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RELATIONS AND FUNCTIONS · Uttarakhand Board (UBSE) STD 12 Science (Uttarakhand - English Medium). 31 questions.

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Question 31 Mark
Let A = {1, 2, 3}, B = {4, 5, 6, 7} and let f = {(1, 4), (2, 5), (3, 6)} be a function from A to B. State whether f is one-one or not.
Answer
f is a one-one function.
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Question 51 Mark
Let * be a binary operation on N given by a * b = HCF (a, b), a, b $\in$ N. Write the value of 22 * 4.
Answer
2.
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Question 71 Mark
Let * be a binary operation, on the set of all non-zero real numbers, given by a * b = $\frac{\text{ab}}{5}$ for all a, b $\in$ R - {0}. Find the value of x, given that 2 * (x * 5) = 10.
Answer
x = 25.
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Question 81 Mark
Let * be a 'binary' operation on N given by a * b = LCM (a, b) for all a, b $\in$ N. Find 5 * 7.
Answer
35.
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Question 91 Mark
State the reason for the relation R in the set {1, 2, 3} given by R = {(1, 2), (2, 1)} not to be transitive.
Answer
(1, 2) $\in$ R, (2, 1) $\in$ R but (1, 1) $\notin$ R.
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Question 111 Mark
If the binary operation $^{\ast}$ on the set of integers Z, is defined by $a^{\ast} b = a + 3b^{2},$ then find the value of $2^{\ast} 4.$
Answer
Given $a^{\ast} b = a + 3b^{2} $ $ \forall a,b \in z$$\therefore 2^{\ast} 4 = 2 + 3 \times 4^{2} = 2 + 48 = 50.$
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Question 121 Mark
If a * b denotes the larger of ‘a’ and ‘b’ and if a o b = (a * b) + 3, then write the value of (5) o (10), where * and o are binary operations.
Answer
a o b = (a * b) + 3
5 o 10 = (5 * 10) + 3
= 50 + 3 = 53.
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Question 131 Mark
If f(x) = x + 1, find $\frac{\text{d}}{\text{dx}}(\text{fof})(\text{x}).$
Answer
Given: f(x) = x + 1
fof(x) = (x + 1) + 1 = x + 2
$\frac{\text{d}}{\text{dx}}(\text{fof})(\text{x})$
$=\frac{\text{d}}{\text{dx}}(\text{x}+2)$
$=1$
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Question 141 Mark
If $\text{f(x)}=\text{x}+7$ and $\text{g(x)}=\text{x}-7,\ \text{x}\in\text{R},$ then find $=\frac{\text{d}}{\text{dx}}(\text{fog})(\text{x}).$
Answer
Given: f(x) = x + 7
g(x) = x - 7
(fog)(x) = f(g(x))
= f(x - 7)
= (x - 7) + 7
= x
$\frac{\text{d}}{\text{dx}}(\text{fof})(\text{x})$
$=\frac{\text{d}}{\text{dx}}(\text{x})=1$
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Question 151 Mark
Let * be the binary operation on N given by a * b = L.C.M. of a and b. Find:
Is * commutative?
Answer
a * b = L.C.M. of a and b = L.C.M. of b and a = b * a
Therefore, operation * is commutative.
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Question 161 Mark
Let * be the binary operation on N given by a * b = L.C.M. of a and b. Find:
5 * 7, 20 * 16
Answer
a * b = L.C.M. of a and b
5 * 7 = L.C.M. of 5 and 7 = 35
20 * 16 = L.C.M. of 20 and 16 = 80
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Question 171 Mark
Find which of the binary operations are commutative and which are associative.
State whether the following statements are true or false. Justify
For an arbitrary binary operation * on a set N, $\text{a}*\text{a}=\text{a}\ \forall\text{a}\in\text{N}.$
Answer
* being a binary operation on N, is defined as $\text{a}*\text{a}=\text{a}\ \forall\text{a}\in\text{N}.$
Hence operation * is not defined, therefore, the given statement is false.
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Question 181 Mark
Let * be the binary operation on N given by a * b = L.C.M. of a and b. Find:
Is * associative?
Answer
a * (b * c) = a * (L.C.M. of b and c) = L.C.M. of (a and L.C.M. of b and c) = L.C.M. of a, b and c.
Similarly, (a * b) * c = L.C.M. of a, b and c
Thus, a * (b * c) = (a * b) * c
Therefore, the operation is associative.
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Question 191 Mark
Determine whether or not each of the definition of $*$ given below gives a binary operation. In the event that $*$ is not a binary operation, give justification for this.
On $Z^+$, define $*$ by $a * b = ab$
Answer
on $Z^+ = {1, 2, 3,.....}, a * b = ab$
Let $a = 2, b = 4$
$\therefore\ \text{a}*\text{b}=2\times4=8\in\text{Z}^{+}$
Therefore, operation $*$ is a binary operation on $Z^+.$
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Question 201 Mark
Let L be the set of all lines in XY plane and R be the relation in L defined as $R$ = {$(L_1, L_2): L_1$ is parallel to $L_2$}. Show that R is an equivalence relation. Find the set of all lines related to the line $y = 2x + 4$.
