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- Question 2, Exercise 8.1 @math-11-nbf:sol:unit08
- , Islamabad, Pakistan. ===== Question 2(a)===== Find the exact value of $\cos 15^{\circ}$ by using $\c... \cos 15^{\circ}=\dfrac{\sqrt{3}+1}{2\sqrt{2}}$ to find $\cos 165^{\circ}$ by using $\cos \left(180^{\cir... \cos 15^{\circ}=\dfrac{\sqrt{3}+1}{2\sqrt{2}}$ to find $\cos 345^{\circ}$ by using $\cos \left(360^{\cir... circ = \dfrac{\sqrt{3}+1}{2\sqrt{2}}$. Now, let's find $\cos 345^\circ$. \begin{align*} \cos 345^\circ &
- Question 9, Exercise 1.2 @math-11-nbf:sol:unit01
- oard, Islamabad, Pakistan. ====Question 9(i)==== Find real and imaginary parts of $(2+4 i)^{-1}$. **So... ac{1}{5}. \end{align} GOOD ====Question 9(ii)==== Find real and imaginary parts of $(3-\sqrt{-4})^{-2}$... \(\dfrac{12}{169}\). GOOD ====Question 9(iii)==== Find real and imaginary parts of $\left(\dfrac{7+2 i}... t is \(\dfrac{-13}{53}\). ====Question 9(iv)==== Find real and imaginary parts of $\left(\dfrac{4+2 i}
- Question 1, Exercise 2.1 @math-11-nbf:sol:unit02
- slamabad, Pakistan. ===== Question 1(i) ===== Find the order of the following matrix $A=\left[\begin... times 3\end{align} ===== Question 1(ii) ===== Find the order of the following matrix $B=\left[\begin... = 3\times 2\end{align} ===== Question 1(iii)===== Find the order of the following matrix $C=\left[\begin... = 3\times 1\end{align} ===== Question 1(iv) ===== Find the order of the following matrix $D=\left[\begin
- Question 4, Exercise 2.1 @math-11-nbf:sol:unit02
- d, Islamabad, Pakistan. =====Question 4(i)===== Find the transpose of the following matrix and identif... & 6 & 9 \end{bmatrix}$$ =====Question 4(ii)===== Find the transpose of the following matrix and identif... \end{array}\right] $$ =====Question 4(iii)===== Find the transpose of the following matrix and identif... & 2 \end{array}\right]$$ =====Question 4(iv)===== Find the transpose of the following matrix and identif
- Question 4, Exercise 2.2 @math-11-nbf:sol:unit02
- d, Islamabad, Pakistan. =====Question 4(i)===== Find $A$ if \begin{align}\left[\begin{array}{cc} 2 & 1... y}\right]. \end{align*} =====Question 4(ii)===== Find $X$ if $$\begin{bmatrix}3 & 2 \\ 0 & 1 \\ 2 & 0\... t[\begin{array}{ll}2 & 14\end{array}\right]$ then find a non-zero matrix $C$ such that $A C=B C$. ** So... {array}{ll}8 & z \\ t & 6\end{array}\right]$ then find the values of $z, t$ and $x^{2}+y^{2}$. ** Solut
- Question 2, Exercise 2.3 @math-11-nbf:sol:unit02
- = 3\), \(a_{12} = 2\), and \(a_{13} = 3\). Now we find their corresponding cofactors. \begin{align*} A &... = -6 \end{align*} Now, we use these cofactors to find the determinant: \begin{align*} \det(A) &= a_{11}... 2\), \(a_{12} = 3\), and \(a_{13} = -1\). Now we find their corresponding cofactors. \begin{align*} A &... = -1 \end{align*} Now, we use these cofactors to find the determinant: \begin{align*} \det(A) &= a_{11}
- Question 1, Exercise 4.2 @math-11-nbf:sol:unit04
- , Islamabad, Pakistan. =====Question 1(i)===== Find the first four terms of the arithmetic sequence w... a_3=10$, $a_4=13$. GOOD =====Question 1(ii)===== Find the first four terms of the arithmetic sequence w... 3=17$, $a_4=22$. GOOD =====Question 1(iii)===== Find the first four terms of each arithmetic sequence.... _3=12$, $a_4=10$. GOOD =====Question 1(iv)===== Find the first four terms of the arithmetic sequence.
