Let

Question: LetA= {a,b,c} and the relationRbe defined onAas follows:R= {(a,a), (b,c), (a,b)}. Then, write minimum number of ordered pairs to be added inRto make it reflexive and transitive.[NCERT EXEMPLAR] Solution: We have, A= {a,b,c} andR= {(a,a), (b,c), (a,b)}Rcan be a reflexive relation only when elements (b,b) and (c,c) are added to itRcan be a transitive relation only when the element (a,c) isadded to itSo, the minmum number of ordered pairs to be added inRis 3....

Solve the following

Question: A sequence $x_{1}, x_{2}, x_{3}, \ldots$ is defined by letting $x_{1}=2$ and $x_{k}=\frac{x_{k-1}}{k}$ for all natural numbers $k, k \geq 2 .$ Show that $x_{n}=\frac{2}{n !}$ for all $n \in \mathbf{N}$. Solution: Given : A sequence $x_{1}, x_{2}, x_{3}, \ldots$ is defined by letting $x_{1}=2$ and $x_{k}=\frac{x_{k-1}}{k}$ for all natural numbers $k, k \geq 2$. Let $\mathrm{P}(n): x_{n}=\frac{2}{n !}$ for all $n \in \mathbf{N}$. Step I: For $n=1$, $\mathrm{P}(1): x_{1}=\frac{2}{1 !}=2$ ...

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Question: If $\sqrt{7}=2.646$ then $\frac{1}{\sqrt{7}}=?$ (a) $0.375$ (b) $0.378$ (c) $0.441$ (d) None of these Solution: $\frac{1}{\sqrt{7}}=\frac{1}{\sqrt{7}} \times \frac{\sqrt{7}}{\sqrt{7}}=\frac{\sqrt{7}}{7}$ Given that $\sqrt{7}=2.646$ So, $\frac{\sqrt{7}}{7}=\frac{2.646}{7}=0.378$ Hence, the correct answer is option (b)....

Question: LetA= {a,b,c} and the relationRbe defined onAas follows:R= {(a,a), (b,c), (a,b)}. Then, write minimum number of ordered pairs to be added inR to make it reflexive and transitive. [NCERT EXEMPLAR] Solution: We have, A= {a,b,c} andR= {(a,a), (b,c), (a,b)}Rcan be a reflexive relation only when elements (b,b) and (c,c) are added to itRcan be a transitive relation only when the element (a,c) isadded to itSo, the minmum number of ordered pairs to be added inRis 3....

The number of students absent in a class were recorded every day

Question: The number of students absent in a class were recorded every day for 120 days and the information is given in the following frequency table: Find the mean number of students absent per day. Solution: Let the assume mean be $A=4$. We know that mean, $\bar{X}=A+\frac{1}{N} \sum_{i=1}^{n} f_{i} d_{i}$ Now, we have $N=\sum f_{i}=120, \sum f_{i} d_{i}=-57$ and $A=4$. Putting the values in the above formula, $\bar{X}=A+\frac{1}{N} \sum_{i=1}^{n} f_{i} d_{i}$ $=4+\frac{1}{120} \times(-57)$ $=...

Let

Question: LetA= {1, 2, 3} andR= {(1, 2), (1, 1), (2, 3)} be a relation onA. What minimum number of ordered pairs may be added toRso that it may become a transitive relation onA. Solution: We have, A= {1, 2, 3} andR= {(1, 2), (1, 1), (2, 3)}To makeRa transitive relation onA, (1, 3) must be added to it.So, the minimum number of ordered pairs that may be added to R to make it a transitive relation is 1....

Let

Question: LetA= {1, 2, 3} andR= {(1, 2), (1, 1), (2, 3)} be a relation onA. What minimum number of ordered pairs may be added toRso that it may become a transitive relation onA. Solution: We have, A= {1, 2, 3} andR= {(1, 2), (1, 1), (2, 3)}To makeRa transitive relation onA, (1, 3) must be added to it.So, the minimum number of ordered pairs that may be added to R to make it a transitive relation is 1....

