Exercise 1.6 - Chapter 1 Relations & Functions 10th Maths Guide Samacheer Kalvi Solutions - SaraNextGen [2024-2025]

Updated By SaraNextGen

On April 24, 2024, 11:35 AM

Ex $1.6$
Question $1 .$
If $n(A \times B)=6$ and $A=\{1,3\}$ then $n(B)$ is
(1) 1
(2) 2
(3) 3
(4) 6
Answer:
(3) 3
Hint:
If $\mathrm{n}(A \times B)=6$
$\mathrm{A}=\{1,1\}, \mathrm{n}(\mathrm{A})=2$
$\mathrm{n}(\mathrm{B})=3$

Question $2 .$
$A=\{a, b, p\}, B=\{2,3\}, C=\{p, q, r, s)$
then $n[(A \cup C) \times B]$ is
(1) 8
(2) 20
(3) 12
(4) 16
Answer:
(3) 12

Question $3 .$
If $\mathrm{A}=\{1,2\}, \mathrm{B}=\{1,2,3,4\}, \mathrm{C}=\{5,6\}$ and $\mathrm{D}=\{5,6,7,8\}$ then state which of the following statement is true.
(1) $(\mathrm{A} \times \mathrm{C}) \subset(\mathrm{B} \times \mathrm{D})$
$(2)(B \times D) \subset(A \times C)$
(3) $(A \times B) \subset(A \times D)$
(4) $(\mathrm{D} \times \mathrm{A}) \subset(\mathrm{B} \times \mathrm{A})$
Answer:
(1) $(\mathrm{A} \times \mathrm{C}) \subset(\mathrm{B} \times \mathrm{D})]$
 

Question $4 .$
If there are 1024 relations from a set $\mathrm{A}=\{1,2,3,4,5\}$ to a set $\mathrm{B}$, then the number of elements in $B$ is ....................
(1) 3
(2) 2
(3) 4
(4) 8
Answer:
(2) 2
Hint: $n(A)=5$
$\mathrm{n}(\mathrm{A} \times \mathrm{B})=10$
(consider 1024 as 10 )
$\mathrm{n}(\mathrm{A}) \times \mathrm{n}(\mathrm{B})=10$
$5 \times \mathrm{n}(\mathrm{B})=10$
$\mathrm{n}(\mathrm{B})=\frac{10}{5}=2$
$\mathrm{n}(\mathrm{B})=2$
 

Question $5 .$
The range of the relation $\mathrm{R}=\left\{\left(\mathrm{x}, \mathrm{x}^{2}\right) \mid \mathrm{x}\right.$ is a prime number less than 13$\}$ is
(1) $\{2,3,5,7\}$
(2) $\{2,3,5,7,11\}$
(3) $\{4,9,25,49,121\}$
(4) $\{1,4,9,25,49,121\}$
Answer:
(3) $\{4,9,25,49,121\}]$

Question $6 .$
If the ordered pairs $(a+2,4)$ and $(5,2 a+6)$ are equal then $(a, b)$ is
(1) $(2,-2)$
(2) $(5,1)$
(3) $(2,3)$
(4) $(3,-2)$
Answer:
(4) $(3,-2)$

Question 7 .
Let $n(A)=m$ and $n(B)=n$ then the total number of non-empty relations that can be defined from $A$ to $\mathrm{B}$ is
(1) $m^{n}$
(2) $n^{m}$
(3) $2^{\mathrm{mn}}-1$
(4) $2^{\mathrm{mn}}$
Answer:
(4) $2^{\mathrm{mn}}$

Question 8 .
If $\{(a, 8),(6, b)\}$ represents an identity function, then the value of a and 6 are respectively
(1) $(8,6)$
(2) $(8,8)$
(3) $(6,8)$
(4) $(6,6)$
Answer:
(1) $(8,6)$
Hint: $f=\{\{a, 8)(6,6)\}$. In an identity function each one is the image of it self.
$\therefore \mathrm{a}=8, \mathrm{~b}=6$
 

