Find the direction cosines of a line which makes equal angles with the coordinate axes.
Let the direction cosines of the line make an angle α with each of the coordinate axes.
∴ l = cos α, m = cos α, n = cos α
l2+m2+n2 =1
⇒ cos2α + cos2α + cos2α = 1
⇒ 3cos2α =1
\begin{align}\Rightarrow cos^2α = \frac{1}{3}\end{align}
\begin{align}\Rightarrow cosα = \pm\frac{1}{\sqrt 3}\end{align}
Thus, the direction cosines of the line, which is equally inclined to the coordinate axes, are
\begin{align} \pm\frac{1}{\sqrt 3},\pm\frac{1}{\sqrt 3},and \pm\frac{1}{\sqrt 3}\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.
Maximise Z = 3x + 4y
Subject to the constraints:x + y ≤ 4, x ≥ 0, y ≥ 0
Find the area of the region bounded by the curve y2 = x and the lines x = 1, x = 4 and the x-axis.
Show that the Signum Function f : R → R, given by
is neither one-one nor onto.
Let f : N → N be defined by
State whether the function f is bijective. Justify your answer.
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