Class 12 Relations And Functions · Case Study5 Marksmixed_relations_and_functions
Class 12 Maths · Relations And Functions · Case Study #66
Saplings: parallel lines and functions
Part (1) 1 Marks
Let \(R=\{(L_1,L_2):L_1\parallel L_2,\ L_1,L_2\in L\}\). Then \(R\) is a relation which is
Answer: Option (A): Equivalence
- Having the same direction is reflexive, symmetric and transitive, so the relation is an equivalence relation under the stated convention.
Part (2) 1 Marks
Let \(R=\{(L_1,L_2):L_1\perp L_2,\ L_1,L_2\in L\}\). Which statement is true?
Answer: Option (A): R is symmetric but neither reflexive nor transitive
- Perpendicularity is symmetric. No line is perpendicular to itself, so reflexivity fails. Two perpendicular lines give a two-step chain back to the first line, which is not perpendicular to itself, so transitivity also fails.
Part (3) 1 Marks
The function \(f:\mathbb{R}\to\mathbb{R}\) defined by \(f(x)=x-4\) is
Answer: Option (A): Bijective
- Each real \(y\) has the unique preimage \(y+4\).
Part (4) 1 Marks
Let \(f:\mathbb{R}\to\mathbb{R}\) be defined by \(f(x)=x-4\). The range of \(f\) is
Answer: Option (A): \(\mathbb{R}\)
- For every real \(y\), the real input \(y+4\) maps to it; the range is \(\mathbb{R}\).
Part (5) 1 Marks
Let \(L_1\) be the line \(y=x-4\). Which of the following equations defines a line \(L_2\) parallel to \(L_1\)?
Answer: Option (A): \(2x-2y+5=0\)
- \(2x-2y+5=0\) has slope 1, matching \(y=x-4\). The others have slopes -2 or -1.