Let
Question: Let $A=\{2,3,4,5\}$ and $B=\{1,3,4\}$. If $R$ is the relation from $A$ to $B$ given by a $R b$ if "a is a divisor of $b$ ". Write $R$ as a set of ordered pairs. Solution: Since $R=\{(a, b): a, b \in N: a$ is a divisor of $b\}$ So, $R=\{(2,4),(3,3),(4,4)\}$...
Read More →Using factor theorem, show that g(x) is a factor of p(x), when
Question: Using factor theorem, show that g(x) is a factor of p(x), when $p(x)=x^{4}-x^{2}-12, g(x)=x+2$ Solution: Let: $p(x)=x^{4}-x^{2}-12$ Here, $x+2=0 \Rightarrow x=-2$ By the factor theorem, (x+ 2) is a factor ofthe given polynomial ifp(-2) = 0.Thus, we have: $p(-2)=\left[(-2)^{4}-(-2)^{2}-12\right]$ $=(16-4-12)$ $=0$ Hence,(x+ 2) is a factor of the given polynomial....
Read More →A = {1, 2, 3, 4, 5, 6, 7, 8}
Question: $A=\{1,2,3,4,5,6,7,8\}$ and if $R=\{(x, y): y$ is one half of $x, x, y \in A\}$ is a relation on $A$, then write $R$ as a set of ordered pairs. Solution: Since $R=\{(x, y): y$ is one half of $x ; x, y \in A\}$ So, $R=\{(2,1),(4,2),(6,3),(8,4)\}$...
Read More →If
Question: If $A=\{3,5,7\}$ and $B=\{2,4,9\}$ and $R$ is a relation given by "is less than", write $R$ as a set ordered pairs. Solution: Since, $R=\{(x, y): x, y \in N$ and $xy\}$ $\mathrm{R}=\{(3,4),(3,9),(5,9),(7,9)\}$...
Read More →Define an equivalence relation.
Question: Define an equivalence relation. Solution: A relationRon setAis said to be an equivalence relation iff(i) it is reflexive,(ii) it is symmetric and(iii) it is transitive.RelationRon setAsatisfying all the above three properties is an equivalence relation....
Read More →Using factor theorem, show that g(x) is a factor of p(x), when
Question: Using factor theorem, show that g(x) is a factor of p(x), when $p(x)=2 x^{4}+9 x^{3}+6 x^{2}-11 x-6, g(x)=x-1$ Solution: Let: $p(x)=2 x^{4}+9 x^{3}+6 x^{2}-11 x-6$ Here, $x-1=0 \Rightarrow x=1$ By the factor theorem, (x-1) is a factor ofthe given polynomial ifp(1) = 0.Thus, we have: $p(1)=\left(2 \times 1^{4}+9 \times 1^{3}+6 \times 1^{2}-11 \times 1-6\right)$ $=(2+9+6-11-6)$ $=0$ Hence, $(x-1)$ is a factor of the given polynomial....
Read More →Define a transitive relation.
Question: Define a transitive relation. Solution: A relationRon a setAis said to be transitive iff $(a, b) \in R$ and $(b, c) \in R$ $\Rightarrow(a, c) \in R$ for all $a, b, c \in R$ i. e. $a R b$ and $b R c$ $\Rightarrow a R c$ for all $a, b, c \in R$...
Read More →The following table gives the number of boys of a particular age in a class of 40 students.
Question: The following table gives the number of boys of a particular age in a class of 40 students. Calculate the mean age of the students 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_...
Read More →Define a symmetric relation.
Question: Define a symmetric relation. Solution: A relationRon a setAis said to be symmetric iff $(a, b) \in R$ $\Rightarrow(b, a) \in R$ for all $a, b \in A$ i. e. $a R b \Rightarrow b R a$ for all $a, b \in A$...
Read More →Define a symmetric relation.
Question: Define a symmetric relation. Solution: A relationRon a setAis said to be symmetric iff $(a, b) \in R$ $\Rightarrow(b, a) \in R$ for all $a, b \in A$ i. e. $a R b \Rightarrow b R a$ for all $a, b \in A$...
Read More →Using factor theorem, show that g(x) is a factor of p(x), when
Question: Using factor theorem, show that g(x) is a factor of p(x), when $p(x)=2 x^{3}+7 x^{2}-24 x-45, g(x)=x-3$ Solution: Let: $p(x)=2 x^{3}+7 x^{2}-24 x-45$ Now, $x-3=0 \Rightarrow x=3$ By the factor theorem, (x-3) is a factor of the given polynomialifp(3) = 0.Thus, we have: $p(3)=\left(2 \times 3^{3}-7 \times 3^{2}-24 \times 3-45\right)$ $=(54+63-72-45)$ $=0$ Hence,(x-3) is a factor of the given polynomial....
Read More →Define a reflexive relation.
Question: Define a reflexive relation. Solution: A relationRonAis said to be reflexive iff every element ofAis related to itself. i.e. $R$ is reflexive $\Leftrightarrow(a, a) \in R$ for all $a \in A$...
Read More →Using factor theorem, show that g(x) is a factor of p(x), when
Question: Using factor theorem, show that g(x) is a factor of p(x), when $p(x)=x^{3}-8, g(x)=x-2$ Solution: Let: $p(x)=x^{3}-8$ Now, $g(x)=0 \Rightarrow x-2=0 \Rightarrow x=2$ By the factor theorem, (x 2) is a factor of the given polynomialifp(2) = 0.Thus, we have: $p(2)=\left(2^{3}-8\right)=0$ Hence, $(x-2)$ is a factor of the given polynomial....
