Let `z_1 and z_2`q, be two complex numbers with `alpha and beta` as their principal arguments such that `alpha+beta` then principal `arg(z_1z_2)` is given by:
A. `alpha+beta+pi`
B. `alpha+beta-pi`
C. `alpha+beta-2pi`
D. `alpha+beta`
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If `A(alpha, beta)=[("cos" alpha,sin alpha,0),(-sin alpha,cos alpha,0),(0,0,e^(beta))]`, then `A(alpha, beta)^(-1)` is equal to
Correct Answer - A We have, `A(alpha, beta)^(-1)=1/e^(beta) [(e^(beta) cos alpha,-e^(beta) sin alpha,0),(e^(beta) sin alpha,e^(beta) cos alpha,0),(0,0,1)]` `=A(-alpha, -beta)`
2 Answers 2 views
If `alpha, beta, gamma` are three real numbers and `A=[(1,cos (alpha-beta),cos(alpha-gamma)),(cos (beta-alpha),1,cos (beta-gamma)),(cos (gamma-alpha),
Correct Answer - A::B::D `A=[(cos alpha,sin alpha,0),(cos beta,sin beta,0),(cos gamma,sin gamma,0)][(cos alpha,cos beta,cos gamma),(sin alpha,sin beta,sin gamma),(0,0,0)]` Clearly, A is symmetric and `|A|=0`, hence, singular and not invertiable. Also, `A A^(T)...
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The value of `(alpha^3)/2cos e c^2(1/2tan^(-1)alpha/beta)+(beta^3)/2sec^2(1/2tan^(-1)(beta/alpha))i se q u a lto` `(alpha+beta)(alpha^2+beta^2)` (b) `
Correct Answer - C `(alpha^(3))/(2) cosec^(2) ((1)/(2) tan^(-1). (alpha)/(beta)) + (beta^(3))/(2) sec^(2) ((1)/(2) tan^(-1).(beta)/(alpha))` `= alpha^(3) (1)/(1 - cos(tan^(-1) ((alpha)/(beta)))) + beta^(3) (1)/(1 + cos (tan^(-1).(beta)/(alpha)))` `= alpha^(3) (1)/(1 -cos (cos^(-1)...
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Let `alpha,beta` be two real numbers satisfying the following relations `alpha^2+beta^2=5, 3(alpha^5+beta^5)=11(alpha^3+beta^3)1.` Possible value of `
Correct Answer - A `alpha^(2)+beta^(2)=5` `3(alpha^(5)+beta^(5))=11(alpha^(3)+beta^(3))` `(alpha^(5)+beta^(5))/(alpha^(3)+beta^(3))=(11)/(3)` ` :. ((alpha^(3)+beta^(3))(alpha^(2)+beta^(2))-(alpha^(2)beta^(2)(alpha+beta)))/(alpha^(3)+beta^(3))=(11)/(3)` `:. alpha^(2)+beta^(2)-(alpha^(2)beta^(2)(alpha+beta))/((alpha+beta)(alpha^(2)+beta^(2)-alphabeta))=(11)/(3)` ` :. 5-(alpha^(2)beta^(2))/(5-alphabeta)=(11)/(3)` ltbrlt `:. (25-5alphabeta-alpha^(2)beta^(2))/(5-alphabeta)=(11)/(3)` Let `alphabeta=t` `(25-5t-t^(2))/(5-t)=(11)/(3)` `75-15t-3t^(2)=55-11 t` `75-15t-3t^(2)-55+11t=0` `-3t^(2)-4t+20=0` `(t-2)(3t+10)=0` ` :. t=2` or `(-10)/(3)` So `alpha...
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Let `alpha`, `beta` be two real numbers satisfying the following relations `alpha^(2)+beta^(2)=5`, `3(alpha^(5)+beta^(5))=11(alpha^(3)+beta^(3))` Poss
Correct Answer - B `alpha^(2)+beta^(2)=5` `3(alpha^(5)+beta^(5))=11(alpha^(3)+beta^(3))` `(alpha^(5)+beta^(5))/(alpha^(3)+beta^(3))=(11)/(3)` ` :. ((alpha^(3)+beta^(3))(alpha^(2)+beta^(2))-(alpha^(2)beta^(2)(alpha+beta)))/(alpha^(3)+beta^(3))=(11)/(3)` `:. alpha^(2)+beta^(2)-(alpha^(2)beta^(2)(alpha+beta))/((alpha+beta)(alpha^(2)+beta^(2)-alphabeta))=(11)/(3)` ` :. 5-(alpha^(2)beta^(2))/(5-alphabeta)=(11)/(3)` ltbrlt `:. (25-5alphabeta-alpha^(2)beta^(2))/(5-alphabeta)=(11)/(3)` Let `alphabeta=t` `(25-5t-t^(2))/(5-t)=(11)/(3)` `75-15t-3t^(2)=55-11 t` `75-15t-3t^(2)-55+11t=0` `-3t^(2)-4t+20=0` `(t-2)(3t+10)=0` ` :. t=2` or `(-10)/(3)` So `alpha...
