Thirty children were asked about the number of hours they watched TV programmes in the previous week.
Question: Thirty children were asked about the number of hours they watched TV programmes in the previous week. The results were found as under:8, 4, 8, 5, 1, 6, 2, 5, 3, 12, 3, 10, 4, 12, 2, 8, 15, 1, 6, 17, 5, 8, 2, 3, 9, 6, 7, 8, 14, 12.(i) Make a grouped frequency distribution table for this data, taking class width 5 and one of the class interval as 5 10.(ii) How many children watched television for 15 or more hours a week? Solution: (i) (ii) As we can see from the table, there are 2 childr...
Read More →If the coefficients of
Question: If the coefficients of $x^{2}$ and $x^{3}$ are both zero, in the expansion of the expression $\left(1+a x+b x^{2}\right)(1-3 x)^{15}$ in powers of $\mathrm{x}$, then the ordered pair $(\mathrm{a}, \mathrm{b})$ is equal to :(1) $(28,861)$(2) $(-54,315)$(3) $(28,315)$(4) $(-21,714)$Correct Option: , 3 Solution: Given expression is $\left(1+a x+b x^{2}\right)(1-3 x)^{15}$ Co-efficient of $x^{2}=0$ $\Rightarrow{ }^{15} \mathrm{C}_{2}(-3)^{2}+a \cdot{ }^{15} \mathrm{C}_{1}(-3)+b \cdot{ }^{1...
Read More →Following data gives the number of children in 40 families:
Question: Following data gives the number of children in 40 families:1,2,6,5,1,5,1,3,2,6,2,3,4,2,0,4,4,3,2,2,0,0,1,2,2,4,3,2,1,0,5,1,2,4,3,4,1,6,2,2.Represent it in the form of a frequency distribution, taking classes 02, 14, etc. Solution: The minimum observation is 0 and the maximum observation is 8.Therefore, classes of the same size covering the given data are 0-2, 2-4, 4-6 and 6-8. Frequency distribution table:...
Read More →In the sixth period, the orbitals that are filled are :
Question: In the sixth period, the orbitals that are filled are : $6 s, 4 f, 5 d, 6 p$$6 s, 5 d, 5 f, 6 p$$6 s, 5 f, 6 d, 6 p$$6 s, 6 p, 6 d, 6 f$Correct Option: 1 Solution: Thus, order of orbitals filled are $6 s4 f5 d6 p$...
Read More →Three coins are tossed 30 times. Each time the number of heads occurring was noted down as follows:
Question: Three coins are tossed 30 times. Each time the number of heads occurring was noted down as follows:0, 1, 2, 2, 1, 2, 3, 1, 3, 0, 1, 3, 1, 1, 2, 2, 0, 1, 2, 1, 0, 3, 0, 2, 1, 1, 3, 2, 0, 2.Prepare a frequency distribution table. Solution:...
Read More →The blood groups of 30 students of a class are recorded as under:
Question: The blood groups of 30 students of a class are recorded as under:A, B, O, O, AB, O, A, O, A, B, O, B, A, O, O, A, AB, O, A, A, O, O, AB, B, A, O, B, A, B, O.(i) Represent this data in the form of a frequency distribution table.(ii) Find out which is the most common and which is the rarest blood group among these students. Solution: (i) (ii) AB is rarest and O is most common....
Read More →If some three consecutive coefficients in the binomial expansion
Question: If some three consecutive coefficients in the binomial expansion of $(x+1)^{\mathrm{n}}$ in powers of $x$ are in the ratio $2: 15: 70$, then the average of these three coefficients is:(1) 964(2) 232(3) 227(4) 625Correct Option: , 2 Solution: Given ${ }^{n} C_{r-1}:{ }^{n} C_{r}:{ }^{n} C_{r+1}=2: 15: 70$ $\Rightarrow \frac{{ }^{n} C_{r-1}}{{ }^{n} C_{r}}=\frac{2}{15}$ and $\frac{{ }^{n} C_{r}}{{ }^{n} C_{r+1}}=\frac{15}{70}$ $\Rightarrow \frac{r}{n-r+1}=\frac{2}{15}$ and $\frac{r+1}{n-...
