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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.
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.
Class 12 Maths Chapter 1 exemplar solutions Exercise: 1.3 Page number: 11-17 Total questions: 62 |
Question:1
Let A = {a, b, c} and the relation R be defined on A as follows:
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
(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
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
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
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
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
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
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
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
(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
(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
(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
$\\ (a) 9 \\(b) 14 \\(c) 5 \\(d) None of these \\$
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
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
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
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
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
Answer:
False.
Here, the given relation in the question is reflexive and transitive. However, it is not symmetric
Question:57
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: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.
This chapter is divided into various sub-topics that are related to relations and functions and its operations. They are mentioned below:
NCERT Exemplar solutions for Class 12 Maths chapter 1, students will learn about the basic concept of functions and relations.
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Here are the subject-wise links for the NCERT solutions of class 12:
Given below are the subject-wise NCERT Notes of class 12 :
Here are some useful links for NCERT books and the NCERT syllabus for class 12:
Given below are the subject-wise exemplar solutions of class 12 NCERT:
Yes, NCERT exemplar solutions for Class 12 Maths chapter 1 are designed well to prepare you for board examinations.
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.
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