In deriving the single slit diffraction pattern,
Question: In deriving the single slit diffraction pattern, it was stated that the intensity is zero at angles ofn/a. Justify this by suitably dividing the slit to bring out the cancellation. Solution: Consider that a single slit of widthdis divided intonsmaller slits. $\therefore$ Width of each slit, $d^{\prime}=\frac{d}{n}$ Angle of diffraction is given by the relation, $\theta=\frac{\frac{d}{d^{\prime}} \lambda}{d}=\frac{\lambda}{d^{\prime}}$ Now, each of these infinitesimally small slit sends...
Read More →Given a G.P. with a = 729 and 7th term 64, determine S7.
Question: Given a G.P. with $a=729$ and $7^{\text {th }}$ term 64 , determine $S_{7}$. Solution: $a=729$ $a_{7}=64$ Let $r$ be the common ratio of the G.P. It is known that, $a_{n}=a r^{n-1}$ $a_{7}=a r^{7-1}=(729) r^{6}$ $\Rightarrow 64=729 r^{6}$ $\Rightarrow r^{6}=\frac{64}{729}$ $\Rightarrow r^{6}=\left(\frac{2}{3}\right)^{6}$ $\Rightarrow r=\frac{2}{3}$ Also, it is known that, $S_{n}=\frac{a\left(1-r^{n}\right)}{1-r}$ $\therefore S_{7}=\frac{729\left[1-\left(\frac{2}{3}\right)^{7}\right]}{1...
Read More →For the following bond cleavages, use curved-arrows to show the electron flow and classify each as homolysis or heterolysis.
Question: For the following bond cleavages, use curved-arrows to show the electron flow and classify each as homolysis or heterolysis. Identify reactive intermediate produced as free radical, carbocation and carbanion. (a) (b) (c) (d) Solution: (a)The bond cleavage using curved-arrows to show the electron flow of the given reaction can be represented as It is an example of homolytic cleavage as one of the shared pair in a covalent bond goes with the bonded atom. The reaction intermediate formed ...
Read More →Find the inverse of each of the matrices (if it exists).
Question: Find the inverse of each of the matrices (if it exists). $\left[\begin{array}{lll}2 1 3 \\ 4 -1 0 \\ -7 2 1\end{array}\right]$ Solution: Let $A=\left[\begin{array}{lll}2 1 3 \\ 4 -1 0 \\ -7 2 1\end{array}\right]$. We have, $\begin{aligned}|A| =2(-1-0)-1(4-0)+3(8-7) \\ =2(-1)-1(4)+3(1) \\ =-2-4+3 \\ =-3 \end{aligned}$ Now, $A_{11}=-1-0=-1, A_{12}=-(4-0)=-4, A_{13}=8-7=1$ $A_{21}=-(1-6)=5, A_{22}=2+21=23, A_{23}=-(4+7)=-11$ $A_{31}=0+3=3, A_{32}=-(0-12)=12, A_{33}=-2-4=-6$ $\therefore \o...
Read More →The sum of first three terms of a G.P. is 16 and the sum of the next three terms is 128.
Question: The sum of first three terms of a G.P. is 16 and the sum of the next three terms is 128.Determine the first term, the common ratio and the sum ton terms of the G.P. Solution: Let the G.P. be $a, a r, a r^{2}, a r^{3}, \ldots$ According to the given condition, $a+a r+a r^{2}=16$ and $a r^{3}+a r^{4}+a r^{5}=128$ $\Rightarrow a\left(1+r+r^{2}\right)=16 \ldots(1)$ $a r^{3}\left(1+r+r^{2}\right)=128 \ldots(2)$ Dividing equation (2) by (1), we obtain $\frac{a r^{3}\left(1+r+r^{2}\right)}{a\...
Read More →Answer the following questions:
Question: Answer the following questions: (a)When a low flying aircraft passes overhead, we sometimes notice a slight shaking of the picture on our TV screen. Suggest a possible explanation. (b)As you have learnt in the text, the principle of linear superposition of wave displacement is basic to understanding intensity distributions in diffraction and interference patterns. What is the justification of this principle? Solution: (a)Weak radar signals sent by a low flying aircraft can interfere wi...
Read More →A parallel beam of light of wavelength 500 nm falls on a narrow slit and the resulting diffraction pattern is observed on a screen 1 m away.