Read More →The difference between the radii of
Question: The difference between the radii of $3^{\mathrm{rd}}$ and $4^{\text {th }}$ orbits of $\mathrm{Li}^{2+}$ is $\Delta \mathrm{R}_{1}$. The difference between the radii of $3^{\mathrm{rd}}$ and $4^{\text {th }}$ orbits of $\mathrm{He}^{+}$is $\Delta \mathrm{R}_{2}$. Ratio $\Delta \mathrm{R}_{1}: \Delta \mathrm{R}_{2}$ is :8 : 33 : 82 : 33 : 2Correct Option: , 3 Solution: For $\mathrm{Li}^{2+}$ $\left(r_{\mathrm{Li}^{2+}}\right)_{n=4}-\left(r_{\mathrm{Li}^{2+}}\right)_{n=3}=\frac{0.529}{3}...
Read More →Explain the meaning of each of the following terms:
Question: Explain the meaning of each of the following terms:(i) Variate(ii) Class interval(iii) Class size(iv) Class mark(v) Class limit(vi) True class limits(vii) Frequency of a class(viii) Cumulative frequency of a class Solution: (i) Variate : Any character which is capable of taking several different values is called a variant or a variable.(ii) Class interval : Each group into which the raw data is condensed is called class interval .(iii) Class size: The difference between the true upper ...
Read More →The difference between the radii of
Question: The difference between the radii of $3^{\mathrm{rd}}$ and $4^{\text {th }}$ orbits of $\mathrm{Li}^{2+}$ is $\Delta \mathrm{R}_{1}$. The difference between the radii of $3^{\mathrm{rd}}$ and $4^{\text {th }}$ orbits of $\mathrm{He}^{+}$is $\Delta \mathrm{R}_{2}$. Ratio $\Delta \mathrm{R}_{1}: \Delta \mathrm{R}_{2}$ is :8 : 33 : 82 : 33 : 2Correct Option: , 3 Solution: (c) $r=0.529 \frac{n^{2}}{Z} \AA$ For $\mathrm{Li}^{2+}$ $\left(r_{\mathrm{Li}^{2+}}\right)_{n=4}-\left(r_{\mathrm{Li}^...
Read More →What are primary data and secondary data? Which of the two is more reliable and why?
Question: What are primary data and secondary data? Which of the two is more reliable and why? Solution: Primary data: The data collected by the investigator himself with a definite plan in mind are known as primary data.Secondary data: The data collected by someone other than the investigator are known as secondary data.Primary data are highly reliable and relevant because they are collected by the investigator himself with a definite plan in mind, whereas secondary data are collected with a pu...
Read More →If the fourth term in the Binomial expansion
Question: If the fourth term in the Binomial expansion of $\left(\frac{2}{x}+x^{\log _{8} x}\right)^{6}(x0)$ is $20 \times 8^{7}$, then a value of $x$ is:(1) $8^{3}$(2) $8^{2}$(3) 8(4) $8^{-2}$Correct Option: , 3 Solution: $\because \mathrm{T}_{4}=20 \times 8^{7}$ $\Rightarrow{ }^{6} C_{3}\left(\frac{2}{x}\right)^{3} \times\left(x^{\log _{8} x}\right)^{3}=20 \times 8^{7}$ $\Rightarrow 8 \times 20 \times\left(\frac{x^{\log _{8} x}}{x}\right)^{3}=20 \times 8^{7}$ $\Rightarrow \frac{x^{\log _{8} x}...
Read More →Define some fundamental characteristics of statistics.
Question: Define some fundamental characteristics of statistics. Solution: The fundamental characteristics of data (statistics) are as follows:(i) Numerical facts alone constitute data.(ii) Qualitative characteristics like intelligence and poverty, which cannot be measured numerically, do not form data.(iii) Data are aggregate of facts. A single observation does not form data.(iv) Data collected for a definite purpose may not be suited for another purpose.(v) Data in different experiments are co...
Read More →Define statistics as a subject.
Question: Define statistics as a subject. Solution: Statistics is the science which deals with the collection, presentation, analysis and interpretation of numerical data....
