Let a, be the roots of the equation `x^2+x+1=0`. The equation whose roots are `alpha^19` and `beta^7` are:
A. `x^(2) - x - 1 = 0`
B. `x^(2) - x + 1 = 0`
C. `x^(2) + x - 1 = 0`
D. `x^(2) + x + 1 = 0`
1 Answers
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Given that `alpha`, `gamma` are roots of the equation `Ax^(2)-4x+1=0` and `beta`, `delta` are roots of the equation `Bx^(2)-6x+1=0`. If `alpha`, `beta
Correct Answer - B::C `(b,c)` `alpha,beta,gamma,delta` are in `H.P.` `implies(1)/(alpha)`, `(1)/(beta)`, `(1)/(gamma)`, `(1)/(delta)` are in `A.P.` Let `d` be the common difference of the `A.P.` Since `alpha`, `gamma` are roots of...
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There are two sets of parallel lines, their equations being `x cos alpha+y sin alpha=p` and `x sin alpha- y cos alpha=p` , `p=1,2,3,….n` and `alpha in
Correct Answer - C `(c )` `"^(n)C_(2)*^(n)C_(2)=225impliesn=6`
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A positive integer `n` is of the form `n=2^(alpha)3^(beta)`, where `alpha ge 1`, `beta ge 1`. If `n` has `12` positive divisors and `2n` has `15` posi
Correct Answer - B `(b)` `n=2^(alpha)*3^(beta)` No. of divisors `=(alpha+1)(beta+1)=12` `2n=2^(alpha+1)3^(beta)` No.of divisors `=(alpha+2)(beta+1)=15` `implies(alpha+2)/(alpha+1)=(5)/(4)` `implies4alpha+8=5alpha+5impliesalpha=3` `impliesbeta=2implies3n=2^(3)3^(3)` No. of divisors `=(3+1)(3+1)=16`
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If `alpha`, `beta`, `gamma` are roots of the equation `x^(2)(px+q)=r(x+1)`, then the value of determinant `|{:(1+alpha,1,1),(1, 1+beta,1),(1,1,1+gamma
Correct Answer - C `(c )` The given equation is `px^(3)+qx^(2)-rx-r=0` `alpha+beta+gamma=(-q)/(p)`, `alphabeta+betagamma+gammaalpha=-(r )/(p)`, `alphabetagamma=(r )/(p)` `D=(1+alpha)(1+beta)(1+gamma)+1+1-(1+alpha)-(1+beta)-(1+gamma)` `=alphabetagamma+alphabeta+alphagamma+betagamma` `=(r )/(p)-(r )/(p)=0`
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If `alpha` is a root of `x^4 = 1` with negative principal argument then the principal argument of `Delta(alpha) = |(1,1,1), (alpha^n, alpha^(n+1), alp
Correct Answer - B `(b)` Clearly `alpha=-i` where `i^(2)=-1` So `Delta(alpha)=0+alpha^(2)+1-1-0-alpha^(3)` `=(-i)^(2)+1-1-(-i)^(3)` =-1+1-1-i=-1-i` So, principal argument of `Delta(alpha)` is `-(3pi)/(4)`.
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If `alpha, beta. gamma` are the roots of `x^3 + px^2 + q = 0,` where `q=0,` ther `Delta=[(1/alpha,1/beta,1/gamma),(1/beta,1/gamma,1/alpha),(1/gamma,1/
Correct Answer - D `(d)` We have `betaalpha+gammaalpha+alphabeta=0` `Delta=(1)/(alpha^(3)beta^(3)gamma^(3))|{:(betagamma,gammaalpha,alphabeta),(gammaalpha,alphabeta,betagamma),(alphabeta,betagamma,gammaalpha):}|` `=(1)/(alpha^(3)beta^(3)gamma^(3))|{:(betagamma+gammaalpha+alphabeta,gammaalpha,alphabeta),(gammaalpha+alphabeta+betagamma,alphabeta,betagamma),(alphabeta+betagamma+gammaalpha,betagamma,gammaalpha):}|` [using `C_(1)toC_(1)+C_(2)+C_(3)`] `=(1)/(alpha^(3)beta^(3)gamma^(3))|{:(0,gammaalpha,alphabeta),(0,alphabeta,betagamma),(0,betagamma,gammaalpha):}|=0` [all zero property].
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Suppose a nucleus initally at rest undergoes `alpha` decay according to equation `._(92)^(225)X rarr Y +alpha` At `t=0`, the emitted `alpha`-particles
Correct Answer - `[(a) ((q_(alpha)E_(0))/m_(alpha)t)i+v_(0) cos theta hat(j)-v_(0) sin theta hat(k)` where `theta=omegat` and `omega=(q_(alpha)B)/m_(alpha)`, (b) `10^(7) m//s` (c) `8.00 MeV]`
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A quadratic equation whose roots are 2 moe than the roots of the quadratic equation `2x^(2) +3x + 5 = 0`, can be obtained by substituting `"_____"` fo
Correct Answer - `(x-2)`
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If `alpha, beta` are the roots of the quadratic equation `lx^(2) + mx + n = 0`,then evaluate the following expressions. (a) `alpha^(2) + beta^(2)` (b)
Correct Answer - (a) `(m^(2)-2ln)/(l^(2))`, (b) `(m^(2) - 2ln)/(ln)`, (c ) `(3lmn - m^(3))/(n^(3))`
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Find the equation of the line for which `(i)p=3 and alpha=45^(@)` (ii)`p=5 and alpha=135^(@)` (iii)`p=8 alpha=150^(@)` (iv) `p=3 and alpha=225^(@)` (v
Correct Answer - `(i)x+y-3sqrt2=0 (ii)x-y+5sqrt2=0 (iii)sqrt3x-y+16=0` `(iv)x+y+3sqrt2=0 (v)x-sqrt3y-4=0 (vi)x+4=0`
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