Relations and Functions

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New answer posted•a year ago

Relations and Functions Maths NCERT Exemplar Solutions Class 12th Chapter One

Let $n$ be a fixed positive integer. Define a relation $R$ in $Z$ as follows: $\forall a, b \in Z, a R b$ if and only if $a - b$ is divisible by $n$ . Show that $R$ is an equivalence relation.

5 Views|Posted a year ago

Asked by Alok Kumar Singh

Answered by

Alok Kumar Singh | Contributor-Level 10

a year ago

This is a Short Answer Type Question as classified in NCERT Exemplar

Sol: $\begin{array}{l} \begin{array}{l} \begin{array}{l} \begin{array}{l} \end{array} \end{array} \end{array} \end{array}$

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New answer posted•a year ago

Relations and Functions Maths NCERT Exemplar Solutions Class 12th Chapter One

Let $f : R \to R$ be the function defined by $f (x) = \frac{1}{2 - c o s x}, \forall x \in R$ . Then, find the range of $f$ .

5 Views|Posted a year ago

Asked by Alok Kumar Singh

Answered by

Alok Kumar Singh | Contributor-Level 10

a year ago

This is a Short Answer Type Question as classified in NCERT Exemplar

Sol:

$\begin{array}{l} f (x) \frac{1}{2 - c o s x}, \forall x \in R \\ R a n g e o f c o s x i s [- 1, 1] \\ L e t f (x) = y = \frac{1}{2 - c o s x} \\ \Rightarrow 2 y - y c o s x = 1 \Rightarrow y c o s x = 2 y - 1 \\ \Rightarrow c o s x = \frac{2 y - 1}{y} = 2 - \frac{1}{y} \\ N o w - 1 \leq c o s x \leq 1 \\ \Rightarrow - 1 \leq 2 - \frac{1}{y} \leq 1 \Rightarrow - 1 - 2 \leq - \frac{1}{y} \leq 1 - 2 \\ \Rightarrow - 3 \leq - \frac{1}{y} \leq - 1 \Rightarrow 3 \geq \frac{1}{y} \geq 1 \Rightarrow \frac{1}{3} \leq y \leq 1 \\ ∴ t h e r a n g e o f f = [\frac{1}{3}, 1] . \end{array}$

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New answer posted•a year ago

Relations and Functions Maths NCERT Exemplar Solutions Class 12th Chapter One

If functions $f : A \to B$ and $g : B \to A$ satisfy $g ? f = I_{A}$ , then show that $f$ is one-one and $g$ is onto.

4 Views|Posted a year ago

Asked by Alok Kumar Singh

Answered by

Alok Kumar Singh | Contributor-Level 10

a year ago

This is a Short Answer Type Question as classified in NCERT Exemplar

Sol:

$\begin{array}{l} \end{array} L e t x_{1}, x_{2} \in g o f$

$g o f {f (x_{1})} = g o f {f (x_{2})}$

$\Rightarrow g (x_{1}) = g (x_{2}) \begin{array}{l} \end{array}$

$∴$
$\begin{array}{l} x_{1} = x_{2} \end{array}$

$\begin{array}{l} H e n c e, f i s o n e - o n e a n d g i s o n t o . \end{array}$

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New answer posted•a year ago

Relations and Functions Maths NCERT Exemplar Solutions Class 12th Chapter One

Let $X = {1, 2, 3}$ and $Y = {4, 5}$ . Find whether the following subsets of $X \times Y$ are functions from $X$ to $Y$ or not:

$I. f = {(1, 4), (1, 5), (2, 4), (3, 5)}$

$II. g = {(1, 4), (2, 4), (3, 4)}$

$III. h = {(1, 4), (2, 5), (3, 5)}$

$IV. k = {(1, 4), (2, 5)}$ .

