If the mean of the following distribution is 27, find the value of p.
Question: If the mean of the following distribution is 27, find the value ofp. Solution: Given: Mean = 27 Let the assumed meanA= 25 andh= 10. We know that mean, $\bar{X}=A+h\left(\frac{1}{N} \sum f_{i} u_{i}\right)$ Now, we have $\sum f_{i}=43+p, \sum f_{i} u_{i}=17-p, h=10$ and $A=25$ Putting the values in the above formula, we have $27=25+10\left(\frac{1}{43+p} \times(17-p)\right)$ $\frac{2}{10}=\left(\frac{(17-p)}{43+p}\right)$ $43+p=85-5 p$ $6 p=42$ $p=7$ Thus, the value ofpis 7....
Read More →If the mean of the following distribution is 27, find the value of p.
Question: If the mean of the following distribution is 27, find the value ofp. Solution: Given: Mean = 27 Let the assumed meanA= 25 andh= 10. We know that mean, $\bar{X}=A+h\left(\frac{1}{N} \sum f_{i} u_{i}\right)$ Now, we have $\sum f_{i}=43+p, \sum f_{i} u_{i}=17-p, h=10$ and $A=25$ Putting the values in the above formula, we have $27=25+10\left(\frac{1}{43+p} \times(17-p)\right)$ $\frac{2}{10}=\left(\frac{(17-p)}{43+p}\right)$ $43+p=85-5 p$ $6 p=42$ $p=7$ Thus, the value ofpis 7....
Read More →Give an example of a statement P(n) which is true for all
Question: Give an example of a statementP(n) which is true for alln 4 butP(1),P(2) andP(3) are not true. Justify your answer. Solution: LetP(n) be the statement 3n n!. For n= 1,3n= 3 1 = 3n! = 1! = 1Now, 3 1 So,P(1) is not true. Forn= 2,3n= 3 2 = 6n! = 2! = 2Now, 6 2So,P(2) is not true.Forn= 3,3n= 3 3 = 9n! = 3! = 6Now, 9 6So,P(3) is not true. Forn= 4,3n= 3 4 = 12n! = 4! = 24Now, 12 24So,P(4) is true.Forn= 5,3n= 3 5 = 15n! = 5! = 120 Now, 15 120So,P(5) is true.Similarly, it can be verified that ...
Read More →The following distribution shows the daily pocket allowance given to the children of a multistorey building.
Question: The following distribution shows the daily pocket allowance given to the children of a multistorey building. The average pocket allowance is Rs 18.00. Find the missing frequency. Solution: Given: Mean = 18 Suppose the missing frequency isx.Let the assumed meanA= 18 andh= 2. We know that mean, $\bar{X}=A+h\left(\frac{1}{N} \sum f_{i} u_{i}\right)$ Now, we have $\sum f_{i}=44+x, \sum f_{i} u_{i}=x-20, h=2$ and $A=18$. Putting the values in the above formula, we have $18=18+2\left(\frac{x...
Read More →If P (n) is the statement
Question: IfP(n) is the statement "n2n+ 41 is prime", prove thatP(1),P(2) andP(3) are true. Prove also thatP(41) is not true. Solution: $P(n): n^{2}-n+41$ is prime. Now, $P(1)=1^{2}-1+41=41$ (prime) $P(2)=2^{2}-2+41=4-2+41=43$ (prime) $P(3)=3^{2}-3+41=9-3+41=47$ (prime) $P(41)=41^{2}-41+41=1681$ (not prime) Thus, we can say that $P(1), P(2)$ and $P(3)$ are true, but $P(41)$ is not true....
Read More →m is said to be related to n if m and n are integers and m − n is divisible by 13. Does this define an equivalence relation?
Question: mis said to be related tonifmandnare integers andmnis divisible by 13. Does this define an equivalence relation? Solution: We observe the following properties of relationR. Let $R=\{(m, n): m, n \in Z: m-n$ is divisible by 13$\}$ Relexivity : Let $m$ be an arbitrary element of $Z$. Then, $m \in R$ $\Rightarrow m-m=0=0 \times 13$ $\Rightarrow m-m$ is divisible by 13 $\Rightarrow(m, m)$ is reflexive on $Z$. Symmetry: Let $(m, n) \in R$. Then, $m-n$ is divisible by 13 $\Rightarrow m-n=13 ...
