The distribution below gives the weight of 30 students in a class.
Question: The distribution below gives the weight of 30 students in a class. Find the median weight of students: Solution: We prepare the cumulative frequency table, as given below. We have, $N=30$ So, $\frac{N}{2}=15$ Now, the cumulative frequency just greater than 15 is 19 and the corresponding class is $55-60$. Therefore, $55-60$ is the median class. Here, $l=55, f=6, F=13$ and $h=5$ We know that Median $=l+\left\{\frac{\frac{N}{2}-F}{f}\right\} \times h$ $=55+\left\{\frac{15-13}{6}\right\} \...
Read More →Evaluate
Question: Evaluate $\left(\frac{81}{49}\right)^{\frac{-3}{2}}$ Solution: $\left(\frac{81}{49}\right)^{\frac{-3}{2}}=\left(\frac{49}{81}\right)^{\frac{3}{2}}$ $=\left(\frac{7^{2}}{9^{2}}\right)^{\frac{3}{2}}$ $=\left[\left(\frac{7}{9}\right)^{2}\right]^{\frac{3}{2}}$ $=\left(\frac{7}{9}\right)^{3}$ $=\frac{7^{3}}{9^{3}}$ $=\frac{343}{729}$ Hence, $\left(\frac{81}{49}\right)^{\frac{-3}{2}}=\frac{343}{729}$....
Read More →Let O be the origin.
Question: LetObe the origin. We define a relation between two pointsPandQin a$\Rightarrow O P=O Q$ and $O Q=O R$ $\Rightarrow O P=O Q=O R$ $\Rightarrow O P=O R$ $\Rightarrow(P, R) \in R$plane ifOP=OQ. Show that the relation, so defined is an equivalence relation. Solution: LetAbe the set of all points in a plane such that $A=\{P: P$ is a point in the plane $\}$ Let $R$ be the relation such that $R=\{(P, Q): P, Q \in A$ and $O P=O Q$, where $O$ is the origin $\}$ We observe the following properti...
Read More →Evaluate the following:
Question: Evaluate the following: (i) $i^{457}$ (ii) $i^{528}$ (iii) $\frac{1}{i^{58}}$ (iv) $i^{37}+\frac{1}{i^{67}}$ (v) $\left(i^{41}+\frac{1}{i^{257}}\right)^{9}$ (vi) $\left(i^{77}+i^{70}+i^{87}+i^{414}\right)^{3}$ (vii) $i^{30}+i^{40}+i^{60}$ (viii) $i^{49}+i^{68}+i^{89}+i^{110}$ Solution: (i) $i^{457}=i^{4 \times 114+1}$ $=\left(i^{4}\right)^{114} \times i$ $=i \quad\left(\because i^{4}=1\right)$ (ii) $i^{528}=i^{4 \times 132}$ $=\left(i^{4}\right)^{132}$ $=1 \quad\left(\because i^{4}=1\r...
Read More →Evaluate the following:
Question: Evaluate the following: (i) $i^{457}$ (ii) $i^{528}$ (iii) $\frac{1}{i^{58}}$ (iv) $i^{37}+\frac{1}{i^{67}}$ (v) $\left(i^{41}+\frac{1}{i^{257}}\right)^{9}$ (vi) $\left(i^{77}+i^{70}+i^{87}+i^{414}\right)^{3}$ (vii) $i^{30}+i^{40}+i^{60}$ (viii) $i^{49}+i^{68}+i^{89}+i^{110}$ Solution: (i) $i^{457}=i^{4 \times 114+1}$ $=\left(i^{4}\right)^{114} \times i$ $=i \quad\left(\because i^{4}=1\right)$ (ii) $i^{528}=i^{4 \times 132}$ $=\left(i^{4}\right)^{132}$ $=1 \quad\left(\because i^{4}=1\r...
