Question

If A = $\left(\begin{array}{l l l }1 & 0 & 2 \\ 0 & 2 & 1 \\ 2 & 0 & 3\end{array}\right)$, prove that A³ – 6A² + 7A + 2I = 0

Find the particular solution of the differential equation $\left(1+x^{2}\right){\frac{d y}{d x}}=\left(e^{{\mathrm{mtan}^{\perp}x}}-y\right)$ given that y = 1 when x = 0.

If A and B are two independent events such that P(A' ∩ B)= 2/15 and P(A ∩ B')= 1/6, then find P(A) and P(B).

LetA = {x ∈ Z:0≤ x ≤12}. Show that R={(a,b):a,b ∈ A, |a-b| is divisible by 4} is an equivalence relation. Find the set of all elements related to 1. Also write the equivalence class [2].

Find the particular solution of the differential equation $\left(\tan^{-1}y-x\right)d y=(1+y^{2})\,d x,$ given that when x = 0, y = 0.

Consider $f:R-\left\{-{\frac{4}{3}}\right\}\to R-\left\{{\frac{4}{3}}\right\}$ given by f(x) = ${\frac{4x+3}{3x+4}}\,.$ Show that f is bijective. Find the inverse of f and hence find f⁻¹(0) and x such that f⁻¹(x) = 2.

Show that the function f : R {x ∈ R – 1 < x < 1} defined by f(x) =${\frac{x}{1+\left|\,x\right|}},$ x ∈ R is one-one and onto function. Hence find f⁻¹(x).