Read More →Given an example of a statement P (n) such that it is true for all
Question: Given an example of a statementP(n) such that it is true for allnN. Solution: Proved: $P(n)=n^{2}+n$ is even for $P(n)$ and $P(n+1)$. Therefore, $\frac{n^{2}+n}{2}$ is also even for all $n \in N$. [Dividing an even number by 2 gives an even number.] Thus, we have : $P(n)=1+2+\ldots+n$ $=\frac{n(n+1)}{2} \quad($ Even for all $n \in \mathrm{N})$...
Read More →The mean of the following frequency distribution is 62.8 and the sum of all the frequencies is 50.
Question: The mean of the following frequency distribution is $62.8$ and the sum of all the frequencies is 50 . Compute the missing frequency $f_{1}$ and $f_{2}$. Solution: It is given that mean $=62.8$ and $N=50$. Let the assumed meanA= 50 andh= 20. $\sum f_{i}=50$ $30+f_{1}+f_{2}=50$ $f_{1}=20-f_{2}$......(1) We know that mean, $\bar{X}=A+h\left(\frac{1}{N} \sum f_{i} u_{i}\right)$ Now, we have $\sum f_{i}=30+f_{1}+f_{2}, \sum f_{t} u_{t}=28-f_{1}+f_{2}, h=20$ and $A=50$. Putting the values in...
Read More →If P (n) is the statement
Question: IfP(n) is the statement "n2+nis even", and ifP(r) is true, thenP(r+ 1) is true. Solution: $P(n): n^{2}+n$ is even. Also, $P(r)$ is true. Thus, $r^{2}+r$ is even. To prove: $P(r+1)$ is true. Now, $P(r+1)=(r+1)^{2}+r+1$ $=r^{2}+1+2 r+r+1$ $=r^{2}+3 r+2$ $=r^{2}+r+2 r+2$ $=P(r)+2(r+1)$ $P(r)$ is even. Also, $2(r+1)$ is even, $a s$ it is a multiple of 2 . Therefore, $P(r+1)$ is even and true....
Read More →If P (n) is the statement
Question: IfP(n) is the statement "2n 3n" and ifP(r)is true, prove thatP(r+ 1) is true. Solution: We have: $P(n): 2^{n} \geq 3 n$ Also, $P(r)$ is true. $\therefore 2^{r} \geq 3 r$ To Prove: $P(r+1)$ is true. We have: $2^{r} \geq 3 r$ $\Rightarrow 2^{r} \times 2 \geq 3 r \times 2 \quad$ [Multiplying both sides by 2] $\Rightarrow 2^{r+1} \geq 6 r$ $\therefore 2^{r+1} \geq 3 r+3 \quad[6 r \geq 3 r+3$ for every $r \in N .]$ Hence, $P(r+1)$ is true....
Read More →If P (n) is the statement
Question: IfP(n) is the statement "2n 3n" and ifP(r)is true, prove thatP(r+ 1) is true. Solution: We have: $P(n): 2^{n} \geq 3 n$ Also, $P(r)$ is true. $\therefore 2^{r} \geq 3 r$ To Prove: $P(r+1)$ is true. We have: $2^{r} \geq 3 r$ $\Rightarrow 2^{r} \times 2 \geq 3 r \times 2 \quad$ [Multiplying both sides by 2] $\Rightarrow 2^{r+1} \geq 6 r$ $\therefore 2^{r+1} \geq 3 r+3 \quad[6 r \geq 3 r+3$ for every $r \in N .]$ Hence, $P(r+1)$ is true....
Read More →The following table shows the marks scored by 140 students in an examination of a certain paper.