Read More →If the fourth term in the binomial expansion of
Question: If the fourth term in the binomial expansion of $\left(\sqrt{\frac{1}{x^{1+\log _{10} x}}}+x^{\frac{1}{12}}\right)^{6}$ is equal to 200, and $x1$, then the value of $x$ is:(1) 100(2) 10(3) $10^{3}$(4) $10^{4}$Correct Option: , 2 Solution: $\therefore$ fourth term is equal to 200 . $T_{4}={ }^{6} C_{3}\left(\sqrt{x\left(\frac{1}{1+\log _{10} x}\right)}\right)^{3}\left(x^{\frac{1}{12}}\right)^{3}=200$ $\Rightarrow 20 x^{\frac{3}{2\left(1+\log _{10} x\right)}} \cdot x^{\frac{1}{4}}=200$ $...
Read More →The shortest wavelength of Hatom in the Lyman series is
Question: The shortest wavelength of Hatom in the Lyman series is $\lambda_{1}$. The longest wavelength in the Balmer series is $\mathrm{He}^{+}$is :$\frac{36 \lambda_{1}}{5}$$\frac{5 \lambda_{1}}{9}$$\frac{9 \lambda_{1}}{5}$$\frac{27 \lambda_{1}}{5}$Correct Option: Solution: Shortest wavelength $\rightarrow$ Max. energy $(\infty \rightarrow 1)$ For Lyman series of H atom, $\frac{1}{\lambda_{1}}=R_{\mathrm{H}}(1)^{2}\left[\frac{1}{1}-0\right]$ $\Rightarrow \frac{1}{\lambda_{1}}=R_{\mathrm{H}} \R...
Read More →The sum of the series
Question: The sum of the series $2 \cdot{ }^{20} \mathrm{C}_{0}+5 \cdot{ }^{20} \mathrm{C}_{1}+8 \cdot{ }^{20} \mathrm{C}_{2}+11 \cdot{ }^{.20} \mathrm{C}_{3}$ $+\ldots+62 \cdot{ }^{20} \mathrm{C}_{20}$ is equal to :(1) $2^{26}$(2) $2^{25}$(3) $2^{23}$(4) $2^{24}$Correct Option: , 2 Solution: $2 \cdot{ }^{20} \mathrm{C}_{0}+5 \cdot{ }^{20} \mathrm{C}_{1}+8 \cdot{ }^{20} \mathrm{C}_{2}+\ldots \ldots+62 \cdot{ }^{20} \mathrm{C}_{20}$ $=\sum_{r=0}^{20}(3 r+2){ }^{20} C_{r}=3 \sum_{r=0}^{20} r \cdot...
Read More →The region in the electromagnetic spectrum
Question: The region in the electromagnetic spectrum where the Balmer series lines appear is :VisibleMicrowaveInfraredUltravioletCorrect Option: 1 Solution: In hydrogen spectrum maximum lines of Balmer series lies in visible region....
Read More →The sum of the co-efficients of all even degree terms
Question: The sum of the co-efficients of all even degree terms in $x$ in the expansion of $\left(x+\sqrt{x^{3}-1}\right)^{6}+\left(x-\sqrt{x^{3}-1}\right)^{6},(x1)$ is equal to :(1) 29(2) 32(3) 26(4) 24Correct Option: , 4 Solution: $\left(x+\sqrt{x^{3}-1}\right)^{6}+\left(x-\sqrt{x^{3}-1}\right)^{6}$ $=2\left[{ }^{6} C_{0} x^{6}+{ }^{6} C_{2} x^{4}\left(x^{3}-1\right)+{ }^{6} C_{4} x^{2}\left(x^{3}-1\right)^{2}\right.$ $\left.+{ }^{6} C_{6}\left(x^{3}-1\right)^{3}\right]$ $=2\left[x^{6}+15 x^{7...
