between two vectors. We are given\\ $$|\vec{a} \times \ve
- Question 15 Exercise 4.2 @math-11-kpk:sol:unit04
- a^{n+1}+b^{n+1}}{a^n+b^n}$ is the arithmetic mean between $a$ and $b$. Where $a$ and $b$ are not zero simul... on==== Suppose $A$ represents the arithmetic mean between $a$ and $b$, then $$ A=\dfrac{a+b}{2}. --- (1) $$
- Question 17 Exercise 4.2 @math-11-kpk:sol:unit04
- ==Question 17===== There are $n$ arithmetic means between 5 and 32 such that the ratio of the 3rd and 7th m... 1, A_2, A_3, \ldots, A_n$ be $n$ arithmetic means between 5 and 32. Then $5, A_1, A_2, A_3, \ldots, A_n, 32