Tamil Nadu Board Solutions for Chapter: Algebra, Exercise 35: Unit Exercise

Complete the factor tree for the given polynomials $f\left(x\right)$ and $g\left(x\right)$. Hence find their GCD and LCM.

GCD $\left[f\right(x)$ and $g\left(x\right)]=$________

LCM $\left[f\right(x)$ and $g\left(x\right)]=$________

Find the least common multiple of $xy\left(k^2+1\right)+k\left(x^2+y^2\right)$ and $xy\left(k^2-1\right)+k\left(x^2-y^2\right)$

Simplify $\frac{\frac{1}{p}+\frac{1}{q+r}}{\frac{1}{p}-\frac{1}{q+r}}\times \left(1+\frac{q^2+r^2-p^2}{2qr}\right)$

The number of seats in a row is equal to the total number of rows in a hall. The total number of seats in the hall will increase by $375$if the number of rows is doubled and the number of seats in each row is reduced by $5$. Find the number of rows in the hall at the beginning.

$\alpha$ and $\beta$ are the roots of the polynomial $f\left(x\right)=x^2-2x+3,$ If the polynomial whose roots are $\alpha +2, \beta +2$ is $x^2-6x+k$, find the value of $k$.
 

$\alpha$ and $\beta$ are the roots of the polynomial $f\left(x\right)=x^2-2x+3,$ If the polynomial whose roots are $\frac{\alpha -1}{\alpha +1} \mathrm{and} \frac{\beta -1}{\beta +1}$ is $k{x}^2-2x+1$, find the value of $k$.

For $\cos \theta \left[\begin{array}{cc}\cos \theta & \sin \theta \\ -\sin \theta & \cos \theta \end{array}\right]+\sin \theta \left[\begin{array}{cc}x& -\cos \theta \\ \cos \theta & x\end{array}\right]=I_2$ if $x=k\theta$.

Then find the value of $k$.

$A=\left[\begin{array}{cc}3& 0\\ 4& 5\end{array}\right],B=\left[\begin{array}{cc}6& 3\\ 8& 5\end{array}\right],C=\left[\begin{array}{cc}3& 6\\ 1& 1\end{array}\right]$, find the matrix $D$, such that $CD-AB=0$.