Read More →A particle A of mass ' m ' and charge ' q ' is accelerated by
Question: A particle $\mathrm{A}$ of mass ' $\mathrm{m}$ ' and charge ' $\mathrm{q}$ ' is accelerated by a potential difference of $50 \mathrm{v}$ Another particle B of mass ' $4 \mathrm{~m}$ ' and charge ' $q$ ' is accelerated by a potential differnce of $2500 \mathrm{~V}$. The ratio of de-Broglie wavelength $\frac{\lambda_{A}}{\lambda_{B}}$ is(1) $10.00$(2) $0.07$(3) $14.14$(4) $4.47$Correct Option: 3, Solution: (3) de Broglie wavelength $(\lambda)$ is given by $\mathrm{K}=\mathrm{qV}$ $\lambd...
Read More →Consider the hypothetical situation where the azimuthal quantum number,
Question: Consider the hypothetical situation where the azimuthal quantum number, $l$, takes values $0,1,2, \ldots . . n+1$, where $n$ is the principal quantum number. Then, the element with atomic number:9 is the first alkali metal13 has a half-filled valence subshell8 is the first noble gas 6 has a $2 p$-valence subshellCorrect Option: , 2 Solution: Under the given situation for $n=1, l=0,1,2$ $n=2, l=0,1,2,3$ $n=3, l=0,1,2,3,4$ According to $(n+l)$ rule of order of filling of subshells will b...
Read More →In the expansion of
Question: In the expansion of $\left(\frac{x}{\cos \theta}+\frac{1}{x \sin \theta}\right)^{16}$, if $l_{1}$ is the least value of the term independent of $x$ when $\frac{\pi}{8} \leq \theta \leq \frac{\pi}{4}$ and $l_{2}$ is the least value of the term independent of $x$ when $\frac{\pi}{16} \leq \theta \leq \frac{\pi}{8}$, then the ratio $l_{2}: l_{1}$ is equal to : (1) $1: 8$(2) $16: 1$(3) $8: 1$(4) $1: 16$Correct Option: , 2 Solution: General term of the given expansion $T_{r+1}={ }^{16} C_{r...
Read More →The work function of sodium metal is
Question: The work function of sodium metal is $4.41 \times 10^{-19} \mathrm{~J}$. If photons of wavelength $300 \mathrm{~nm}$ are incident on the metal, the kinetic energy of the ejected electrons will be $\left(h=6.63 \times 10^{-34} \mathrm{~J} \mathrm{~s} ; c=3 \times 10^{8} \mathrm{~m} / \mathrm{s}\right)$ Solution: (222) $\mathrm{E}=\mathrm{E}_{0}+(\mathrm{KE})_{\max }$ $\frac{h c}{\lambda}=4.41 \times 10^{-19}+\mathrm{KE}$ $\frac{6.63 \times 10^{-34} \times 3 \times 10^{8}}{300 \times 10^...
Read More →The coefficient of x{4} in the expansion
Question: The coefficient of $x^{4}$ in the expansion of $\left(1+x+x^{2}\right)^{10}$ is Solution: General term of the expansion $=\frac{10 !}{\alpha ! \beta ! \gamma !} x^{\beta+2 \gamma}$ For coefficient of $x^{4} ; \beta+2 \gamma=4$ Here, three cases arise Case-1: When $\gamma=0, \beta=4, \alpha=6$ $\Rightarrow \frac{10 !}{6 ! 4 ! 0 !}=210$ Case-2 : When $\gamma=1, \beta=2, \alpha=7$ $\Rightarrow \frac{10 !}{7 ! 2 ! 1 !}=360$ Case-3: When $\gamma=2, \beta=0, \alpha=8$ $\Rightarrow \frac{10 !...
Read More →A particle of mass $m$ moves in a circular orbit in a central potential field
Question: A particle of mass $m$ moves in a circular orbit in a central potential field $\mathrm{U}(\mathrm{r})=\frac{1}{2} \mathrm{kr}^{2}$. If Bohr's quantization conditions are applied, radii of possible orbitls and energy levels vary with quantum number $n$ as:(1) $\mathrm{r}_{\mathrm{n}} \propto \sqrt{\mathrm{n}}, \mathrm{E}_{\mathrm{n}} \propto \mathrm{n}$(2) $r_{n} \propto \sqrt{n}, E_{n} \propto \frac{1}{n}$(3) $\mathrm{r}_{\mathrm{n}} \propto \mathrm{n}, \mathrm{E}_{\mathrm{n}} \propto ...
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