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Let `alpha`, `beta` be two real numbers satisfying the following relations `alpha^(2)+beta^(2)=5`, `3(alpha^(5)+beta^(5))=11(alpha^(3)+beta^(3))` Quad
Correct Answer - D `alpha^(2)+beta^(2)=5` `3(alpha^(5)+beta^(5))=11(alpha^(3)+beta^(3))` `(alpha^(5)+beta^(5))/(alpha^(3)+beta^(3))=(11)/(3)` ` :. ((alpha^(3)+beta^(3))(alpha^(2)+beta^(2))-(alpha^(2)beta^(2)(alpha+beta)))/(alpha^(3)+beta^(3))=(11)/(3)` `:. alpha^(2)+beta^(2)-(alpha^(2)beta^(2)(alpha+beta))/((alpha+beta)(alpha^(2)+beta^(2)-alphabeta))=(11)/(3)` ` :. 5-(alpha^(2)beta^(2))/(5-alphabeta)=(11)/(3)` ltbrlt `:. (25-5alphabeta-alpha^(2)beta^(2))/(5-alphabeta)=(11)/(3)` Let `alphabeta=t` `(25-5t-t^(2))/(5-t)=(11)/(3)` `75-15t-3t^(2)=55-11 t` `75-15t-3t^(2)-55+11t=0` `-3t^(2)-4t+20=0` `(t-2)(3t+10)=0` ` :. t=2` or `(-10)/(3)` So `alpha...
2 Answers 1 views
For any complex numbers `z_1,z_2 and z_3, z_3 Im(bar(z_2)z_3) +z_2Im(bar(z_3)z_1) + z_1 Im(bar(z_1)z_2)` is
Correct Answer - A `(a)` `z_(1)((barz_(2)z_(3)-z_(2)barz_(3))/(2i))+z_(2)((barz_(3)z_(1)-z_(3)barz_(1))/(2i))+z_(3)((barz_(1)z_(2)-z_(1)barz_(2))/(2i))=(1)/(2i)xx0=0`
2 Answers 1 views
If `|z_1|=|z_2|=|z_3|=1` then value of `|z_1-z_3|^2+|z_3-z_1|^2+|z_1-z_2|^2` cannot exceed
Correct Answer - B `(b)` Let `y=|z_(1)-z_(2)|^(2)+|z_(2)-z_(3)|^(2)+|z_(3)-z_(1)|^(2)` `=(z_(1)-z_(2))(barz_(1)-barz_(2))+(z_(2)-z_(3))(barz_(2)-barz_(3))+(z_(3)-z_(1))(barz_(3)-barz_(1))` `=6-(z_(1)barz_(2)+z_(2)barz_(1)+z_(2)barz_(3)+barz_(2)z_(3)+z_(3)barz_(1)+z_(1)barz_(3))`...........`(i)` Now we know `|z_(1)+z_(2)+z_(3)|^(2) ge 0` `implies 3+(z_(1)barz_(2)+z_(2)barz_(1)+z_(1)barz_(3)+z_(3)barz_(1)+z_(2)barz_(3)+barz_(2)z_(3)) ge 0`.........`(ii)` From `(i)` and `(ii)`, `y le 9`
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Let `arg(z_(k))=((2k+1)pi)/(n)` where `k=1,2,………n`. If `arg(z_(1),z_(2),z_(3),………….z_(n))=pi`, then `n` must be of form `(m in z)`
Correct Answer - B `(b)` `arg(z_(1),z_(2),z_(3)……….z_(n))=pi` `impliesarg(z_(1))+arg(z_(2))+….+arg(z_(n))=pi+-2mpi`, `m in I` `implies (pi)/(n)[3+5+7+….+(2n+1)]=pi+-2mpi` `implies(pi)/(n)[(n)/(2)[6+2(n-1)]]=pi+-2mpi` `implies3+n-1=1+-2m` `impliesn=-1=1+-2m`
2 Answers 1 views
If `z_1,z_2,z_3` are any three roots of the equation `z^6=(z+1)^6,` then `arg((z_1-z_3)/(z_2-z_3))` can be equal to
Correct Answer - A::B `(a,b)` Given equation `z^(6)=(z+1)^(6)` `implies|z^(6)|=|(z+1)^(6)|` `implies|z|=|z+1|` `implies` Roots are collinear. `impliesarg((z_(1)-z_(3))/(z_(2)-z_(3)))=0` or `pi`
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