What is the correct sequence of reagents used for converting nitrobenzene into m dibromobenzene?
Question: What is the correct sequence of reagents used for converting nitrobenzene into m dibromobenzene? Correct Option: , 4 Solution:...
Read More →For the frequency distribution:
Question: For the frequency distribution: Variate $(x): \quad x_{1} \quad x_{2} \quad x_{3} \ldots x_{15}$ Frequency $(f): \begin{array}{lll}f_{1} f_{2} f_{3} \ldots f_{15}\end{array}$ where $0x_{1}x_{2}x_{3}\ldotsx_{15}=10$ and $\sum_{i=1}^{15} f_{i}0$, the standard deviation cannot be :(1) 4(2) 1(3) 6(4) 2Correct Option: , 3 Solution: If variate varries from $a$ to $b$ then variance $\operatorname{var}(x) \leq\left(\frac{b-a}{2}\right)^{2}$ $\Rightarrow \operatorname{var}(x)\left(\frac{10-0}{2...
Read More →If the variance of the terms in an increasing
Question: If the variance of the terms in an increasing A.P., $b_{1}, b_{2}, b_{3}, \ldots . ., b_{11}$ is 90 , then the common difference of this A.P. is________. Solution: Variance $=\frac{\sum_{i=1}^{11} b_{i}^{2}}{11}-\left(\frac{\sum_{i=1}^{11} b_{i}}{11}\right)^{2}$ Let common difference of A.P. be $d$ $=\frac{\sum_{r=0}^{10}\left(b_{1}+r d\right)^{2}}{11}-\left(\frac{\sum_{r=0}^{10}\left(b_{1}+r d\right)}{11}\right)^{2}$ $=\frac{11 b_{1}^{2}+2 b_{1} d\left(\frac{10 \times 11}{2}\right)+d^...
Read More →The diazonium salt of which of the following
Question: The diazonium salt of which of the following compounds will form a coloured dye on reaction with $\beta$-Naphthol in $\mathrm{NaOH}$ ?Correct Option: , 3 Solution: Orange bright dye....
Read More →Prove the following
Question: Let $X=\{x \in \mathbf{N}: 1 \leq x \leq 17\}$ and $Y=\{a x+b: x \in X$ and $a, b \in \mathbf{R}, a0\} .$ If mean and variance of elements of $Y$ are 17 and 216 respectively then $a+b$ is equal to :(1) 7(2) $-7$(3) $-27$(4) 9Correct Option: , 2 Solution: $\because \bar{x}=\frac{1+2+3+\ldots .+17}{17}=\frac{17 \times 18}{17 \times 2}=9$ $\bar{y}=a \bar{x}+b=\frac{a(1+2+3+\ldots . .+17)}{17}+b=17$ $\Rightarrow \frac{a \cdot(17 \cdot 18)}{17 \cdot 2}+b=17 \Rightarrow 9 a+b=17 \ldots(\math...
Read More →If the velocity-time graph has the shape AMB,
Question: If the velocity-time graph has the shape $\mathrm{AMB}$, what would be the shape of the corresponding acceleration-time graph? (1) (2) (3) (4) Correct Option: 1 Solution: $a=\frac{d v}{d t}=$ slope of $(v-t)$ curve If $m=+$ ve, then equation of straight line isc $y=m x+c \Rightarrow v=m t+c$ (for MB) If $m=-$ ve, then equation of straight line is $y=-m x+c \Rightarrow v=-m t+c($ for $A M)$ If we differentiate equation (1) and (2), we get $\mathrm{a}_{\mathrm{MB}}=+\mathrm{ve}=\mathrm{m...
Read More →Prove the following
Question: Let $X_{1}, \quad X_{2}, \ldots \ldots . . . X_{18}$ be eighteen observation such that $\sum_{i=1}^{18}\left(X_{i}-\alpha\right)=36$ and $\sum_{i=1}^{18}\left(X_{i}-\beta\right)^{2}=90$, where $\alpha$ and $\beta$ are distinct real numbers. If the standard deviation of these observations is 1 , then the value of $|\alpha-\beta|$ is Solution: Given, $\sum_{i=1}^{18}\left(X_{i}-\alpha\right)=36$ $\Rightarrow \sum x_{i}-18 \alpha=36$ $\Rightarrow \sum x_{i}-18(\alpha+2) \ldots(1)$ Also, $...
Read More →' A and 'B' in the following reactions are :
Question: $' A^{\prime}$ and ' $B$ ' in the following reactions are : Correct Option: , 3 Solution:...
Read More →$' A^{prime}$ and ' $B$ ' in the following reactions are :
Question: $' A^{\prime}$ and ' $B$ ' in the following reactions are : Correct Option: , 3 Solution:...
