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- Question 4 Exercise 6.4 @math-11-kpk:sol:unit06
- space of the given problem is: \begin{align}S&=(HHII,HHT.HTH.HTT.THII.THT.TTH,TT7),\\ \text{then} n(S)&=2^3=8\end{align} When all heads. Let $$A=\{H H H\... dfrac{n(A)}{n(S)}=\dfrac{1}{8}$. =====Question 4(ii)===== Three unbiased coins are tossed. What is th... space of the given problem is: \begin{align}S&=(HHII,HHT.HTH.HTT.THII.THT.TTH,TT7),\\ \text{then} n(S
- Question 1, Exercise 1.3 @math-11-kpk:sol:unit01
- hat \begin{align}z-4w&=3i …(i)\\ 2z+3w&=11-5i …(ii)\end{align} Multiply $2$ by (i), we get\\ \begin{align}2z-8w&=6i …(iii)\end{align} Subtract (iii) from (ii), we get\\ \[\begin{array}{cccc} 2z&-8w&=6i \\ \mathop+\limits_{-}2z&\mathop+\limits_{-
- Question 1 Exercise 5.2 @math-11-kpk:sol:unit05
- n=1.2^2+2.2^3+3.2^4+4.2^5+\ldots +n \cdot 2^n....(ii)\end{align} Suburacting the (ii) from (i), we get \begin{align} (1-2) S_n&=1 \cdot 2+(2-1) 2^2+(3-2) ... S_n& =2+(n-1) 2^{n+1}\end{align} =====Question 1(ii)===== Sum up to $n$ terms the series $1+4 x+7 x^2... =x+4 x^2+7 x^3+10 x^4+\ldots +(3 n-2) x^{4 t}....(ii)\end{align} Subtracting the (ii) from (i), we get
- Question 1 Review Exercise 7 @math-11-kpk:sol:unit07
- id="a1" collapsed="true">(a): $2520$</collapse> ii. How many two digits odd numbers can be formed fo... e id="a2" collapsed="true">(c): $28$ </collapse> iii. How many six digits number can be formed from t... id="a6" collapsed="true">(d): $4775$</collapse> vii. The number of all possible matrices of order $3 ... id="a7" collapsed="true">(c): $512$</collapse> viii. How many diagonals can be drawn in plane figure
- Question, Exercise 10.1 @math-11-kpk:sol:unit10
- os \beta =-\dfrac{12}{13}$, $\alpha $in Quadrant III and $\beta $in Quadrant II, find the exact value of $\sin \left( \alpha -\beta \right)$. ====Solut... ght)&=\frac{33}{65}.\end{align} =====Question 4(ii)===== If $\sin \alpha =-\dfrac{4}{5}$ and $\cos \beta =-\dfrac{12}{13}$, $\alpha $in Quadrant III and $\beta $in Quadrant II, find the exact valu
- Question 7, Exercise 10.2 @math-11-kpk:sol:unit10
- \sec 2\theta }=R.H.S.\end{align} =====Question 7(ii)===== Prove the identity $\tan \dfrac{\theta }{2}... e\, angle\, identity)\end{align} =====Question 7(iii)===== Prove the identity $\dfrac{1+\cos 2\theta ... t 2\theta =R.H.S.\end{align} =====Question 7(vii)===== Prove the identity $\dfrac{{{\cos }^{3}}\th... 2\theta }{2}=R.H.S.\end{align} =====Question 7(viii)===== Prove the identity $\dfrac{2{{\cos }^{3}}\
- Question 5(iii) & 5(iv) Exercise 3.5 @math-11-kpk:sol:unit03
- ====== Question 5(iii) & 5(iv) Exercise 3.5 ====== Solutions of Question 5(iii) & 5(iv) of Exercise 3.5 of Unit 03: Vectors. Th... TB or KPTBB) Peshawar, Pakistan. =====Question 5(iii)===== Let $\vec{a}=a_1 \hat{i}+a_2 \hat{j}+a_3 \... olution==== We have already calculated L.H.S in (ii) that\\ \begin{align}|\vec{a} \times \vec{b}|^2&=
- Question 5 & 6 Exercise 4.5 @math-11-kpk:sol:unit04
- _{2 n}&=\dfrac{a_1(r^{2 n}-1)}{r-1} \ldots . . . (ii)\\ \text { and } S_{3 n}&=\dfrac{a_1(r^{3 n}-1)}{r-1}...(iii)\end{align}\\ Puting (i),(ii) and (iii) in L.H.S of the given, we get \begin{align}S_n(S_{3 n}-S_{2 n})&=\dfrac{a_1(r^n-1)}{r-1}[\
