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A compound made up of A and B atoms have a crystalline structure ,in which A forms Hexagonal close packed structure and B occupies $\;\large\frac{2}{3}\;$ of octahedral holes .What will be the simplest molecular formula ?
answered Jun 7, 2014
Answer : $A_{3}B_{3} $Explanation :Effective no of A atoms forming HCP=6Effective no of octahedral h...
0 votes
potassium crystallizes in a bode centred cubic lattice .What is the approximate number unit cells in 4.0 g of potassium ? Atomic mass of potassium =39.
answered Jun 7, 2014
Answer : $3.09 \times 10^{22}$Explanation :No of atoms per unit $=8 \times \large\frac{1}{8}+\norma...
0 votes
A compound formed by elements A and B has a cubic structure in which A atoms are at the corners of the cube and B atoms are at face centres .Derive the formula of the compound
answered Jun 7, 2014
Answer : $\;AB_{3}$Explanation :As 'A' atom are present at the 8 corners of the cube therefore no o...
0 votes
Which one of the graphs below best illustrates the relationship between internal energy U and temperature T of the gas in K ?
answered Apr 22, 2014
Answer : https://clay6.com/mpaimg/5_figure1.PNGExplanation :$U = nC_{V}T$As T increases U also incr...
0 votes
$\;C_{p}\;$ and $\;C_{v}\;$ denote the molar specific heat capacities of a gas at constant pressure and constant volume respectively . Then
answered Apr 22, 2014
Answer : $\;C_{p}+C_{v}\;$ is larger for a diatomic gas than a monoatomic gas.Explanation :$C_{v} ...
0 votes
Number of solutions of the equation $\;z^{2}+|z|^{2} =0\;$ is
answered Apr 19, 2014
Answer : $\;infinitely \;many $Explanation :$\;z^{2}+|z|^{2} =0\; ,z \neq 0$$x^{2}-y^{2} + 2 i xy +...
0 votes
The amplitude of $\; sin \large\frac{\pi}{5} + i\; \normalsize cos (1-\large\frac{\pi}{5})\;$ is
answered Apr 19, 2014
Answer : $\;\large\frac{\pi}{10}$Explanation :We have , $\; sin \large\frac{\pi}{5} + i\; \normal...
0 votes
The equation $\;|z+1-i|=|z-1+i|\;$ represents a
answered Apr 19, 2014
Answer : straight lineExplanation :$\;|z+1-i|=|z-1+i|\;$$|z-(-1+i)|=|z-(1-i)|$PA=PA , where A denote...
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The area of the triangle on the complex plane formed by the complex number $\;z \;,iz \;,$ and $\;z+iz\;$ is :
answered Apr 19, 2014
Answer : $\;\large\frac{|z|^{2}}{2}$Explanation :Let $\;z=x+iy\;.$ Then $\;-iz=y-ix$.Therefore , ...
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If the complex number $\;z=x+iy\;$ satisfies the condition $\;|z+1|=1\;$ , then z lies on
answered Apr 18, 2014
Answer : Circle with centre (-1,0) and radius 1Explanation :$|z+1| =1$$|(x+1)+iy| =1$$(x+1)^{2} + ...
0 votes
$1+i+i^{2}+i^{3} + i^{4}+i^{5}+i^{6}---+i^{2n}\;$ is
answered Apr 18, 2014
Answer : Can not be evaluatedExplanation :$1+i+i^{2}+i^{3} + i^{4}+i^{5}+i^{6}---+i^{2n}\; =1-1+1-1-...
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If $\;1-i\;$ is a root of the equation $\;x^{2} +ax + b=0\;$ where $\;a,b \in R\;$ , then find the values of a and b .
answered Apr 18, 2014
Answer : -2 , 2Explanation :Sum of roots $\; \large\frac{-a}{1} = (1-i)+(1+i)$$a=-2$(Since non rea...
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What is the locus of z , if amplitude of $\;z-2-3i\;$ is $\;\large\frac{\pi}{4}\;$ ?
answered Apr 18, 2014
Answer : Straight lineExplanation :Let $\; z=x+iy$Then , $\;z-2-3i\; = (x-2)+i(y-3)$ Let $\;\theta\...