Question: The following table shows the marks scored by 140 students in an examination of a certain paper. Calculate the average marks by using all the three methods: direct method, assumed mean deviation and shortcut method. Solution: We may prepare the table as shown: (i)Direct method We know that mean, $\bar{X}=\frac{\sum f_{i} x_{i}}{\sum f_{i}}$ $=\frac{3620}{140}$ $=25.857$ Hence, the mean is 25.857. (ii)Short-cut methodLet the assumed meanA= 25. We know that mean, $\bar{X}=A+\left(\frac{1...
Read More →If P (n) is the statement "n3 + n is divisible by 3",
Question: IfP(n) is the statement "n3 +nis divisible by 3", prove thatP(3) is true butP(4) is not true. Solution: We have: $P(n): n^{3}+n$ is divisible by 3 . Thus, we have : $P(3)=3^{3}+3=27+3=30$ It is divisible by $3 .$ Hence, $P(3)$ is true. Now, $P(4)=4^{3}+4=64+4=68$ It is not divisible by 3 . Hence, $P(4)$ is not true....
Read More →If P (n) is the statement "n3 + n is divisible by 3",
Question: IfP(n) is the statement "n3 +nis divisible by 3", prove thatP(3) is true butP(4) is not true. Solution: We have: $P(n): n^{3}+n$ is divisible by 3 . Thus, we have : $P(3)=3^{3}+3=27+3=30$ It is divisible by $3 .$ Hence, $P(3)$ is true. Now, $P(4)=4^{3}+4=64+4=68$ It is not divisible by 3 . Hence, $P(4)$ is not true....
Read More →Let Z be the set of integers. Show that the relation
Question: LetZbe the set of integers. Show that the relationR= {(a,b) :a,bZanda+bis even}is an equivalence relation onZ. Solution: We observe the following properties ofR. Reflexivity: Let $a$ be an arbitrary element of $Z$. Then, $a \in R$ Clearly, $a+a=2 a$ is even for all $a \in Z$. $\Rightarrow(a, a) \in R$ for all $a \in Z$ So, $R$ is reflexive on $Z$. Symmetry: Let $(a, b) \in R$ $\Rightarrow a+b$ is even $\Rightarrow b+a$ is even $\Rightarrow(b, a) \in R$ for all $a, b \in Z$ So, $R$ is s...
Read More →If P (n) is the statement "n(n + 1) is even",
Question: IfP(n) is the statement "n(n+ 1) is even", then what isP(3)? Solution: We have: P(n):n(n+ 1) is even. Now, P(3) = 3(3 + 1) = 12 (Even) Therefore,P(3) is even....
Read More →If P(n):
Question: If $\mathrm{P}(n): 2 \times 4^{2 n+1}+3^{3 n+1}$ is divisible by $\lambda$ for all $n \in \mathbf{N}$ is true, then find the value of $\lambda$. Solution: For $n=1$, $\mathrm{P}(1)=2 \times 4^{2+1}+3^{3+1}=2 \times 4^{3}+3^{4}=128+81=209$ For $n=2$, $\mathrm{P}(2)=2 \times 4^{4+1}+3^{6+1}=2 \times 4^{5}+3^{7}=2048+2187=4235$ As, $\mathrm{HCF}(209,4235)=11$ So, $2 \times 4^{2 n+1}+3^{3 n+1}$ is divisible by 11 . Hence, the value of $\lambda$ is 11 ....
Read More →The weekly observations on cost of living index in a certain
Question: The weekly observations on cost of living index in a certain city for the year 2004 2005 are given below. Compute the weekly cost of living index. Solution: Let the assumed mean beA= 1650 andh= 100. We know that mean, $\bar{X}=A+h\left(\frac{1}{N} \sum f_{i} u_{i}\right)$ Now, we have $N=\sum f_{i}=52, \sum f_{\mu} u_{i}=7, h=100$ and $A=1650$. Putting the values in the above formula, we have $\bar{X}=A+h\left(\frac{1}{N} \sum f_{i} u_{i}\right)$ $=1650+100\left(\frac{1}{52} \times(7)\...
Read More →State the second principle of mathematical induction.