Read More →Simplify
Question: Simplify $(32)^{\frac{1}{5}}+(-7)^{0}+(64)^{\frac{1}{2}}$ Solution: $(32)^{\frac{1}{5}}+(-7)^{0}+(64)^{\frac{1}{2}}=\left(2^{5}\right)^{\frac{1}{5}}+1+\left(2^{6}\right)^{\frac{1}{2}}$ $=2+1+2^{3}$ $=2+1+8$ $=11$ Hence, $(32)^{\frac{1}{5}}+(-7)^{0}+(64)^{\frac{1}{2}}=11$...
Read More →The following table gives the literacy rate (in percentage) of 35 cities. Find the mean literacy rate.
Question: The following table gives the literacy rate (in percentage) of 35 cities. Find the mean literacy rate. Solution: Let the assumed meanA= 70 andh= 10. We know that mean, $\bar{X}=A+h\left(\frac{1}{N} \sum f_{i} u_{i}\right)$ Now, we have $N=\sum f_{i}=35, \sum f_{i} u_{i}=-2, h=10$ and $A=70$ Putting the values in the above formula, we have $\bar{X}=A+h\left(\frac{1}{N} \sum f_{i} u_{i}\right)$ $=70+10\left(\frac{1}{35} \times(-2)\right)$ $=70-\frac{20}{35}$ $=70-0.571$ $=69.428$...
Read More →Solve for x:
Question: Solve for $x:\left(\frac{2}{5}\right)^{2 x-2}=\frac{32}{3125}$ Solution: $\left(\frac{2}{5}\right)^{2 x-2}=\frac{32}{3125}$ $\Rightarrow\left(\frac{2}{5}\right)^{2 x-2}=\frac{2^{5}}{5^{5}}$ $\Rightarrow\left(\frac{2}{5}\right)^{2 x-2}=\left(\frac{2}{5}\right)^{5}$ $\Rightarrow 2 x-2=5$ $\Rightarrow 2 x=5+2$ $\Rightarrow x=\frac{7}{2}$ Hence, $x=\frac{7}{2}$....
Read More →Show that the relation R, defined on the set A of all polygons as
Question: Show that the relationR, defined on the setAof all polygons as $R=\left\{\left(P_{1}, P_{2}\right): P_{1}\right.$ and $P_{2}$ have same number of sides $\}$, is an equivalence relation. What is the set of all elements inArelated to the right angle triangleTwith sides 3, 4 and 5? Solution: We observe the following properties onR. Reflexivity: Let $P_{1}$ be an arbitrary element of $A$. Then, polygon $P_{1}$ and $P_{1}$ have the same number of sides, since they are one and the same. $\Ri...
Read More →Rationalise
Question: Rationalise $\frac{1}{\sqrt{3}+\sqrt{2}}$. Solution: $\frac{1}{\sqrt{3}+\sqrt{2}}=\frac{1}{\sqrt{3}+\sqrt{2}} \times \frac{\sqrt{3}-\sqrt{2}}{\sqrt{3}-\sqrt{2}}$ $=\frac{\sqrt{3}-\sqrt{2}}{(\sqrt{3})^{2}-(\sqrt{2})^{2}}$ $=\frac{\sqrt{3}-\sqrt{2}}{3-2}$ $=\frac{\sqrt{3}-\sqrt{2}}{1}$ $=\sqrt{3}-\sqrt{2}$ Hence, the rationalised form is $\sqrt{3}-\sqrt{2}$....
Read More →A class teacher has the following absentee record of 40 students of a class for the whole term.
Question: A class teacher has the following absentee record of 40 students of a class for the whole term. Find the mean number of days a student was absent. Solution: Let the assume meanA= 17. We know that mean, $\bar{X}=A+\frac{1}{N} \sum_{i=1}^{n} f_{i} d_{i}$ Now, we have $N=\sum f_{i}=40, \sum f_{i} d_{i}=181$ and $A=17$. Putting the values in the above formula, we have $\bar{X}=A+\frac{1}{N} \sum_{i=1}^{n} f_{i} d_{i}$ $=17+\frac{1}{40} \times(-181)$ $=17-\frac{181}{40}$ $=17-4.525$ $=12.47...
