NCERT Exemplar Solutions
Class 12 - Mathematics - Chapter 1: RELATIONS AND FUNCTIONS

Short Answer (S.A.)

Let \(A = \{a, b, c\}\) and the relation \(R\) be defined on \(A\) as follows: \(R = \{(a,a),(b,c),(a,b)\}\). Then, write minimum number of ordered pairs to be added in \(R\) to make \(R\) reflexive and transitive.

Let \(D\) be the domain of the real valued function \(f\) defined by \(f(x) = \sqrt{25 - x^2}\). Then write \(D\).

Let \(f, g : \mathbb{R} \to \mathbb{R}\) be defined by \(f(x) = 2x + 1\) and \(g(x) = x^2 - 2\). Find \(g \circ f\).

Let \(f : \mathbb{R} \to \mathbb{R}\) be defined by \(f(x) = 2x - 3\). Write \(f^{-1}\).

If \(A = \{a,b,c,d\}\) and the function \(f = \{(a,b),(b,d),(c,a),(d,c)\}\), write \(f^{-1}\).

If \(f : \mathbb{R} \to \mathbb{R}\) is defined by \(f(x) = x^2 - 3x + 2\), write \(f(f(x))\).

Is \(g = \{(1,1),(2,3),(3,5),(4,7)\}\) a function? If \(g(x) = \alpha x + \beta\), then what are the values of \(\alpha\) and \(\beta\)?

Are the following sets of ordered pairs functions? If so, examine whether the mapping is injective or surjective.
(i) \(\{(x,y): x \text{ is a person}, y \text{ is the mother of } x\}\)
(ii) \(\{(a,b): a \text{ is a person}, b \text{ is an ancestor of } a\}\)

If mappings \(f\) and \(g\) are given by \(f = \{(1,2),(3,5),(4,1)\}\) and \(g = \{(2,3),(5,1),(1,3)\}\), write \(g \circ f\).

Let \(C\) be the set of complex numbers. Prove that the mapping \(f:C \to \mathbb{R}\) defined by \(f(z)=|z|\) is neither one-one nor onto.

Let the function \(f:\mathbb{R} \to \mathbb{R}\) be defined by \(f(x) = \cos x\). Show that \(f\) is neither one-one nor onto.

Let \(X = \{1,2,3\}\) and \(Y = \{4,5\}\). Determine whether the following sets of \(X \times Y\) are functions from \(X\) to \(Y\):
(i) \(f = \{(1,4),(1,5),(2,4),(3,5)\}\)
(ii) \(g = \{(1,4),(2,4),(3,4)\}\)
(iii) \(h = \{(1,4),(2,5),(3,5)\}\)
(iv) \(k = \{(1,4),(2,5)\}\)

If functions \(f:A \to B\) and \(g:B \to A\) satisfy \(g \circ f = I_A\), show that \(f\) is one-one and \(g\) is onto.

Let \(f:\mathbb{R} \to \mathbb{R}\) be defined by \(f(x) = \dfrac{1}{2 - \cos x}\). Find the range of \(f\).

Let \(n\) be a fixed positive integer. Define a relation \(R\) in \(\mathbb{Z}\) as follows: for \(a,b \in \mathbb{Z}\), \(aRb\) iff \(a-b\) is divisible by \(n\). Show that \(R\) is an equivalence relation.