Solve the following

Question: A sequence $a_{1}, a_{2}, a_{3}, \ldots$ is defined by letting $a_{1}=3$ and $a_{k}=7 a_{k-1}$ for all natural numbers $k \geq 2$. Show that $a_{n}=3 \cdot 7^{n-1}$ for all $n \in \mathbf{N}$. Solution: Let $\mathrm{P}(n): a_{n}=3 \cdot 7^{n-1}$ for all $n \in \mathbf{N}$. Step I: For $n=1$, $P(1)$ : $a_{1}=3 \cdot 7^{1-1}=3 \cdot 1=3$ So, it is true for $n=1$. Step II : For $n=k$, Let $\mathrm{P}(k): a_{k}=3 \cdot 7^{k-1}$ be true for some $k \in \mathbf{N}$ and $k \geq 2$. Step III :...

solve this

Question: If $\sqrt{2}=1.41$ then $\frac{1}{\sqrt{2}}=?$ (a) $0.075$ (b) $0.75$ (c) $0.705$ (d) $7.05$ Solution: $\frac{1}{\sqrt{2}}=\frac{1}{\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}}=\frac{\sqrt{2}}{2}$ Given : $\sqrt{2}=1.41$ So, $\frac{\sqrt{2}}{2}=\frac{1.41}{2}=0.705$ Hence, the correct answer is option (c)....

Given the relation

Question: Given the relationR= {(1, 2), (2, 3)} on the setA= {1, 2, 3}, add a minimum number of ordered pairs so that the enlarged relation is symmeteric, transitive and reflexive. Solution: We have, R= {(1, 2), (2, 3)}Rcan be a transitive only when the elements (1, 3) is addedRcan be a reflexive only when the elements (1, 1), (2, 2), (3, 3) are addedRcan be a symmetric only when the elements (2, 1), (3, 1) and (3, 2) are addedSo, the required enlarged relation,R' ={(1, 1), (1, 2), (1, 3), (2, 1...

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Question: If $x=(7+4 \sqrt{3})$ then $\left(x+\frac{1}{x}\right)=?$ (a) $8 \sqrt{3}$ (b) 14 (c) 49 (d) 48 Solution: Given: $x=(7+4 \sqrt{3})$ $\frac{1}{x}=\frac{1}{7+4 \sqrt{3}}=\frac{1}{7+4 \sqrt{3}} \times \frac{7-4 \sqrt{3}}{7-4 \sqrt{3}}=\frac{7-4 \sqrt{3}}{49-48}=7-4 \sqrt{3}$ $\left(x+\frac{1}{x}\right)=7+4 \sqrt{3}+(7-4 \sqrt{3})=14$ Hence, the correct answer is option (b)....

Prove that the number of subsets of a set containing n distinct elements is 2

Question: Prove that the number of subsets of a set containing $n$ distinct elements is $2^{n}$, for all $n \in \mathbf{N}$.[NCERT EXEMPLAR] Solution: Let the given statement be defined as $\mathrm{P}(n)$ : The number of subsets of a set containing $n$ distinct elements $=2^{n}$, for all $n \in \mathbf{N}$. Step I: For $n=1$, LHS $=$ As, the subsets of a set containing only 1 element are : $\phi$ and the set itself. i. e. the number of subsets of a set containing only 1 element $=2$ RHS $=2^{1}=...

Give an example of a relation which is

Question: Give an example of a relation which is(i) reflexive and symmetric but not transitive;(ii) reflexive and transitive but not symmetric;(iii) symmetric and transitive but not reflexive;(iv) symmetric but neither reflexive nor transitive.(v) transitive but neither reflexive nor symmetric. Solution: SupposeAbe the set such thatA= {1, 2, 3} (i) LetRbe the relation onAsuch thatR= {(1, 1), (2, 2), (3, 3), (1, 2), (2, 1), (2, 3)}Thus,Ris reflexive and symmetric, but not transitive. (ii) LetRbe ...

