Let C be the set of all complex numbers and C0 be the set of all no-zero complex numbers.
Question: Let $C$ be the set of all complex numbers and $C_{0}$ be the set of all no-zero complex numbers. Let a relation $R$ on $C_{0}$ be defined as $z_{1} R z_{2} \Leftrightarrow \frac{z_{1}-z_{2}}{z_{1}+z_{2}}$ is real for all $z_{1}, z_{2} \in C_{0}$. Show thatRis an equivalence relation. Solution: (i) Test for reflexivity: Since, $\frac{z_{1}-z_{1}}{z_{1}+z_{1}}=0$, which is a real number. So, $\left(z_{1}, z_{1}\right) \in R$ Hence,Ris relexive relation. (ii) Test for symmetric: Let $\lef...
Read More →Calculate the median from the following data:
Question: Calculate the median from the following data: Solution: First we prepare the following cummulative table to compute the median. Here, $N=250$ So, $\frac{N}{2}=125$ Thus, the cumulative frequency just greater than 125 is 127 and the corresponding class is $50-60$. Therefore, $50-60$ is the median class. Here, $l=50, f=31, F=96$ and $h=10$ We know that Median $=l+\left\{\frac{\frac{N}{2}-F}{f}\right\} \times h$ $=50+\left\{\frac{125-96}{31}\right\} \times 10$ $=50+\frac{29 \times 10}{31}...
Read More →Simplify
Question: Simplify $\left[\left\{(256)^{-\frac{1}{2}}\right\}^{\frac{1}{4}}\right]^{2}$ Solution: $\left[\left\{(256)^{-\frac{1}{2}}\right\}^{\frac{1}{4}}\right]^{2}$ $=\left\{(256)^{-\frac{1}{2}}\right\}^{\frac{1}{2}}$ $=(256)^{-\frac{1}{4}}$ $=\left(4^{4}\right)^{-\frac{1}{4}}$ $=4^{-1}$ $=\frac{1}{4}$...
Read More →Evaluate
Question: Evaluate $\frac{2^{n}+2^{n-1}}{2^{n+1}-2^{n}}$ Solution: $\frac{2^{n}+2^{n-1}}{2^{n+1}-2^{n}}$ $=\frac{2^{n}\left(1+2^{-1}\right)}{2^{n}(2-1)}$ $=\frac{\left(1+\frac{1}{2}\right)}{1}$ $=\frac{2+1}{2}$ $=\frac{3}{2}$...
Read More →If R and S are transitive relations on a set A, then prove that R ∪ S may not be a transitive relation on A.
Question: IfRandSare transitive relations on a setA, then prove thatRSmay not be a transitive relation onA. Solution: LetA= {a,b,c} andRandSbe two relations onA,given by R= {(a,a), (a,b), (b,a), (b,b)} andS= {(b,b), (b,c), (c,b), (c,c)} Here, the relationsRandSare transitive onA. $(a, b) \in R \cup S$ and $(b, c) \in R \cup S$ But $(a, c) \notin R \cup S$ Hence, $R \cup S$ is not a transitive relation on $A$....
Read More →find the value
Question: If $10^{x}=64$, find the value of $10^{\left(\frac{x}{2}+1\right)}$ Solution: We have, $10^{x}=64$ Taking square root from both sides, we get $\sqrt{10^{x}}=\sqrt{64}$ $\Rightarrow\left(10^{x}\right)^{\frac{1}{2}}=8$ $\Rightarrow 10^{\left(\frac{x}{2}\right)}=8$ Multiplying both sides by 10 , we get $10^{\left(\frac{x}{2}\right)} \times 10=8 \times 10$ $\therefore 10^{\left(\frac{x}{2}+1\right)}=80$...
Read More →Simplify
Question: Simplify $(2 \sqrt{5}+3 \sqrt{2})^{2}$. Solution: $(2 \sqrt{5}+3 \sqrt{2})^{2}$ $=(2 \sqrt{5})^{2}+(3 \sqrt{2})^{2}+2(2 \sqrt{5})(3 \sqrt{2}) \quad\left[(a+b)^{2}=a^{2}+b^{2}+2 a b\right]$ $=20+18+12 \sqrt{10}$ $=38+12 \sqrt{10}$...
