Find the value of p for the following distribution whose mean is 16.6.
Question: Find the value ofpfor the following distribution whose mean is 16.6. Solution: Given: Mean=16.6 First of all prepare the frequency table in such a way that its first column consist of the values of the variate $\left(x_{i}\right)$ and the second column the corresponding frequencies $\left(f_{i}\right)$. Thereafter multiply the frequency of each row with corresponding values of variable to obtain third column containing $\left(f_{i} x_{i}\right)$. Then, sum of all entries in the column ...
Read More →Write the identity relation on set
Question: Write the identity relation on setA= {a,b,c}. Solution: Identity set ofAisI= {(a,a), (b,b), (c,c)}Every element of this relation is related to itself....
Read More →If
Question: If $R=\left\{(x, y): x^{2}+y^{2} \leq 4 ; x, y \in Z\right\}$ is a relation on $Z$, write the domain of $R$. Solution: Domain ofRis the set of values ofxsatisfying the relationR.Asxmust be an integer, we get the given values ofx: $0, \pm 1, \pm 2$ Thus, Domain of $R=\{0, \pm 1, \pm 2\}$...
Read More →If the mean of the following data is 15, find p.
Question: If the mean of the following data is 15, findp. Solution: Given: Also, mean = 15 First of all prepare the frequency table in such a way that its first column consist of the values of the variate $\left(x_{i}\right)$ and the second column the corresponding frequencies $\left(f_{i}\right)$. Thereafter multiply the frequency of each row with corresponding values of variable to obtain third column containing $\left(f_{i} x_{i}\right)$. Then, sum of all entries in the column second and deno...
Read More →The polynomials
Question: The polynomials $\left(2 x^{3}+x^{2}-a x+2\right)$ and $\left(2 x^{3}-3 x^{2}-3 x+a\right)$ when divided by $(x-2)$ leave the same remainder. Find the value of $a$. Solution: Let $f(x)=2 x^{3}+x^{2}-a x+2$ and $g(x)=2 x^{3}-3 x^{2}-3 x+a$ By remainder theorem, when $f(x)$ is divided by $(x-2)$, then the remainder $=f(2)$. Puttingx= 2 inf(x), we get $f(2)=2 \times 2^{3}+2^{2}-a \times 2+2=16+4-2 a+2=-2 a+22$ By remainder theorem, wheng(x) isdivided by (x 2), then the remainder =g(2).Put...
Read More →Write the domain of the relation R defined on the set Z of integers as follows:
Question: Write the domain of the relationRdefined on the setZof integers as follows: $(a, b) \in R \Leftrightarrow a^{2}+b^{2}=25$ Solution: $0, \pm 3, \pm 4, \pm 5$ Thus, Domain of $R=\{0, \pm 3, \pm 4, \pm 5\}$...
Read More →The relation R
Question: The relationR= {(1, 2,), (1, 3)} on setA= [1, 2, 3] is _________________ only. Solution: Given: ArelationRon the set {1, 2, 3} be defined byR=R = {(1, 2,), (1, 3)}. R = {(1, 2,), (1, 3)} Since, $(1,1) \notin R$ Therefore, It is not reflexive. Since, $(1,2) \in R$ but $(2,1) \notin R$ Therefore, It is not symmetric. But there is no counter example to disapprove transitive condition.Therefore, it is transitive. Hence, The relationR= {(1, 2,), (1, 3)} on setA= {1, 2, 3} istransitiveonly....
Read More →If the mean of the following data is 20.6. Find the value of p.
Question: If the mean of the following data is 20.6. Find the value ofp. Solution: Given: Also, mean $=20.6$ First of all prepare the frequency table in such a way that its first column consist of the values of the variate $\left(x_{i}\right)$ and the second column the corresponding frequencies $\left(f_{i}\right)$. Thereafter multiply the frequency of each row with corresponding values of variable to obtain third column containing $\left(f_{i} x_{i}\right)$. Then, sum of all entries in the colu...
Read More →Using the remainder theorem, find the remainder, when p(x) is divided by g(x), where
Question: Using the remainder theorem, find the remainder, whenp(x) is divided byg(x), where $p(x)=x^{3}-a x^{2}+6 x-a, g(x)=x-a$ Solution: $p(x)=x^{3}-a x^{2}+6 x-a$ $g(x)=x-a$ By remainder theorem, when $p(x)$ is divided by $(x-a)$, then the remainder $=p(a)$. Putting $x=a$ in $p(x)$, we get $p(a)=a^{3}-a \times a^{2}+6 \times a-a=a^{3}-a^{3}+6 a-a=5 a$ $\therefore$ Remainder $=5 a$ Thus, the remainder when $p(x)$ is divided by $g(x)$ is $5 a$....