Question: A parallel beam of light of wavelength 500 nm falls on a narrow slit and the resulting diffraction pattern is observed on a screen 1 m away. It is observed that the first minimum is at a distance of 2.5 mm from the centre of the screen. Find the width of the slit. Solution: Wavelength of light beam,= 500 nm = 500 109m Distance of the screen from the slit,D= 1 m For first minima,n= 1 Distance between the slits =d Distance of the first minimum from the centre of the screen can be obtaine...
Read More →Find the inverse of each of the matrices (if it exists).
Question: Find the inverse of each of the matrices (if it exists). $\left[\begin{array}{lll}1 0 0 \\ 3 3 0 \\ 5 2 -1\end{array}\right]$ Solution: Let $A=\left[\begin{array}{ccc}1 0 0 \\ 3 3 0 \\ 5 2 -1\end{array}\right]$ We have, $|A|=1(-3-0)-0+0=-3$ Now, $A_{11}=-3-0=-3, A_{12}=-(-3-0)=3, A_{13}=6-15=-9$ $A_{21}=-(0-0)=0, A_{22}=-1-0=-1, A_{23}=-(2-0)=-2$ $A_{31}=0-0=0, A_{32}=-(0-0)=0, A_{33}=3-0=3$ $\therefore \operatorname{adj} A=\left[\begin{array}{ccc}-3 0 0 \\ 3 -1 0 \\ -9 -2 3\end{array}...
Read More →Two towers on top of two hills are 40 km apart.
Question: Two towers on top of two hills are 40 km apart. The line joining them passes 50 m above a hill halfway between the towers. What is the longest wavelength of radio waves, which can be sent between the towers without appreciable diffraction effects? Solution: Distance between the towers,d= 40 km Height of the line joining the hills,d= 50 m. Thus, the radial spread of the radio waves should not exceed 50 km. Since the hill is located halfway between the towers, Fresnels distance can be ob...
Read More →How many terms of G.P. 3, 32, 33, … are needed to give the sum 120?
Question: How many terms of G.P. $3,3^{2}, 3^{3}, \ldots$ are needed to give the sum $120 ?$ Solution: The given G.P. is $3,3^{2}, 3^{3}$ Let $n$ terms of this G.P. be required to obtain the sum as 120 . $S_{n}=\frac{a\left(r^{n}-1\right)}{r-1}$ Here, $a=3$ and $r=3$ $\therefore S_{n}=120=\frac{3\left(3^{n}-1\right)}{3-1}$ $\Rightarrow 120=\frac{3\left(3^{n}-1\right)}{2}$ $\Rightarrow \frac{120 \times 2}{3}=3^{n}-1$ $\Rightarrow 3^{n}-1=80$ $\Rightarrow 3^{n}=81$ $\Rightarrow 3^{n}=3^{4}$ $\ther...
Read More →What is the relationship between the members of following pairs of structures?
Question: What is the relationship between the members of following pairs of structures? Are they structural or geometrical isomers or resonance contributors? (a) (b) (c) Solution: (a)Compounds having the same molecular formula but with different structures are called structural isomers. The given compounds have the same molecular formula but they differ in the position of the functional group (ketone group). In structure I, ketone group is at the C-3 of the parent chain (hexane chain) and in st...
Read More →Find the inverse of each of the matrices (if it exists)
Question: Find the inverse of each of the matrices (if it exists) $\left[\begin{array}{lll}1 2 3 \\ 0 2 4 \\ 0 0 5\end{array}\right]$ Solution: Let $A=\left[\begin{array}{lll}1 2 3 \\ 0 2 4 \\ 0 0 5\end{array}\right]$. We have. $|A|=1(10-0)-2(0-0)+3(0-0)=10$ Now, $A_{11}=10-0=10, A_{12}=-(0-0)=0, A_{13}=0-0=0$ $A_{21}=-(10-0)=-10, A_{22}=5-0=5, A_{23}=-(0-0)=0$ $A_{31}=8-6=2, A_{32}=-(4-0)=-4, A_{33}=2-0=2$ $\therefore a d j A=\left[\begin{array}{ccc}10 -10 2 \\ 0 5 -4 \\ 0 0 2\end{array}\right]...