Prove that the number of subsets of a set containing n distinct elements is 2

Question: Prove that the number of subsets of a set containing $n$ distinct elements is $2^{n}$, for all $n \in \mathbf{N}$.[NCERT EXEMPLAR] Solution: Let the given statement be defined as $\mathrm{P}(n)$ : The number of subsets of a set containing $n$ distinct elements $=2^{n}$, for all $n \in \mathbf{N}$. Step I: For $n=1$, LHS $=$ As, the subsets of a set containing only 1 element are : $\phi$ and the set itself. i. e. the number of subsets of a set containing only 1 element $=2$ RHS $=2^{1}=...

Prove that the number of subsets of a set containing n distinct elements is 2

Question: Prove that the number of subsets of a set containing $n$ distinct elements is $2^{n}$, for all $n \in \mathbf{N}$.[NCERT EXEMPLAR] Solution: Let the given statement be defined as $\mathrm{P}(n)$ : The number of subsets of a set containing $n$ distinct elements $=2^{n}$, for all $n \in \mathbf{N}$. Step I: For $n=1$, LHS $=$ As, the subsets of a set containing only 1 element are : $\phi$ and the set itself. i. e. the number of subsets of a set containing only 1 element $=2$ RHS $=2^{1}=...

The marks obtained out of 50, by 102 students in a Physics test are given in the frequency table below:

Question: The marks obtained out of 50, by 102 students in a Physics test are given in the frequency table below: Find the average number of marks. Solution: Let the assume mean be $A=25$. We know that mean, $\bar{X}=A+\frac{1}{N} \sum_{i=1}^{n} f_{i} d_{i}$ Now, we have $N=\sum f_{i}=102, \sum f_{i} d_{i}=110$ and $A=25$. Putting the values in the above formula, we get $\bar{X}=A+\frac{1}{N} \sum_{i=1}^{n} f_{i} d_{i}$ $=25+\frac{1}{102} \times(110)$ $=25+\frac{110}{102}$ $=25+1.078$ $=26.078$ ...

Solve this

Question: $\frac{1}{(3+2 \sqrt{2})}=?$ (a) $\frac{3-2 \sqrt{2}}{17}$ (b) $\frac{(3-2 \sqrt{2})}{13}$ (c) $(3-2 \sqrt{2})$ (d) None of these Solution: $\frac{1}{(3+2 \sqrt{2})}=\frac{1}{(3+2 \sqrt{2})} \times \frac{(3-2 \sqrt{2})}{(3-2 \sqrt{2})}=\frac{(3-2 \sqrt{2})}{9-8}=(3-2 \sqrt{2})$ Hence, the correct answer is option (c)....

Show that the relation

Question: Show that the relation '' on the setRof all real numbers is reflexive and transitive but not symmetric. Solution: Let $R$ be the set such that $R=\{(a, b): a, b \in R ; a \geq b\}$ Reflexivity: Let $a$ be an arbitrary element of $R$. $\Rightarrow a \in R$ $\Rightarrow a=a$ $\Rightarrow a \geq a$ is true for $a=a$ $\Rightarrow(a, a) \in R$ Hence, $R$ is reflexive. Symmetry: Let $(a, b) \in R$ $\Rightarrow a \geq b$ is same as $b \leq a$, but not $b \geq a$ Thus, $(b, a) \notin R$ Hence,...

The following table gives the number of children of 150 families in a village

Question: The following table gives the number of children of 150 families in a village Find the average number of children per family Solution: Let the assume mean be $A=3$. We know that mean, $\bar{X}=A+\frac{1}{N} \sum_{i=1}^{n} f_{i} d_{i}$ Now, we have $N=\sum f_{i}=150, \sum f_{i} d_{i}=-98$ and $A=3$ Putting the values above in formula, we get $\bar{X}=A+\frac{1}{N} \sum_{i=1}^{n} f_{i} d_{i}$ $=3+\frac{1}{150} \times(-98)$ $=3-0.653$ $=2.347$ $\approx 2.35$ ( approximate) Hence, the aver...

solve this

Question: If $x=2+\sqrt{3}$ then $\left(x+\frac{1}{x}\right)$ equals (a) $-2 \sqrt{3}$ (b) 2 (c) 4 (d) $4-2 \sqrt{3}$ Solution: $x=2+\sqrt{3}$ $\frac{1}{x}=\frac{1}{2+\sqrt{3}}=\frac{1}{2+\sqrt{3}} \times \frac{2-\sqrt{3}}{2-\sqrt{3}}=\frac{2-\sqrt{3}}{4-3}=2-\sqrt{3}$ $x+\frac{1}{x}=2+\sqrt{3}+2-\sqrt{3}=4$ Hence, the correct answer is option (c)....