Read More →If the variance of 10 natural numbers
Question: If the variance of 10 natural numbers $1,1,1, \ldots, 1, k$ is less than 10 , then the maximum possible value of $k$ is Solution: $\sigma^{2}=\frac{\Sigma X^{2}}{n}-\left(\frac{\Sigma x}{n}\right)^{2}$ $\sigma^{2}=\frac{\left(9+k^{2}\right)}{10}-\left(\frac{9+k}{10}\right)^{2}10$ $\left(90+k^{2}\right) 10-\left(81+k^{2}+8 k\right)1000$ $90+10 k^{2}-k^{2}-18 k-811000$ $9 k^{2}-18 k+91000$ $(k-1)^{2}\frac{1000}{9} \Rightarrow k-1\frac{10 \sqrt{10}}{3}$ $k\frac{10 \sqrt{10}}{3}+1$ Maximum...
Read More →Let in a series of 2n observations,
Question: Let in a series of $2 \mathrm{n}$ observations, half of them are equal to a and remaining half are equal to -a. Also by adding a constant b in each of these observations, the mean and standard deviation of new set become 5 and 20 , respectively. Then the value of $a^{2}+b^{2}$ is equal to:(1) 425(2) 650(3) 250(4) 925Correct Option: 1 Solution: Let observations are denoted by $x i$ for $1 \leq i$ $2 \mathrm{n}$ $\bar{x}=\frac{\sum x_{i}}{2 n}=\frac{(a+a+\ldots+a)-(a+a+\ldots+a)}{2 n}$ $...
Read More →Solve the following
Question: $\mathrm{C}\mathrm{A}\mathrm{B}$$\mathrm{B}\mathrm{C}\mathrm{A}$$ACB$$\mathrm{C}\mathrm{B}\mathrm{A}$Correct Option: , 4 Solution:...
Read More →The mean age of 25 teachers in a school is 40 years.
Question: The mean age of 25 teachers in a school is 40 years. A teacher retires at the age of 60 years and a new teacher is appointed in his place. If the mean age of the teachers in this school now is 39 years, then the age (in years) of the newly appointed teacher is Solution: $\frac{\sum x_{i}}{25}=40 \ \frac{\sum x_{i}-60+N}{25}=39$ Let age of newly appointed teacher is $\mathrm{N}$ $\Rightarrow 1000-60+\mathrm{N}=975$ $\Rightarrow \mathrm{N}=35$ years...
Read More →An organic compound "A" on treatment with benzene sulphonyl chloride gives compound B $.
Question: An organic compound "A" on treatment with benzene sulphonyl chloride gives compound $\mathrm{B}$. B is soluble in dil. $\mathrm{NaOH}$ solution. Compound $\mathrm{A}$ is:$\mathrm{C}_{6} \mathrm{H}_{5}-\mathrm{N}-\left(\mathrm{CH}_{3}\right)_{2}$$\mathrm{C}_{6} \mathrm{H}_{5}-\mathrm{NHCH}_{2} \mathrm{CH}_{3}$$\mathrm{C}_{6} \mathrm{H}_{5}-\mathrm{CH}_{2} \mathrm{NHCH}_{3}$Correct Option: , 4 Solution: Hinsberg reagent (Benzene sulphonyl chloride) gives reaction product with $1^{\circ}$...
Read More →Consider a set of 3 n numbers having
Question: Consider a set of $3 \mathrm{n}$ numbers having variance $4 .$ In this set, the mean of first $2 \mathrm{n}$ numbers is 6 and the mean of the remaining numbers is 3 . A new - set is constructed by adding 1 into each of first $2 \mathrm{n}$ numbers, and subtracting 1 from each of the remaining $\mathrm{n}$ numbers. If the variance of the new set is $\mathrm{k}$, then $9 \mathrm{k}$ is equal to__________. Solution: Let number be $\mathrm{a}_{1}, \mathrm{a}_{2}, \mathrm{a}_{3}, \ldots \ld...
Read More →Consider the statistics of two sets of observations as follows:
Question: Consider the statistics of two sets of observations as follows: If the variance of the combined set of these two observations is $\frac{17}{9}$, then the value of $\mathbf{n}$ is equal to___________. Solution: $\sigma^{2}=\frac{\mathrm{n}_{1} \sigma_{1}^{2}+\mathrm{n}_{2} \sigma_{2}^{2}}{\mathrm{n}_{1}+\mathrm{n}_{2}}+\frac{\mathrm{n}_{1} \mathrm{n}_{2}}{\left(\mathrm{n}_{1}+\mathrm{n}_{2}\right)}\left(\overline{\mathrm{x}}_{1}-\overline{\mathrm{x}}_{2}\right)^{2}$ $\mathrm{n}_{1}=10, ...
Read More →In the reaction of hypobromite with amide,
Question: In the reaction of hypobromite with amide, the carbonyl carbon is lost as :$\mathrm{CO}_{3}^{2-}$$\mathrm{HCO}_{3}^{-}$$\mathrm{CO}_{2}$$\mathrm{CO}$Correct Option: 1 Solution:...
