NCERT Exemplar Class 12 Maths Solutions Chapter 1 Relations and Functions

NCERT Exemplar Class 12 Maths Solutions Chapter 1 Relations and Functions

Komal MiglaniUpdated on 31 Mar 2025, 03:06 AM IST

Throughout our everyday encounters, we relate objects, people, or ideas together. For example, a student relates to his or her school, a country relates to its capital, or a person relates to the date he or she was born. Those are what we call relations in math. A relation is a way of relating elements of one set to elements of another and is a subset of the Cartesian product of two sets. A function is a relation in which each element of the first set (domain) relates to one element of the second set (codomain). Functions have various applications in real life, thereby allowing us to forecast weather patterns, design computer programs, and many other applications.

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  1. NCERT Exemplar Class 12 Maths Solutions Chapter 1
  2. Subtopics Covered in NCERT Exemplar Solutions for Class 12 Maths Chapter 1
  3. NCERT Exemplar Class 12 Maths Solutions
  4. Importance of NCERT Exemplar Class 12 Maths Solutions Chapter 1
  5. NCERT solutions of class 12 - Subject-wise
  6. NCERT Notes of class 12 - Subject Wise
  7. NCERT Books and NCERT Syllabus
  8. NCERT Exemplar Class 12 Solutions - Subject Wise

It is easy to understand relations and functions through NCERT Class 12 solutions. The solutions provide precise step-by-step solutions for various kinds of relations such as reflexive, symmetric, transitive, and equivalence, and functions, function composition, and inverse functions. Practicing NCERT problems on a daily basis will not just improve your concept clarity but will also increase your logical reasoning and problem-solving ability. Practice problems in NCERT textbooks form a great base for board and competitive examinations. So, step into the world of relations and functions through NCERT solutions and learn how mathematical relationships are involved in numerous aspects of life and education.

NCERT Exemplar Class 12 Maths Solutions Chapter 1

Class 12 Maths Chapter 1 exemplar solutions Exercise: 1.3
Page number: 11-17
Total questions: 62
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Question:1

Let A = {a, b, c} and the relation R be defined on A as follows:

R = {(a, a), (b, c), (a, b)}.

Then, write the minimum number of ordered pairs to be added in R to make R reflexive and transitive.

Answer:

Here, R = {(a, a), (b, c), (a, b)}

The minimum number of ordered pairs to be added to make R as reflexive is (b, b) and (c, c) to R. Whereas, to make R transitive, the minimum number of ordered pairs to be added is (a, c) to R.

Therefore, we need 3 ordered pairs to add to R to make it reflexive and transitive.

Question:2

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

Answer:

$f(x) = \sqrt {25 - x^{2}}$
Therefore, the function can be defined as:
$\\25 - x^{2} \geq 0\\ $

$or, x^{2} \leq 25\\ -5 \leq x \leq 5\\$

Therefore, the given function has a domain of [-5, 5].

Question:3

Let f, g:$R \rightarrow R$ be defined by f(x) = 2x + 1 and $g (x) = x^{2} - 2, \forall x \in R,$ respectively. Then, find gof.

Answer:

Here,

f(x) = 2x + 1
$g (x) = x^{2} -2, \forall x \in R\\$
Therefore,

$gof= g(f(x)) = g (2x + 1)$

$ = (2x + 1)^{2} -2 $

$= 4x^{2} + 4x + 1 -2 $

$= 4x^{2} + 4x - 1\\$

Question:4

. Let f: $R \rightarrow R$ be the function defined by $f (x) = 2x - 3, \forall x \in R$. write $f^{-1}$.

Answer:

Here, $f (x) = 2x - 3, \forall x \in R\\$
Let’s say,
$y = 2x - 3\\$

$ or, x = (y + 3)/ 2$

Therefore,
$f^{-1}(x) = (x + 3)/ 2\\$

Question:5

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

Answer:
Here, $A = \{ a, b, c, d \}$ and the function $f = \{ (a, b), (b, d), (c, a), (d, c) \} \\$
Therefore, $f^{-1} = \{ (b, a), (d, b), (a, c), (c, d) \} \\$

Question:6

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

Answer:

Here, $f (x) = x^{2} - 3x + 2$

Therefore,
$\\ f(f(x)) = f(x^{2} - 3x + 2)\\ = (x^{2} - 3x + 2)^{2} - 3(x^{2} - 3x + 2) + 2,\\$

$ = x^{4} + 9x^{2} + 4 - 6x^{3} + 4x^{2} - 12x - 3x^{2} + 9x - 6 + 2\\$

$ = x^{4} - 6x^{3} + 10x^{2} - 3x\\$
Similarly,
$\\f(f(x)) = x^{4} - 6x^{3} + 10x^{2} - 3x\\$

Question:7

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

Answer:

Here,$g = \{ (1, 1), (2, 3), (3, 5), (4, 7) \}$
Here, each and every element of a domain has a unique image. Therefore, g is a function.
Also, $g (x) = \alpha x + \beta$
Therefore,
$\\g(1) = \alpha (1) + \beta = 1\\ \alpha + \beta = 1 \ldots \ldots .. (1)\\$
Similarly,
$g(2) = \alpha (2) + \beta = 2 \alpha + \beta = 3 \ldots \ldots .. (2)\\$
By solving (1) and (2), we get
$\alpha = 2 and \beta = -1\\$
Therefore, $g (x) = 2x - 1$

Question:8

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

Answer:
(i)Here, $\{ (x, y): \text{x is a person, y is the mother of x} \} .\\$
Therefore, each person (x) has only one biological mother.
Hence, the given set of ordered pairs makes a function.
Therefore, there are more than one person who may have the same mother. Hence, the function is many-one and surjective.
(ii) Here, $\{ (a, b): \text{a is a person, b is an ancestor of a} \} .\\$
It’s seen that any person ‘a’ has more than one ancestor.
Therefore, it is not a function.