Read More →Let R be a relation on the set Z of all integers defined as
Question: Let $R$ be a relation on the set $Z$ of all integers defined as $(x, y) \in \mathrm{R} \Leftrightarrow x-y$ is divisible by 2 . Then, the equivalence class [1] is_________________. Solution: Given: $R$ is the equivalence relation on the set $Z$ of integers defined as $(x, y) \in R \Leftrightarrow x-y$ is divisible by 2 . To findthe equivalence class [1], we puty= 1 in the given relation and find all the possible values ofx. Thus, $R=\{(x, 1): x-1$ is divisible by 2$\}$ $\Rightarrow x-1...
Read More →Using the remainder theorem, find the remainder, when p(x) is divided by g(x), where
Question: Using the remainder theorem, find the remainder, whenp(x) is divided byg(x), where $p(x)=2 x^{3}+3 x^{2}-11 x-3, g(x)=\left(x+\frac{1}{2}\right)$ Solution: $p(x)=2 x^{3}+3 x^{2}-11 x-3$ $g(x)=\left(x+\frac{1}{2}\right)=\left[x-\left(-\frac{1}{2}\right)\right]$ By remainder theorem, when $p(x)$ is divided by $\left(x+\frac{1}{2}\right)$, then the remainder $=p\left(-\frac{1}{2}\right)$. Putting $x=-\frac{1}{2}$ in $p(x)$, we get $p\left(-\frac{1}{2}\right)=2 \times\left(-\frac{1}{2}\rig...
Read More →Find the mean of the following data:
Question: Find the mean of the following data: Solution: Given: First of all prepare the frequency table in such a way that its first column consist of the values of the variate $\left(x_{i}\right)$ and the second column the corresponding frequencies $\left(f_{i}\right)$. Thereafter multiply the frequency of each row with corresponding values of variable to obtain third column containing $\left(f_{i} x_{i}\right)$. Then, sum of all entries in the column second and denoted by $\sum f_{i}$ and in ...
Read More →Using the remainder theorem, find the remainder, when p(x) is divided by g(x), where
Question: Using the remainder theorem, find the remainder, whenp(x) is divided byg(x), where $p(x)=x^{3}-6 x^{2}+2 x-4, g(x)=1-\frac{3}{2} x$ Solution: $p(x)=x^{3}-6 x^{2}+2 x-4$ $g(x)=1-\frac{3}{2} x=-\frac{3}{2}\left(x-\frac{2}{3}\right)$ By remainder theorem, when $p(x)$ is divided by $\left(1-\frac{3}{2} x\right)$, then the remainder $=p\left(\frac{2}{3}\right)$. Putting $x=\frac{2}{3}$ in $p(x)$, we get $p\left(\frac{2}{3}\right)=\left(\frac{2}{3}\right)^{3}-6 \times\left(\frac{2}{3}\right)...
Read More →Let R be the equivalence relation on the set Z of integers given by
Question: LetRbe the equivalence relation on the setZof integers given byR= {(a, b): 3 dividesa-b}. Then the equivalence class [0] is equal to ____________________. Solution: Given:Ris the equivalence relation on the setZof integers given byR= {(a, b): 3 dividesa b}.To findthe equivalence class [0], we putb= 0 in the given relation and find all the possible values ofa. Thus, $R=\{(a, 0): 3$ divides $a-0\}$ $\Rightarrow a-0$ is a multiple of 3 $\Rightarrow a$ is a multiple of 3 $\Rightarrow a=3 n...
Read More →Calculate the mean for the following distribution :
Question: Calculate the mean for the following distribution : Solution: Given: First of all prepare the frequency table in such a way that its first column consist of the values of the variate $\left(x_{i}\right)$ and the second column the corresponding frequencies $\left(f_{i}\right)$. Thereafter multiply the frequency of each row with corresponding values of variable to obtain third column containing $\left(f_{i} x_{i}\right)$. Then, sum of all entries in the column second and denoted by $\sum...
Read More →The largest equivalence relation on the set
Question: The largest equivalence relation on the setA= {1, 2, 3} is ___________________. Solution: Given:A= {1, 2, 3}Thelargest equivalence relation contains all the possible ordered pairs.Therefore,R={(1, 1), (2, 2), (3, 3), (1, 2), (2, 1), (2, 3), (3, 2), (1, 3), (3, 1)} isthe largest equivalence relation.Hence,the largest equivalence relation on the setA= {1, 2, 3} is{(1, 1), (2, 2), (3, 3), (1, 2), (2, 1), (2, 3), (3, 2), (1, 3), (3, 1)}....
Read More →The smallest equivalence relation on the set
Question: The smallest equivalence relation on the setA= {a, b, c, d} is _________________________. Solution: Given:A= {a, b, c, d}Identity relation is the smallest equivalence relation. Therefore,R={(a,a), (b,b), (c,c)} isthe smallest equivalence relation.Hence, the smallest equivalence relationon the setA= {a, b, c, d} is{(a,a),(b,b),(c,c)}....