0 votes
What is the polar form of the complex number $\;(i^{25})^{3} \;$ ?
answered Apr 18, 2014
Answer : $\; cos \large\frac{\pi}{2} - i sin \large\frac{\pi}{2}$Explanation :$\;z=(i^{25})^{3} \; ...
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What is the principle value of amplitude of $\;1-i\;$ ?
answered Apr 18, 2014
Answer : $\;-\large\frac{\pi}{4}$Explanation :Let $\;\theta\;$ be the principle value of amplitude...
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What is the conjugate of $\; \large\frac{\sqrt{5+12 i } +\sqrt{5-12 i } } {\sqrt{5+12 i } -\sqrt{5-12 i } }$
answered Apr 18, 2014
Answer : $\;0 + \large\frac{3}{2} i$Explanation :Let $\; z= \large\frac{\sqrt{5+12 i } +\sqrt{5-1...
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If $\;z_{1} = \sqrt{3} + i \sqrt{3}\;$ and $\;z_{2} =\sqrt{3} + i\;$ , then find the quadrant in which $\;(\large\frac{z_{1}}{z_{2}})\;$ lies
answered Apr 18, 2014
Answer : $\;(\large\frac{3+ \sqrt{3}}{4})+(\large\frac{3- \sqrt{3}}{4})i$Explanation :$\large\frac{...
0 votes
What is the reciprocal of $\;3+ \sqrt{7} i$ ?
answered Apr 18, 2014
Answer : $\;\large\frac{3}{16} - \large\frac{\sqrt{7}}{16} i$Explanation :Reciprocal of $\;z = \l...
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What is the smallest positive integer n , for which $\; (1+i)^{2n} = (1-i)^{2n}$
answered Apr 18, 2014
Answer : 2Explanation :n=2 , because $\; (1+i)^{2n} = (1-i)^{2n}$$(\large\frac{1+i}{1-i})^{2n} =1$...
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What is the value of $\; \large\frac{i^{4n+1} - i^{4n-1}}{2}\;$ ?
answered Apr 18, 2014
Answer : $\;i$Explanation :$\; \large\frac{i^{4n+1} - i^{4n-1}}{2}\; = \large\frac{i^{4n} . i - i^{...
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State true or false for the following : If n is a positive integer , then the value of $\;i^{n} + i^{n+1} + i^{n+2} + i^{n+3} \;$ is 0
answered Apr 18, 2014
Answer : TrueExplanation :True , because $\;i^{n} + i^{n+1} + i^{n+2} + i^{n+3} \;$ $= i^{n}(1 +...
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State true or false for the following : If three complex numbers $\;z_{1} , z_{2}\;$ and $\;z_{3}\;$ are in A.P. then they lie on a circle in the complex plane .
answered Apr 18, 2014
Answer : FalseExplanation :False , because if $\;z_{1} , z_{2}\;$ and $\;z_{3}\;$ are in A.P. th...
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State true or false for the following : The points representing the complex number z for which $\;|z+1| < |z-1|\;$ lies in the interior of a circle
answered Apr 18, 2014
Answer : FalseExplanation :False , because $\;|x+iy+1| < |x+iy-1|\;$$|(x+1)^{2} + y^{2}| < ...
0 votes
State true or false for the following : The argument of the complex number $\;z=(1+ i\sqrt{3})(1+i)(cos \theta + i sin \theta)\;$ is $\; \large\frac{7 \pi}{12} + \theta$
answered Apr 18, 2014
Answer : TrueExplanation :True , because $\;arg(z)=arg(1+ i\sqrt{3})+ arg(1+i) +arg(cos \theta + i...
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State true or false for the following : If a complex number coincides with its conjugate , then the number must lie on imaginary axis .
answered Apr 18, 2014
Answer : FalseExplanation : False , because $\;x+iy=x-iy\;$ => $\;y=0\;$ => number lies on ...
0 votes
State true or false for the following : The complex number $\;cos \theta + i sin \theta\;$ can be zero for some $\;\theta$
answered Apr 18, 2014
Answer : FalseExplanation :False , because $\;cos \theta + i sin \theta =0\;$ => $\;cos \theta...
0 votes
State true or false for the following : Multiplication of non-zero complex number by i rotates it through a right angle in the anti clock - wise direction .
answered Apr 18, 2014
Answer : TrueExplanation :Let $\;z=2+3i\;$ be a complex number represented by OP . then $\;iz=-3+2i...
0 votes
If $\;(2+i)(2+2i)(2+3i)....(2+ni)=x+iy\;$ , then$\;5\;.8\;.13....(4+n^{2})=$----
answered Apr 18, 2014
Answer : $\;x^{2}+y^{2} $Explanation :Given that , $\;(2+i)(2+2i)(2+3i)....(2+ni)=x+iy\;$---(1)$\o...
0 votes
If a complex number lies in the third quadrant , then its conjugate lies in the ----
answered Apr 18, 2014
Answer : Second quadrantExplanation :Conjugate of a complex number is the image of the complex num...
0 votes
The conjugate of the complex number $\;\large\frac{1-i}{1+i}\;$ is
answered Apr 17, 2014
Answer : iExplanation :$ \large\frac{1-i}{1+i} = \large\frac{1-i}{1+i} \times \large\frac{1-i}{1-i}...
0 votes
The value of $\;(-\sqrt{-1})^{4n-3}\;$ , where $\;n \in N \;$ is
answered Apr 17, 2014
Answer : $\;-i$Explanation :$\;(-\sqrt{-1})^{4n-3}\; = (-i)^{4n-3} = (-i)^{4n} \; . (-i)^{-3}$...
0 votes
The locus of $z$ satisfying $\;arg(z) = \large\frac{\pi}{3}\;$ is
answered Apr 17, 2014
Answer : $\;\sqrt{3} x$Explanation :Let $\;z=x+iy\;$ then its polar form is $\;z=r(cos \theta + i s...
0 votes
If $\;|z|=2\;$ and $\;arg(z) = \large\frac{\pi}{4}\;$ , then $\;z=$
answered Apr 17, 2014
Answer : $\;\sqrt{2}(1+i)$Explanation :$z=|z| (cos \large\frac{\pi}{4} + i sin\large\frac{\pi}{4})$...
0 votes
The real value of $' a '$ for which $\;3i^{3}-2ai^{2}+(1-a)i+5\;$ is real is
answered Apr 17, 2014
Answer : $-\;2$Explanation :$\;3i^{3}-2ai^{2}+(1-a)i+5\; =-3i+2a+5+(1-a)i $$=2a+5+i(-a-2)\;$, which...
0 votes
If $\;z_{1}\;$ and $\;z_{2}\;$ both satisfy $\;z+\overline{z} =2|z-1|\;arg(z_{1}-z_{2}) = \large\frac{\pi}{4}\;,$ then find $\;Im(z_{1}+z_{2})\;.$
answered Apr 17, 2014
Answer : 2Explanation :Let $\;z=x+iy\;,z_{1}=x_{1}+iy_{1}\;$ and $\;z_{2} = x_{2}+iy_{2}$Then , $\;...
0 votes
Find the value of $k$ if for the complex numbers $\;z_{1}\;$ and $\;z_{2}\;,|1-\overline{z_{1}}z_{2}|^{2} - |z_{1} - z_{2}|^{2}= k (1-|z_{1}|^{2}) (1-|z_{2}|^{2})$
answered Apr 17, 2014
Answer : 1Explanation :L.H.S = $|1-\overline{z_{1}}z_{2}|^{2} - |z_{1} - z_{2}|^{2}$$= (1- \overlin...
0 votes
Find the value of a such that the sum of the squares of the roots of the equation $\;x^{2}-(a-2)x-(a+1)=0\;$ is least .
answered Apr 17, 2014
Answer : 1Explanation :Let $\;\alpha \;, \beta\;$ be the roots of the equationTherefore , $\;\alpha...
0 votes
Find the value of P such that the difference of the roots of the equation $\;x^{2}-Px+8=0\;$ is $2$
answered Apr 17, 2014
Answer : $\;\pm6$Explanation :Let $\;\alpha \;, \beta\;$ be the roots of the equation $\;x^{2}-Px...
0 votes
Find the value of $\;2x^{4} + 5 x^{3} +7x^{2} - x+41\;,$ when $\;x=-2-\sqrt{3} i$
answered Apr 17, 2014
Answer : $\;6$Explanation : $\;x=-2-\sqrt{3} i$$\;x+2=-\sqrt{3} i$ => $\;x^{2} +4x+7=0$$\;2x^{4...
0 votes
Locate the points for which $\; 3 < |z| < 4$
answered Apr 17, 2014
Answer : $\;x^{2}+y^{2}=9 \;and\;x^{2}+y^{2} =16$Explanation :$|z| < 4 \; => \; x^{2}+y^{2...
0 votes
If a complex number z lies on the interior or on the boundary of a circle of radius 3 units and centre (-4,0) find the greatest and least values of $\;|z+1| \;.$
answered Apr 17, 2014
Answer : $\;6\;and\;0$Explanation :Distance of the point representing z from the centre of the circ...
0 votes
If $\;z_{1} , z_{2} ,z_{3}\;$ are complex numbers such that $\;|z_{1}|=|z_{2}|=|z_{3}|=|\large\frac{1}{z_{1}} +\large\frac{1}{z_{2}} + \large\frac{1}{z_{3}}|=1\;$ , then find the value of $\;|z_{1}+ z_{2}+z_{3}|\; .$
answered Apr 17, 2014
Answer : $\;1$Explanation :$|z_{1}|=|z_{2}|=|z_{3}| =1$$|z_{1}|^{2}=|z_{2}|^{2}=|z_{3}|^{2} =1$$z...
0 votes
Let $\;z_{1}\;$ and $\;z_{2}\;$ be two complex numbers such that $\;|z_{1}+z_{2}| = |z_{1}|+|z_{2}|\;$ Then show that $\;arg(z_{1}) - arg(z_{2}) =0$
answered Apr 17, 2014
Answer : $\;0$Explanation :Let $\;z_{1}= r_{1}(cos \theta_{1} + i sin \theta_{1}) \;$ and $\;z_{2}...
0 votes
Let $\;z_{1}\;$ and $\;z_{2}\;$ be two complex numbers such that $\;\overline{z_{1} }+ i\overline{z_{2}}=0\;$ and $\;arg(z_{1} z_{2}) = \pi\;$ . Then find $\;arg(z_{1})$
answered Apr 17, 2014
Answer : $\;\large\frac{3 \pi}{4}$Explanation :Given that , $\;\overline{z_{1} }+i \overline{z_{2...
0 votes
If $\;|z^{2}-1| = |z|^{2}+1\;$, then show that $z$ lies on imaginary axis
answered Apr 17, 2014
Answer : z lies on imaginary axisExplanation :Let $\;z=x+iy$Then , $\;|z^{2}-1| = |z|^{2}+1\;$$|...
0 votes
If the imaginary part of $\;\large\frac{2z+1}{iz+1}\;$ is $-2$ , then show that the locus of the point representing $z$ in the argand plane is a straight line .
answered Apr 17, 2014
Answer : $\;x+2y-2=0$Explanation :Let $\; z=x+iy\;$ Then , $\;\large\frac{2z+1}{iz+1}\;= \large\f...
0 votes
Solve the equation $\;z^{2} = \overline{z}\;$ where $\;z=x+iy$
answered Apr 17, 2014
Answer : $\;-\large\frac{1}{2} \pm \large\frac{\sqrt{3}}{2}$Explanation : $\;z^{2} = \overline{z}\;...
0 votes
If $\;(x+iy)^{\large\frac{1}{3}}= a+ib\;$ where $\;x\;,y\;,a\;,b \in R\;$ show that $\; \large\frac{x}{a} - \large\frac{y}{b} = -2 (a^{2} + b^{2})$
answered Apr 17, 2014
Answer : $\;-2(a^{2}+b^{2})$Explanation :$\;(x+iy)^{\large\frac{1}{3}}= a+ib\;$$\;(x+iy)^{\large\fr...
0 votes
Evaluate : $\; (1+i)^{6} +(1-i)^{3}$
answered Apr 17, 2014
Answer : $\;-2-10i$Explanation :$\; (1+i)^{6} = ((1+i)^{2})^{3}$$= (1^{2} + i^{2} + 2i )^{3}$$= (...
0 votes
Convert the complex number $\;z=\large\frac{i-1}{cos \large\frac{\pi}{3} + i sin \large\frac{\pi}{3}}\;$ in the polar form
answered Apr 16, 2014
Answer : $\;\sqrt{2} (cos \large\frac{5 \pi}{12} + i sin \large\frac{5 \pi}{12})$Explanation :We ha...
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