Question:9

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 $fog.\\$

Answer:
Here,

$f = \{ (1, 2), (3, 5), (4, 1) \}$ and $g = \{ (2, 3), (5, 1), (1, 3) \}$
Therefore,
$\\fog (2) = f(g(2)) = f(3) = 5\\ $

$fog (5) = f(g(5)) = f(1) = 2\\$

$ fog (1) = f(g(1)) = f(3) = 5\\$
So, we can write that:
$\\fog = \{ (2, 5), (5, 2), (1, 5) \} \\$

Question:10

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

Answer:

Here, $f: C \rightarrow R$ is given by $f (z) = \vert z \vert , \forall z \in C$
If we assume $z = 4 + 3i$
Then,
$f(4 + 3i) = \vert 4 + 3i \vert = \sqrt (4^{2} + 3^{2})= \sqrt 25 = 5\\$
Similarly, for $z = 4 - 3i$
$f(4 - 3i) = \vert 4 - 3i \vert = \sqrt (4^{2} + 3^{2}) = \sqrt 25 = 5\\$
Therefore, it is clear that f(z) is many-one.
So, $\vert z \vert \geq 0, \forall z \in C,\\$
However, in the question R is the co-domain given.
Hence, f(z) is not onto. So, f(z) is neither one-one nor onto.

Question:11

Let the function $f: R \rightarrow R$ be defined by $f (x) = cos x, \forall x \in R.$ Show that f is neither one-one nor onto.

Answer:
It is given that, $f: R \rightarrow R, f(x) = cos x, \forall x \in R\\$
So, we can write:
$f(x_{1}) = f(x_{2})\\ or, cos x_{1}= cos x_{2}\\$
Hence, $x_{1 }= 2n \pi \pm x_{2}, where n \in Z\\$
It is understandable that for any value of $x_{1 }and\: \: x_{2}$, the above equation has an infinite number of solutions.
Therefore, f(x) is a many one function.
We know the range of cos x is [-1, 1] and it is a subset of the given co-domain R.
Hence, the given function is not onto.

Question:12

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

Answer:

Here, $X = \{ 1, 2, 3 \} and Y = \{ 4, 5 \} \\$
Therefore, $X \times Y = \{ (1, 4), (1, 5), (2, 4), (2, 5), (3, 4), (3, 5) \} \\$
(i) $f = \{ (1, 4), (1, 5), (2, 4), (3, 5) \} \\$
Here, f(1) = 4 and again f(1) = 5.
Therefore, f is not a function here.
As a result, there is no unique of pre- image ‘1’.
(ii) $g = \{ (1, 4), (2, 4), (3, 4) \} \\$
We can clearly see that g is a function. Here in g, each element of the given domain has a unique image at the given range
(iii) $h = \{ (1,4), (2, 5), (3, 5) \} \\$
It’s clear that h is a function of each pre-image that has a unique image.
Again, h(2) = h(3) = 5
Therefore, the function h is also many-one.
(iv) $k = \{ (1, 4), (2, 5) \} \\$
Here, ‘3’ does not have any image under the mapping. Therefore, k is not a function.

Question:13

If functions $f: A \rightarrow B \: \: and\: \: g: B \rightarrow A \text{ satisfies } g of = IA$, then show that f is one-one and g is onto.

Answer:

Here, it is given:
$f: A \rightarrow B \: \: and\: \: g: B \rightarrow A \text{ satisfies } g of = IA$
It’s clear here that the function ‘g’ is inverse of ‘f’.
So, ‘f’ has to be both one-one as well as onto.
As a result, ‘g’ is both one-one and onto.

Question:14

Let $f: R \rightarrow R$ be the function defined by $f(x) = 1/(2 - cos x) \forall x \in R$. Then, find the range off.

Answer:

Here, $f(x) = 1/(2 - cos x) \forall x \in R$
Let’s say,
$\\y = 1/(2 - cos x)\\ \therefore2y - ycos x = 1\\ Or, cos x = (2y - 1)/ y\\ Or, cos x = 2 - 1/y\\$
As the range of cos x is, $-1 \leq cos x \leq 1\\$
Therefore,
$\\-1 \leq 2 - 1/y \leq 1\\ -3 \leq - 1/y \leq -1\\ 1 \leq - 1/y \leq 3\\ 1/3 \leq y \leq 1\\$
Hence, the range of the given function is [1/3, 1].

Question:15

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

Answer:

Here, we have to a relation R in Z as follows: $\forall a, b \in Z$, aRb if and only if a - b is divisible by n.
Here, aRa $\Rightarrow (a - a)$ is divisible by n and this is true for all integers.
Therefore, R is reflective.
For aRb, $aRb \Rightarrow (a - b)$ is also divisible by n.
or, - (b - a) is divisible by n.
or, (b - a) is divisible by n
Hence, we can write it as bRa.
Therefore, R is symmetric.
For aRb , (a - b) is divisible by n.
For bRc, (b - c) is divisible by n.
Hence, (a - b) + (b-c) is divisible by n.
Or, (a-c) is divisible by n. This can be expressed as aRc.
Therefore, R is transitive.
So, R is an equivalence relation.

Question:16

If A = {1, 2, 3 }, define relations on A which have properties of being:

(a) reflexive, transitive but not symmetric

(b) symmetric but neither reflexive nor transitive

(c) reflexive, symmetric, and transitive.

Answer:

Here, $A = \{ 1, 2, 3 \} .\\$
(i) Assume $R_{1}= \{ (1, 1), (1, 2), (1, 3), (2, 3), (2, 2), (1, 3), (3, 3) \} \\$
Here, (1, 1), (2, 2) and (3, 3) $\in \ R_{1}. R_{1}$ is reflexive.
$(1, 2) \in R_{1}, (2, 3) \in R_{1} \Rightarrow (1, 3) \in R_{1}$
. Hence,$R_{1}$ is transitive.
Now, $(1, 2) \in R_{1 } \Rightarrow (2, 1) \notin R_{1}.$
Therefore, $R_{1}$ is not symmetric.
(ii) Let say, $R_{2}= \{ (1, 2), (2, 1) \}$
So, $(1, 2) \in R_{2}, (2, 1) \in R_{2}\\$
Therefore, $R_{2 }$ is symmetric,
$(1, 1) \notin R_{2}$. Therefore, $R_{2 }$ is not reflexive.
$(1, 2) \in R_{2}$, $(2, 1) \in R_{2 }$ but $(1, 1) \notin R_{2}$. Hence, $R_{2 }$ is not transitive.
(iii) Let $R_{3 }$ = $\{ (1, 1), (1, 2), (1, 3), (2, 1), (2, 2), (2, 3), (3, 1), (3, 2), (3, 3) \} \\$
$R_{3 }$ is reflexive as $(1, 1) (2, 2) and (3, 3) \in R_{3 }\\$
$R_{3 }$ is symmetric as $(1, 2), (1, 3), (2, 3) \in R_{3 }$ $\Rightarrow (2, 1), (3, 1), (3, 2) \in R_{3 }\\$
Therefore, $R_{3 }$ is reflexive, symmetric and transitive.

