please solve these questions fast its urgent
4. Let α and βbe the roots of the equation x^ + x + 1 = 0. Find the equation whose roots
are α^10, β^7?
6. If α. and β are the roots of the equation 3x^2 - 2x + 4 = 0, find the equations whose roots
are (i) α^2 - 1,β^2- 1 (ii) 2α+B, α+2β.
7. If the roots of the equation x2 + kx + 7 = 0 are denoted by αand β, and α^2 +β^2 = 22,
show that k = -/+6.
please give the correct answer only . the person who gives the right answer for all the three questions i will mark as brainlist.
Answers
Answer 4
Given that,
x
2
−2x+3=0
Let the roots be α,β
If we consider the roots of required equation at A,B
A=α+2B=β+2
Weknow,
αβ=3
α+β=2
Re plating α,β by x and A,Bby x.
x=x−2
In equation:
(α−2)
2
−2(α−2)+3=0
x
2
+4−4x−2x+4+3=0
x
2
−6x+11=0
Required equation.
Answer 6
α+β=a,αβ=b
let equation whose roots are 2α+1,2β+1
be x
2
+Ax+B=0
then −A=(2α+1)+(2β+1)=0
−A=2(a)+2=2(a+1)
B=(2α+1)(2β+1)
=4αβ+2α+2β+1
=4b+2a+1
∴ required equation is
x
2
−2(a+1)x+4b+2a+1=0
Answer 7
k
2
+kx−12=0
a,a+1 are the roots. Therefore,
2a+1=−k
or a=
2
−k−1
Also, a
2
+a=−12
or
4
(k+1)
2
−
2
(k+1)
=−12
or (k+1)
2
+2(k+1)=−48
or (k+1)
2
+2(k+1)+48=0
or (k+1−8)(k+1+6)=0
or (k−7)(k+7)=0
I have given all answer correct.
I hope this help you.