- Question 1, Review Exercise 10 @math-11-kpk:sol:unit10
- collapsed="true">(B): $\dfrac{1}{2}$</collapse> ii. If $\tan {{15}^{\circ }}=2-\sqrt{3}$, then the v... collapsed="true">(B): $\dfrac{1}{2}$</collapse> iii. If $\tan \left( \alpha +\beta \right)=\dfrac{1... collapsed="true">(B): $\dfrac{1}{2}$</collapse> vii. A point is in Quadrant-III and on the unit circle. If its x-coordinate is $-\dfrac{4}{5},$ what is t
- Question 1, Exercise 3.2 @math-11-kpk:sol:unit03
- \\ &=-\hat{i}+\hat{j}\end{align} =====Question.1(ii)===== If $\vec{a}=3\hat{i}-5\hat{j}$ and $\vec{b}... &=13\hat{i}-21\hat{j}\end{align} =====Question.1(iii)===== If $\vec{a}=3\hat{i}-5\hat{j}$ and $\vec{b... } …(i)\\ |\vec{b}|&=\sqrt(-2)^2+(3)^2=\sqrt{13} …(ii)\end{align} Subtracting (i) from (ii). We get $$|\hat{a}|-|\hat{b}|=\sqrt{34}-\sqrt{13}$$ =====Quest
- Question 5(i) & 5(ii) Exercise 3.5 @math-11-kpk:sol:unit03
- ====== Question 5(i) & 5(ii) Exercise 3.5 ====== Solutions of Question 5(i) & 5(ii) of Exercise 3.5 of Unit 03: Vectors. This is uni... a} \times \vec{b}\perp \vec{b}$. =====Question 5(ii)===== Let $\vec{a}=a_1 \hat{i}+a_2 \hat{j}+a_3 \h... "><btn type="success">[[math-11-kpk:sol:unit03:ex3-5-p4|Question 5(iii) & 5(iv) >]]</btn></text>
- Question 1 Review Exercise 3 @math-11-kpk:sol:unit03
- pse id="a1" collapsed="true">(a): $0$</collapse> ii. The vectors $3 \hat{i}+5 \hat{j}+2 \hat{k}$, $2 ... 2" collapsed="true">(a): Equilateral </collapse> iii. Two vectors $\hat{i}-2 \hat{i}+\hat{j}+3 \hat{k... ed="true">(a): $\alpha=-2, \beta=-3$</collapse> vii. If $\vec{a} \cdot \vec{b} . \vec{c}$ are positio... se id="a7" collapsed="true">(a): $0$</collapse> viii. If $\theta$ be the angel between any two vector
- Question 1 and 2 Exercise 4.1 @math-11-kpk:sol:unit04
- quence whose last term is $50 $. =====Question 1(ii)===== Classify into finite and infinite sequences... in this sequence, we don't know. =====Question 1(iii)===== Classify into finite and infinite sequence... of the sequence are $1,3,6, 10$. =====Question 2(ii)==== Find first four terms of the sequence with t... the sequence are $4,-8,16,-32$. =====Question 2(iii)==== Find first four terms of the sequence with
- Question 6 & 7 Exercise 4.4 @math-11-kpk:sol:unit04
- ^{\mathbf{A 5}}=n\end{align} Multiplying (i) and (iii), we get\\ \begin{align}a_{10} \cdot a_{16}&=\ln... \quad \ln &=m^2 \because m=a_1 r^{12} \text { by (ii) }\end{align} Hence showed that $\ln =m^2$. ====... , \dfrac{1}{a_1 r^{n-1}}$,\\ General term of the (ii) sequence is:\\ \begin{align}a_n&=\dfrac{1}{a_1 r... endent of $n$, \\ which implies that sequence in (ii) is also geometric sequence with common ratio $\d
- Question 4 Exercise 4.5 @math-11-kpk:sol:unit04
- . \overline{8}=\dfrac{8}{9}$$.\\ =====Question 4(ii)===== Convert each decimal to common fraction $1 ... 7}{11} \ldots \ldots \ldots \ldots . . . \text { (ii) }\end{align} Putting (ii) in (i), we get\\ $$1.63=1+\dfrac{7}{11}=\dfrac{18}{11} \text {. }$$\\ =====Question 4(ii)===== Convert each decimal to common fraction $2