Question:17

Let R be relation defined on the set of natural number N as follows:
$R = \{ (x, y): x \in N, y \in N, 2x + y = 41 \}$ . Find the domain and range of the relation R. Also verify whether R is reflexive, symmetric, and transitive.

Answer:

Here, $R = \{ (x, y): x \in N, y \in N, 2x + y = 41 \}$
So, the domain $D= \{ 1, 2, 3, \ldots .., 20 \} \\$
And the Range $= \{ 1, 3, 5, \ldots .., 39 \} \\$
Here, $(2, 2) \notin \: \: R\: \: as\: \: 2 \times 2 + 2 \neq 41$. Therefore, R is not reflexive.
Again, $(1, 39) \in R \: \: but\: \: (39, 1) \: \: \notin R$. So, R is not symmetric.
Again, $(11, 19) \notin R, (19, 3) \notin R; but (11, 3) \notin R$. So, Further R is not transitive.
Therefore, R is neither reflexive nor symmetric and nor transitive.

Question:18

Given A = {2, 3, 4}, B = {2, 5, 6, 7}. Construct an example of each of the following:

(a) an injective mapping from A to B

(b) a mapping from A to B which is not injective

(c) a mapping from B to A.

Answer:

Here, $A = \{ 2, 3, 4 \} , B = \{ 2, 5, 6, 7 \} \\$
(i) Assume, $f: A \rightarrow B \text{ denote a mapping}$
Then, $f = \{ (x, y): y = x + 3 \} or\\$
$f = \{ (2, 5), (3, 6), (4, 7) \}$ , this is an injective mapping.
(ii) Again assume, $g: A \rightarrow B$ denotes a mapping: $g = \{ (2, 2), (3, 2), (4, 5) \}$ . Hence, it is not an injective mapping.
(iii) Again assume, h: $B \rightarrow A$ denotes a mapping: $h = \{ (2, 2), (5, 3), (6, 4), (7, 4) \}$ . Hence, it is a mapping from B to A.

Question:19

Give an example of a map

(i) which is one-one but not onto

(ii) which is not one-one but onto

(iii) which is neither one-one nor onto.
Answer:
(i) Say, $f: N \rightarrow N,$ be a mapping defined by $f (x) = x^{2}\\$
If, $f(x_{1}) = f (x_{2})\\$
Then, $x_{1}^{2}= x_{2}^{2}\\$
So,$x_{1 }= x_{2 }(as, x_{1 }+ x_{1 }\text{cannot be } 0)$
$f(x_{1 }) = f(x_{2 })$, hence, f(x) is one-one.
However, ‘f’ is not onto, as for $1 \in N$, therefore, there is no existence of x in N : f(x) = 2x + 1.
(ii) Let $f: R \rightarrow [0, \infty )$, is a mapping that is defined by $f(x) = \vert x \vert \\$
Then, we can conclude that f(x) is not a one-one as f(2) and f(-2) are the same here.
But $\vert x \vert \geq 0$, so the range is $[0, \infty ].\\$
Therefore, f(x) is onto.
(iii) Assume, $f: R \rightarrow R$ , be a mapping which is defined by $f(x) = x^{2}\\$
Then we can say that f(x) is not one-one as f(1) and f (-1) are the same.
The range of f(x) is $[0, \infty )$.
Therefore, f (x) is neither one-one nor onto.

Question:20

Let $A = R - \{ 3 \} , B = R - \{ 1 \}$ . Let $f : A \rightarrow B$ be defined by $f(x) = x - 2/ x - 3 \forall x \in A$ . Then show that f is bijective.

Answer:

Here,

$A = R - \{ 3 \} , B = R - \{ 1 \} \\$
$f : A \rightarrow B$ be defined by $f (x) = x - 2/ x - 3 \forall x \in A\\$
Hence, $f (x) = (x - 3 + 1)/ (x - 3) = 1 + 1/ (x - 3)\\$
Say $f(x1) = f (x2)\\$
$1+\frac{1}{x_{1}-3}=1+\frac{1}{x_{2}-3}$
$\\\frac{1}{x_{1}-3}=\frac{1}{x_{2}-3} \\\\ x_{1}=x_{2}$
So, f (x) is an injective function.
$\\If, y = (x - 2)/ (x -3)\\$

$ Or, x - 2 = xy - 3y\\$

$ Or, x(1 - y) = 2 - 3y\\$

$ Or, x = (3y - 2)/ (y - 1)\\$

$ Or, y \in R - \{ 1 \} = B\\$
So, f (x) is onto or subjective.
Therefore, f(x) is a bijective function.

Question:21

Let A = [-1, 1]. Then, discuss whether the following functions defined on A are one-one, onto, or bijective:
(i) $f(x) = x/2$ (ii) $g(x) = \vert x \vert \\$
(iii) $h(x) = x \vert x \vert$ (iv) $k(x) = x^{2}\\$

Answer:

Here, A = [-1, 1]
(i) $f: [-1, 1] \rightarrow [-1, 1],$

$ f (x) = x/2\\$
If, $f(x_{1}) = f(x_{2})\\$
$x_{1}/2 = x_{2}/2\\$
So, f(x) is one-one.
Also, $x \in [-1, 1]\\$
$x/2 = f(x) = [-1/2, 1/2]\\$
Hence, the range is a subset of co-domain A
So, f(x) is not onto.
Therefore, f (x) is not bijective.
(ii)
$\\g(x) = \vert x \vert \\$

$ \text{Let} g(x_{1}) = g (x_{2})\\ $

$\vert x_{1} \vert = \vert x_{2} \vert \\ $

$x_{1}= \pm x_{2}\\$
So, g(x) is not one-one.
Also, $g(x) = \vert x \vert \geq 0, \text{for all real x}\\$
Hence, the range is [0, 1], which is subset of co-domain ‘A’
So, f(x) is not onto.
Therefore, f(x) is not bijective.
(iii)
$\\\\h(x) = x \vert x \vert \\$

$ \text{Let } h(x_{1}) = h(x_{2})\\ x_{1} \vert x_{1} \vert = x_{2} \vert x_{2} \vert \\ $

$\text{If }x_{1}, x_{2}> 0\\ x_{1}^{2}= x_{2}^{2}\\ x_{1}^{2}- x_{2}^{2}= 0\\ (x_{1} - x_{2})(x_{1} + x_{2}) = 0\\ x_{1 }= x_{2 }(as x_{1}+ x_{2} \neq 0)\\$

$ \text{Again }, x_{1}, x_{2 }< 0, and x_{1}= x_{2}\\$

$ \text{Therefore }, x_{1 }and x_{2 }\text{of opposite sign}, x_{1 } \neq x_{2 }.\\$
Hence, f(x) is one-one.
$\\\text{For x} \in [0, 1], f(x) = x_{2} \in [0, 1]\\$

$ \text{For x} < 0, f(x) = -x_{2} \in [-1, 0)\\$
Hence, the range is [-1, 1].
So, h(x) is onto.
Therefore, h(x) is bijective.
(iv)
$\\k(x) = x^{2}\\$

$ \text{Let }k (x_{1}) = k (x_{2})\\$

$ x_{1}^{2}= x_{2}^{2}\\ $

$x_{1 }= \pm x_{2}\\$
Therefore, k(x) is not one-one.