Read More →Find the value of
Question: Find the value of $\frac{21 \sqrt{12}}{10 \sqrt{27}}$. Solution: $\frac{21 \sqrt{12}}{10 \sqrt{27}}=\frac{21 \sqrt{2 \times 2 \times 3}}{10 \sqrt{3 \times 3 \times 3}}$ $=\frac{21 \times 2 \sqrt{3}}{10 \times 3 \sqrt{3}}$ $=\frac{3 \times 7 \times 2}{5 \times 2 \times 3}$ $=\frac{7}{5}$ Hence, the value of $\frac{21 \sqrt{12}}{10 \sqrt{27}}$ is $\frac{7}{5}$....
Read More →Find an irrational number between 5 and 6.
Question: Find an irrational number between 5 and 6. Solution: A number which is non terminating and non recurring is known as irrational number. There are infinitely many irrational numbers between 5 and 6 . One of the example is $5.40430045000460000 \ldots .$...
Read More →To find out the concentration of SO2 in the air (in parts per million, i.e., ppm),
Question: To find out the concentration of $\mathrm{SO}_{2}$ in the air (in parts per million, i.e., ppm), the data was collected for 30 localities in a certain city and is presented below. Find the mean of concentration of $\mathrm{SO}_{2}$ in the air. Solution: Let the assumed meanA= 0.1 andh= 0.04. We know that mean, $\bar{X}=A+h\left(\frac{1}{N} \sum f_{i} u_{i}\right)$ Now, we have $N=\sum f_{i}=30, \sum f_{i} u_{i}=-1, h=0.04$ and $A=0.10$. Putting the values in the above formula, we have ...
Read More →Simplify
Question: Simplify $6 \sqrt{36}+5 \sqrt{12}$. Solution: $6 \sqrt{36}+5 \sqrt{12}$ $=6 \times 6+5 \sqrt{4 \times 3}$ $=36+10 \sqrt{3}$...
Read More →Find the value of
Question: Find the value of $(1296)^{0.17} \times(1296)^{0.08}$. Solution: $(1296)^{0.17} \times(1296)^{0.08}$ $=(1296)^{0.17+0.08}$ $=(1296)^{0.25}$ $=(1296)^{\frac{1}{4}}$ $=\sqrt[4]{1296}$ $=6$...
Read More →The table below shows the daily expenditure on food of 25 household in a locality
Question: The table below shows the daily expenditure on food of 25 household in a locality Find the mean daily expenditure on food by a suitable method. Solution: Let the assumed meana= 225 andh= 50 Now, $\sum f_{i}=25$ $\sum f_{i} u_{i}=-7$ Mean, $\bar{x}=a+\left(\frac{\sum f_{i} u_{i}}{\sum f_{i}}\right) \times h$ $=225+\left(\frac{-7}{25}\right) \times(50)$ $=225-14$ $=211$ Therefore, mean daily expenditure on food is Rs 211....
Read More →The number
Question: The number $\frac{665}{625}$ will terminate after how many decimal places? Solution: $\frac{665}{625}=\frac{5 \times 19 \times 7}{5^{4}}=\frac{19 \times 7}{5^{3}}=\frac{19 \times 7 \times 2^{3}}{5^{3} \times 2^{3}}=\frac{1064}{1000}=1.064$ So, $\frac{665}{625}$ will terminate after 3 decimal places....