Read More →find the value
Question: If $\sqrt{10}=3.162$, find the value of $\frac{1}{\sqrt{10}} .$ Solution: The value of $\frac{1}{\sqrt{10}}=\frac{1}{\sqrt{10}} \times \frac{\sqrt{10}}{\sqrt{10}}=\frac{\sqrt{10}}{10}=\frac{3.162}{10}=0.3162$...
Read More →If R and S are relations on a set A, then prove that
Question: IfRandSare relations on a setA, then prove that(i)RandSare symmetric ⇒RSandRSare symmetric(ii)Ris reflexive andSis any relation ⇒RSis reflexive. Solution: (i)RandSare symmetric relations on the setA. $\Rightarrow R \subset A \times A$ and $S \subset A \times A$ $\Rightarrow R \cap S \subset A \times A$ Thus, $R \cap S$ is a relation on $A$. Let $a, b \in A$ such that $(a, b) \in R \cap S$. Then, $(a, b) \in R \cap S$ $\Rightarrow(a, b) \in R$ and $(a, b) \in S$ $\Rightarrow(b, a) \in R...
Read More →Write the reciprocal of
Question: Write the reciprocal of $(2+\sqrt{3})$. Solution: The reciprocal of $(2+\sqrt{3})$ $=\frac{1}{(2+\sqrt{3})}$ $=\frac{1}{(2+\sqrt{3})} \times \frac{(2-\sqrt{3})}{(2-\sqrt{3})}$ $=\frac{(2-\sqrt{3})}{(2+\sqrt{3})(2-\sqrt{3})}$ $=\frac{(2-\sqrt{3})}{2^{2}-(\sqrt{3})^{2}} \quad\left[(a+b)(a-b)=a^{2}-b^{2}\right]$ $=\frac{(2-\sqrt{3})}{4-3}$ $=\frac{(2-\sqrt{3})}{1}$ $=(2-\sqrt{3})$...
Read More →Give an example of a number x such that x2 is an irrational number and x3 is a rational number.
Question: Give an example of a numberxsuch thatx2is an irrational number andx3is a rational number. Solution: The cube roots of natural numbers which are not perfect cubes are all irrational numbers. Let $x=\sqrt[3]{2}=2^{\frac{1}{3}}$ Now, $x^{2}=\left(2^{\frac{1}{3}}\right)^{2}=2^{\frac{2}{3}}=\left(2^{2}\right)^{\frac{1}{3}}=4^{\frac{1}{3}}$, which is an irrational number Also, $x^{3}=\left(2^{\frac{1}{3}}\right)^{3}=2^{3 \times \frac{1}{3}}=2$, which is a rational number...
Read More →Let Z be the set of all integers and Z0 be the set of all non-zero integers. Let a relation R on Z × Z0 be defined as
Question: LetZbe the set of all integers andZ0be the set of all non-zero integers. Let a relationRonZZ0be defined as $(a, b) R(c, d) \Leftrightarrow a d=b c$ for all $(a, b),(c, d) \in Z \times Z_{0}$ Prove thatRis an equivalence relation onZZ0. Solution: We observe the following properties ofR. Reflexivity: Let $(a, b)$ be an arbitrary element of $\mathrm{Z} \times \mathrm{Z}_{0}$. Then, $(a, b) \in \mathrm{Z} \times \mathrm{Z}_{0}$ $\Rightarrow a, b \in \mathrm{Z}, \mathrm{Z}_{0}$ $\Rightarrow...
Read More →Is the product of a rational and an irrational number always irrational?
Question: Is the product of a rational and an irrational number always irrational? Give an example. Solution: Yes, the product of a rational and an irrational number is always an irrational number. Example: 2 is a rational number and $\sqrt{3}$ is an irrational number. Now, $2 \times \sqrt{3}=2 \sqrt{3}$, which is an irrational number....
Read More →Give an example of two irrational numbers whose sum as well as product is rational.