Question:22

Each of the following defines a relation on N:
(i) x is greater than y, $x, y \in N\\$
(ii) x + y = 10, $x, y \in N\\$
(iii) x y is square of an integer $x, y \in N\\$
(iv) x + 4y = 10 $x, y \in N\\$
Determine which of the above relations are reflexive, symmetric, and transitive.

Answer:

(i)Here, x is greater than y; $x, y \in N\\$
If $(x, x) \in R$, then x > x, that does not satisfy for any $x \in N$.\\
Therefore, R is not reflexive.
Say,$(x, y) \in R$
$\\Or, xRy\\ $

$\\ Or, x > y\\$
$\Rightarrow y > x \\$ For any $x, y \in N$, the above condition is not true.
Hence, R is not symmetric.
Again, xRy and yRz
$\\Or, x > y and y > z\\ $

$Or, x > z\\$

$ Or, xRz\\$
Hence, R is transitive.
(ii) x + y = 10;$x, y \in N$
Thus,
$\\R = \{ (x, y); x + y = 10, x, y \in N \} \\ $

$R = \{ (1, 9), (2, 8), (3, 7), (4, 6), (5, 5), (6, 4), (7, 3), (8, 2), (9, 1) \} \\$
Therefore, $(1, 1) \notin R\\$
So, R is not reflexive.
Again, $(x, y) \in R$

$ \Rightarrow (y, x) \in R\\$
Therefore, R is symmetric.
And, $(1, 9) \in R, (9, 1) \in R, but (1, 1) \notin R\\$
Therefore, R is not transitive.
(iii)Here, xy is square of an integer $x, y \in N$
$R = \{ (x, y) : \text{xy is a square of an integer} x, y \in N \} \\$
So, $(x, x) \in R, \forall x \in N\\$
For any $x \in N, x^{2 }$ is an integer.
Thus, R is reflexive.
If $(x, y) \in R $

$\Rightarrow (y, x) \in R\\$
So, R is symmetric.
Again, if xy and yz both are square of an integer.
Then, $xy = m^{2} \: \: and \: \: yz = n^{2} \text{for some m, n} \in Z\\$
$\\x = m^{2} /y and z = n^{2} /y\\ xz = m^{2} n^{2} / y^{2}$, this must be the square of an integer.
Therefore, R is transitive.
(iv) x + 4y = 10; $x, y \in N\\$
$\\R = \{ (x, y): x + 4y = 10; x, y \in N \} \\$

$ R = \{ (2, 2), (6, 1) \} \\ Here, (1, 1) \notin R\\$
Hence, R is not symmetric.
$\\(x, y) \in R$

$ \Rightarrow x + 4y = 10\\ And, (y, z) \in R $

$\Rightarrow y + 4z = 10\\$

$ Or,\ x - 16z = -30\\$

$ Or,\ (x, z) \notin R\\$
Therefore, R is not transitive.

Question:23

Let A = {1, 2, 3, … 9} and R be the relation in $A \times A$ defined by (a, b) R (c, d) if a + d = b + c for (a, b), (c, d) in $A \times A$. Prove that R is an equivalence relation and also obtain the equivalent class [(2, 5)].

Answer:

$\\Here, A = \{ 1, 2, 3, \ldots 9 \} \: \: and \: \: (a, b) R (c, d)\: \:$

$ if a + d = b + c for (a, b), (c, d) \in A \times A.\\ Say (a, b) R (a, b)\\ $

$Therefore, a + b = b + a, \forall a, b \in A .$
This must be true for any $a, b \in A.\\$
Hence, R is reflexive.
Say, (a, b) R (c, d)
Then,
$\\a + d = b + c\\ c + b = d + a\\ (c, d) R (a, b)\\$
Therefore, R is symmetric.
Let $\\(a, b) R(c, d) ,and, (c, d) R(e, f)$
$\\ a + d = b + c ,and, c + f = d + e\\ a + d = b + c ,and, d + e = c + f\\$

$ Or, (a + d) - (d + e = (b + c) - (c + f)\\ $

$Or, a - e = b - f\\ $

$Or, a + f = b + e\\ $

$Or, (a, b) R(e, f)\\$
So, R is transitive.
Therefore, R is an equivalence relation.

Question:24

Using the definition, prove that the function $f: A \rightarrow B$ is invertible if and only if f is both one-one and onto.

Answer:

Say, $f: A \rightarrow B$ be many-one functions.
If, $f(a) = p \text{ and } f(b) = p\\$
Then $f^{-1}(p) = a \: \: and \: \: f^{-1}(p) = b\\$
In this case, we have two images ‘a and b’ for one pre-image ‘p’. This is because the inverse function is not defined here.
However, to be one-one, f must be invertible.
Say, $f: A \rightarrow B$ is not onto function.
$B = \{ p, q, r \} and \{ p, q \}$ is the range of f.
There is no pre-image for the image r, which will have no image in set A.
And, f must be onto to be invertible.
Thus, to be both one-one and onto f must be invertible
If f is a bijective function, then $f = X \rightarrow Y$ is invertible.