7 Views|Posted a year ago

Asked by Alok Kumar Singh

Answered by

Alok Kumar Singh | Contributor-Level 10

a year ago

This is a Short Answer Type Question as classified in NCERT Exemplar

Sol:

$G i v e n t h a t,$

$\begin{array}{l} X = {1, 2, 3} a n d Y = {4, 5} \end{array}$

$\begin{array}{l} ∴ X \times ? Y = {(1, 4), (1, 5), (2, 4), (2, 5), (3, 4), (3, 5)} \end{array}$

$\begin{array}{l} (i) f = {(1, 4), (1, 5), (2, 4), (3, 5)} \end{array}$

$\begin{array}{l} f i s n o t a f u n c t i o n b e c a u s e f h a s n o u n i q u e i m a g e . \end{array}$

$\begin{array}{l} (i i) g = {(1, 4), (2, 4), (3, 4)} \end{array}$

$\begin{array}{l} S i n c e, g i s a f u n c t i o n a s e a c h e l e m e n t o f t h e d o m a i n h a s a u n i q u e i m a g e . \end{array}$

$\begin{array}{l} (i i i) h = {(1, 4), (2, 5), (3, 5)} \end{array}$

$\begin{array}{l} I t i s a f u n c t i o n a s e a c h e l e m e n t o f t h e d o m a i n h a s a u n i q u e i m a g e . \end{array}$

$\begin{array}{l} (i v) k = {(1, 4), (2, 5)} \end{array}$

$\begin{array}{l} k i s n o t a f u n c t i o n a s 3 d o e s n o t h a v e a n y i m a g e u n d e r t h e m a p p i n g . \end{array}$

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New answer posted•a year ago

Relations and Functions Maths NCERT Exemplar Solutions Class 12th Chapter One

Let $C$ be the set of complex numbers. Prove that the mapping $f : C \to R$ given by $f (z) = ∣ z ∣, \forall z \in C$ , is neither one-one nor onto.

7 Views|Posted a year ago

Asked by Alok Kumar Singh

Answered by

Alok Kumar Singh | Contributor-Level 10

a year ago

This is a Short Answer Type Question as classified in NCERT Exemplar

Sol:

$\begin{array}{l} f (z) = | z | \forall z \in C \\ f (1) = | 1 | = 1 \\ f (- 1) = | - 1 | = 1 \\ f (1) = f (- 1) \\ B u t 1 \neq - 1 \\ T h e r e f o r e, i t i s n o t o n e - o n o . \\ N o w, l e t f (z) = y = | z | . \\ H e r e, t h e r e i s n o; p r e - i m a g e o f n e g a t i v e n u m b e r s . \\ H e n c e, i t i s n o t o n t o . \end{array}$

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New answer posted•a year ago

Relations and Functions Maths NCERT Exemplar Solutions Class 12th Chapter One

If the mappings $f$ and $g$ are given by

$f = {(1, 2), (3, 5), (4, 1)}$ and $g = {(2, 3), (5, 1), (1, 3)}$ , write $f ? g$ .

3 Views|Posted a year ago

Asked by Alok Kumar Singh

Answered by

Alok Kumar Singh | Contributor-Level 10

a year ago

This is a Short Answer Type Question as classified in NCERT Exemplar

Sol:

$G i v e n t h a t,$
$\begin{array}{l} f = {(1, 2), (3, 5), (4, 1)} \end{array}$

$a n d$
$\begin{array}{l} g = {(2, 3), (5, 1), (1, 3)} \end{array}$

$\begin{array}{l} N o w, f o g (2) = f {g (2)} = f (3) = 5 \end{array}$

$\begin{array}{l} f o g (5) = f {g (5)} = f (1) = 2 \end{array}$

$\begin{array}{l} f o g (1) = f {g (1)} = f (3) = 5 \end{array}$

$\begin{array}{l} f o g = {(2, 5), (5, 2), (1, 5)} \end{array}$

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New answer posted•a year ago

Relations and Functions Maths NCERT Exemplar Solutions Class 12th Chapter One

Are the following set of ordered pairs functions? If so, examine whether the mapping is injective or surjective.

$I. {(x, y) : xisaperson, y isthemonthofx}$ .

