Make the correct alternative in the following question:
Question: Make the correct alternative in the following question: If $P(n): 49^{n}+16^{n}+\lambda$ is divisible by 64 for $n \in N$ is true, then the least negative integral value of $\lambda$ is (a) $-3$ (b) $-2$ (c) $-1$ (d) $-4$ Solution: We have, $\mathrm{P}(n): 49^{n}+16^{n}+\lambda$ is divisible by 64 for all $n \in \mathbf{N}$. For $n=1$, $\mathrm{P}(1)=49^{1}+16^{1}+\lambda=65+\lambda$ As, the nearest value of $\mathrm{P}(1)$ which is divisible by 64 is 64 itself. $\Rightarrow 65+\lambda...
Read More →Make the correct alternative in the following question:
Question: Make the correct alternative in the following question: If $P(n): 49^{n}+16^{n}+\lambda$ is divisible by 64 for $n \in N$ is true, then the least negative integral value of $\lambda$ is (a) $-3$ (b) $-2$ (c) $-1$ (d) $-4$ Solution: We have, $\mathrm{P}(n): 49^{n}+16^{n}+\lambda$ is divisible by 64 for all $n \in \mathbf{N}$. For $n=1$, $\mathrm{P}(1)=49^{1}+16^{1}+\lambda=65+\lambda$ As, the nearest value of $\mathrm{P}(1)$ which is divisible by 64 is 64 itself. $\Rightarrow 65+\lambda...
Read More →Show that the relation R defined by
Question: Show that the relation $R$ defined by $R=\{(a, b): a-b$ is divisible by $3 ; a, b \in Z\}$ is an equivalence relation. Solution: We observe the following relations of relationR. Reflexivity: Let $a$ be an arbitrary element of $R$. Then, $a-a=0=0 \times 3$ $\Rightarrow a-a$ is divisible by 3 $\Rightarrow(a, a) \in R$ for all $a \in Z$ So, $R$ is reflexive on $Z$. Symmetry: Let $(a, b) \in R$ $\Rightarrow a-b$ is divisible by 3 $\Rightarrow a-b 3 p$ for some $p \in Z$ $\Rightarrow b-a=3(...
Read More →Assertion:
Question: Assertion: $\sqrt{3}$ is an irrational number. Reason:Square root a positive integer which is not a perfect square is an irrational number.(a) Both Assertion and Reason are true and Reasom is a correct explanation of Assertion.(b) Both Assertion and Reason and Reasom are true but Reasom is not a correct explanation of Assertion.(c) Assertion is true and Reasom is false.(d) Assertion is false and Reasom is true. Solution: (a) Both Assertion and Reason are true, and Reason is the correct...
Read More →Thirty women were examined in a hospital by a doctor and the number
Question: Thirty women were examined in a hospital by a doctor and the number of heart beats per minute recorded and summarised as follows. Find the mean heat beats per minute for these women, choosing a suitable method. Solution: Let the assumed mean beA= 75.5 andh= 3. We know that mean, $\bar{X}=A+h\left(\frac{1}{N} \sum f_{i} u_{i}\right)$ Now, we have $N=\sum f_{i}=30, \sum f_{i} u_{i}=4, h=3$ and $A=75.5$. Putting the values in the above formula, we have $\bar{X}=A+h\left(\frac{1}{N} \sum f...
Read More →Make the correct alternative in the following question:
Question: Make the correct alternative in the following question: A student was asked to prove a statement P(n) by induction. He proved P(k+1) is truewhenever P(k) is true for allk 5Nand also P(5) is true. On the basis of this he couldconclude that P(n) is true. (a) for all $n \in \mathbf{N}$ (b) for alln 5 (c) for alln5 (d) for alln 5 Solution: As, P(5) is true and $\mathrm{P}(k+1)$ is true whenever $\mathrm{P}(k)$ is true for all $k5 \in \mathbf{N}$. By the definition of the priniciple of math...
Read More →Assertion: Three rational numbers between
Question: Assertion: Three rational numbers between $\frac{2}{5}$ and $\frac{3}{5}$ are $\frac{9}{20}, \frac{10}{20}$ and $\frac{11}{20}$. Reason: A rational number between two rational numbers $p$ and $q$ is $\frac{1}{2}(p+q)$. (a) Both Assertion and Reason are true and Reasom is a correct explanation of Assertion.(b) Both Assertion and Reason and Reasom are true but Reasom is not a correct explanation of Assertion.(c) Assertion is true and Reasom is false.(d) Assertion is false and Reasom is t...