Question:25

Functions $f, g : R \rightarrow R$ are defined, respectively, by $f(x) = x^{2} + 3x + 1, g(x) = 2x - 3$, find
(i) fog (ii) gof (iii) fof (iv) gog

Answer:

Here,
$f(x) = x^{2} + 3x + 1, g (x) = 2x - 3\\$

$\\(i)fog = f(g(x))\\ = f(2x - 3)\\ = (2x - 3)^{2} + 3(2x - 3) + 1\\ $$= 4x^{2}\ + 9 - 12x + 6x - 9 + 1\\ = 4x^{2}\ - 6x + 1\\ $

$(ii) gof = g(f(x))\\ = g(x^{2}\ + 3x + 1)\\ = 2(x^{2}\ + 3x + 1) - 3\\$

$ = 2x^{2}\ + 6x - 1\\$

$ (iii) fof = f(f(x))\\ = f(x^{2}\ + 3x + 1)\\ = (x^{2}\ + 3x + 1)2 + 3(x^{2} + 3x + 1) + 1\\ $

$= x4 + 9x^{2}\ + 1 + 6x^{3} + 6x + 2x^{2}\ + 3x^{2}\ + 9x + 3 + 1\\$

$ = x4 + 6x^{3} + 14x^{2}\ + 15x + 5\\ (iv) gog = g(g(x))\\ $

$= g(2x - 3)\\ = 2(2x - 3) - 3\\ = 4x - 6 - 3\\ = 4x - 9\\$

Question:26

Let $\ast$ be the binary operation defined on Q. Find which of the following binary operations are commutative
$\\(i) a \ast b = a - b\: \: \forall a, b \in Q\\ (ii) a \ast b = a^2 + b^2 \: \: \forall a, b \in Q\\ (iii) a \ast b = a + ab \: \: \forall a, b \in Q\\ (iv) a \ast b = (a - b)^{2}\: \: \forall a, b \in Q\\$

Answer:

Here, $\ast$ is a binary operation defined on Q.
(i)
$\\a \ast b = a - b, \forall a, b \in Q \: \: and \: \: b \ast a = b - a\\ So, a \ast b \neq b \ast a\\$
Hence, $\ast$ is not commutative.
(ii)
$\\a \ast b = a^{2}+ b^{2}\\ b \ast a = b^{2}+ a^{2}\\$
Therefore, $\ast$ is commutative.
(iii)
$\\a \ast b = a + ab\\ b \ast a = b + ab\\ Hence, a + ab \neq b + ab\\$
Therefore, $\ast$ is not commutative.
(iv)
$\\a \ast b = (a - b)^{2}, \forall a, b \in Q\\ b \ast a = (b -a)^{2}\\ As, (a - b)^{2} = (b - a)^{2}\\$
Therefore, $\ast$ is commutative.

Question:27

Let $\ast$ be binary operation defined on R by $a \ast b = 1 + ab, \forall a, b \in R$. Then the operation $\ast$ is
(i) commutative but not associative
(ii) associative but not commutative
(iii) neither commutative nor associative
(iv) both commutative and associative

Answer:

(i) Here,

$\ast$ is a binary operation defined on R by $a \ast b = 1 + ab, \forall a, b \in R$
So, $a \ast b = ab + 1 = b \ast a\\$
Therefore, $\ast$ is a commutative binary operation.
$\\Now, a \ast (b \ast c) = a \ast (1 + bc) = 1 + a (1 + bc) = 1 + a + abc\\ $

$Again, (a \ast b) \ast c = (1 + ab) \ast c = 1 + (1 + ab) c = 1 + c + abc\\$

$ Therefore, a \ast (b \ast c) \neq (a \ast b) \ast c\\$
Hence, $\ast$ is not associative.
Therefore, $\ast$ is commutative but not associative.

Question:28

Let T be the set of all triangles in the Euclidean plane, and let a relation R on T be defined as aRb if a is congruent to $b \forall a, b \in T$. Then R is
(A) reflexive but not transitive (B) transitive but not symmetric
(C) equivalence (D) none of these

Answer:

(C) equivalence
Here aRb, if a is congruent to b, $\forall a, b \in T.\\$
So, in aRa, a is congruent to a. This must always be true.
Therefore, R is reflexive.
Say, $aRb \Rightarrow a \sim b\\$
$b \sim a \Rightarrow bRa\\$
Therefore, R is symmetric.
Say, aRb and bRc
$\\a \sim b \: \: and\: \: b \sim c\\ a \sim c \Rightarrow aRc\\$
Therefore, R is transitive.
Therefore, R is an equivalence relation.

Question:29

Consider the non-empty set consisting of children in a family and a relation R defined as aRb if a is brother of b. Then R is

(A) symmetric but not transitive (B) transitive but not symmetric

(C) neither symmetric nor transitive (D) both symmetric and transitive

Answer:

(B) transitive but not symmetric

If, aRb means a is brother of b.

Then, it does not mean b is also a brother of a. Because, b can be a sister of a too.

Therefore, R is not symmetric.

If, aRb implies that a is the brother of b.

and bRc implies that b is the brother of c.

Therefore, a must be the brother of c.

Hence, R is transitive.

Question:30

The maximum number of equivalence relations on the set A = {1, 2, 3} are

(A) 1 (B) 2

(C) 3 (D) 5

Answer:

(D) 5

Given, set $A = \{ 1, 2, 3 \} \\$
Now, the number of equivalence relations as follows
$\\R1 = \{ (1, 1), (2, 2), (3, 3) \} \\$

$ R2 = \{ (1, 1), (2, 2), (3, 3), (1, 2), (2, 1) \} \\$

$ R3 = \{ (1, 1), (2, 2), (3, 3), (1, 3), (3, 1) \} \\$

$ R4 = \{ (1, 1), (2, 2), (3, 3), (2, 3), (3, 2) \} \\$

$ R5 = \{ (1, 2, 3) \Leftrightarrow A \times A = A^2 \} \\$
Thus, the maximum number of equivalence relations is ‘5’.

Question:31

If a relation R on the set {1, 2, 3} be defined by R = {(1, 2)}, then R is

(A) reflexive (B) transitive

(C) symmetric (D) none of these

Answer:

(D) none of these

If, R be defined on the set {1, 2, 3} by R = {(1, 2)}

Then, we can say that R is not reflexive, transitive, and symmetric.

Question:32

Let us define a relation R in R as aRb if $a \geq b.$ Then R is

(a) an equivalence relation (b) reflexive, transitive but not symmetric (c) symmetric, transitive but not reflexive (d) neither transitive nor reflexive but symmetric.

Answer:

(b).
The defined relation R in R as aRb if $a \geq b.\\$
Similarly, aRa implies $a \geq a$ which is true.
Therefore, it is reflexive.
Let $aRb \rightarrow a \geq b, but, b \ngtr a.\\$
Therefore, we can’t write it as Rba
Hence, R is not symmetric.
Now, $a \geq b, b \geq c$. So, $a \geq c$, and this is true.
Therefore, R is transitive.

