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- Unit 03: Vectors (Solutions)
- definitions using analytical representation. * Find a unit vector in the direction of another given vector. * Find the position vector of a point which divides the ... * Give analytic representation of a vector. * Find magnitude of a vector. * Repeat all fundamental... * Express dot product in terms of components. * Find the condition for orthogonality of two vectors.
- Question 7 Exercise 6.4 @math-11-kpk:sol:unit06
- ion 7(i)===== Two dice are thrown simultaneously. Find the probability of getting doublet of even number... on 7(ii)===== Two dice are thrown simultaneously. Find the probability of getting a sum less than $6$. ... n 7(iii)===== Two dice are thrown simultaneously. Find the probability of getting a sum more than $7.$ =... on 7(iv)===== Two dice are thrown simultaneously. Find the probability of getting a sum greater than $10
- Question 2 Exercise 4.3 @math-11-kpk:sol:unit04
- components $a_1, a_n, n, d$ and $S_n$ are given. Find the one that is missing: $a_1=2, n=17, d=3$. GOOD... ution==== Given: $a_1=2, n=17, d=3$ \\ We need to find $a_{17}$ and $S_{17}$. As we know $$a_{n}=a_1+(n-... components $a_1, a_n, n, d$ and $S_n$ are given. Find the one that are missing $a_1=-40, S_{21}=210$. G... $S_{21}=210$.\\ So we have $n=21$ and we have to find $a_{21}$ and $d$. As \begin{align}&S_{21}=\dfrac{
- Question 1, Exercise 3.2 @math-11-kpk:sol:unit03
- -5\hat{j}$ and $\vec{b}=-2\hat{i}+3\hat{j}$, then find $\vec{a}+2\vec{b}$. ====Solution==== \begin{ali... -5\hat{j}$ and $\vec{b}=-2\hat{i}+3\hat{j}$, then find $3\vec{a}-2\vec{b}$. ====Solution==== \begin{al... -5\hat{j}$ and $\vec{b}=-2\hat{i}+3\hat{j}$, then find $2(\vec{a}-\vec{b})$. ====Solution==== First we... -5\hat{j}$ and $\vec{b}=-2\hat{i}+3\hat{j}$, then find $|\vec{a}+\vec{b}|$. ====Solution==== We have,
- Unit 04: Sequence and Series (Solutions)
- sequences. * Define an arithmetic sequence. * Find the nth or general term of an arithmetic sequence... n arithmetic series. * Establish the formula to find the sum to n terms of an arithmetic series. * S... etic series. * Define a geometric sequence. * Find the $n$th or general term of a geometric sequence... n two numbers. * Define a geometric series. * Find the sum of $n$ terms of a geometric series. * F
- Question 13 Exercise 6.2 @math-11-kpk:sol:unit06
- BB) Peshawar, Pakistan. =====Question 13(i)===== Find the number of permutation of word "Excellence." H... \ &=15,120 \end{align} =====Question 13(ii)===== Find the number of permutation of word "Excellence." H... \\ &=3360 \end{align} =====Question 13(iii)===== Find the number of permutation of word "Excellence." H... \\ &=10,080\end{align} =====Question 13(iv)===== Find the number of permutation of word "Excellence." H
- Question 4 Exercise 6.4 @math-11-kpk:sol:unit06
- Solution==== First we construct a tree diagram to find out the sample space, when the coin is tossed th... Solution==== First we construct a tree diagram to find out the sample space, when the coin is tossed th... Solution==== First we construct a tree diagram to find out the sample space, when the coin is tossed t... Solution==== First we construct a tree diagram to find out the sample space, when the coin is tossed th
- Question 6, Exercise 10.2 @math-11-kpk:sol:unit10
