Question

Find the particular solution of the differential equation $\log\left(\frac{d y}{d x}\right)=3x+4y,$ given that y = 0, when x = 0.

If A = $\left(\begin{array}{l l }1 & -1 \\ 2 & -1\end{array}\right)$ , B = $\left(\begin{array}{l l }a & 1 \\ b & -1\end{array}\right)$ and (A + B)² = A²+ B², then find the values of a and b.

Let f : N → N be defined as $f(n)={\left\{\begin{array}{l l}{\displaystyle{\frac{n+1}{2}},{\mathrm{when~}}n{\mathrm{~is~odd}}}\\ {\displaystyle{\frac{n}{2}},{\mathrm{when~}}n{\mathrm{iseven}}}\end{array}\right.}$ for all n ∈ N. State whether the function f is bijective. Justify your answer.

Find the equations of tangents to the curve y = (x²– 1) (x – 2) at the points, where the curve cuts the X-axis.

Let A = R – {3}, B = R – {1}. Let f : A → B be defined by f(x) =${\frac{x-2}{x-3}},~~\forall~x\in~A.$ Show that f is bijective. Also, find (i) x, if $f^{-1}(x)$ = 4 (ii) $f^{-1}(7)$

Find the equation of tangent to the curve y = cos (x + y), -2π ≤ x ≤ 0, that is parallel to the line x + 2y = 0.

Form the differential equation representing family of ellipses having foci on X-axis and centre at the origin.