Read More →Answer the following questions:
Question: Answer the following questions: (a)In a single slit diffraction experiment, the width of the slit is made double the original width. How does this affect the size and intensity of the central diffraction band? (b)In what way is diffraction from each slit related to the interference pattern in a double-slit experiment? (c)When a tiny circular obstacle is placed in the path of light from a distant source, a bright spot is seen at the centre of the shadow of the obstacle. Explain why? (d)...
Read More →The sum of first three terms of a G.P. is and their product is 1. Find the common ratio and the terms.
Question: The sum of first three terms of a G.P. is $\frac{39}{10}$ and their product is 1 . Find the common ratio and the terms. Solution: Let $\frac{a}{r}, a, a r$ be the first three terms of the G.P. $\frac{a}{r}+a+a r=\frac{39}{10}$ $\ldots(1)$ $\left(\frac{a}{r}\right)(a)(a r)=1$ $\ldots(2)$ From (2), we obtain $a^{3}=1$ $\Rightarrow a=1$ (Considering real roots only) Substituting $a=1$ in equation (1), we obtain $\frac{1}{r}+1+r=\frac{39}{10}$ $\Rightarrow 1+r+r^{2}=\frac{39}{10} r$ $\Righ...
Read More →In double-slit experiment using light of wavelength 600 nm,
Question: In double-slit experiment using light of wavelength 600 nm, the angular width of a fringe formed on a distant screen is 0.1. What is the spacing between the two slits? Solution: Wavelength of light used,= 6000 nm = 600 109m Angular width of fringe, $\theta=0.1^{\circ}=0.1 \times \frac{\pi}{180}=\frac{3.14}{1800} \mathrm{rad}$ Angular width of a fringe is related to slit spacing (d) as: $\theta=\frac{\lambda}{d}$ $d=\frac{\lambda}{\theta}$ $=\frac{600 \times 10^{-9}}{\frac{3.14}{1800}}=...
Read More →Find the inverse of each of the matrices (if it exists).
Question: Find the inverse of each of the matrices (if it exists). $\left[\begin{array}{ll}-1 5 \\ -3 2\end{array}\right]$ Solution: Let $A=\left[\begin{array}{ll}-1 5 \\ -3 2\end{array}\right]$ we have, $|A|=-2+15=13$ Now, $A_{11}=2, A_{12}=3, A_{21}=-5, A_{22}=-1$ $\therefore \operatorname{adj} A=\left[\begin{array}{ll}2 -5 \\ 3 -1\end{array}\right]$ $\therefore A^{-1}=\frac{1}{|A|}$ adjA $=\frac{1}{13}\left[\begin{array}{ll}2 -5 \\ 3 -1\end{array}\right]$...
Read More →For sound waves, the Doppler formula for frequency shift differs slightly between the two situations:
Question: For sound waves, the Doppler formula for frequency shift differs slightly between the two situations: (i) source at rest; observer moving, and (ii) source moving; observer at rest. The exact Doppler formulas for the case of light waves in vacuum are, however, strictly identical for these situations. Explain why this should be so. Would you expect the formulas to be strictly identical for the two situations in case of light travelling in a medium? Solution: No Sound waves can propagate ...
Read More →Classify the following reactions in one of the reaction type studied in this unit.
Question: Classify the following reactions in one of the reaction type studied in this unit. (a) $\mathrm{CH}_{3} \mathrm{CH}_{2} \mathrm{Br}+\mathrm{HS}^{-} \rightarrow \mathrm{CH}_{3} \mathrm{CH}_{2} \mathrm{SH}+\mathrm{Br}^{-}$ (b) $\left(\mathrm{CH}_{3}\right)_{2} \mathrm{C}=\mathrm{CH}_{2}+\mathrm{HCl} \rightarrow\left(\mathrm{CH}_{3}\right)_{2} \mathrm{ClC}-\mathrm{CH}_{3}$ (c) $\mathrm{CH}_{3} \mathrm{CH}_{2} \mathrm{Br}+\mathrm{HO}^{-} \rightarrow \mathrm{CH}_{2}=\mathrm{CH}_{2}+\mathrm{...