Read More →If R is a relation from
Question: If $R$ is a relation from $A=\{11,12,13\}$ to $B=\{8,1012\}$ defined by $y=x-3$, then $R^{-1}=$ Solution: Given: $R=\{(x, y): y=x-3, x \in A$ and $y \in B\}$, where $A=\{11,12,13\}$ and $B=\{8,1012\}$. R = {(11, 8), (13, 10)} Therefore, $R^{-1}=\{(8,11),(10,13)\}$ Hence, $R^{-1}=\{(\underline{8}, 11),(\underline{10}, 13)\}$....
Read More →If R s a relation defined on set
Question: If $R$ s a relation defined on set $A=\{1,2,3\}$ by the rule $\left(a_{t} b\right) \in R \Leftrightarrow\left|a^{2}-b^{2}\right| \leq 5$, then $R^{-1}=$ Solution: Given: $R=\left\{(a, b):\left|a^{2}-b^{2}\right| \leq 5\right\}$, where $A=\{1,2,3\}$ and $a, b \in A$. R = {(1, 1), (1, 2), (2, 1), (2, 2), (2, 3), (3, 2), (3, 3)} Therefore, $R^{-1}=\{(1,1),(2,1),(1,2),(2,2),(3,2),(2,3),(3,3)\}=R$ Hence, if $R$ is a relation defined on set $A=\{1,2,3\}$ by the rule $(a, b) \in R \Leftrighta...
Read More →The following table gives the daily income of 50 workers of a factory:
Question: The following table gives the daily income of 50 workers of a factory: Find the mean, mode and median of the above data. Solution: Consider the following table. Here, the maximum frequency is 14 so the modal class is 120140. Therefore, $l=120$ $h=20$ $f=14$ $f_{1}=12$ $f_{2}=8$ $F=12$ Mean $=\frac{\sum f_{i} x_{i}}{\sum f}$ $=\frac{7260}{50}$ Mean $=145.20$ Thus, the mean daily income of the workers is Rs 145.20. Median $=l+\frac{\frac{N}{2}-F}{f} \times h$ $=120+\frac{25-12}{14} \time...
Read More →Let A = {1, 2, 3, 4, 5} The domain of the relation on A defined by R
Question: Let $A=\{1,2,3,4,5\}$ The domain of the relation on $A$ defined by $R=\{(x, y): y=2 x-1\}$, is Solution: Given: $R=\{(x, y): y=2 x-1\}$, where $A=\{1,2,3,4,5\}$ and $x, y \in A$. R= {(1, 1), (2, 3), (3, 5)}Therefore, Domain of R = {1, 2, 3}. Hence,the domain of therelation onAdefined byR={(x, y):y= 2x1}, is{1, 2, 3}....
Read More →Find the mean, median and mode of the following data:
Question: Find the mean, median and mode of the following data: Solution: Therefore, $l=150$ $h=50$ $f=6$ $f_{1}=5$ $f_{2}=5$ $F=10$ Mean $=\frac{\sum f_{i} x_{i}}{\sum f}$ $=\frac{4225}{25}$ Mean = 169 Thus, the mean of the data is 169. Median $=l+\frac{\frac{N}{2}-F}{f} \times h$ $=150+\frac{12.5-10}{6} \times 50$ $=150+\frac{2.5}{6} \times 50$ $=150+\frac{125}{6}$ Median $=170.83$ Thus, the median of the data is 170.83. Mode $=l+\frac{f-f_{1}}{2 f-f_{1}-f_{2}} \times h$ $=150+\frac{6-5}{12-5-...
Read More →Let A = {1, 2, 3, 4} and R be the relation on A defined by
Question: Let $A=\{1,2,3,4\}$ and $R$ be the relation on A defined by $\{(a, b): a, b \in A, a \times b$ is an even number $\}$, then the range of $R$ is Solution: Given:R= {(a, b):a, bA, a bis an even number}, whereA= {1, 2, 3, 4}.R= {(1, 2), (1, 4), (2, 1), (2, 2), (2, 3), (2, 4), (3, 2), (3, 4), (4, 1), (4, 2), (4, 3), (4, 4)}Therefore, Range of R = {1, 2, 3, 4}Hence,the range ofRis{1, 2, 3, 4}....
Read More →The number of relations on a finite set having 5 elements is
Question: The number of relations on a finite set having 5 elements is Solution: LetRbe a relation onA, whereAcontains 5 elements.Ris a subset ofAA.Number of elements inAA =5 5 = 25 Number of relations $=$ Number of subsets of $A \times A=2^{25}$ Hence, the number of relations on a finite set having 5 elements is $2^{25}$....
Read More →Let R be a relation in N defined by
Question: Let $R$ be a relation in $N$ defined by $R=\{(x, y): x+2 y=8\}$, then the range of $R$ is Solution: Given:R= {(x, y):x+ 2y= 8} wherex,yNR= {(6, 1), (4, 2), (2, 3)}Therefore, Range of R = {1, 2, 3}Hence,the range ofRis{1, 2, 3}....
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