Maths

S = $\underset{n\to \infty }{Lt}{\displaystyle \sum _{r=1}^n\frac{n^2}{\left(n^2+r^2\right)\left(n+r\right)}}$

$\frac{\pi }{8}+\frac{1}{4}\mathcal{l}n2$

$l={\displaystyle \underset{-\frac{\pi }{2}}{\overset{\frac{\pi }{2}}{\int }}\frac{dx}{\left(1+e^x\right)\left(si{n}^{6}x+co{s}^{6}x\right)}}$ ……. (i)

$=2{\displaystyle \underset{0}{\overset{\infty }{\int }}\frac{dt}{4+t^2}=2{\left(ta{n}^{-1}\left(\frac{t}{2}\right)\right)}_0^{\infty }}=\pi$

$f\left(x\right)=\{\begin{array}{l}\frac{sin\left(x+2\right)}{x+2},x\in \left(-2,-1\right)\\ 0,x\in \left(-1,0\right]\\ 2x,x\in \left(0,1\right)\\ 1,otherwise\end{array}$

$LHD=\underset{h\to 0}{Lt}\frac{f\left(0-h\right)-f\left(0\right)}{-h}=0$

$RHD=\underset{h\to 0}{Lt}\frac{f\left(0+h\right)-f\left(0\right)}{h}=2$

Hence f (x) is not differentiable at x = 1, 0, 1

$\therefore m=2,n=3$

 $x_{1}y>0and{x}^3{y}^2=2^{15}$

$AM\ge GM$

$\frac{3x+2y}{5}\ge {\left(x^3{y}^2\right)}^{\frac{1}{5}}$

$\Rightarrow 3x+2y\ge 40$

x + y + az = 2………… (i)

3x + y + z = 4 ………… (ii)

x + 2z = 1 ……………. (iii)

x = 1, y = 1, z = 0

(for unique solution $a\ne -3$  

$\left(\alpha ,-1\right),\left(1,\alpha \right)and\left(1,-1\right)$ are collinear.

$\frac{\alpha -1}{1-\alpha }=\frac{-1-\alpha }{1-1}\Rightarrow 1-{\alpha }^2=0\Rightarrow \alpha =\pm 1$                

Sum of absolute value = 2

$e^{2x}-4=0or6e^{2x}-5e^x+1=0$

$\Rightarrow x=\mathcal{l}n2x=\mathcal{l}n\left(\frac{1}{3}\right),\mathcal{l}n\left(\frac{1}{2}\right)$

= $-\mathcal{l}n3,-\mathcal{l}n2$

$\left(x^2+1\right)*1=x*2$

$\Rightarrow {\left(x^2+1\right)}^2+1=x^2+8\Rightarrow x^2=2$

$2si{n}^{-1}\left(\frac{x^4+x^2-2}{x^4+x^2+2}\right)=2si{n}^{-1}\left(\frac{1}{2}\right)=\frac{\pi }{3}$

Apart from upper and lower triangular matrices, we have the unit triangular matrix, strictly-triangular matrix and an atomic-triangular matrix.

A few types of matrices are row and column matrices, singleton, null matrix, square, diagonal, scalar, identity, equal, triangular (both upper and lower), singular and non-singular matrices, symmetric, skew-symmetric, hermitian, and orthogonal matrices. NCERT excercise on matrices chapter covers questions related to this topic

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.