Question:33

Let A = {1, 2, 3} and consider the relation R = {(1, 1), (2, 2), (3, 3), (1, 2), (2, 3), (1, 3)}, then R is

(a) reflexive but not symmetric (b) reflexive but not transitive (c) symmetric and transitive (d) neither symmetric nor transitive.

Answer:

(a)

Given, R = {(1, 1), (2, 2), (3, 3), (1, 2), (2, 3), (1, 3)}

Therefore, it can be written as: 1R1, 2R2 and 3R3.

Therefore, R is reflexive.

Here, 1R2 is not the same as 2R1 and 2R3 is not the same as 3R2.

Therefore, R is not symmetric.

Again, 1R1 and $1R2 \rightarrow 1R3\\$

Therefore, R is transitive.

Question:34

The identity element for the binary operation * defined on Q ~ {0} as a * b = 2 ab a, b Q ~ {0} is

(a) 1 (b) 0 (c) 2 (d) None of these

Answer:

(c)
Here: $a \ast b = ab/2 a, b Q \sim \{ 0 \} \\$
Assume ‘e’ as the identity element.
Therefore, $a \ast e = ae/2 = a \rightarrow e = 2 \\$

Question:35

If the set A contains 5 elements and set B contains 6 elements, then the number of one-one and onto mapping from A to B is

(a) 720 (b) 120 (c) 0 (d) None of these

Answer:

Let, the number of elements in A and B set are m and n respectively. Therefore, one-one and onto mapping

from A to B is n! when m = n

and, 0 if m ≠ n

It is given that, m = 5 and n = 6.

As,5 ≠ 6 So, from A to B mapping = 0

Question:36

Let $A = \{ 1, 2, 3, ..., n \}$ and $B = \{ a, b \}$ . Then the number of surjections from A to B is
(a) $^{n}P_{2}$
(b) $2^{n} - 2$
(c)$2^{n} - 1$
(d) None of these

Answer:

(d)
It is given that, $A = \{ 1, 2, 3, ..., n \}$ and $B = \{ a, b \} \\$
Say, the number of elements in set A and B are m and n respectively.
Therefore, $^{n}C_{m} \times m!$ is the number of surjections from A to B, for $n \ngeq m \\$
Given, m=2.
Therefore, the number of surjections from A to B
$= nC_2 \times 2! = n!/[2! \times (n-2)!] = n(n - 1) = n^{2} - n \\$

Question:37

Let $f : R \rightarrow R$ be defined by $f(x) = 1/x \forall x \in R$ then f is
(a) one-one (b) onto (c) bijective (d) f is not defined

Answer:

(d)
Here, f(x) = 1/x
Say, x = 0. Then f(x) = 1/0 = undefined
Hence, f(x) is not defined.

Question:38

Let $f : R \rightarrow R$ be defined by $f(x) = 3x^{2} - 5$ and $g : R \rightarrow R$ by $g(x) = x/(x^{2}+1)$, then gof is
$\\(a) (3x^{2}-5)/(9x^{4}-30x^{2}+ 26)\\ (b) (3x^{2}-5)/(9x^{4}-6x^{2}+ 26) \\(c) 3x^{2}/(x^{4}+2x^{2}-4) \\(d) (3x^{2})/(9x^{4}-30^{2}+2)\\$

Answer:

(a)

Given, $f(x) = 3x^{2} - 5 \ and \ g(x) = x/(x^{2}+1)\\$
Therefore,
$\\gof = gof (x) = g(3x^{2}-5)$

$ = (3x^{2}-5)/[(3x^{2}-5)^{2}+1]\\$

$ = (3x^{2}-5)/(9x^{4}-30x^{2}+ 26)\\$

Question:39

Which of the following functions from Z to Z are bijections?
$\\(a) f(x) = x^{3 }\\(b) f(x) = x + 2 \\(c) f(x) = 2x + 1 \\(d) f(x) = x^{2} + 1 \\$

Answer:

(b)
Given that $f : Z \rightarrow Z \\$
Say, $x_{1}, x_{2} \in f(x) \rightarrow f(x_{1}) = x_{1}+ 2; f(x_{2}) = x_{2}+ 2;\\$
For, $f(x_{1}) = f(x_{2})\ \rightarrow x_{1}+ 2 = x_{2}+\ 2 \\$
or, $x_{1 }= x_{2}\\$
Therefore, the function f(x) is one-one.
Say, y = x + 2
or, $x = y - 2 \forall y \in Z \\$
Hence, f(x) is onto.
Therefore, the function f(x) is bijective.

Question:40

Let $f : R \rightarrow R$ be the functions defined by $f(x) = x^{3} + 5$. Then $f^{-1}(x) is$
$\\(a) (x + 5)^{1/3}\\ (b) (x - 5)^{1/3}\\ (c) (5 - x)^{1/3}\\ (d) 5 - x \\$

Answer:

(b)Here,

$\\f(x) = x^{3} + 5\\ $

$Say, y = x^{3} + 5\\ $

$or, x^{3} = y - 5 \\ $

$or, x = (y - 5)^{1/3}\ f^{-1}1(x) = (x - 5)^{1/3} \\$

Question:41

Let $f : A \rightarrow B$ and $g : B \rightarrow C$ be the bijective functions. Then$(gof)^{-1}$ is
$\\(a) f^{-1 }og^{-1} \\(b) fog \\(c) g^{-1}of^{-1}\\ (d) gof \\$
Answer:

(a)
Given that, $f : A \rightarrow B \text{ and } g : B \rightarrow C \\$
Then, $(gof)^{-1} = f^{-1}og^{-1}\\$

Question:42

Let $f: \{ R-3/5 \} \rightarrow R$ be defined by $f(x) = (3x + 2)/(5x - 3)$ then
$\\(a) f(x) = f(x); \\(b) f^{-1}(x) = - f(x) \\(c) (fof)x = - x \\(d) f^{-1}(x) = f(x)/19\\$

Answer:

(a)

Here,$f(x) = (3x + 2)/(5x - 3) \forall x \neq 3/5\\$
$\\So, y = (3x + 2)/(5x - 3)\\ $