Read More →Make the correct alternative in the following question:
Question: Make the correct alternative in the following question: Let P(n): 2n (123...n). Then the smallest positive integer for which P(n) is true (a) 1 (b) 2 (c) 3 (d) 4 Solution: As,2n (123...n) is possible only whenn 4 So, the smallest positive integer for which P(n) is true, is 4. Hence, the correct alternative is option (d)....
Read More →Make the correct alternative in the following question:
Question: Make the correct alternative in the following question: If $10^{n}+3 \times 4^{n+2}+\lambda$ is divisible by 9 for all $n \in \mathbf{N}$, then the least positive integral value of $\lambda$ is (a) 5 (b) 3 (c) 7 (d) 1 Solution: Let $\mathrm{P}(n): 10^{n}+3 \times 4^{n+2}+\lambda$ be divisible by 9 for all $n \in \mathbf{N}$. For $n=1$, $\mathrm{P}(1)=10^{1}+3 \times 4^{1+2}+\lambda$ $=10+3 \times 4^{3}+\lambda$ $=10+192+\lambda$ $=202+\lambda$ As, the least value of $P(1)$ which is div...
Read More →Make the correct alternative in the following question:
Question: Make the correct alternative in the following question: For all $n \in \mathbf{N}, 3 \times 5^{2 n+1}+2^{3 n+1}$ is divisible by (a) 19 (b) 17 (c) 23 (d) 25 Solution: Let $\mathrm{P}(n)=3 \times 5^{2 n+1}+2^{3 n+1}$, for all $n \in \mathbf{N}$. For $n=1$, $\mathrm{P}(1)=3 \times 5^{2+1}+2^{3+1}$ $=3 \times 5^{3}+2^{4}$ $=375+16$ $=391$ $=17 \times 23$ For $n=2$, $\mathrm{P}(2)=3 \times 5^{4+1}+2^{6+1}$ $=3 \times 5^{5}+2^{7}$ $=9375+128$ $=9503$ $=17 \times 13 \times 43$ As, $\operator...
Read More →Solve this
Question: If $x=3+\sqrt{8}$ then $\left(x^{2}+\frac{1}{x^{2}}\right)=?$ (a) 34 (b) 56 (c) 28 (d) 63 Solution: Given: $x=3+\sqrt{8}$ $\frac{1}{x}=\frac{1}{3+\sqrt{8}}=\frac{1}{3+\sqrt{8}} \times \frac{3-\sqrt{8}}{3-\sqrt{8}}=\frac{3-\sqrt{8}}{9-8}=\frac{3-\sqrt{8}}{1}=3-\sqrt{8}$ $x+\frac{1}{x}=(3+\sqrt{8})+(3-\sqrt{8})=6$ $\left(x+\frac{1}{x}\right)^{2}=x^{2}+\frac{1}{x^{2}}+2 \times x \times \frac{1}{x}=x^{2}+\frac{1}{x^{2}}+2$ $\Rightarrow 6^{2}=x^{2}+\frac{1}{x^{2}}+2$ $\Rightarrow 36=x^{2}+\...
Read More →Consider the following distribution of daily wages of 50 workers of a factory.
Question: Consider the following distribution of daily wages of 50 workers of a factory. Find the mean daily wages of the workers of the factory by using an appropriate method. Solution: Let the assumed mean beA= 120 andh= 20. We know that mean, $\bar{X}=A+h\left(\frac{1}{N} \sum f_{i} u_{i}\right)$ Now, we have $N=\sum f_{i}=50, \sum f_{i} u_{i}=-12, h=20$ and $A=150$. Putting the values in the above formula, we have $\bar{X}=A+h\left(\frac{1}{N} \sum f_{i} u_{i}\right)$ $=150+20\left(\frac{1}{...
Read More →Make the correct alternative in following question:
Question: Make the correct alternative in following question: If $x^{n}-1$ is divisible by $x-\lambda$, then the least positive integral value of $\lambda$ is (a) 1 (b) 2 (c) 3 (d) 4 Solution: Let $\mathrm{P}(n): x^{n}-1$ is divisible by $(x-\lambda)$. As, for $n=1$, $\mathrm{P}(1)=x^{1}-1=x-1$ As, $\mathrm{P}(1)$ must be divisible by $(x-\lambda)$. $\Rightarrow(x-1)$ must be divisible by $(x-\lambda)$. So, $\lambda=1$ Hence, the correct alternative is option (a)....