- Question 20, 21 and 22, Exercise 4.3 @math-11-nbf:sol:unit04
- ard, Islamabad, Pakistan. =====Question 20===== Find the first three terms of the arithmetic series. $... ven $a_{1}=7$, $a_{n}=139$, $S_{n}=876$. First we find $n$ and $d$.\\ As \begin{align} &S_n=\frac{n}{2}[... $a_2=19$, $a_3=31$. =====Question 21===== Find the first three terms of each arithmetic series. ... Given $n=14$, $a_{n}=53$, $S_{n}=378$. First we find $a_1$ and $d$.\\ As \begin{align} &S_n=\frac{n}{2
- Question 5 and 6, Exercise 4.5 @math-11-nbf:sol:unit04
- ard, Islamabad, Pakistan. =====Question 5===== Find the sum of the geometric series. $a_{1}=7, r=2, n... $a_1 = 7$, $r = 2$ and $n = 14$. The formula to find the sum of $n$ terms of a geometric series is\\ $... e required sum is $114681$. =====Question 6===== Find the sum of the geometric series. $a_{1}=12, a_{5}... = -3$.\\ Now, we can use \(r = -3\) (as given) to find the number of terms, \(n\): To find $n$, we use $
- Question 14, Exercise 4.5 @math-11-nbf:sol:unit04
- , Islamabad, Pakistan. =====Question 14(i)===== Find fractional notation for the infinite geometric se... infty} =\dfrac{4}{9}$. =====Question 14(ii)===== Find fractional notation for the infinite geometric se... e $S_{\infty}= 1 $. =====Question 14(iii)===== Find fractional notation for the infinite geometric se... nfty}= \frac{5}{9}.$ =====Question 14(iv)===== Find fractional notation for the infinite geometric se
- Question 1, Exercise 8.1 @math-11-nbf:sol:unit08
- , Islamabad, Pakistan. ===== Question 1(i)===== Find the value of $\cos (\alpha \pm \beta), \sin (\alp... ) =-\tan 60^\circ$ GOOD ===== Question 1(ii)===== Find the value of $\cos (\alpha \pm \beta), \sin (\alp... qrt{3} \end{align*} ===== Question 1(iii)===== Find the value of $\cos (\alpha \pm \beta), \sin (\alp... t{3}}{3} \end{align*} ===== Question 1(iv)===== Find the value of $\cos (\alpha \pm \beta), \sin (\alp
- Question 3, Exercise 8.1 @math-11-nbf:sol:unit08
- , Islamabad, Pakistan. ===== Question 3(a)===== Find the exact value of $\cos 120^{\circ}$ by using $\... 1}{2}. \end{align*} GOOD ===== Question 3(b)===== Find the exact value of $\sin 120^{\circ}$ and then $\... -\sqrt{3}. \end{align*} ===== Question 3(c)===== Find the exact value of $\cos 75^{\circ}$ by using $\c... \cos 75^{\circ}=\dfrac{\sqrt{3}-1}{2\sqrt{2}}$ to find $\cos 105^{\circ}$ by using $\cos \left(180^{\cir
- Question 4 Exercise 8.2 @math-11-nbf:sol:unit08
- d, Islamabad, Pakistan. =====Question 4(i)===== Find (a) $\sin 2 \theta$ (b) $\cos 2 \theta$ %%(c)%% $... 1}{2}} \end{align*} GOOD =====Question 4(ii)===== Find (a) $\sin 2 \theta$ (b) $\cos 2 \theta$ %%(c)%% $... {3}{2}}. \end{align*} =====Question 4(iii)===== Find (a) $\sin 2 \theta$ (b) $\cos 2 \theta$ %%(c)%% $... ac{1}{7}}. \end{align*} =====Question 4(iv)===== Find (a) $\sin 2 \theta$ (b) $\cos 2 \theta$ %%(c)%% $
- Question 2, Exercise 9.1 @math-11-nbf:sol:unit09
- d, Islamabad, Pakistan. =====Question 2(i)===== Find the maximum and minimum values of the reciprocal ... m) = \dfrac{1}{7}$. GOOD =====Question 2(ii)===== Find the maximum and minimum values of the reciprocal ... 't have any maximum or minimum value so asking to find out the maximum and minimum values is not appropriate. =====Question 2(iii)===== Find the maximum and minimum values of the reciprocal
- Question 3, Exercise 9.1 @math-11-nbf:sol:unit09
- d, Islamabad, Pakistan. =====Question 3(i)===== Find domain and range: $y=7 \cos 4x$ ** Solution. ** ... {R}$ Range $=[-7,7]$. =====Question 3(ii)===== Find domain and range: $y=\cos \frac{x}{3}$ ** Soluti... R}$ Range $=[-1,1]$. =====Question 3(iii)===== Find domain and range: $y=\sin \frac{2 x}{3}$ ** Solu... }$ Range $=[-1, 1]$. =====Question 3(iv)===== Find domain and range: $y=7 \cot \frac{\pi}{2} x$ **