Read More →Let L be the set of all lines in XY-plane and R be the relation in L defined as R
Question: Let $L$ be the set of all lines in $X Y$-plane and $R$ be the relation in $L$ defined as $R=\left\{L_{1}, L_{2}\right): L_{1}$ is parallel to $\left.L_{2}\right\}$. Show that $R$ is an equivalence relation. Find the set of all lines related to the line $y=2 x+4$. Solution: We observe the following properties ofR. Reflexivity: Let $L_{1}$ be an arbitrary element of the set $L$. Then, $L_{1} \in L$ $\Rightarrow L_{1}$ is parallel to $L_{1}$ [Every line is parallel to itself] $\Rightarrow...
Read More →Solve
Question: Solve $(3-\sqrt{11})(3+\sqrt{11})$ Solution: $(3-\sqrt{11})(3+\sqrt{11})$ $=\left(3^{2}-(\sqrt{11})^{2}\right)$ $=9-11$ $=-2$...
Read More →What can you say about the sum of a rational number and an irrational number?
Question: What can you say about the sum of a rational number and an irrational number? Solution: Sum of a rational number and an irrational number is an irrational number. Example: $4+\sqrt{5}$ represents sum of rational and an irrational number where 4 is rational and $\sqrt{5}$ is irrational....
Read More →Match the following columns:
Question: Match the following columns: (a) ......(b) ......(c) ......(d) ...... Solution: (a) $\left((81)^{-2}\right)^{\frac{1}{4}}$ $=\left((9)^{-4}\right)^{\frac{1}{4}}=(9)^{-4 \times \frac{1}{4}}=(9)^{-1}$ $=\frac{1}{9}$ $=\frac{1}{9}$ (b) $\left(\frac{a}{b}\right)^{x-2}=\left(\frac{b}{a}\right)^{x-4}$ $\Rightarrow\left(\frac{b}{a}\right)^{2-x}=\left(\frac{b}{a}\right)^{x-4}$ $\Rightarrow 2-x=x-4$ $\Rightarrow 2 x=6$ $\Rightarrow x=3$ (c) $x=9+4 \sqrt{5}$ and $\frac{1}{x}=\frac{1}{9+4 \sqrt{5...
Read More →Show that the relation R on the set
Question: Show that the relation $R$ on the set $A=\{x \in Z ; 0 \leq x \leq 12\}$, given by $R=\{(a, b): a=b\}$, is an equivalence relation. Find the set of all elements related to 1 . Solution: We observe the following properties ofR. Reflexivity: Letabe an arbitrary element ofA. Then, $a \in R$ $\Rightarrow a=a$ [Since, every element is equal to itself] $\Rightarrow(a, a) \in R$ for all $a \in A$ So, $R$ is reflexive on $A$. Symmetry: Let $(a, b) \in R$ $\Rightarrow a b$ $\Rightarrow b=a$ $\R...
Read More →In a retail market, fruit vendors were selling mangoes kept in packing boxes.
Question: In a retail market, fruit vendors were selling mangoes kept in packing boxes. These boxes contained varying number of mangoes. The following was the distribution of mangoes according to the number of boxes. Find the mean number of mangoes kept in a packing box. Which method of finding the mean did you choose? Solution: The given series is an inclusive series. Firstly, make it an exclusive series. Let the assumed mean beA= 57 andh= 3. We know that mean, $\bar{X}=A+h\left(\frac{1}{N} \su...
Read More →Let R be a relation on the set A of ordered pair of integers defined by
Question: LetRbe a relation on the setAof ordered pair of integers defined by (x,y)R(u,v) ifxv=yu. Show thatRis an equivalence relation. Solution: We observe the following properties ofR. Reflexivity: Let $(a, b)$ be an arbitrary element of the set A. Then, $(a, b) \in A$ $\Rightarrow a b=b a$ $\Rightarrow(a, b) R(a, b)$ Thus, $R$ is reflexive on $A$. Symmetry : Let $(x, y)$ and $(u, v) \in A$ such that $(x, y) R(u, v)$. Then, $x v=y u$ $\Rightarrow v x=u y$ $\Rightarrow u y=v x$ $\Rightarrow(u,...
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