If a line makes angles 90° and 60° respectively with the positive directions of x and y axes, find the angle which it makes with the positive direction of z-axis.
Here, α = 90° , β = 60°
Let γ = θº, where 0 ≤ θº ≤ π
Now cos2 α + cos2 β + cos2 γ = 1
⇒ cos2 90° + cos2 60° + cos2 θº = 1
⇒ 02 + (½)2 + cos2 θº = 1
cos2 θ = 1- 1/4 = 3/4
cos θ = ± √3 / 2
θ = π / 6 or 5π / 6
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.
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
A balloon, which always remains spherical on inflation, is being inflated by pumping in 900 cubic centimetres of gas per second. Find the rate at which the radius of the balloon increases when the radius is 15 cm.
y = x2 + 2x + C : y' - 2x - 2 = 0