$or, y (5x-3) = (3x + 2)\\$

$ or, 5xy-3y = 3x + 2\\ $

$or, x = (3y+2)/(5y-3)\\ $

$Therefore, f^{-1}(x) = (3x+2)/(5x-3)\\$

$ Hence, f^{-1}(x) = f(x)\\$

Question:43

Let $f : [0, 1] \rightarrow [0, 1]$ be defined by

f(x) = { x, if is rational

{ 1-x , if is irrational
Then (fof)x is

(a) constant

(b) 1 + x

(c) x

(d) None of these

Answer:

(c)
Here, $f : [0, 1] \rightarrow [0, 1]\\$
$\\f = f^{-1} \\ Therefore, (fof)x = x\\$

Question:44

Let $f : [2, \infty )$ and R be the function defined by $f(x) = x^{2} - 4x + 5$, then the range of f is
$\\(a) R\\ (b) [1, \infty ) \\(c) [4, \infty ) \\(d) [5, \infty ) \\$
Answer:

(b)

$\\Here, f(x) = x^{2} - 4x + 5 \\$

$ Assume, y = x^{2}\ - 4x + 5 \\ $

$or, x^{2}\ - 4x + 5 - y = 0 \\$

$ or, x = [-4 \pm \sqrt \{ (-4)^{2}- 4(-4)(5-y) \} ]/2x = 2 \pm \sqrt (y-1)\\ $

$\text{When x is real}, y-1 \geq 0\\$

$ Therefore, y \geq 1\\ \text{Hence the range is}[1, \infty )\\$

Question:45

Let $f : N \rightarrow R$ be the function defined by $f(x) = (2x-1)/2$ and $g : Q \rightarrow R$ be another function defined by $g(x) = x + 2$ then, gof(3/2) is
$\\(a) 1 \\(b) - 1 \\(c) 7/2 \\(d) None of these \\$

Answer:

(d)
Given that, $f(x) = (2x-1)/2 \: \: and\: \: g(x) = x + 2 \\$
Therefore, $gof(x) = g[(f(x)] = f(x) + 2 = (2x-1)/2 = (2x+3)/2\\$
Hence, $gof(3/2) = 3 \\$

Question:46

Let f : $R \rightarrow R$ be defined by

\\f(x) = \{2x, when x> 3 \\ \ \ \ \: \: \: \: \: \: \: \: \ \ \ \ \ \{ x\textsuperscript{2}, when 1< x \leq 3\\ \ \ \ \ \ \ \ \ \ \ \ \{ 3x, when x \leq 1\\ then f(- 1) + f(2) + f(4) is\\

$\\ (a) 9 \\(b) 14 \\(c) 5 \\(d) None of these \\$

Answer:

(a)
$Here, f(- 1) + f(2) + f(4) = 3(- 1) + (2)^{2} + 2(4) = - 3 + 4 + 8 = 9 \\$

Question:47

If $f : R \rightarrow R$ be given by $f(x) = tan x$, then f-1 is
$\\(a) \frac{\pi}{4} \\ (b) (n \pi + \pi /4) where\ n \in Z \\(c) \ does \ not \ exist\\ (d) None\ of \ these \\$
Answer:

(a)

$\\Here, f(x) = tan x \\$

$ Assume, f(x) = y = tan x\\ $

$Then,\ \ x = tan^{-1} (y)\\ $

$or, f^{-1}(x) = tan^{-1}\ (x) \\ $

$or, f^{-1}(1) = tan^{-1} (1)\\$

$ or,\ f^{-1}(1) = tan^{-1} tan ( \pi /4)=\frac{\pi}{4}\\$

Question:48

Let the relation R be defined in N by aRb if 2a + 3b = 30. Then R = ............

Answer:

Here, aRb : 2a + 3b = 30

Or, 3b = 30 – 2a

Or, b = (30 - 2a)/ 3

for a = 3, b = 8

a = 6, b = 6

a = 9, b = 4

a = 12, b = 2

Therefore, R = {(3, 8), (6, 6), (9, 4), (12, 2)}

Question:49

Let the relation R be defined on the set $A = \{ 1, 2, 3, 4, 5 \}$ by $R = \{ (a, b) : \vert a^{2} - b^{2} \vert < 8 \}$. Then R is given by ............

Answer:

Here
$\\A = \{ 1, 2, 3, 4, 5 \} and R = \{ (a, b) : \vert a^{2} - b^{2} \vert < 8 \} \\$

$ \text{Therefore, we can say}, $

$R = \{ (1, 1), (1, 2), (2, 1), (2, 2), (2, 3), (3, 2), (4, 3) (3, 4), (4, 4), (5, 5) \} \\$

Question:50

Let f = {(1, 2), (3, 5), (4, 1)} and g = {(2, 3), (5, 1), (1, 3)}. Then gof = ................ and fog = ...................

Answer:

$\\Here, f = \{ (1, 2), (3, 5), (4, 1) \} $

$and, g = \{ (2, 3), (5, 1), (1, 3) \} \\ $

$Then, gof(1) = g[f(1)] = g(2) = 3\\$

$ Again, gof(3) = g[f(3)] = g(5) = 1 \\$

$ Again, gof(4) = g[f(4)] = g(1) = 3\\ $

$Therefore, gof = \{ (1, 3), (3, 1), (4, 3) \} \\$

$ Again, fog(2) = f[g(2)] = f(3) = 5\\ $

$Again, fog(5) = f[g(5)] = f(1) = 2\\ $

$Again, fog(1) = f[g(1)] = f(3) = 5\\$

$ Hence, fog = \{ (2, 5), (5, 2), (1, 5) \} \\$

Question:51

Let $f : R \rightarrow R$ be defined by $f(x) = x/ \sqrt (1+x^{2})$ then $(fofof)(x) = ................. \\$

Answer:

$(\text { fofof })(x) =f[f(x))] \\ =f\left[\begin{array}{l} f\left(\frac{x}{\sqrt{1+x^{2}}}\right) \end{array}\right] \\ $

$=f\left(\frac{\frac{x}{\sqrt{1+x^{2}}}}{\sqrt{1+\frac{x^{2}}{1+x^{2}}}}\right) \\$

$ =f\left(\frac{x}{\sqrt{1+2 x^{2}}}\right)$
$\\\text { fofof }(x) =\frac{\frac{x}{\sqrt{1+2 x^{2}}}}{\sqrt{1+\frac{x^{2}}{1+2 x^{2}}}} \\ =\frac{x}{\sqrt{1+3 x^{2}}}$

Question:52

If, $f(x) = [4 - (x - 7)^{3}]$, then $f^{-1}$(x) = .............