Question: Give an example of two irrational numbers whose sum as well as product is rational. Solution: Let the two irrational numbers be $2+\sqrt{3}$ and $2-\sqrt{3}$. Sum of these irrational numbers $=(2+\sqrt{3})+(2-\sqrt{3})=4$, which is rational Product of these irrational numbers $=(2+\sqrt{3})(2-\sqrt{3})=2^{2}-(\sqrt{3})^{2}=4-3=1$, which is rational...
Read More →Simplify
Question: Simplify $\left(\frac{3125}{243}\right)^{\frac{4}{5}}$ Solution: $\left(\frac{3125}{243}\right)^{\frac{4}{5}}$ $=\left(\frac{5 \times 5 \times 5 \times 5 \times 5}{3 \times 3 \times 3 \times 3 \times 3}\right)^{\frac{4}{5}}$ $=\left(\frac{5^{5}}{3^{5}}\right)^{\frac{4}{5}}$ $=\left[\left(\frac{5}{3}\right)^{5}\right]^{\frac{4}{5}}$ $\left[\left(\frac{x}{y}\right)^{a}=\frac{x^{a}}{y^{a}}\right]$ $\left[\left(x^{a}\right)^{b}=x^{a b}\right]$ $=\left(\frac{5}{3}\right)^{5 \times \frac{4}{...
Read More →Calculate the median from the following data:
Question: Calculate the median from the following data: Solution: First we prepare the following cummulative table to compute the median. Here, $N=140$ So, $\frac{N}{2}=70$ Thus, the cumulative frequency just greater than 70 is 98 and the corresponding class is $55-65$. Therefore, $55-65$ is the median class. Here, $l=55, f=40, F=58$ and $h=10$ We know that Median $=l+\left\{\frac{\frac{N}{2}-F}{f}\right\} \times h$ $=55+\left\{\frac{70-58}{40}\right\} \times 10$ $=55+\frac{12 \times 10}{40}$ $=...
Read More →Express the following complex numbers in the standard form a + i b:
Question: Express the following complex numbers in the standard forma+i b: (i) $(1+i)(1+2 i)$ (ii) $\frac{3+2 i}{-2+i}$ (iii) $\frac{1}{(2+i)^{2}}$ (iv) $\frac{1-i}{1+i}$ (v) $\frac{(2+i)^{3}}{2+3 i}$ (vi) $\frac{(1+i)(1+\sqrt{3} i)}{1-i}$ (vii) $\frac{2+3 i}{4+5 i}$ (viii) $\frac{(1-i)^{3}}{1-i^{3}}$ (ix) $(1+2 i)^{-3}$ (x) $\frac{3-4 i}{(4-2 i)(1+i)}$ (xi) $\left(\frac{1}{1-4 i}-\frac{2}{1+i}\right)\left(\frac{1-4 i}{5+i}\right)$ (xii) $\frac{5+\sqrt{2} i}{1-2 \sqrt{i}}$(i) $(1+i)(1+2 i)$ Solu...
Read More →Let S be a relation on the set R of all real numbers defined by
Question: LetSbe a relation on the setRof all real numbers defined by $S=\left\{(a, b) \in R \times R: a^{2}+b^{2}=1\right\}$ Prove thatSis not an equivalence relation onR. Solution: We observe the following properties ofS. Reflexivity : Let $a$ be an arbitrary element of $R$. Then, $a \in R$ $\Rightarrow a^{2}+a^{2} \neq 1 \forall a \in R$ $\Rightarrow(a, a) \notin S$ So, $S$ is not reflexive on $R$. Symmetry : Let $(a, b) \in R$ $\Rightarrow a^{2}+b^{2}=1$ $\Rightarrow b^{2}+a^{2}=1$ $\Rightar...
Read More →Express the following complex numbers in the standard form a + i b:
Question: Express the following complex numbers in the standard forma+i b: (i) $(1+i)(1+2 i)$ (ii) $\frac{3+2 i}{-2+i}$ (iii) $\frac{1}{(2+i)^{2}}$ (iv) $\frac{1-i}{1+i}$ (v) $\frac{(2+i)^{3}}{2+3 i}$ (vi) $\frac{(1+i)(1+\sqrt{3} i)}{1-i}$ (vii) $\frac{2+3 i}{4+5 i}$ (viii) $\frac{(1-i)^{3}}{1-i^{3}}$ (ix) $(1+2 i)^{-3}$ (x) $\frac{3-4 i}{(4-2 i)(1+i)}$ (xi) $\left(\frac{1}{1-4 i}-\frac{2}{1+i}\right)\left(\frac{1-4 i}{5+i}\right)$ (xii) $\frac{5+\sqrt{2} i}{1-2 \sqrt{i}}$(i) $(1+i)(1+2 i)$ Solu...