Question 9 .
Let $A=\{1,2,3,4\}$ and $B=\{4,8,9,10\}$. A function $f: A \rightarrow B$ given by $f=\{(1,4),(2,8),(3,9),(4$,
10)\} is a
(1) Many-one function
(2) Identity function
(3) One-to-one function
(4) Into function
Answer:
(3) One-to one function

Question 10 .
If $f(x)=2 x^{2}$ and $g(x)=\frac{1}{3 x}$, Then fog is
(1) $\frac{3}{2 x^{2}}$
(2) $\frac{2}{3 x^{2}}$

(3) $\frac{2}{9 x^{2}}$
(4) $\frac{1}{6 x^{2}}$
Answer:
(3) $\frac{2}{9 x^{2}}$
Hint:
$\begin{aligned}
&f(x)=2 x^{2} \\
&g(x)=\frac{1}{3 x} \\
&f o g=f(g(x))=f\left(\frac{1}{3 x}\right)=2\left(\frac{1}{3 z}\right)^{2} \\
&=2 \times \frac{1}{9 x^{2}}=\frac{2}{9 x^{2}}
\end{aligned}$

Queston $11 .$
If $\mathrm{A}: \mathrm{A} \rightarrow \mathrm{B}$ is a bijective function and if $n(B)=7$, then $n(A)$ is equal to
(1) 7
(2) 49
(3) 1
(4) 14
Answer:
(1) 7

Question 12 .
Let $f$ and $g$ be two functions given by $f=\{(0,1),(2,0),(3,-4),(4,2),(5,7)\} g=\{(0,2),(1,0),(2$, $4),(-4,2),(7,0)\}$ then the range of fog is
(1) $\{0,2,3,4,5\}$
(2) $\{-4,1,0,2,7\}$
(3) $\{1,2,3,4,5\}$
(4) $\{0,1,2\}$
Answer:
(4) $\{0,1,2\}$
Hint:
gof $=g(f(x))$
$f \circ g=f(g(x))$
$=\{(0,2),(1,0),(2,4),(-4,2),(7,0)\}$
Range of fog $=\{0,1,2\}$

Question $13 .$
Let $f(x)=\sqrt{1+x^{2}}$ then
(1) $f(x y)=f(x) f(y)$
(2) $f(x y) \geq f(x) f(y)$
(3) $f(x y) \leq f(x)$. f(y)
(4) None of these
Answer:
(3) $f(x y) \leq f(x) \cdot f(y)$

Question 14 .
If $g=\{(1,1),(2,3),(3,5),(4,7)\}$ is a function given by $g(x)=\alpha x+\beta$ then the values of $\alpha$ and $\beta$ are
(1) $(-1,2)$

(2) $(2,-1)$
(3) $(-1,-2)$
(4) $(1,2)$
Answer:
(2) $(2,-1)$
Hint:
$\begin{aligned}
&g(x)=\alpha x+\beta \\
&\alpha=2 \\
&\beta=-1 \\
&g(x)=2 x-1 \\
&g(1)=2(1)-1=1 \\
&g(2)=2(2)-1=3 \\
&g(3)=2(3)-1=5 \\
&g(4)=2(4)-1=7
\end{aligned}$

Question 15 .
$f(x)=(x+1)^{3}-(x-1)^{3}$ represents a function which is
(1) linear
(2) cubic
(3) reciprocal
(4) quadratic
Answer:
(4) quadratic
Hint: $f(x)=(x+1)^{3}-(x-1)^{3}$
$\left[\right.$ using $\left.\mathrm{a}^{3}-\mathrm{b}^{3}=(\mathrm{a}-\mathrm{b})^{3}+3 \mathrm{ab}(\mathrm{a}-\mathrm{b})\right]$
$=(x+1-x+1)^{3}+3(x+1)(x-1)$
$(x+1-x+1)$
$=8+3\left(x^{2}-1\right)^{2}$
$=8+6\left(x^{2}-1\right)$
$=8+6 x^{2}-6$
$=6 x^{2}+2$
It is quadratic polynomial