Read More →Let us list some of the factors, which could possibly influence the speed of wave propagation:
Question: Let us list some of the factors, which could possibly influence the speed of wave propagation: (i)Nature of the source. (ii)Direction of propagation. (iii)Motion of the source and/or observer. (iv)Wave length. (v)Intensity of the wave. On which of these factors, if any, does (a)The speed of light in vacuum, (b)The speed of light in a medium (say, glass or water), depend? Solution: (a)Thespeed of light in a vacuum i.e., 3 108m/s (approximately) is a universal constant. It is not affecte...
Read More →Find the inverse of each of the matrices (if it exists).
Question: Find the inverse of each of the matrices (if it exists). $\left[\begin{array}{rr}2 -2 \\ 4 3\end{array}\right]$ Solution: Let $A=\left[\begin{array}{rr}2 -2 \\ 4 3\end{array}\right]$. we have. $|A|=6+8=14$ Now, $A_{11}=3, A_{12}=-4, A_{21}=2, A_{22}=2$ $\therefore \operatorname{adj} A=\left[\begin{array}{ll}3 2 \\ -4 2\end{array}\right]$ $\therefore A^{-1}=\frac{1}{|A|}$ adjA $=\frac{1}{14}\left[\begin{array}{ll}3 2 \\ -4 2\end{array}\right]$...
Read More →Evaluate $sum_{k=1}^{11}left(2+3^{k} ight)$
Question: Evaluate $\sum_{k=1}^{11}\left(2+3^{k}\right)$ Solution: $\sum_{k=1}^{11}\left(2+3^{k}\right)=\sum_{k=1}^{11}(2)+\sum_{k=1}^{11} 3^{k}=2(11)+\sum_{k=1}^{11} 3^{k}=22+\sum_{k=1}^{11} 3^{k}$ $\ldots(1)$ $\sum_{k=1}^{11} 3^{k}=3^{1}+3^{2}+3^{3}+\ldots+3^{11}$ The terms of this sequence $3,3^{2}, 3^{3}, \ldots$ forms a G.P. $S_{n}=\frac{a\left(r^{n}-1\right)}{r-1}$ $\Rightarrow \mathrm{S}_{\|}=\frac{3\left[(3)^{\|}-1\right]}{3-1}$ $\Rightarrow S_{11}=\frac{3}{2}\left(3^{11}-1\right)$ $\the...
Read More →Identify the reagents shown in bold in the following equations as nucleophiles or electrophiles:
Question: Identify the reagents shown in bold in the following equations as nucleophiles or electrophiles: Solution: Electrophiles are electron-deficient species and can receive an electron pair. On the other hand, nucleophiles are electron-rich species and can donate their electrons. Here, $\mathrm{HO}^{-}$acts as a nucleophile as it is an electron-rich species, i.e., it is a nucleus-seeking species. Here, ${ }^{-} \mathrm{CN}$ acts as a nucleophile as it is an electron-rich species, i.e., it i...
Read More →You have learnt in the text how Huygens’ principle leads to the laws of reflection and refraction.
Question: You have learnt in the text how Huygens principle leads to the laws of reflection and refraction. Use the same principle to deduce directly that a point object placed in front of a plane mirror produces a virtual image whose distance from the mirror is equal to the object distance from the mirror. Solution: Let an object at O be placed in front of a plane mirror MO at a distancer(as shown in the given figure). A circle is drawn from the centre (O) such that it just touches the plane mi...
Read More →Explain how Corpuscular theory predicts the speed of light in a medium, say,
Question: Explain how Corpuscular theory predicts the speed of light in a medium, say, water, to be greater than the speed of light in vacuum. Is the prediction confirmed by experimental determination of the speed of light in water? If not, which alternative picture of light is consistent with experiment? Solution: No; Wave theory Newtons corpuscular theory of light states that when light corpuscles strike the interface of two media from a rarer (air) to a denser (water) medium, the particles ex...
Read More →Find the sum to n terms in the geometric progression
Question: Find the sum tonterms in the geometric progression$x^{3}, x^{5}, x^{7} \ldots($ if $x \neq \pm 1)$ Solution: The given G.P. is $x^{3}, x^{5}, x^{7}, \ldots$ Here, $a=x^{3}$ and $r=x^{2}$ $S_{n}=\frac{a\left(1-r^{n}\right)}{1-r}=\frac{x^{3}\left[1-\left(x^{2}\right)^{n}\right]}{1-x^{2}}=\frac{x^{3}\left(1-x^{2 n}\right)}{1-x^{2}}$...
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