Find the value of $k$ if $f(x) = x^{3} - kx^{2} + 11x - 6$ and $(x - 1)$ is a factor of $f(x)$.

When $4x^3 - ax^2 + bx - 4$ is divided by $x - 2$ and $x + 1$,the respective remainders are $20$ and $-13$. Find the values of $a$ and $b$.

Resolve into factors: $9(3 x+5 y)^{2}-12(3 x+5 y)(2 x+3 y)+4(2 x+3 y)^{2}$

Resolve into factors: $(a+b)^{2}-2(a^{2}-b^{2})+(a-b)^{2}$

Factorize: $45 a^{3} b + 5 a b^{3} - 30 a^{2} b^{2}$

Let $f(x) = a_{0}x^{n} + a_{1}x^{n-1} + a_{2}x^{n-2} + \ldots + a_{n-1}x + a_{n}$,where $a_{0}, a_{1}, a_{2}, \ldots, a_{n}$ are constants. If $f(x)$ is divided by $ax - b$,then the remainder is: