Question: Consider the force $\mathrm{F}$ on a charge ' $q$ ' due to a uniformly charged spherical shell of radius $R$ carrying charge $Q$ distributed uniformly over it. Which one of the following statements is true for F, if ' $q$ ' is placed at distance $r$ from the centre of the shell?(1) $\mathrm{F}=\frac{1}{4 \pi \varepsilon_{0}} \frac{Q q}{R^{2}}$ for $rR$(2) $\frac{1}{4 \pi \varepsilon_{0}} \frac{Q q}{R^{2}}\mathrm{F}0$ for $rR$(3) $\mathrm{F}=\frac{1}{4 \pi \varepsilon_{0}} \frac{Q q}{R^...

Find the roots of each of the following equations, if they exist, by applying the quadratic formula:

Question: Find the roots of each of the following equations, if they exist, by applying the quadratic formula: $2 x^{2}+x-4=0$ Solution: The given equation is $2 x^{2}+x-4=0$. Comparing it with $a x^{2}+b x+c=0$, we get a= 2,b= 1 andc= 4 $\therefore$ Discriminant, $D=b^{2}-4 a c=(1)^{2}-4 \times 2 \times(-4)=1+32=330$ So, the given equation has real roots. Now, $\sqrt{D}=\sqrt{33}$ $\therefore \alpha=\frac{-b+\sqrt{D}}{2 a}=\frac{-1+\sqrt{33}}{2 \times 2}=\frac{-1+\sqrt{33}}{4}$ $\beta=\frac{-b-...

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Question: If $\alpha=\cos ^{-1}\left(\frac{3}{5}\right), \beta=\tan ^{-1}\left(\frac{1}{3}\right)$, where $0\alpha, \beta\frac{\pi}{2}$, then $\alpha-\beta$ is equal to : (1) $\tan ^{-1}\left(\frac{9}{5 \sqrt{10}}\right)$(2) $\cos ^{-1}\left(\frac{9}{5 \sqrt{10}}\right)$(3) $\tan ^{-1}\left(\frac{9}{14}\right)$(4) $\sin ^{-1}\left(\frac{9}{5 \sqrt{10}}\right)$Correct Option: , 4 Solution: $\because \cos \alpha=\frac{3}{5}$, then $\sin \alpha=\frac{4}{5}$ $\Rightarrow \tan \alpha=\frac{4}{3}$ and...

Solve this

Question: $x^{2}-6 x+4=0$ Solution: Given : $x^{2}-6 x+4=0$ On comparing it with $a x^{2}+b x+c=0$, we get: $a=1, b=-6$ and $c=4$ Discriminant $D$ is given by : $D=\left(b^{2}-4 a c\right)$ $=(-6)^{2}-4 \times 1 \times 4$ $=36-16$ $=200$ Hence, the roots of the equation are real. Roots $\alpha$ and $\beta$ are given by : $\alpha=\frac{-b+\sqrt{D}}{2 a}=\frac{-(-6)+\sqrt{20}}{2 \times 1}=\frac{6+2 \sqrt{5}}{2}=\frac{2(3+\sqrt{5})}{2}=(3+\sqrt{5})$ $\beta=\frac{-b-\sqrt{D}}{2 a}=\frac{-(-6)-\sqrt{...

If S is the sum of the first 10 terms of the series

Question: If $S$ is the sum of the first 10 terms of the series $\tan ^{-1}\left(\frac{1}{3}\right)+\tan ^{-1}\left(\frac{1}{7}\right)+\tan ^{-1}\left(\frac{1}{13}\right)+\tan ^{-1}\left(\frac{1}{21}\right)+\ldots .$ then $\tan (S)$ is equal to:(1) $\frac{5}{6}$(2) $\frac{5}{11}$(3) $-\frac{6}{5}$(4) $\frac{10}{11}$Correct Option: 1 Solution: $S=\tan ^{-1}\left(\frac{1}{3}\right)+\tan ^{-1} \frac{1}{7}+\tan ^{-1} \frac{1}{13}+\ldots .$ upto 10 terms $=\tan ^{-1}\left(\frac{2-1}{1+2 \cdot 1}\righ...

The calculated magnetic moments (spin only value) for species

Question: The calculated magnetic moments (spin only value) for species $\left[\mathrm{FeCl}_{4}\right]^{2-},\left[\mathrm{Co}\left(\mathrm{C}_{2} \mathrm{O}_{4}\right)_{3}\right]^{3-}$ and $\mathrm{MnO}_{4}^{2-}$ respectively are :$5.92,4.90$ and $0 \mathrm{BM}$$5.82, \mathrm{O}$ and $0 \mathrm{BM}$$4.90,0$ and $1.73 \mathrm{BM}$$4.90,0$ and $2.83 \mathrm{BM}$Correct Option: , 3 Solution: $\left[\mathrm{FeCl}_{4}\right]^{2-} \mathrm{Fe}^{2+} 3 \mathrm{~d}^{6} \rightarrow 4$ unpaired electron. a...

Two identical electric point dipoles have dipole moments

Question: Two identical electric point dipoles have dipole moments $\overrightarrow{\mathrm{P}_{1}}=\mathrm{P} \hat{i}$ and $\overrightarrow{\mathrm{P}_{2}}=-\mathrm{P} \hat{i}$ and are held on the $x$ axis at distance 'a' from each other. When released, they move along $x$-axis with the direction of their dipole moments remaining unchanged. If the mass of each dipole is ' $\mathrm{m}$ ', their speed when they are infinitely far apart is :(1) $\frac{\mathrm{P}}{a} \sqrt{\frac{1}{\pi \varepsilon_...

Find the roots of each of the following equations, if they exist, by applying the quadratic formula:

Question: Find the roots of each of the following equations, if they exist, by applying the quadratic formula: $x^{2}-4 x-1=0$ Solution: Given : $x^{2}-4 x-1=0$ On comparing it with $a x^{2}+b x+c=0$, we get : $a=1, b=-4$ and $c=-1$ Discriminant $D$ is given by : $D=\left(b^{2}-4 a c\right)$ $=(-4)^{2}-4 \times 1 \times(-1)$ $=16+4$ $=20$ $=200$ Hence, the roots of the equation are real. Roots $\alpha$ and $\beta$ are given by : $\alpha=\frac{-b+\sqrt{D}}{2 a}=\frac{-(-4)+\sqrt{20}}{2 \times 1}=...

is equal to:

Question: $2 \pi-\left(\sin ^{-1} \frac{4}{5}+\sin ^{-1} \frac{5}{13}+\sin ^{-1} \frac{16}{65}\right)$ is equal to:(1) $\frac{\pi}{2}$(2) $\frac{5 \pi}{4}$(3) $\frac{3 \pi}{2}$(4) $\frac{7 \pi}{4}$Correct Option: , 3 Solution: $2 \pi-\left(\sin ^{-1} \frac{4}{5}+\sin ^{-1} \frac{5}{13}+\sin ^{-1} \frac{16}{65}\right)$ $=2 \pi-\left(\tan ^{-1} \frac{4}{3}+\tan ^{-1} \frac{5}{12}+\tan ^{-1} \frac{16}{63}\right)$ $\left[\because \sin ^{-1} \frac{4}{5}=\tan ^{-1} \frac{4}{3}\right]$ $=2 \pi-\left\{\...

The total number of unpaired electrons present in the complex

Question: The total number of unpaired electrons present in the complex $\mathrm{K}_{3}\left[\mathrm{Cr}(\text { oxalate })_{3}\right]$ is __________________ . Solution: (3) $\mathrm{K}_{3}\left[\mathrm{Cr}(\text { oxalate })_{3}\right]$Chromium is in $+3$ oxidation state. Number of unpaired electrons in $\mathrm{Cr}^{+3}$ will be 3 ....

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Question: If $y=\sum_{k=1}^{6} k \cos ^{-1}\left\{\frac{3}{5} \cos k x-\frac{4}{5} \sin k x\right\}$, then $\frac{d y}{d x}$ at $x=0$ is_____________. Solution: $y=\sum_{k=1}^{6} k \cos ^{-1}\left\{\frac{3}{5} \cos k x-\frac{4}{5} \sin k x\right\}$ Let $\cos a=\frac{3}{5}$ and $\sin a=\frac{4}{5}$ $\therefore y=\sum_{k=1}^{6} k \cos ^{-1}\{\cos a \cos k x-\sin a \sin k x\}$ $=\sum_{k=1}^{6} k \cos ^{-1}(\cos (k x+a))=\sum_{k=1}^{6} k(k x+a)=\sum_{k=1}^{6}\left(k^{2} x+a k\right)$ $\therefore \fr...

The correct structures of trans

Question: The correct structures of trans-[NiBr $\left.{ }_{2}\left(\mathrm{PPh}_{3}\right)_{2}\right]$ and meridonial-[ $\mathrm{Co}(\mathrm{NH}$$\left.3)_{3}\left(\mathrm{NO}_{2}\right)_{3}\right]$, respectively, areCorrect Option: , 4 Solution:...

Charges Q1 and Q2 are at points A and B of a right angle triangle OAB (see figure).

Question: Charges $Q_{1}$ and $Q_{2}$ are at points A and B of a right angle triangle OAB (see figure). The resultant electric field at point $\mathrm{O}$ is perpendicular to the hypotenuse, then $Q_{1} / Q_{2}$ is proportional to : (1) $\frac{x_{1}{ }^{3}}{x_{2}{ }^{3}}$(2) $\frac{x_{2}}{x_{1}}$(3) $\frac{x_{1}}{x_{2}}$(4) $\frac{x_{2}^{2}}{x_{1}^{2}}$Correct Option: , 3 Solution: (3) Electric field due charge $Q_{2}, E_{2}=\frac{k Q_{2}}{x_{2}^{2}}$ Electric field due charge $Q_{1}, E_{1}=\fra...

Find the discriminant of each of the following equations:

Question: Find the discriminant of each of the following equations: (i) $2 x^{2}-7 x+6=0$ (ii) $3 x^{2}-2 x+8=0$ (iii) $2 x^{2}-5 \sqrt{2} x+4=0$ (iv) $\sqrt{3} x^{2}+2 \sqrt{2} x-2 \sqrt{3}=0$ (v) $(x-1)(2 x-1)=0$ (vi) $1-x=2 x^{2}$ Solution: (i) $2 x^{2}-7 x+6=0$ Here, $a=2$ $b=-7$ $c=6$ Discriminant $D$ is diven by : $D=b^{2}-4 a c$ $=(-7)^{2}-4 \times 2 \times 6$ $=49-48$ $=1$ (ii) $3 x^{2}-2 x+8=0$ Here, $a=3$, $b=-2$, $c=8$ Discriminant $D$ is given by : $D=b^{2}-4 a c$ $=(-2)^{2}-4 \times...

if 0<a , b<1, and

Question: If $0a, b1$, and $\tan ^{-1} a+\tan ^{-1} b=\frac{\pi}{4}$, then the value of $(a+b)-\left(\frac{a^{2}+b^{2}}{2}\right)+\left(\frac{a^{3}+b^{3}}{3}\right)-\left(\frac{a^{4}+b^{4}}{4}\right)+\ldots$ is :(1) $\log _{e} 2$(2) $\log _{e}\left(\frac{e}{2}\right)$(3) $e$(4) $e^{2}-1$Correct Option: 1, Solution: $\tan ^{-1}\left(\frac{a+b}{1-a b}\right)=\frac{\pi}{4} \Rightarrow a+b=1-a b \Rightarrow(1+a)(1+b)=2$ Now, $(a+b)-\left(\frac{a^{2}+b^{2}}{2}\right)+\left(\frac{a^{3}+b^{3}}{3}\right...