Read More →Let
Question: Let $A=\{3,5,7\}, B=\{2,6,10\}$ and $R$ be a relation from $A$ to $B$ defined by $R=\{(x, y): x$ and $y$ are relatively prime $\}$. Then, write $R$ and $R^{-1}$. Solution: R= {(x,y) :xandyare relatively prime}Then, R= {(3, 2), (5, 2), (7, 2), (3, 10), (7, 10), (5, 6), (7, 6)} So, $R^{-1}=\{(2,3),(2,5),(2,7),(10,3),(10,7),(6,5),(6,7)\}$...
Read More →If p(x)=2x3−11x2−4x+5 and g
Question: If $p(x)=2 x^{5}-11 x^{2}-4 x+5$ and $g(x)=2 x+1$, show that $g(x)$ is not a factor of $p(x)$. Solution: $p(x)=2 x^{3}-11 x^{2}-4 x+5$ $g(x)=2 x+1=2\left(x+\frac{1}{2}\right)=2\left[x-\left(-\frac{1}{2}\right)\right]$ Putting $x=-\frac{1}{2}$ in $p(x)$, we get $p\left(-\frac{1}{2}\right)=2 \times\left(-\frac{1}{2}\right)^{3}-11 \times\left(-\frac{1}{2}\right)^{2}-4 \times\left(-\frac{1}{2}\right)+5=-\frac{1}{4}-\frac{11}{4}+2+5=-\frac{12}{4}+7=-3+7=4 \neq 0$ Therefore, by factor theore...
Read More →If
Question: If $A=\{2,3,4\}, B=\{1,3,7\}$ and $R=\{(x, y): x \in A, y \in B$ and $xy\}$ is a relation from $A$ to $B$, then write $R^{-1}$. Solution: Since $R=\{(x, y): x \in A, y \in A$ and $xy\}$, R= {(2, 3), (2, 7), (3, 7), (4, 7)}So, $R^{-1}=\{(3,2),(7,2),(7,3),(7,4)\}$...
Read More →If p(x)=x3−5x2+4x−3 and g
Question: If $p(x)=x^{3}-5 x^{2}+4 x-3$ and $g(x)=x-2$, show that $p(x)$ is not a multiple of $g(x)$. Solution: $p(x)=x^{3}-5 x^{2}+4 x-3$ $g(x)=x-2$ Puttingx= 2 inp(x), we get $p(2)=2^{3}-5 \times 2^{2}+4 \times 2-3=8-20+8-3=-7 \neq 0$ Therefore, by factor theorem, (x 2) is not a factor ofp(x).Hence,p(x) is not a multiple ofg(x)....
Read More →Let
Question: Let $R=\left\{(x, y):\left|x^{2}-y^{2}\right|1\right)$ be a relation on set $A=\{1,2,3,4,5\}$. Write $R$ as a set of ordered pairs. Solution: Ris the set of ordered pairs satisfying the above relation. Also, no two different elements can satisfy the relation; only the same elements can satisfy the given relation. So, R = {(1, 1), (2, 2), (3, 3), (4, 4), (5, 5)}...
Read More →Find the value of p, if the mean of the following distribution is 20.
Question: Find the value ofp, if the mean of the following distribution is 20. Solution: Given: Mean $=20$ 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 →If R is a symmetric relation on a set A, then write a relation between
Question: If $R$ is a symmetric relation on a set $A$, then write a relation between $R$ and $R^{-1}$ Solution: Here,Ris symmetric on the setA. Let $(a, b) \in R$ $\Rightarrow(b, a) \in R$ [Since $R$ is symmetric] $\Rightarrow(a, b) \in R^{-1}$ [Bydefinitionofinverserelation $\Rightarrow R \subset R^{-1}$ Let $(x, y) \in R^{-1}$ $\Rightarrow(y, x) \in R$ [By definition of inverse relation] $\Rightarrow(x, y) \in R$ [Since $R$ is symmetric] $\Rightarrow R^{-1} \subset R$ Thus, $R=R^{-1}$...
Read More →The polynomial p(x)
Question: The polynomial $p(x)=x^{4}-2 x^{3}+3 x^{2}-a x+b$ when divided by $(x-1)$ and $(x+1)$ leaves the remainders 5 and 19 respectively. Find the values of $a$ and $b$. Hence, find the remainder when $p(x)$ is divided by $(x-2)$. Solution: Let $p(x)=x^{4}-2 x^{3}+3 x^{2}-a x+b$ Now, When $p(x)$ is divided by $(x-1)$, the remainder is $p(1)$. When $p(x)$ is divided by $(x+1)$, the remainder is $p(-1)$ Thus, we have: $p(1)=\left(1^{4}-2 \times 1^{3}+3 \times 1^{2}-a \times 1+b\right)$ $=(1-2+3...
Read More →Find the missing frequency (p) for the following distribution whose mean is 7.68.
Question: Find the missing frequency (p) for the following distribution whose mean is 7.68. Solution: Given: Mean $=7.68$ 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 entrie...
Read More →Find the missing value of p for the following distribution whose mean is 12.58
Question: Find the missing value ofpfor the following distribution whose mean is 12.58 Solution: Given: Mean $=12.58$ 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...
Read More →If
Question: If $R=\{(x, y): x+2 y=8\}$ is a relation on $N$ by, then write the range of $R$. Solution: $R=\{(x, y): x+2 y=8, x, y \in N\}$ Then, the values ofycan be 1, 2, 3 only. Also,y= 4 cannot result inx= 0 becausexis a natural number.Therefore, range ofRis {1, 2, 3}....
Read More →Write the smallest reflexive relation on set
Question: Write the smallest reflexive relation on setA= {1, 2, 3, 4}. Solution: Here,A= {1, 2, 3, 4}Also, a relation is reflexive iff every element of the set is related to itself. So, the smallest reflexive relation on the setAisR= {(1, 1), (2, 2), (3, 3), (4, 4)}...
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