If the fractional part of the number
Question: If the fractional part of the number $\frac{2^{403}}{15}$ is $\frac{\mathrm{k}}{15}$, then $\mathrm{k}$ is equal to:(1) 6(2) 8(3) 4(4) 14Correct Option: , 2 Solution: $2^{403}=2^{400} \cdot 2^{3}$ $=2^{4 \times 100} \cdot 2^{3}$ $=\left(2^{4}\right)^{100} \cdot 8$ $=8\left(2^{4}\right)^{100}=8(16)^{100}$ $=8(1+15)^{100}$ $=8+15 \mu$ When $2^{403}$ is divided by 15 , then remainder is $8 .$ Hence, fractional part of the number is $\frac{8}{15}$ Therefore value of $k$ is 8...
Read More →The radius of the second Bohr orbit, in terms of the Bohr radius,
Question: The radius of the second Bohr orbit, in terms of the Bohr radius, $a_{0}$, in $\mathrm{Li}^{2+}$ is:$\frac{2 a_{0}}{3}$$\frac{4 a_{0}}{9}$$\frac{4 a_{0}}{3}$$\frac{2 a_{0}}{9}$Correct Option: , 3 Solution: $r=\frac{a_{0} n^{2}}{\mathrm{Z}}$ For $\mathrm{Li}^{2+}, r=\frac{a_{0}(2)^{2}}{3}=\frac{4 a_{0}}{3}$...
Read More →In a parallel plate capacitor set up, the plate area of capacitor
Question: In a parallel plate capacitor set up, the plate area of capacitor is $2 \mathrm{~m}^{2}$ and the plates are separated by $1 \mathrm{~m}$. If the space between the plates are filled with a dielectric material of thickness $0.5 \mathrm{~m}$ and area $2 \mathrm{~m}^{2}$ (see fig.) the capacitance of the set-up will be (Dielectric constant of the material $=3.2$ ) (Round off to the Nearest Integer) Solution: (3) $\mathrm{C}=\frac{\varepsilon_{0} \mathrm{~A}}{\frac{\mathrm{d}}{2 \mathrm{~K}...
Read More →The term independent of x in the expansion of
Question: The term independent of $x$ in the expansion of $\left(\frac{1}{60}-\frac{x^{8}}{81}\right) \cdot\left(2 x^{2}-\frac{3}{x^{2}}\right)^{6}$ is equal to :(1) $-72$(2) 36(3) $-36$(4) $-108$Correct Option: , 4 Solution: Given expression is, $\left(\frac{1}{60}-\frac{x^{8}}{81}\right)\left(2 x^{2}-\frac{3}{x^{2}}\right)^{6}$ $=\frac{1}{60}\left(2 x^{2}-\frac{3}{x^{2}}\right)^{6}-\frac{x^{8}}{81}\left(2 x^{2}-\frac{3}{x^{2}}\right)^{6}$ Term independent of $x$, $=$ Coefficient of $x^{\circ}$...
Read More →The term independent of x in the expansion of
Question: The term independent of $x$ in the expansion of $\left(\frac{1}{60}-\frac{x^{8}}{81}\right) \cdot\left(2 x^{2}-\frac{3}{x^{2}}\right)^{6}$ is equal to :(1) $-72$(2) 36(3) $-36$(4) $-108$Correct Option: , 4 Solution: Given expression is,...
Read More →For the Balmer series in the spectrum of
Question: For the Balmer series in the spectrum of $\mathrm{H}$ atom, $\bar{v}=R_{H}\left\{\frac{1}{n_{1}^{2}}-\frac{1}{n_{2}^{2}}\right\}$, the correct statements among (I) to (IV) are: (I) As wavelength decreases, the lines in the series converge (II) The integer $n_{1}$ is equal to 2 (III) The lines of longest wavelength corresponds to $n_{2}=3$ (IV) The ionization energy of hydrogen can be calculated from wave number of these lines(I), (III), (IV)(i), (II), (III)(I), (II), (IV)(II), (III), (...