- c{\theta }{2}=\dfrac{{{30}^{\circ }}}{2}$, we can find $\cos {{15}^{\circ }}$by using half angle identit... heta }{2}=\dfrac{{{135}^{\circ }}}{2}$, so we can find $\tan {{67.5}^{\circ }}$by using half angle ident... heta }{2}=\dfrac{{{225}^{\circ }}}{2}$, so we can find $sin{{112.5}^{\circ }}$by using half angle identi... theta }{2}=\dfrac{\dfrac{\pi }{4}}{2}$, so we can find $\cos \dfrac{\pi }{8}$by using half angle identit
- Question 12 & 13 Exercise 4.2 @math-11-kpk:sol:unit04
- =750$. The given problem is of A.P and we have to find $a_{21}$. As, we have \begin{align} a_{21}&=a_1+... is dollars 18,500. GOOD =====Question 13(i)===== Find the arithmetic mean between $12$ and $18$. GOOD =... etween 12 and 18. GOOD =====Question 13(ii)===== Find the arithmetic mean between $\dfrac{1}{3}$ and $\... }{24}\end{align} GOOD =====Question 13(iii)===== Find the arithmetic mean between $-6,-216$. GOOD ====S
- Question 2 Exercise 4.5 @math-11-kpk:sol:unit04
- 2 r$ and $S_n$ of a geometric sequence are given. Find the ones that are missing $a_1=1, \quad r=-2, \quad a_n=64$. ====Solution==== We first find $n$ and then $S_n$\\ We know $a_n=a_1 r^{n-1}$, t... 2 r$ and $S_n$ of a geometric sequence are given. Find the ones that are missing $r=\dfrac{1}{2}, a_9=1, n=9$ ====Solution==== We first find $a_1$ and then $S_9$.\\ We know $$a_9=a_1 r^8$$\
- Question 9 Exercise 6.5 @math-11-kpk:sol:unit06
- g $\dfrac{1}{7}$ and $\dfrac{1}{5}$ respectively. Find the probability that both will be selected. ====S... g $\dfrac{1}{7}$ and $\dfrac{1}{5}$ respectively. Find the probability that Only one is selected. ====So... g $\dfrac{1}{7}$ and $\dfrac{1}{5}$ respectively. Find the probability that none will be sclected. ====S... nd{align} None will be selected. Here we have to find probability that none of both is selected that is
- Unit 05: Miscellaneous Series (Solutions)
- will be able to * Recognize sigma notation. * Find the sum of * The first $n$ natural numbers. ... ers. * Define arithmetico-geometric series. * Find sum to n terms of the arithmetico-geometric serie... ne method of differences. * Use this method to find the sum of $n$ terms of the series whose differen... geometric sequence. * Use partial fractions to find the sum to $n$ terms and to infinity the series o
- Question 5, Exercise 1.3 @math-11-kpk:sol:unit01
- TBB) Peshawar, Pakistan. =====Question 5(i)===== Find the solutions of the equation ${{z}^{2}}+z+3=0$. ... \dfrac{\sqrt{11}}{2}i$. =====Question 5(ii)===== Find the solutions of the equation ${{z}^{2}}-1=z$.\\ ... frac{1\pm\sqrt{5}}{2}$. =====Question 5(iii)===== Find the solutions of the equation ${{z}^{2}}-2z+i=0$\... s are $1\pm\sqrt{1-i}$. =====Question 5(iv)===== Find the solutions of the equation ${{z}^{2}}+4=0$\\ =
- Question 6, Exercise 1.3 @math-11-kpk:sol:unit01
- TBB) Peshawar, Pakistan. =====Question 6(i)===== Find the solutions of the equation ${{z}^{4}}+{{z}^{2}... \dfrac{\sqrt{3}}{2}i$$ =====Question 6(ii)===== Find the solutions of the equation ${{z}^{3}}=-8$. ===... he required equations. =====Question 6(iii)===== Find the solutions of the equation ${{\left( z-1 \righ... \dfrac{\sqrt{3}i}{2}$. =====Question 6(iv)===== Find the solutions of the equation ${{z}^{3}}=1$. \\ =
- Question 2, Exercise 2.3 @math-11-kpk:sol:unit02
- PTBB) Peshawar, Pakistan. =====Question 2(i)===== Find the inverse of the matrix by using elementary row... d{bmatrix} \end{align} =====Question 2(ii)===== Find the inverse of the matrix by using elementary row... end{matrix} \right]$$ =====Question 2(iii)===== Find the inverse of the matrix by using elementary row... {1}{4} \end{bmatrix}$$ =====Question 2(iv)===== Find the inverse of the matrix by using elementary row