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# 12th Mathematics 2017 Set1 Delhi Board Paper Solution

Question 4

## 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

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Let A = R – {3} and B = R – {1}. Consider the function  f : A → B defined by . Is f one-one and onto? Justify your answer.

• Q:- Show that the relation R in the set R of real numbers, defined as R = {(a, b): a ≤ b2} is neither reflexive nor symmetric nor transitive.
• Q:-

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.

• Q:- Show that the relation R in the set A = {1, 2, 3, 4, 5} given by R = { (a,b) ; |a - b| is even}, is an equivalence relation. Show that all the elements of {1, 3, 5} are related to each other and all the elements of {2, 4} are related to each other. But no element of {1, 3, 5} is related to any element of {2, 4}.
• Q:-

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.