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.

Let N denote the set of all natural numbers and R be the relation on N × N defined by (a, b) R (c, d) if ad(b + c) = bc(a + d). Show that R is an equivalence relation.

Find the equation of tangent and normal to the curve x = 1 – cos θ, y = θ – sin θ at θ = ${\frac{\pi}{4}}\,.$

Find the equation of the tangent line to the curve y =x² -2x + 7 which is (i) parallel to the line 2x-y +9 =0 (ii) perpendicular to the line 5y - 15x = 13.

Show that the normal at any point θ to the curve x = a cos θ + aθ sin θ, y = a sin θ – aθ cos θ is at a constant distance from the origin.

Find the equations of tangents to the curve 3x² – y² = 8, which passes through the point $\left({\frac{4}{3}},\,0\right).$