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- Question 5 Exercise 4.1 @math-11-kpk:sol:unit04
- istan. =====Question 5(i)===== Write each of the following series in expanded form, $\sum_{j=1}^6(2 j-3)$ ==... lign} =====Question 5(ii)===== Write each of the following series in expanded form, $\sum_{k=1}^5(-1)^k 2^{k... ign} =====Question 5(iii)===== Write each of the following series in expanded form, $\sum_{j=1}^{\infty} \df... lign} =====Question 5(iv)===== Write each of the following series in expanded form, $\sum_{k=0}^{\infty}\lef
- Question 3, Exercise 10.1 @math-11-kpk:sol:unit10
- n $0$ and $\dfrac{\pi }{2}$, evaluate each of the following exactly $\cos \left( u+v \right)$ ====Solution==... $0$ and $\dfrac{\pi }{2}$, evaluate each of the following exactly $\tan \left( u-v \right)$ ====Solution===... n $0$ and $\dfrac{\pi }{2}$, evaluate each of the following exactly $\sin \left( u-v \right)$ ====Solution==... n $0$ and $\dfrac{\pi }{2}$, evaluate each of the following exactly $\cos \left( u+v \right)$ ====Solution==
- Question 13, Exercise 10.1 @math-11-kpk:sol:unit10
- an. =====Question 13(i)===== Express each of the following in the form $r\,\,\sin \left( \theta +\phi \righ... n} =====Question 13(ii)===== Express each of the following in the form $r\,\,\sin \left( \theta +\phi \righ... } =====Question 13(iii)===== Express each of the following in the form $r\,\,\sin \left( \theta +\phi \righ... n} =====Question 13(iv)===== Express each of the following in the form $r\,\,\sin \left( \theta +\phi \righ
- Question 2, Exercise 10.2 @math-11-kpk:sol:unit10
- \cos\theta = -\dfrac{12}{13}$$ Thus, we have the following by using double angle identities: \begin{align}\s... \cos\theta = -\dfrac{12}{13}$$ Thus, we have the following by using double angle identities: \begin{align}\c... \cos\theta = -\dfrac{12}{13}$$ Thus, we have the following by using double angle identities: Thus, we have the following by using double angle identities. \begin{align}\s
- Question 7 Exercise 3.5 @math-11-kpk:sol:unit03
- =====Question 7(i)===== For what value of $c$ the following vectors are coplanar $\vec{u}=\hat{i}+2 \hat{j}+3... ====Question 7(ii)===== For what value of $c$ the following vectors are coplanar $\vec{u}=\hat{i}+\hat{j}-\ha... ===Question 7(iii)===== For what value of $c$ the following vectors are coplanar $\vec{u}=\hat{i}+\hat{j}+2 \
- Question 3, Exercise 10.2 @math-11-kpk:sol:unit10
- $\cos \theta =-\dfrac{3}{5}$. Thus, we have the following by using double angle identity: \begin{align}\sin... $\cos \theta =-\dfrac{3}{5}$. Thus, we have the following by using half angle identities: \begin{align}\cos
- Unit 01: Complex Numbers (Solutions)
- e inverse of a complex $z$. * Demonstrate the following properties $|z|=|-z|=|\bar{z}=|-\bar{z}|$ * F
- Unit 02: Matrices and Determinants (Solutions)
- ive inverse of a complex $z$. * Demonstrate the following properties $|z|=|-z|=|\bar{z}=|-\bar{z}|$ * Fin
- Unit 03: Vectors (Solutions)
- metrical representation of a vector. * Give the following fundamental definitions using geometrical represe
- Unit 06: Permutation, Combination and Probability (Solutions)
- ve problems involving combination. * Define the following: * statistical experiment, * sample space
- Question 11 & 12 Exercise 4.5 @math-11-kpk:sol:unit04
- each term is four times the sum of all the terms following it, therefore\\ \begin{align}a_1&=4(a_1 r+a_1 r^2
- Question 2 & 3 Exercise 5.2 @math-11-kpk:sol:unit05
- ion 3===== Find the $n^{\text {th }}$ term of the following arithmetic-geometric series: $\dfrac{0}{1}+\dfrac
- Question 4 & 5 Exercise 5.2 @math-11-kpk:sol:unit05
- kistan. =====Question 4===== Find the sum of the following arithtical geometrical series $5+\dfrac{7}{3}+\df
- Question 11 Review Exercise 6 @math-11-kpk:sol:unit06
- hawar, Pakistan. =====Question 11===== Given the following spinner, determine the probability. ====Solution=
- Question 10 Exercise 7.3 @math-11-kpk:sol:unit07
- TBB) Peshawar, Pakistan. Q10 Find the sum of the following series: (i) $1-\frac{1}{2^2}+\frac{1.3}{2 !} \cdo