Read More →Consider three observations
Question: Consider three observations $a, b$ and $c$ such that $b=a+c$. If the standard deviation of $\mathrm{a}+2 \mathrm{~b}+2, \mathrm{c}+2$ is $\mathrm{d}$, then which of the following is true?(1) $b^{2}=3\left(a^{2}+c^{2}\right)+9 d^{2}$(2) $b^{2}=a^{2}+c^{2}+3 d^{2}$(3) $b^{2}=3\left(a^{2}+c^{2}+d^{2}\right)$(4) $b^{2}=3\left(a^{2}+c^{2}\right)-9 d^{2}$Correct Option: , 4 Solution: For $\mathrm{a}, \mathrm{b}, \mathrm{c}$ mean $=\frac{a+b+c}{3}(=\bar{x})$ $\mathrm{b}=\mathrm{a}+\mathrm{c}$...
Read More →A reaction of 0.1 mole of Benzylamine with bromomethane gave 23 g of Benzyl trimethyl ammonium bromide.
Question: A reaction of $0.1$ mole of Benzylamine with bromomethane gave $23 \mathrm{~g}$ ofBenzyl trimethyl ammonium bromide. The number of moles ofbromomethane consumed in this reaction are$\mathrm{n} \times 10^{-1}$, when $\mathrm{n}=\ldots$ (Round off to the Nearest Integer). (Given : Atomic masses: C: $12.0 \mathrm{u}$,$\mathrm{H}: 1.0 \mathrm{u}, \mathrm{N}: 14.0 \mathrm{u}, \mathrm{Br}: 80.0 \mathrm{u}]$ Solution: (3)...
Read More →Prove the following
Question: If $\mathrm{A}=\{x \in \mathrm{R}:|x|2\}$ and $\mathrm{B}=\{x \in \mathrm{R}:|x-2| \geq 3\} ;$ then $:$ (1) $A \cap B=(-2,-1)$(2) $\mathrm{B}-\mathrm{A}=\mathrm{R}-(-2,5)$(3) $\mathrm{A} \cup \mathrm{B}=\mathrm{R}-(2,5)$(4) $\mathrm{A}-\mathrm{B}=[-1,2)$Correct Option: , 2 Solution: $A=\{x: x \in(-2,2)\}$ $B=\{x: x \in(-\infty,-1] \cup[5, \infty)\}$ $A \cap B=\{x: x \in(-2,-1]\}$ $A \cup B=\{x: x \in(-\infty, 2) \cup[5, \infty)\}$ $A-B=\{x: x \in(-1,2)\}$ $B-A=\{x: x \in(-\infty,-2] \c...
Read More →Prove the following
Question: Let $X=\{n \in N: 1 \leq n \leq 50\} .$ if $A=\{n \in X: n$ is $a$ multiple of 2$\}$ and $B=\{n \in X: n$ is $a$ multiple of 7$\}$, then the number of elements in the smallest subset of $X$ containing both $A$ and $B$ is________. Solution: From the given conditions, $n(A)=25, n(B)=7$ and $n(A \cap B)=3$ $n(A \cup B)=n(A)+n(B)-n(A \cap B)$ $=25+7-3=29$...
Read More →Set A has m elements and set B has n elements.
Question: Set A has $m$ elements and set B has $n$ elements. If the total number of subsets of $A$ is 112 more than the total number of subsets of B, then the value of $m \cdot n$ is___________. Solution: $2^{m}=112+2^{n} \Rightarrow 2^{m}-2^{n}=112$ $\Rightarrow 2^{n}\left(2^{m-n}-1\right)=2^{4}\left(2^{3}-1\right)$ $\therefore m=7, n=4 \Rightarrow m n=28$...
Read More →Consider the above chemical reaction and identify product "A
Question: Consider the above chemical reaction and identify product "ACorrect Option: , 3 Solution:...
Read More →A survey shows that 73%
Question: A survey shows that $73 \%$ of the persons working in an office like coffee, whereas $65 \%$ like tea. If $x$ denotes the percentage of them, who like both coffee and tea, then $x$ cannot be :(1) 63(2) 36(3) 54(4) 38Correct Option: , 2 Solution: Given, $n(C)=73, n(T)=65, n(C \cap T)=x$ $\therefore 65 \geq n(C \cap T) \geq 65+73-100$ $\Rightarrow 65 \geq x \geq 38 \Rightarrow x \neq 36$...
Read More →Prove the following
Question: Let $\bigcup_{i=1}^{50} X_{i}=\bigcup_{i=1}^{n} Y_{i}=T$, where each $X_{i}$ contains 10 elements and each $Y_{i}$ contains 5 elements. If each element of the set $T$ is an element of exactly 20 of sets $X_{i}^{\prime}$ s and exactly 6 of sets $Y_{i}^{\prime}$ s, then $n$ is equal to(1) 15(2) 50(3) 45(4) 30Correct Option: , 4 Solution: $\bigcup_{i=1}^{50} X_{i}=\bigcup_{i=1}^{n} Y_{i}=T$ $\because \quad n\left(X_{i}\right)=10, n\left(Y_{i}\right)=5$ So, $\bigcup_{i=1}^{50} X_{i}=500, \...
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