Read More →Following is the distribution of I.Q. of 100 students. Find the median I.Q.
Question: Following is the distribution of I.Q. of 100 students. Find the median I.Q. Solution: Here, the frequency table is given in inclusive form. Transforming the given table into exclusive form and prepare the cumulative frequency table. Here, $N=100$ So, $\frac{N}{2}=50$ Thus, the cumulative frequency just greater than 50 is 67 and the corresponding class is 94.5104.5. Therefore, 94.5104.5 is the median class. Here, $I=94.5, f=33, F=34$ and $h=9$ We know that, Median $=l+\left\{\frac{\frac...
Read More →then find the value of
Question: If $a=1, b=2$ then find the value of $\left(a^{b}+b^{a}\right)^{-1}$. Solution: Fora= 1andb = 2, $\left(a^{b}+b^{a}\right)^{-1}$ $=\left(1^{2}+2^{1}\right)^{-1}$ $=(1+2)^{-1}$ $=3^{-1}$ $=\frac{1}{3} \quad\left(x^{-a}=\frac{1}{x^{a}}\right)$ Thus, the value of $\left(a^{b}+b^{a}\right)^{-1}$ when $a=1$ and $b=2$ is $\frac{1}{3}$....
Read More →The following is the distribution of height of students of a certain class in a certain city:
Question: The following is the distribution of height of students of a certain class in a certain city: Find the median height. Solution: First we prepare the following cummulative table to compute the median. Now, $N=420$ $\therefore \frac{N}{2}=210$ Thus, the cumulative frequency just greater than 210 is 275 and the corresponding class is $166-168$. Therefore, $166-168$ is the median class. $I=166, f=142, F=133$ and $h=2$ We know that, Median $=l+\left\{\frac{\frac{N}{2}-F}{f}\right\} \times h...
Read More →Let R be the relation defined on the set
Question: LetRbe the relation defined on the setA= {1, 2, 3, 4, 5, 6, 7} byR= {(a,b) : bothaandbare either odd or even}. Show thatRis an equivalence relation. Further, show that all the elements of the subset {1, 3, 5, 7} are related to each other and all the elements of the subset {2, 4, 6} are related to each other, but no element of the subset {1, 3, 5, 7} is related to any element of the subset {2, 4, 6}. Solution: We observe the following properties ofR. Reflexivity: Let $a$ be an arbitrary...
Read More →Simplify
Question: Simplify $\sqrt[4]{81 x^{8} y^{4} z^{16}}$ Solution: $\sqrt[4]{81 x^{8} y^{4} z^{16}}$ $=\left(3^{4} \times x^{8} \times y^{4} \times z^{16}\right)^{\frac{1}{4}}$ $=\left(3^{4}\right)^{\frac{1}{4}} \times\left(x^{8}\right)^{\frac{1}{4}} \times\left(y^{4}\right)^{\frac{1}{4}} \times\left(z^{16}\right)^{\frac{1}{4}} \quad\left[(x \times y \times z \times \ldots)^{a}=x^{a} \times y^{a} \times z^{a} \times \ldots\right]$ $=\left(3^{4 \times \frac{1}{4}}\right) \times\left(x^{8 \times \frac...
Read More →Following are the lives in hours of 15 pieces of the components of aircraft engine.
Question: Following are the lives in hours of 15 pieces of the components of aircraft engine. Find the median:715, 724, 725, 710, 729, 745, 694, 699, 696, 712, 734, 728, 716, 705, 719 Solution: First of all arranging the data in ascending order of magnitude, we have $694,696,699,705,710,712,715,716,719,724,725,728,729,734,745$ Here, $N=15$, which is an odd number Therefore, median is the value of $\left(\frac{N+1}{2}\right)=\frac{15+1}{2}$ $=8^{\text {th }}$ observation $=716$...
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