Answer:

$\\Here, f(x) = [4 - (x - 7)^{3}]\\$

$ Say, y = [4 - (x - 7)^{3}]\\ $

$Or, (x - 7)^{3} = (4 - y)\\ $

$Or, (x-7) = (4-y)^{1/3}\\$

$ Or, x = 7 + (4-y)^{1/3}\\$

$ Therefore, f^{-1}(x) = 7 + (4 - x)^{1/3}\\$

Question:53

Let R = {(3,1), (1,3), (3,3)} be a relation defined on the set A = {1, 2, 3}. Then R is symmetric, transitive and not reflexive.

Answer:

False Here in the question, R = {(3,1), (1,3), (3,3)} which is defined on the set A = {1, 2, 3}

As (1 ,1) R, R is not a reflexive one.

As (3, 1) R and (1, 3) belongs to R, then R is symmetric.

Again, (1, 3) R, (3, 1) R. However (1, 1) does not belong to R. Then R is not transitive.

Question:54

Let f: R→R be the function defined by $f(x) = sin(3x+2)$, x R. Then f is invertible.

Answer:

False
Here, $f(x) = sin(3x+2)$, x R. This function is not a one-one onto function for all $x \in R.\\$
Again, $f(x) = sin(3x+2) = 0\\$
$or, 3x + 2 = n\pi, n \in \ Z$.
Hence, the function is not invertible.

Question:55

Every relation which is symmetric and transitive is also reflexive.

Answer:

False

Assume, a relation R : R= {(1,2), (2,1), (2,2)} on the set A = {1,2}

Therefore, it is clear that (1,1) ∉ R. Hence, it is not reflexive.

Question:56

An integer m is said to be related to another integer n, if m is an integral multiple of n. This relation Z is reflexive, symmetric, and transitive.

Answer:

False.

Here, the given relation in the question is reflexive and transitive. However, it is not symmetric

Question:57

Let $A = \{ 0,1 \}$ and N be the set of natural numbers. Then the mapping $f: N \rightarrow A$ defined by $f(2n - 1) = 0, f(2n) = 1, n \in N, \\$is onto.

Answer:

True
Here, $A = \{ 0,1 \} , and\: \: f(2n - 1) = 0, f(2n) = 1, \forall A n \in N\\$
Therefore, the range of f is $\{ 0, 1 \} \\$
Hence, $f: N \rightarrow A$ mapping is onto.

Question:58

The relation R on the set A = {1, 2, 3} defined as R = {(1,1), (1,2), (2,1), (3,3)}

Answer:

False

As, $R = \{ (1,1), (1,2), (2,1), (3,3) \} \\$

Therefore, $(2,2) \notin R\\$

Hence, R is not reflexive.

Question:60

The composition of function is associative.

Answer:

True.

fo(goh)(x) = (fog)oh

Question:61

Every function is invertible.

Answer:

False.

Only bijective functions are invertible.

Question:62

A binary operation on a set always has an identity element.

Answer:

False

‘+’ is a binary operation on the set N but it has no identity element.

Subtopics Covered in NCERT Exemplar Solutions for Class 12 Maths Chapter 1

This chapter is divided into various sub-topics that are related to relations and functions and its operations. They are mentioned below:

  • Introduction
  • Types of relations
  • Types of functions
  • Composition of functions & invertible functions
  • Binary operations
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NCERT Exemplar Class 12 Maths Solutions

Importance of NCERT Exemplar Class 12 Maths Solutions Chapter 1

NCERT Exemplar solutions for Class 12 Maths chapter 1, students will learn about the basic concept of functions and relations.

  • The function is a relation, but is used for obtaining the output from a set of inputs. Learning the relations and functions at the very start of 12th Class will help in setting one's roots in calculus mathematics for cracking entrance exams and higher education.
  • A variety of questions with solutions are available to develop a strong base. Practicing these questions will help you to gain confidence and to score good marks in your exams.

NCERT solutions of class 12 - Subject-wise

Here are the subject-wise links for the NCERT solutions of class 12:

NCERT Notes of class 12 - Subject Wise

Given below are the subject-wise NCERT Notes of class 12 :

NCERT Books and NCERT Syllabus

Here are some useful links for NCERT books and the NCERT syllabus for class 12:

NCERT Exemplar Class 12 Solutions - Subject Wise

Given below are the subject-wise exemplar solutions of class 12 NCERT:


Frequently Asked Questions (FAQs)

Q: Are these solutions helpful for board examinations?
A:

Yes, NCERT exemplar solutions for Class 12 Maths chapter 1  are designed well to prepare you for board examinations.

Q: What are the important topics of this chapter?
A:

The important topics of this NCERT Exemplar Class 12 Maths solutions chapter 1 include Types of Relations, Types of Functions, Binary Operations and Composition of Functions and Invertible Function.

Q: How to remember the formulae well?
A:

The best ways to remember formulae is to keep revising them and solving related problems. Other methods include reading them on a daily basis. You can also prepare charts to stick near your study table or bed to go through in your leisure time.

Q: How many questions are there in this chapter?
A:

There are a total 70 questions in NCERT Exemplar Class 12 Maths solutions chapter 1 based on different concepts.

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Yes, in the case that you appeared for the 2025 improvement exam and your roll number is different from what was on the previous year’s marksheet, the board will usually release a new migration certificate. This is because the migration certificate will reflect the most recent exam details, roll number and passing year. You can apply to get it from your board using the process prescribed by them either online or through your school/college.

Yes, if you miss the 1st CBSE exam due to valid reasons, then you can appear for the 2nd CBSE compartment exam.

From the academic year 2026, the board will conduct the CBSE 10th exam twice a year, while the CBSE 12th exam will be held once, as per usual. For class 10th, the second phase exam will act as the supplementary exam. Check out information on w hen the CBSE first exam 2026 will be conducted and changes in 2026 CBSE Board exam by clicking on the link .

If you want to change your stream to humanities after getting a compartment in one subject in the CBSE 12th Board Exam , you actually have limited options to qualify for your board exams. You can prepare effectively and appear in the compartment examination for mathematics again. If you do not wish to continue with the current stream, you can take readmission in the Humanities stream and start from Class 11th again, and continue studying for two more years to qualify for the 12th examination.

The GUJCET Merit List is prepared based on the Class 12th marks and GUJCET marks received by the students. CBSE students who are not from the Gujarat board can definitely compete with GSEB students, as their eligibility is decided based on the combined marks scored by them in GUJCET and the 12th board. The weightage of the GUJCET score is 40% and the weightage of the class 12 scores is 60%.