Answer
It is given that the relation L defined as
$R=\left\{\left(L_1, L_2\right): L_1 \text { is parallel to } L_2\right\}$
$R$ is reflexive as any line $L_1$ is parallel to itself
$\Rightarrow\left(\mathrm{L}_1, \mathrm{~L}_2\right) \in \mathbf{R}$
Now, Let $\left(\mathrm{L}_1, \mathrm{~L}_2\right) \in \mathrm{R}$
$\Rightarrow L_1$ is parallel to $L_2$
$\Rightarrow L_2$ is parallel to $L_1$
$\Rightarrow\left(\mathrm{L}_2, \mathrm{~L}_1\right) \in \mathbf{R}$
Therefore, R is symmetric.
Now, Let $\left(\mathrm{L}_1, \mathrm{~L}_2\right),\left(\mathrm{L}_2, \mathrm{~L}_3\right) \in \mathrm{R}$
$\Rightarrow L_1$ is parallel to $L_2$. Also $L_2$ is parallel to $L_3$
$\Rightarrow L_1$ is parallel to $L_3$.
$\Rightarrow(\text{L}_1,\text{L}_3)\in\text{R}$
Therefore, R is transitive.
Therefore, R is an equivalence relation.
The set of all lines related to the line y = 2x + 4 is the set of all lines that are parallel to the line y = 2x + 4
Slope of the line y = 2x + 4 is m = 2
We know that parallel lines have the same slopes.
The line parallel to the given line is of the form y = 2x + c where, $\text{c}\in\text{R}.$
Therefore, the set of all lines related to the given line by y = 2x + c, where $\text{c}\in\text{R}.$
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Question 211 Mark
Write the smallest equivalence relation on the set A = {1, 2, 3}.
Answer
The smallest equivalence relation on the set A = {1, 2, 3} is R = {(1, 1), (2, 2), (3, 3)}.
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Question 221 Mark
Let * be the binary operation on N given by a * b = L.C.M. of a and b. Find:Find the identity of * in N
Answer
Identity of * in N = 1 because a * 1 = L.C.M. of a and 1 = a
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Question 231 Mark
Consider a binary operation * on the set {1, 2, 3, 4, 5} given by the following multiplication table (Table 1.2).
Is * commutative?
(Hint: use the following table)
*
1
2
3
4
5
1
1
1
1
1
1
2
1
2
1
2
1
3
1
1
3
1
1
4
1
2
1
4
1
5
1
1
1
1
5
Answer
2 * 3 = 1 and 3 * 4 = 1
2 * 3 = 3 * 2 and other element of the given set.
Hence the operation is commutative.
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Question 241 Mark
g: {5, 6, 7, 8} → {1, 2, 3, 4} with
g = {(5, 4), (6, 3), (7, 4), (8, 2)}
Answer
g = {(5, 4), (6, 3), (7, 4), (8, 2)}
It is many-one function, therefore g has no inverse.
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Question 251 Mark
h: {2, 3, 4, 5} → {7, 9, 11, 13} with
h = {(2, 7), (3, 9), (4, 11), (5, 13)}
Answer
h = {(2, 7), (3, 9), (4, 11), (5, 13)}
h is one-one onto function, therefore, h has an inverse.
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Question 261 Mark
State with reason whether following functions have inverse
f: {1, 2, 3, 4} → {10} with
f = {(1, 10), (2, 10), (3, 10), (4, 10)}
Answer
f = {(1, 10), (2, 10), (3, 10), (4, 10)}
It is many-one function, therefore f has no inverse.
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Question 271 Mark
Determine whether or not each of the definition of $*$ given below gives a binary operation. In the event that $*$ is not a binary operation, give justification for this.
On $R,$ define $*$ by $a * b = ab^2$
Answer
on $R($set of real numbers$) a * b = ab$
Let $a = 5.2, b = 3$
$\therefore\ \text{a}*\text{b}=5.2(3)^2=46.8\in\text{R}$
Therefore, operation $*$ is a binary operation on $R.$
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Question 281 Mark
Define a binary operation on a set.
Answer
Let A be a non-empty set. An operation * is called a binary operation on A, if and only if $\text{a}\times\text{b}\in\text{A},\forall\text{a},\text{b}\in\text{A}$
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Question 291 Mark
Let * be the binary operation on N given by a * b = L.C.M. of a and b. Find:Which elements of N are invertible for the operation *?
Answer
Only the element 1in N is invertible for the operation * because $1*\frac{1}{1}=1.$
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Question 301 Mark
Determine whether or not each of the definition of $*$ given below gives a binary operation. In the event that $*$ is not a binary operation, give justification for this.
On $Z^+,$ define $*$ by $a * b = a - b$
Answer
on $Z^+ = \{1, 2, 3,.....\}, a * b = a - b$
Let $a = 1, b = 3$
$\therefore\ \text{a}*\text{b}=1-3=-2\notin\text{Z}^{+}$
Therefore, operation $*$ is not a binary operation on $Z^+.$
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Question 311 Mark
Consider a binary operation * on the set {1, 2, 3, 4, 5} given by the following multiplication table (Table 1.2).
Compute (2 * 3) * 4 and 2 * (3 * 4)
(Hint: use the following table)
*
1
2
3
4
5
1
1
1
1
1
1
2
1
2
1
2
1
3
1
1
3
1
1
4
1
2
1
4
1
5
1
1
1
1
5
Answer
2 * 3 = 1 and 3 * 4 = 1
Now (2 * 3) * 4 = 1 * 4 = 1 and 2 * (3 * 4) = 2 * 1 = 1
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