Maths

An upper triangular matrix is a square matrix where all the elements above the diagonal are non-zero, and below it is zero. A lower triangular matrix is a square matrix where all the elements above the diagonal are zero.

The elements of a matrix may be real or complex numbers. If all the elements of a matrix are real, then the matrix is called a real matrix.

  $\left(1,1\right)\left(1,4\right)\left(4,1\right)\left(2,4\right)\left(4,2\right)\left(3,4\right)\left(4,3\right)\left(4,4\right)$  all have only one image.

(2, 1) (1, 2), (2, 2) each element has 3 choice.

(3, 2) (2, 3) (3, 1) (1, 3) (3, 3) each element has two choices.

$\therefore$ total function = 3 * 3 * 2 * 2 * 2 = 72

Case I

None of the pre image have 3 as image, total functions = 2 * 2 * 1 * 1 * 1 = 4

Case II

None of the pre images have 2 as image then number of function = 2⁵ = 32

Case III

None of the pre image have either 3 or 2 as image

Total function = 1⁵ = 1

$\therefore$ Total number of onto function

= 72 – 4 – 32 + 1 = 37

$\tilde{z}=i{z}^2+z^2-z$

$z+\overline{Z}=z^2\left(i+1\right)$

$z+\overline{Z}=z^2\left(i+1\right)$ Let z be equal to (x + iy)

(x + iy) + (x – iy) = (x + iy)² (i + 1)

$2x=\left(x^2-y^2+2ixy\right)\left(i+1\right)$               

Equating the real & in eg part.

$\left(x^2-y^2+2ixy\right)=0.........\left(i\right)$

$\left(x^2-y^2-2xy\right)=\left(2x\right)..............\left(ii\right)$               

(i) & (ii)

 4xy = -2x Þ x = 0 or y = $\left(\frac{-1}{2}\right)$  

(for x = 0, y = 0)

For y = $\frac{-1}{2}$  

x²   $-\frac{1}{4}+2\left(\frac{-1}{2}\right)x=0$

x =   $\frac{4\pm \sqrt[]{16+16}}{2.4}$

=  $\left(\frac{1+\sqrt[]{2}}{2}\right)or\left(\frac{1-\sqrt[]{2}}{2}\right)$  

$\therefore sum$ of   ${\left|z\right|}^2={\left(\frac{1+\sqrt[]{2}}{2}\right)}^2+\frac{1}{4}+{\left(\frac{1-\sqrt[]{2}}{2}\right)}^2+\frac{1}{4}+0^2+O^2$

=   $\frac{3}{4}+\frac{\sqrt[]{2}}{2}+\frac{1}{4}+\frac{3}{4}-\frac{\sqrt[]{2}}{2}+\frac{1}{4}=\frac{3}{2}+\frac{1}{2}=2$

$\left(\begin{array}{cc}\hfill 1+i\hfill & \hfill 1\hfill \\ \hfill -i\hfill & \hfill 0\hfill \end{array}\right)$

$A^2=\left(\begin{array}{cc}\hfill 1+i\hfill & \hfill 1\hfill \\ \hfill -i\hfill & \hfill 0\hfill \end{array}\right)\left(\begin{array}{cc}\hfill 1+i\hfill & \hfill 1\hfill \\ \hfill -i\hfill & \hfill 0\hfill \end{array}\right)=\left(\begin{array}{cc}\hfill i\hfill & \hfill 1+i\hfill \\ \hfill 1-i\hfill & \hfill -i\hfill \end{array}\right)$

$A^4=\left(\begin{array}{cc}\hfill i\hfill & \hfill 1+i\hfill \\ \hfill 1-i\hfill & \hfill -i\hfill \end{array}\right)\left(\begin{array}{cc}\hfill i\hfill & \hfill 1+i\hfill \\ \hfill 1-i\hfill & \hfill -i\hfill \end{array}\right)=\left(\begin{array}{cc}\hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 1\hfill \end{array}\right)$