Question: State the second principle of mathematical induction. Solution: Second principle of mathematical induction: Let P(n) be a given statement involving the natural numbern such that (i) The statement is true for n= 1, i.e., P(1) is true (or true for any fixed naturalnumber). This step is known as theBasis step. (ii) If the statement (called Induction hypothesis) is true for 1nk(wherekis a particular but arbitrary naturalnumber), then the statement is also true forn=k + 1, i.e, truth of P(k...
Read More →For the following distribution, calculate mean using all suitable methods:
Question: For the following distribution, calculate mean using all suitable methods: Solution: Direct Method:We may prepare the table as shown: We know that mean, $\bar{X}=\frac{\sum f_{i} x_{i}}{\sum f_{i}}$ $=\frac{848}{64}$ $=13.25$ Hence, the mean is 13.25.Short-Cut Method:We may prepare the table as shown: Let the assumed mean beA= 12.5. We know that mean, $\bar{X}=A+\frac{\sum f_{1} d_{1}}{\sum f_{1}}$ $=12.5+\frac{48}{64}$ $=12.5+0.75$ $=13.25$ Hence, the mean is 13.25. Step-deviation met...
Read More →Write the set of value of n for which the statement P(n):
Question: Write the set of value ofnfor which the statement P(n): 2nn! is true. Solution: As,n! 2nwhenn 3. So, the set of value ofnfor which the statement P(n): 2nn! is true = {nN:n 3}....
Read More →Let n be a fixed positive integer. Define a relation R on Z as follows:
Question: Let $n$ be a fixed positive integer. Define a relation $R$ on $Z$ as follows: $(a, b) \in R \Leftrightarrow a-b$ is divisible by $n$. Show thatRis an equivalence relation onZ. Solution: We observe the following properties ofR.Then,Reflexivity: Let $a \in N$ Here, $a-a=0=0 \times n$ $\Rightarrow a-a$ is divisible by $n$ $\Rightarrow(a, a) \in R$ $\Rightarrow(a, a) \in R$ for all $a \in Z$ So, $R$ is reflexive on $Z$. Symmetry Let $(a, b) \in R$ Here, $a-b$ is divisible by $n$ $\Rightarr...
Read More →Let n be a fixed positive integer. Define a relation R on Z as follows:
Question: Let $n$ be a fixed positive integer. Define a relation $R$ on $Z$ as follows: $(a, b) \in R \Leftrightarrow a-b$ is divisible by $n$. Show thatRis an equivalence relation onZ. Solution: We observe the following properties ofR.Then,Reflexivity: Let $a \in N$ Here, $a-a=0=0 \times n$ $\Rightarrow a-a$ is divisible by $n$ $\Rightarrow(a, a) \in R$ $\Rightarrow(a, a) \in R$ for all $a \in Z$ So, $R$ is reflexive on $Z$. Symmetry Let $(a, b) \in R$ Here, $a-b$ is divisible by $n$ $\Rightarr...
Read More →State the first principle of mathematical induction.
Question: State the first principle of mathematical induction. Solution: Let P(n) be a given statement involving the natural numbern such that (i) The statement is true for n= 1, i.e., P(1) is true (or true for any fixed naturalnumber). This step is known as theBasis step. (ii) If the statement (called Induction hypothesis) is true forn=k(wherek is a particular but arbitrary naturalnumber), then the statement is also true forn=k + 1, i.e, truth of P(k) impliesthe truth of P(k+ 1). This step is k...
Read More →Find the mean of each of the following frequency distributions :
Question: Find the mean of each of the following frequency distributions : Solution: The given series is an inclusive series. Firstly, make it an exclusive series. Let the assumed mean beA= 42 andh= 5. We know that mean, $\bar{X}=A+h\left(\frac{1}{N} \sum f_{i} u_{i}\right)$ Now, we have $N=\sum f_{i}=70, \sum f_{i} u_{i}=-79, h=5$ and $A=42$. Putting the values in the above formula, we have $\bar{X}=A+h\left(\frac{1}{N} \sum f_{i} u_{i}\right)$ $=42+5\left(\frac{1}{70} \times(-79)\right)$ $=42-...
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