Read More →then the ordered pair
Question: If ${ }^{20} \mathrm{C}_{1}+\left(2^{2}\right){ }^{20} \mathrm{C}_{2}+\left(3^{2}\right){ }^{20} \mathrm{C}_{3}+\ldots \ldots \ldots .+\left(20^{2}\right)^{20} \mathrm{C}_{20}=\mathrm{A}\left(2^{\beta}\right)$, then the ordered pair $(A, \beta)$ is equal to :(1) $(420,19)$(2) $(420,18)$(3) $(380,18)$(4) $(380,19)$Correct Option: , 2 Solution: Given, ${ }^{20} C_{1}+2^{2} \cdot{ }^{20} C_{2}+3^{2} \cdot{ }^{20} C_{3}+\ldots+20^{2} \cdot{ }^{20} C_{20}=A\left(2^{\beta}\right)$ Taking L.H...
Read More →The coefficient of
Question: The coefficient of $x^{18}$ in the product $(1+x)(1-x)^{10}\left(1+x+x^{2}\right)^{9}$ is :(1) 84(2) $-126$(3) $-84$(4) 126Correct Option: , 2 Solution: Given expression, $(1-x)^{10}\left(1+x+x^{2}\right)^{9}(1+x)=\left(1-x^{3}\right)^{9}\left(1-x^{2}\right)$ $=\left(1-x^{3}\right)^{9}-x^{2}\left(1-x^{3}\right)^{9}$ $\Rightarrow$ Coefficient of $x^{18}$ in $\left(1-x^{3}\right)^{9}-$ coeff. of $x^{16}$ in $\left(1-x^{3}\right)^{9}$ $={ }^{9} C_{6}-0=\frac{9 !}{6 ! 3 !}=\frac{7 \times 8...
Read More →The number of orbitals associated with quantum numbers
Question: The number of orbitals associated with quantum numbers $n=5, m_{s}=+\frac{1}{2}$ is:11255015Correct Option: , 2 Solution: The possible number of orbitals in a shell in term of ' $n$ ' is $n^{2}$ $\therefore n=5 ; n^{2}=25$...
Read More →Amongst the following statements, that which was not proposed by Dalton was:
Question: Amongst the following statements, that which was not proposed by Dalton was:Chemical reactions involve reorganization of atoms.These are neither created nor destroyed in a chemical reaction.All the atoms of a given element have identical properties including identical mass. Atoms of different elements differ in mass.When gases combine or reproduced in a chemical reaction they do so in a simple ratio by volume provided all gases are at the same T \ P.Matter consists of indivisible atoms...
Read More →(i) Find the class mark of the class 90 – 120.
Question: (i) Find the class mark of the class 90 120.(ii) In a frequency distribution, the mid-value of the class is 10 and width of the class is 6. Find the lower limit of the class.(iii) The width of each of five continuous classes in a frequency distribution is 5 and lower class limit of the lowest class is 10. What is the upper class limit of the highest class?(iv) The class marks of a frequency distribution are 15, 20, 25, ... . Find the class corresponding to the class mark 20.(v) In the ...
Read More →The number of orbitals associated with quantum numbers
Question: The number of orbitals associated with quantum numbers $n=5, m_{s}=+\frac{1}{2}$ is:11255015Correct Option: Solution: The possible number of orbitals in a shell in term of ' $n$ ' is $n^{2}$ $\therefore n=5 ; n^{2}=25$...
Read More →The marks obtained by 17 students in a mathematics test (out of 100) are given below:
Question: The marks obtained by 17 students in a mathematics test (out of 100) are given below:90, 79, 76, 82, 65, 96, 100, 91, 82, 100, 49, 46, 64, 48, 72, 66, 68.Find the range of the above data. Solution: Range $=$ Maximum value $-$ minimu value $=100-46=54$ Thus, the range is 54....
Read More →Make a frequency table from the following:
Question: Make a frequency table from the following: Solution: The frequency table can be presented as below:...
Read More →Given below is a cumulative frequency table:
Question: Given below is a cumulative frequency table: Extract a frequency table from the above. Solution: The frequency table can be presented as given below:...
Read More →Present the following as an ordinary grouped frequency table:
Question: Present the following as an ordinary grouped frequency table: Solution: The grouped frequency table can be presented as given below:...
Read More →Following are the ages (in years) of 360 patients, getting medical treatment in a hospital:
Question: Following are the ages (in years) of 360 patients, getting medical treatment in a hospital: Construct the cumulative frequency table for the above data. Solution: The cumulative frequency table can be presented as given below:...