Read More →Each of the following defines a relation on N:
Question: Each of the following defines a relation onN: (i) $xy, x, y \in \mathbf{N}$ (ii) $x+y=10, x, y \in \mathbf{N}$ (iii) $x y$ is square of an integer, $x, y \in \mathbf{N}$ (iv) $x+4 y=10, x, y \in \mathbf{N}$ Determine which of the above relations are reflexive, symmetric and transitive. [NCERT EXEMPLAR] Solution: (i) We have, $R=\{(x, y): xy, x, y \in \mathbf{N}\}$ As, $x=x \forall x \in \mathbf{N}$ $\Rightarrow(x, x) \notin R$ So, $R$ is not a reflexive relation' Let $(x, y) \in R$ $\R...
Read More →A survey was conducted by a group of students as a part
Question: A survey was conducted by a group of students as a part of their environment awareness programme, in which they collected the following data regarding the number of a plants in 20 houses in a locality. Find the mean number of plants per house. Which method did you use for finding the mean, and why? Solution: We may prepare the table as shown: We know that mean, $\bar{X}=\frac{\sum f_{i} x_{i}}{\sum f_{i}}$ $=\frac{162}{20}$ $=8.1$ Hence, mean = 8.1 Direct method is easier than other me...
Read More →The distributive law from algebra states that for all real numbers
Question: The distributive law from algebra states that for all real numbers $c, a_{1}$ and $a_{2}$, we have $c\left(a_{1}+a_{2}\right)=c a_{1}+c a_{2}$. Use this law and mathematical induction to prove that, for all natural numbers, $n \geq 2$, if $c, a_{1}, a_{2}, \ldots, a_{n}$ are any real numbers, then $c\left(a_{1}+a_{2}+\ldots+a_{n}\right)=c a_{1}+c a_{2}+\ldots+c a_{n}$ Solution: Given: For all real numbers $c, a_{1}$ and $a_{2}, c\left(a_{1}+a_{2}\right)=c a_{1}+c a_{2}$. To prove: For ...
Read More →Find the value
Question: If $\sqrt{2}=1.414$ then $\sqrt{\frac{(\sqrt{2}-1)}{(\sqrt{2}+1)}}=?$ (a) $0.207$ (b) $2.414$ (c) $0.414$ (d) $0.621$ Solution: $\sqrt{\frac{(\sqrt{2}-1)}{(\sqrt{2}+1)}}=\sqrt{\frac{(\sqrt{2}-1)}{(\sqrt{2}+1)} \times \frac{(\sqrt{2}-1)}{(\sqrt{2}-1)}}=\sqrt{\frac{(\sqrt{2}-1)^{2}}{2-1}}$ $=\sqrt{\frac{2+1-2 \sqrt{2}}{1}}$ $=\sqrt{3-2 \sqrt{2}}$ $=\sqrt{3-2 \times 1.414}$ $=\sqrt{3-2.828}$ $=\sqrt{0.172}$ $=0.414$ Hence, the correct answer is option (c)....
Read More →The following tables gives the distribution of total household
Question: The following tables gives the distribution of total household expenditure (in rupees) of manual workers in a city. Find the average expenditure (in rupees) per household Solution: Given: Let the assumed mean beA= 275 andh= 50. We know that mean, $\bar{X}=A+h\left(\frac{1}{N} \sum f_{i} u_{i}\right)$ Now, we have $N=\sum f_{i}=200, \sum f_{i} u_{i}=-35, h=50$ and $A=275$. Putting the values in the above formula, we get $\bar{X}=A+h\left(\frac{1}{N} \sum f_{i} u_{i}\right)$ $=275+50\lef...
Read More →Find the value
Question: The value of $\sqrt{5+2 \sqrt{6}}$ is (a) $\sqrt{5}+\sqrt{6}$ (b) $\sqrt{5}-\sqrt{6}$ (c) $\sqrt{3}+\sqrt{2}$ (d) $\sqrt{3}-\sqrt{2}$ Solution: $5+2 \sqrt{6}=2+3+2 \times \sqrt{3} \times \sqrt{2}$ $=(\sqrt{2})^{2}+(\sqrt{3})^{2}+2 \times \sqrt{3} \times \sqrt{2}$ This is in the form $a^{2}+b^{2}+2 a b=(a+b)^{2}$ So, we have $(\sqrt{2})^{2}+(\sqrt{3})^{2}+2 \times \sqrt{3} \times \sqrt{2}=(\sqrt{2}+\sqrt{3})^{2}$ Thus, $\sqrt{5+2 \sqrt{6}}=\sqrt{(\sqrt{2}+\sqrt{3})^{2}}=\sqrt{2}+\sqrt{3...