$II. {(a, b) : aisaperson, bisanancestorof a}$ .

2 Views|Posted a year ago

Asked by Alok Kumar Singh

Answered by

Alok Kumar Singh | Contributor-Level 10

a year ago

This is a Short Answer Type Question as classified in NCERT Exemplar

Sol:

$\begin{array}{l} (i) G i v e n s e t o f o r d e r e d p a i r i s {(x, y) : x i s a p e r s o n, y i s t h e m o t h e r o f x} . \end{array}$ $\begin{array}{l} I t r e p r e s e n t s a f u n c t i o n . \end{array}$

$\begin{array}{l} H e r e, t h e i m a g e o f d i s t i n c t e l e m e n t s o f x u n d e r f a r e n o t d i s t i n c t, s o i t i s n o t a n i n j e c t i v e b u t i t i s a s u r j e c t i v e . \end{array}$

$\begin{array}{l} (i i) S e t o f o r d e r e d p a i r s = {(a, b) : a i s a p e r s o n, b i s a n a n c e s t o r o f a} \end{array}$

$\begin{array}{l} H e r e, e a c h e l e m e n t o f d o m a i n d o e s n o t h a v e a u n i q u e i m a g e . \end{array}$

$\begin{array}{l} S o, i t d o e s n o t r e p r e s e n t f u n c t i o n . \end{array}$

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New answer posted•a year ago

Relations and Functions Class 12th

Is $g = {(1, 1), (2, 3), (3, 5), (4, 7)}$ a function? If $g$ is described by $g (x) = α x + β$ , then what value should be assigned to $α$ and $β$ .

5 Views|Posted a year ago

Asked by Alok Kumar Singh

Answered by

Alok Kumar Singh | Contributor-Level 10

a year ago

This is a Short Answer Type Question as classified in NCERT Exemplar

Sol:

Given that, g = { (1, 1), (2, 3), (3, 5), (4, 7)}.

Here, each element of domain has unique image. So, g is a function.

Now given that, g (x ) = $α$ x + $β$

For (1,1) g (1) = $α$ + $β$

$α$ + $β$ = 1… (i

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New answer posted•a year ago

Relations and Functions Maths NCERT Exemplar Solutions Class 12th Chapter One

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

4 Views|Posted a year ago

Asked by Alok Kumar Singh

Answered by

Alok Kumar Singh | Contributor-Level 10

a year ago

This is a Short Answer Type Question as classified in NCERT Exemplar

Sol:

$\begin{array}{l} G i v e n t h a t \end{array}$ $\begin{array}{l} f (x) = x^{2} - 3 x + 2 \end{array}$ $\begin{array}{l} f {f (x)} = f (x^{2} - 3 x + 2) \end{array}$

$\begin{array}{l} = {(x^{2} - 3 x + 2)}^{2} - 3 (x^{2} - 3 x + 2) + 2 \end{array}$

$\begin{array}{l} = x^{4} + 9 x^{2} + 4 - 6 x^{3} - 12 x + 4 x^{2} - 3 x^{2} + 9 x - 6 + 2 \end{array}$

$\begin{array}{l} = x^{4} + 10 x^{2} - 6 x^{3} - 3 x \end{array}$

$\begin{array}{l} H e n c e f {f (x)} = x^{4} - 6 x^{3} + 10 x^{2} - 3 x \end{array}$

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New answer posted•a year ago

Relations and Functions Maths NCERT Exemplar Solutions Class 12th Chapter One

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

6 Views|Posted a year ago

Asked by Alok Kumar Singh

Answered by

Alok Kumar Singh | Contributor-Level 10

a year ago

This is a Short Answer Type Question as classified in NCERT Exemplar

Sol:

$\begin{array}{l} G i v e n t h a t, A = {a, b, c, d} \end{array}$

$\begin{array}{l} a n d f = {(a, b), (b, d), (c, a), (d, c)} \end{array}$

$\begin{array}{l} f^{- 1} = {(b, a), (d, b), (a, c), (c, d)} \end{array}$

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