Question

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.

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

Show that the function f:R → R defined by f(x) = ${\frac{x}{{\boldsymbol{x}}^{2}+1}},\ \ \forall\ x\in{\boldsymbol{R}}$ is neither one-one nor onto. Also, if g:R → R is defined as g(x) = 2x – 1, find fog(x).

Find the value of p for which the curves x² = 9p(9 – y) and x² = p(y + 1) cut each other at right angles.

Find the equation of the tangent to the curve $\begin{array}{r c l}{y}&{=}&{{\sqrt{3x-2}}}\end{array}$ which is parallel to the line 4x – 2y + 5 = 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 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.