The following table given the number of branches and number of plants in the garden of a school.

Question: The following table given the number of branches and number of plants in the garden of a school. Solution: Let the assume mean be $A=4$. We know that mean, $\bar{X}=A+\frac{1}{N} \sum_{i=1}^{n} f_{i} d_{i}$ Now, we have $N=\sum f_{i}=200, \sum f_{i} d_{i}=-77$ and $A=4$ Putting the values in the above formula, we get $\bar{X}=A+\frac{1}{N} \sum_{i=1}^{n} f_{i} d_{i}$ $=4+\frac{1}{200} \times(-77)$ $=4-0.385$ $=3.615$ $\approx 3.62$ ( approximate) Hence, the mean number of branches per ...

An integer m is said to be related to another integer n if m is a multiple of n.

Question: An integer m is said to be related to another integernif m is a multiple ofn.Check if the relation is symmetric, reflexive and transitive. Solution: $R=\{(m, n): m, n \in Z, m=k n$, where $k \in N\}$ Reflexivity : Let $m$ be an arbitrary element of $R$. Then, $m=k m$ is true for $k=1$ $\Rightarrow(m, m) \in R$ Thus, $R$ is reflexive. Symmetry : Let $(m, n) \in R$ $\Rightarrow m=k n$ for some $k \in N$ $\rightarrow n=\frac{1}{k} m$ $\Rightarrow(n, m) \notin R$ Thus, $R$ is not symmetric...

Five coins were simultaneously tossed 1000 times,

Question: Five coins were simultaneously tossed 1000 times, and at each toss the number of heads was observed. The number of tosses during which 0, 1, 2, 3, 4 and 5 heads were obtained are shown in the table below: Find the mean number of heads per toss Solution: Let the assume mean be $A=2$. We know that mean $\bar{X}=A+\frac{1}{N} \sum_{i=1}^{n} f_{i} d_{i}$ Now, we have $N=\sum f_{i}=1000, \sum f_{i} d_{i}=470$ and $A=2$ Putting the values above in formula, we have $\bar{X}=A+\frac{1}{N} \sum...

Show by the Principle of Mathematical induction that the sum

Question: Show by the Principle of Mathematical induction that the sum $S_{n}$ of then terms of the series $1^{2}+2 \times 2^{2}+3^{2}+2 \times 4^{2}+5^{2}+2 \times 6^{2}+7^{2}+\ldots$ is given by $S_{n}= \begin{cases}\frac{n(n+1)^{2}}{2}, \text { if } n \text { is even } \\ \frac{n^{2}(n+1)}{2}, \text { if } n \text { is odd }\end{cases}$ [NCERT EXEMPLAR] Solution: Let $\mathrm{P}(n): S_{n}=1^{2}+2 \times 2^{2}+3^{2}+2 \times 4^{2}+5^{2}+\ldots=\left\{\begin{array}{l}\frac{n(n+1)^{2}}{2}, \text...

Rationalisation of the denominator of

Question: Rationalisation of the denominator of $\frac{1}{\sqrt{5}+\sqrt{2}}$ gives (a) $\frac{1}{\sqrt{10}}$ (b) $\sqrt{5}+\sqrt{2}$ (c) $\sqrt{5}-\sqrt{2}$ (d) $\frac{\sqrt{5}-\sqrt{2}}{3}$ Solution: $\frac{1}{\sqrt{5}+\sqrt{2}}$ Rationalisation of denominator gives $\frac{1}{\sqrt{5}+\sqrt{2}} \times \frac{\sqrt{5}-\sqrt{2}}{\sqrt{5}-\sqrt{2}}=\frac{\sqrt{5}-\sqrt{2}}{5-2}=\frac{\sqrt{5}-\sqrt{2}}{3}$ Hence, the correct answer is option (d)....