Match List-I with List-II :

Question: Match List-I with List-II : Choose the correct answer from the options given below :$(\mathrm{a})-(\mathrm{iii}),(\mathrm{b})-(\mathrm{i}),(\mathrm{c})-(\mathrm{ii}),(\mathrm{d})-(\mathrm{iv})$$(\mathrm{a})-(\mathrm{iv}),(\mathrm{b})-(\mathrm{ii}),(\mathrm{c})-(\mathrm{iii}),(\mathrm{d})-(\mathrm{i})$(a)-(ii), (b)-(i), (c)-(iii), (d)-(iv)(a)-(i), (b)-(ii), (c)-(iii), (d)-(iv)Correct Option: 1 Solution:...

Match List-I with List-II :

Question: Match List-I with List-II : Choose the correct answer from the options given below :$(\mathrm{a})-(\mathrm{iii}),(\mathrm{b})-(\mathrm{i}),(\mathrm{c})-(\mathrm{ii}),(\mathrm{d})-(\mathrm{iv})$$(\mathrm{a})-(\mathrm{iv}),(\mathrm{b})-(\mathrm{ii}),(\mathrm{c})-(\mathrm{iii}),(\mathrm{d})-(\mathrm{i})$(a)-(ii), (b)-(i), (c)-(iii), (d)-(iv)(a)-(i), (b)-(ii), (c)-(iii), (d)-(iv)Correct Option: , 3 Solution:...

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Question: If $\frac{\sin ^{-1} x}{a}=\frac{\cos ^{-1} x}{b}=\frac{\tan ^{-1} y}{c} ; 0x1$, then the value of $\cos \left(\frac{\pi c}{a+b}\right)$ is:(1) $\frac{1-y^{2}}{2 y}$(2) $\frac{1-y^{2}}{1+y^{2}}$(3) $1-y^{2}$(4) $\frac{1-y^{2}}{y \sqrt{y}}$Correct Option: , 2 Solution: $\frac{\sin ^{-1} x}{a}=\frac{\cos ^{-1} x}{b}=\frac{\tan ^{-1} y}{c}$ $\frac{\sin ^{-1} x}{a}=\frac{\cos ^{-1} x}{b}=\frac{\sin ^{-1} x+\cos ^{-1} x}{a+b}=\frac{\pi}{2(a+b)}$ Now, $\frac{\tan ^{-1} y}{c}=\frac{\pi}{2(a+b...

Ten charges are placed on the circumference of a circle of radius R with constant angular separation between successive charges.

Question: Ten charges are placed on the circumference of a circle of radius $R$ with constant angular separation between successive charges. Alternate charges $1,3,5,7,9$ have charge $(+q)$ each, while $2,4,6,8,10$ have charge $(-q)$ each. The potential $V$ and the electric field $E$ at the centre of the circle are respectively: (Take $V=0$ at infinity)(1) $V=\frac{10 q}{4 \pi \varepsilon_{0} R} ; E=0$(2) $V=0 ; E=\frac{10 q}{4 \pi \varepsilon_{0} R^{2}}$(3) $V=0 ; E=0$(4) $V=\frac{10 q}{4 \pi \...

is equal to:

Question: $\operatorname{cosec}\left[2 \cot ^{-1}(5)+\cos ^{-1}\left(\frac{4}{5}\right)\right]$ is equal to: (1) $\frac{75}{56}$(2) $\frac{65}{56}$(3) $\frac{56}{33}$(4) $\frac{65}{33}$Correct Option: 2, Solution: $\operatorname{cosec}\left(2 \cot ^{-1}(5)+\cos ^{-1}\left(\frac{4}{5}\right)\right)$ $\operatorname{cosec}\left(2 \tan ^{-1}\left(\frac{1}{5}\right)+\cos ^{-1}\left(\frac{4}{5}\right)\right)$ $=\operatorname{cosec}\left(\tan ^{-1}\left(\frac{{ }^{2}\left(\frac{1}{5}\right)}{1-\left(\f...

