If a line makes angles 90°, 135°, 45° with x, y and z-axes respectively, find its direction cosines.
Let direction cosines of the line be l, m, and n.
\begin{align}l = cos90^0=0\end{align}
\begin{align}m = cos135^0=-\frac{1}{\sqrt{2}}\end{align}
\begin{align}n = cos45^0=\frac{1}{\sqrt{2}}\end{align}
\begin{align}Therefore, the\; direction\; cosines\; of \;the\; line\; are\;0, -\frac{1}{\sqrt{2}}\;and\;\frac{1}{\sqrt{2}}\end{align}
In each of the following cases, state whether the function is one-one, onto or bijective. Justify your answer.
(i) f : R → R defined by f(x) = 3 – 4x
(ii) f : R → R defined by f(x) = 1 + x2
Show that the Modulus Function f : R → R, given by f(x) = |x|, is neither oneone nor onto, where | x | is x, if x is positive or 0 and |x| is – x, if x is negative.
Prove that the Greatest Integer Function f : R → R, given by f(x) = [x], is neither one-one nor onto, where [x] denotes the greatest integer less than or equal to x.
Let f: X → Y be an invertible function. Show that the inverse of f –1 is f, i.e., (f–1)–1 = f.
y = x2 + 2x + C : y' - 2x - 2 = 0
\begin{align} y = xsinx:xy{'}=y +x\sqrt{x^2 -y^2}(x\neq0\; and\; x>y\; or\; x<-y)\end{align}
Find the direction cosines of a line which makes equal angles with the coordinate axes.