Read More →Find the mean from the following frequency distribution of marks at a test in statistics:
Question: Find the mean from the following frequency distribution of marks at a test in statistics: Solution: Let the assumed mean be $A=25$ and $h=5$. We know that mean, $\bar{X}=A+h\left(\frac{1}{N} \sum_{i=1}^{n} f_{i} u_{i}\right)$ Now, we have $N=\sum f_{i}=400, \sum f_{i} u_{i}=-234, h=5$ and $A=25$. Putting the values in the above formula, we get $\bar{X}=A+h\left(\frac{1}{N} \sum_{i=1}^{n} f_{i} u_{i}\right)$ $=25+5\left(\frac{1}{400} \times(-234)\right)$ $=25-\frac{234}{80}$ $=25-2.925$...
Read More →Using principle of mathematical induction, prove that
Question: Using principle of mathematical induction, prove that $\sqrt{n}\frac{1}{\sqrt{1}}+\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}}+\ldots+\frac{1}{\sqrt{n}}$ for all natural numbers $n \geq 2$. Solution: Let $\mathrm{P}(n): \sqrt{n}\frac{1}{\sqrt{1}}+\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}}+\ldots+\frac{1}{\sqrt{n}}$ for all natural numbers $n \geq 2$. Step I: For $n=2$, $\mathrm{P}(2)$ : LHS $=\sqrt{2} \approx 1.414$ $\mathrm{RHS}=\frac{1}{\sqrt{1}}+\frac{1}{\sqrt{2}}=1+\frac{\sqrt{2}}{2} \approx 1...
Read More →Solve this
Question: The value of $\sqrt{3-2 \sqrt{2}}$ is (a) $\sqrt{3}+\sqrt{2}$ (b) $\sqrt{3}-\sqrt{2}$ (c) $\sqrt{2}+1$ (d) $\sqrt{2}-1$ Solution: $3-2 \sqrt{2}=2+1-2 \times \sqrt{2} \times 1$ $=(\sqrt{2})^{2}+1^{2}-2 \times \sqrt{2} \times 1$ This is of the form $a^{2}+b^{2}-2 a b=(a-b)^{2}$ $(\sqrt{2})^{2}+1^{2}-2 \times \sqrt{2} \times 1=((\sqrt{2})-1)^{2}$ So, $\sqrt{3-2 \sqrt{2}}=\sqrt{((\sqrt{2})-1)^{2}}=\sqrt{2}-1$ Hence, the correct answer is option (d)....
Read More →The following distribution gives the number of accidents
Question: The following distribution gives the number of accidents met by 160 workers in a factory during a month. Find the average number of accidents per worker. 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 $N=\sum f_{i}=160, \sum f_{i} d_{i}=-187$ and $A=2$. Putting the values in the above formula, we get $\bar{X}=A+\frac{1}{N} \sum_{i=1}^{n} f_{i} d_{i}$ $=2+\frac{1}{160} \times(-187)$ $=2-\frac{187}{160}$ $=2-1.16...
Read More →Solve the following
Question: A sequence $x_{0}, x_{1}, x_{2}, x_{3}, \ldots$ is defined by letting $x_{0}=5$ and $x_{k}=4+x_{k-1}$ for all natural number $k$.Show that $x_{n}=5+4 n$ for all $n \in \mathbf{N}$ using mathematical induction. Solution: Given : A sequence $x_{0}, x_{1}, x_{2}, x_{3}, \ldots$ is defined by letting $x_{0}=5$ and $x_{k}=4+x_{k-1}$ for all natural number $k$. Let $\mathrm{P}(n): x_{n}=5+4 n$ for all $n \in \mathbf{N}$. Step I: For $n=0$, $\mathrm{P}(0): x_{0}=5+4 \times 0=5$ So, it is true...
Read More →In the first proof reading of a book containing 300
Question: In the first proof reading of a book containing 300 pages the following distribution of misprints was obtained: Find the average number of misprints per page 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 $N=\sum f_{i}=300, \sum f_{i} d_{i}=-381$ and $A=2$. Putting the values in above formula, we have $\bar{X}=A+\frac{1}{N} \sum_{i=1}^{n} f_{i} d_{i}$ $=2+\frac{1}{300} \times(-381)$ $=2-\frac{381}{300}$ $=2-1.2...
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