Maths Matrices

Given a > b

Area common to x2 + y2 $\le a^{2}and\frac{x^2}{a^2}+\frac{y^2}{b^2}\le 1$

is $\pi a^2-\pi ab=30\pi ..............\left(i\right)$

Similarly $\pi ab-\pi b^2=18\pi .................\left(ii\right)$

Equation (i) and equation (ii) $\frac{a}{b}=\frac{5}{3}$

Equation (i) + equation (ii) $a^2-b^2=48$

a² = 75, b² = 27

B = (I – adjA)⁵

^{fffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff}

f : R defined as

$f\left(x\right)=x-1andg\left(x\right):R-\left\{1,-1\right\}\to R$

$g\left(x\right)=\frac{x^2}{x^2-1}$

$\therefore fog\left(x\right)=\frac{x^2}{x^2-1}-1=\frac{1}{x^2-1}$

domain of fog (x) R – {-1, 1} and range   $(-\infty ,-1]\cup (0,\infty )$ $\therefore$ $$

$$ fog (x) is neither one-one nor onto

 $f\left(x\right)=\frac{x-1}{x+1}\Rightarrow f\left(f\left(x\right)\right)=\frac{\frac{x-1}{x+1}-1}{\frac{x-1}{x+1}+1}=-\frac{1}{x}$

$f^3\left(x\right)=-\frac{x+1}{x-1}\Rightarrow f^4\left(x\right)=\frac{\frac{x-1}{x+1}+1}{\frac{x-1}{x+1}-1}=x$

$So,f^6\left(6\right)+f^7\left(7\right)=f^2\left(6\right)+f^3\left(7\right)$

= $-\frac{1}{6}-\frac{7+1}{7-1}=-\frac{9}{6}=-\frac{3}{2}$

 $A=\left[\begin{array}{ccc}\hfill 1\hfill & \hfill 2\hfill & \hfill 4\hfill \\ \hfill \frac{1}{2}\hfill & \hfill 1\hfill & \hfill 2\hfill \\ \hfill \frac{1}{4}\hfill & \hfill \frac{1}{2}\hfill & \hfill 1\hfill \end{array}\right]$

$A^2=$ $\left[\begin{array}{ccc}\hfill 1\hfill & \hfill 2\hfill & \hfill 4\hfill \\ \hfill \frac{1}{2}\hfill & \hfill 1\hfill & \hfill 2\hfill \\ \hfill \frac{1}{4}\hfill & \hfill \frac{1}{2}\hfill & \hfill 1\hfill \end{array}\right]\left[\begin{array}{ccc}\hfill 1\hfill & \hfill 2\hfill & \hfill 4\hfill \\ \hfill \frac{1}{2}\hfill & \hfill 1\hfill & \hfill 2\hfill \\ \hfill \frac{1}{4}\hfill & \hfill \frac{1}{2}\hfill & \hfill 1\hfill \end{array}\right]$

$=3\left[\begin{array}{ccc}\hfill 1\hfill & \hfill 2\hfill & \hfill 4\hfill \\ \hfill \frac{1}{2}\hfill & \hfill 1\hfill & \hfill 2\hfill \\ \hfill \frac{1}{4}\hfill & \hfill \frac{1}{2}\hfill & \hfill 1\hfill \end{array}\right]$

A² = 3A

 A³ = 3A²

A³ = 3²A

A⁴ = 3³A

Aⁿ = 3^n-1A

now, A² + A³ +….+A¹⁰

= 3A + 3² A +…. + 3⁹A

= 3A (1 + 3 +….+ 3⁸

$=3A\cdot \frac{\left(3^9-1\right)}{3-1}$

$=\frac{3^{10}-3}{2}A$

$=\left|\begin{array}{ccc}\hfill 2\hfill & \hfill 1\hfill & \hfill -1\hfill \\ \hfill 1\hfill & \hfill -3\hfill & \hfill 2\hfill \\ \hfill 1\hfill & \hfill 4\hfill & \hfill \delta \hfill \end{array}\right|=0\Rightarrow \delta =-3$

and ${\Delta }_1=\left|\begin{array}{ccc}\hfill 7\hfill & \hfill 1\hfill & \hfill -1\hfill \\ \hfill 1\hfill & \hfill -3\hfill & \hfill 2\hfill \\ \hfill k\hfill & \hfill 4\hfill & \hfill -3\hfill \end{array}\right|=0\Rightarrow k=6$

Set of first 10 prime numbers

= {2,3,5,7,11,13,17,19,23,29,31}

So sample space = 104.

Favourable cases

So required probability

$=\frac{10+4{\times }^{10}\text{?}{C}_2}{10^4}=\frac{10+\frac{4\times 10.9}{2}}{10^4}=\frac{19}{1000}$

NCERT Textbooks helps student to get clear understanding of the fundamental concepts and usage of concepts in various problems. Students must solve NCERT Textbooks before advancing their prep. Candidates also some time get stuck while solving the NCERT questions. Candidates can take help of NCERT So

Class 12th Matrices is very significant chapter in CBSE Boards. There are several important topics in Matrices, students can check few of the m below;

• Types of Matrices: Row Matrix, Column Matrix, Square Matrix, Diagonal Matrix, Scalar Matrix, Identity Matrix, Zero Matrix, Upper and Lower Triangular

Matrices is generally combined with determinants, Matrices and Determinants carry significant weightage in the competitive exams. Matrices carries a weightage of around 6-8% of the total marks in JEE Mains. Students can expect 1-2 questions in JEE Mains from this chapter. Students must have a clear