Solve the following

Question: $\left[\mathrm{Ti}\left(\mathrm{H}_{2} \mathrm{O}\right)_{6}\right]^{3+}$ absorbs light of wavelength $498 \mathrm{~nm}$ during a $\mathrm{d}-\mathrm{d}$ transition. The octahedral splitting energy for the above complex is $\times 10^{-19} \mathrm{~J}$. (Round off to the Nearest Integer). $\mathrm{h}=6.626 \times 10^{-34} \mathrm{~J} \mathrm{~s} ; \mathrm{c}=3 \times 10^{8} \mathrm{~ms}^{-1}$ Solution: (4) $\lambda_{\text {absonbed }}=498 \mathrm{~nm}$ (given) The octahedral spilitting...

A solid sphere of radius R carries a charge Q+q distributed uniformly over its volume.

Question: A solid sphere of radius $R$ carries a charge $Q+q$ distributed uniformaly over its volume. A very small point like piece of it of mass $m$ gets detached from the bottom of the sphere and falls down vertically under gravity. This piece carries charge $q$. If it acquires a speed $v$ when it has fallen through a vertical height $y$ (see figure), then : (assume the remaining portion to be spherical). (1) $v^{2}=y\left[\frac{q Q}{4 \pi \varepsilon_{0} R^{2} y m}+g\right]$(2) $v^{2}=y\left[...

Solve this

Question: $2^{2 x}-3 \cdot 2^{(x+2)}+32=0$ Solution: Given : $2^{2 x}-3.2^{(x+2)}+32=0$ $\Rightarrow\left(2^{x}\right)^{2}-3 \cdot 2^{x} \cdot 2^{2}+32=0$ Let $2^{x}$ be $y$. $\therefore y^{2}-12 y+32=0$ $\Rightarrow y^{2}-8 y-4 y+32=0$ $\Rightarrow y(y-8)-4(y-8)=0$ $\Rightarrow(y-8)=0$ or $(y-4)=0$ $\Rightarrow y=8$ or $y=4$ $\therefore 2^{x}=8$ or $2^{x}=4$ $\Rightarrow 2^{x}=2^{3}$ or $2^{x}=2^{2}$ $\Rightarrow x=2$ or 3 Hence, 2 and 3 are the roots of the given equation....

The equivalents of ethylene diamine required to replace the neutral ligands from

Question: The equivalents of ethylene diamine required to replace the neutral ligands from the coordination sphere of the trans-complex of $\mathrm{CoCl}_{3} \cdot 4 \mathrm{NH}_{3}$ is ___________ . (Round off to the Nearest Integer). Solution: (2) trans $-\mathrm{CoCl}_{3} \cdot 4 \mathrm{NH}_{3}$ Or $\because$ trans $-\left[\mathrm{Co}\left(\mathrm{NH}_{3}\right)_{4} \mathrm{Cl}_{2}\right] \mathrm{C} \ell$ As we know that ethylene diamine is a bidentate ligand and ammonia is a mono dentate li...

A possible value of

Question: A possible value of $\tan \left(\frac{1}{4} \sin ^{-1} \frac{\sqrt{63}}{8}\right)$ is:(1) $\frac{1}{2 \sqrt{2}}$(2) $\frac{1}{\sqrt{7}}$(3) $\sqrt{7}-1$(4) $2 \sqrt{2}-1$Correct Option: , 2 Solution: $\tan \left(\frac{1}{4} \sin ^{-1} \frac{\sqrt{63}}{8}\right)$ Let $\sin ^{-1}\left(\frac{\sqrt{63}}{8}\right)=\theta \quad \sin \theta=\frac{\sqrt{63}}{8}$ $\cos \theta=\frac{1}{8}$ $2 \cos ^{2} \frac{\theta}{2}-1=\frac{1}{8}$ $\cos ^{2} \frac{\theta}{2}=\frac{9}{16}$ $\cos \frac{\theta}{...