Read More →The heights (inAns cm) of 30 students of a class are given below:
Question: The heights (inAns cm) of 30 students of a class are given below:161, 155, 159, 153, 150, 158, 154, 158, 160, 148, 149, 162, 163, 159, 148,153, 157, 151, 154, 157, 153, 156, 152, 156, 160, 152, 147, 155, 155, 157.Prepare a frequency table as well as a cumulative frequency table with 160 165 (165 not included) as one of the class intervals. Solution:...
Read More →The weights (in grams) of 40 oranges picked at random from a basket are as follows:
Question: The weights (in grams) of 40 oranges picked at random from a basket are as follows:40,50,60,65,45,55,30,90,75,85,70,85,75,80,100,110,70,55,30,35,45,70,80,85,95,70,60,70,75,100,65,60,40,100,75,110,30,45,84.Construct a frequency table as well as a cumulative frequency table. Solution: The minimum observation is 30 and the maximum observation is 120. Frequency distribution table: Cumulative frequency table:...
Read More →For changing the capacitance of a given parallel plate capacitor,
Question: For changing the capacitance of a given parallel plate capacitor, a dielectric material of dielectric constant $\mathrm{K}$ is used, which has the same area as the plates of the capacitor. The thickness of the dielectric slab is $\frac{3}{4} \mathrm{~d}$, where ' $\mathrm{d}$ ' is the separation between the plates of parallel plate capacitor. The new capacitance $\left(C^{\prime}\right)$ in terms of original capacitance $\left(\mathrm{C}_{0}\right)$ is given by the following relation :...
Read More →The smallest natural number n,
Question: The smallest natural number $n$, such that the coefficient of $x$ in the expansion of $\left(x^{2}+\frac{1}{x^{3}}\right)^{n}$ is ${ }^{n} C_{23}$, is :(1) 38(2) 58(3) 23(4) 35Correct Option: 1, Solution: $\left(x^{2}+\frac{1}{x^{2}}\right)^{n}$ General term $T_{r+1}={ }^{n} C_{r}\left(x^{2}\right)^{n-r}\left(\frac{1}{x^{3}}\right)^{r}={ }^{n} C_{r} \cdot x^{2 n-}$ $5 r$ To find coefficient of $x, 2 n-5 r=1$ Given ${ }^{n} C_{r}={ }^{n} C_{23} \Rightarrow r=23$ or $n-r=23$ $\therefore ...
Read More →Construct a frequency table with equal class intervals from the following data on the monthly wages (in rupees) of 28 labourers working in a factory
Question: Construct a frequency table with equal class intervals from the following data on the monthly wages (in rupees) of 28 labourers working in a factory, taking one of the class intervals as 210230(230 not included).220,268,258,242,210,268,272,242,311,290,300,320,319,304,302,318,306,292,254,278,210,240,280,316,306,215,256,236. Solution: The minimum observation is 210 and the maximum observation is 330.Therefore, classes of the same size covering the given data are 210-230, 230-250,250-270,...
Read More →Construct a frequency table for the following ages (in years) of 30 students using equal class intervals, one of them being 9−12, where 12 is not included.
Question: Construct a frequency table for the following ages (in years) of 30 students using equal class intervals, one of them being 912, where 12 is not included.18,12,7,6,11,15,21,9,8,13,15,17,22,19,14,21,23,8,12,17,15,6,18,23,22,16,9,21,11,16. Solution: The minimum observation is 6 and the maximum observation is 24.Therefore, classes of the same size covering the given data are 6-9, 9-12, 12-15, 15-18, 18-21 and 21-24.Frequency distribution table:...
Read More →The correct statement about probability density
Question: The correct statement about probability density (except at infinite distance from nucleus) is :It can be zero for $1 s$ orbitalIt can be negative for $2 p$ orbitalIt can be zero for $3 p$ orbitalIt can never be zero for $2 s$ orbitalCorrect Option: , 3 Solution: Radial node $=n-l-1$...
Read More →The marks obtained by 40 students of a class in an examination are given below.
Question: The marks obtained by 40 students of a class in an examination are given below.3,20,13,1,21,13,3,23,16,13,18,12,5,12,5,24,9,2,7,18,20,3,10,12,7,18,2,5,7,10,16,8,16,17,8,23,24,6,23, 15.Present the data in the form of a frequency distribution using equal class size, one such class being 1015(15 not included). Solution: The minimum observation is 0 and the maximum observation is 25.Therefore, classes of the same size covering the given data are 0-5, 5-10, 10-15, 15-20 and 20-25.Frequency ...
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