$\therefore A^5=A$

A⁹ = A

$\therefore forn=1,5,9,.....,97$

total possible values of n = 25

$\begin{array}{cc}\hfill 2x-3y=\gamma +5\hfill & \hfill \alpha x+5y=\left(\beta +1\right)\hfill \end{array}\}infinitelymanysolution$

$\therefore \frac{2}{\alpha }=\frac{-3}{5}=\left(\frac{\gamma +5}{\beta +1}\right)$

(i) $\frac{2}{\alpha }=\frac{-3}{5}=\frac{\gamma +5}{\beta +1}$

$\alpha =\frac{5*2}{-3}$ 5x + 25 = -3β - 3

^{$5\gamma +3\beta =-28$}

^{$\left|9\alpha +5\gamma +3\beta \right|=\left|9*\frac{-10}{3}-28\right|$}

= ^{$\left|-30-28\right|=58$}

$S_n=\frac{n^2}{1-\frac{1}{{\left(n+1\right)}^2}}=\frac{n{\left(n+1\right)}^2}{\left(n+2\right)}={\left(n+1\right)}^2-2\left(n+1\right)+2-\frac{2}{\left(n+2\right)}$

$\frac{1}{26}+{\displaystyle \sum _{n=1}^{50}\left(S_n+\frac{2}{\left(n+1\right)}-\left(n+1\right)\right)=\frac{1}{26}+{\displaystyle \sum _{n=1}^{50}{\left(n+1\right)}^2}-3\left(n+1\right)+2+2\left(\frac{1}{\left(n+1\right)}-\frac{1}{\left(n+2\right)}\right)}$

$=\frac{1}{26}+45525-3*1325+2*50+2\left(\frac{1}{2}-\frac{1}{52}\right)$

$=\frac{1}{26}+41550+100+1-\frac{1}{26}=41651$

$\underset{x\to 1}{lim}\frac{sin\left(3x^2-4x+1\right)-x^2+1}{2x^3-7x^2+ax+b}=-2$

For this limit to be defined 2x³ – 7x² + ax + b should also trend to 0 or x ® 1.

$\therefore 2.{\left(1\right)}^3-7{\left(1\right)}^2+a\cdot 1+b=0$

 2 – 7 + (a + b) = 0

(a + b) = 5 …………….(i)

Now this becomes % form  $\therefore$ we apply L'lopital rule

$\underset{x\to 1}{lim}\frac{\left(3x^2-4x+1\right)-x^2+1}{2x^3-7x^2+ax+b}=\underset{x\to 1}{lim}\frac{cos\left(3x^2-4x+1\right)\left(6x-4\right)-2x}{6x^2-14x+a}$

Now the numerator again ® 0 as x = 1

$\therefore$  6x² – 14x + a ® 0 as x = 1

6 . (1)² – 14 + a = 0

a = 8 …………….(ii)

a + b = 5  $\therefore a-b=8-\left(-3\right)=11$       

(b = -3) ® from (i) & (ii)

  $x^2+y^2-2\sqrt[]{2}x-6\sqrt[]{2}y+14=0$

 centre $\left(\sqrt[]{2},3\sqrt[]{2}\right)$

radius  ${\left({\left(\sqrt[]{2}\right)}^2+{\left(3\sqrt[]{2}\right)}^2-14\right)}^{1/2}$

$={\left(2+18-14\right)}^{1/2}=\left(\sqrt[]{6}\right)$

${\left(x-2\sqrt[]{2}\right)}^2+{\left(y-2\sqrt[]{2}\right)}^2=r^2$

centre  $\left(2\sqrt[]{2},2\sqrt[]{2}\right)$

OA = $\sqrt[]{{\left(2\sqrt[]{2}-\sqrt[]{2}\right)}^2+{\left(2\sqrt[]{2}-3\sqrt[]{2}\right)}^2}=\sqrt[]{2+2}=2$                  

r² = ${\left(\sqrt[]{6}\right)}^2+{\left(2\right)}^2=6+4=10$