α2+β2 = (α+β)2 - 2αβ
α-β = √(α+β)2 - 4αβ
α3-β3 = (α-β)3 + 3αβ(α-β)
α4+β4 = (α2+β2)2 - 2α2β2
Question 1 :
If α and β are the roots of the equation
3x2-5x+2 = 0
then find the values of
(i) (α/β) + (β/α)
(ii) α - β
(iii) (α²/β) + (β²/α)
Solution :
By comparing the given quadratic equation with the general form of quadratic equation, we get
a = 3, b = -5 and c = 2
Sum of roots : α+β = -b/a α+β = -(-5)/3 α+β = 5/3 |
Product of roots : αβ = c/a αβ = 2/3 |
(i) (α/β)+(β/α)
By combining the above fractions, we get
(α/β)+(β/α) = (α2+β2)/αβ -----(1)
α2+β2 = (α+β)2-2αβ
= (5/3)2-2(2/3)
= (25/9)-(4/3)
= (25-12)/9
α2+β2 = 13/9
By applying the values in (1), we get
(α/β) + (β/α) = (α2+β2)/αβ
= (13/9)/(2/3)
(α/β) + (β/α) = 13/6
(ii) α - β
α-β = √(α+β)2 - 4αβ
= √(5/3)2-4(2/3)
= √(25/9)-(8/3)
= √1/9
α-β = ± 1/3
(iii) (α2/β) + (β2/α)
By combining the given fractions, we get
(α2/β) + (β2/α) = (α3+β3)/αβ ----(1)
α3-β3 = (α-β)3 + 3αβ(α-β)
= (5/3)3-3(2/3)(5/3)
= (125/27)-(10/9)
α3-β3 = 95/27
By applying the values in (1), we get
(α2/β) + (β2/α) = (95/27)/(2/3)
(α2/β) + (β2/α) = 95/18
Question 2 :
If α and β are the roots of
3x2-6x+4 = 0
find the value of α2+β2
Solution :
a = 3 b = - 6 and c = 4
α2+β2 = (α+β)2 - 2αβ ----(1)
Sum of roots : α+β = -b/a = -(-6)/3 α+β = 2 |
Product of roots : αβ = c/a αβ = 4/3 |
By applying the values in (1), we get
= 22-2(4/3)
= 4-(8/3)
= 4/3
Question 3 :
If roots of the equation x2 + x + r = 0 are α and β and α3+β3 = -6. Find the value of r.
Solution :
x2 + x + r = 0
a = 1, b = 1 and c = 1
α3+β3 = -6
α3+β3 = (α + β)3 - 3αβ (α + β)
(α + β) = -b/a and α β = c/a
(α + β) = -1/1 and α β = r/1
α3+β3 = (-1)3 - 3(r) (1)
-6 = -1 - 3r
-6 + 1 = 3r
3r = -5
r = -5/3
So, the value of r is -5/3.
Question 4 :
If difference between the roots of the equation x2 - kx + 8 = 0 is 4, then find the value of k.
Solution :
For any quadratic equation, which is in the form ax2 + bx + c = 0 the roots are α and β
Sum of the roots α + β = -b/a
Product of roots α β = c/a
Difference of the roots
α - β = √[(α + β)2 - 4 α β]
x2 - kx + 8 = 0
a = 1, b = -k and c = 8
α + β = k/1 ==> k
α β = 8/1 ==> 8
α - β = k2 - 4 (8)
Applying the value of α - β, we get
4 = √(k2 - 32)
(k2 - 32) = 16
k2 = 32 + 16
k2 = 48
k = √48
k = √2 · 2 · 2 · 2 · 3
k = (2 · 2)√3
k = ± 4√3
k = 4√3 and -4√3
So, the values of k are -4√3 and 4√3.
Question 5 :
If one of the roots of the equation
x2 + px + a = 0 is √3+2
then find the value of p and a is.
Solution :
Every quadratic equation will have two roots α and β. If rational number will be one of the given roots, its conjugate will be the other root.
For example,
if a + √b is one of the roots then its conjugate a - √b will be other root.
Since for the given equation 2 + √3 be one root, then its conjugate 2 - √3 will be the other root.
α = 2 + √3 and β = 2 - √3
Sum of roots :
α + β = 2 + √3 + 2 - √3 ==> 4
Product of roots :
α β = (2 + √3) (2 - √3)
= 22 - √32
= 4 - 3
α β = 1
From the equation,
x2 + px + a = 0
α + β = -p/1 ==> -p
4 = -p
p = -4
So, the value of p is -4
From the equation,
x2 + px + a = 0
α β = a/1 ==> a
1 = a
So, the value of a is 1.
Question 6 :
If one root of the equation
px2 + qx + r = 0
is r then other roots of the equation will be
a) 1/q b) 1/r c) 1/p d) 1/(p + q)
Solution :
px2 + qx + r = 0
One root of the equation is α = r, then β = ?
From the given equation, we get
Sum of roots = α + β = -q/p ---(1)
Product of roots = α β = r/p ----(2)
Applying the value α = r in (1) and (2), we get
r + β = -q/p
r β = r/p
β = 1/p
So, the other roots is 1/p. Option c is correct.
Question 7 :
One root of the equation
x2 - 2(5 + m) x + 3(7 + m) = 0
is reciprocal of the other. Find the value of m.
a) -7 b) 7 c) 1/7 d) -1/7
Solution :
x2 - 2(5 + m) x + 3(7 + m) = 0
x2 - (10 - 2m) x + (21 + 3m) = 0
Sum of roots = α + β = (10 - 2m)/1 ---(1)
Product of roots = α β = (21 + 3m)/1 ----(2)
α and 1/α are the roots.
α (1/α) = (21 + 3m)
1 = 21 + 3m
1 - 21 = 3m
m = -20/3
m = -6.6
Approximately the value of m is -7.
Kindly mail your feedback to v4formath@gmail.com
We always appreciate your feedback.
©All rights reserved. onlinemath4all.com
Aug 31, 24 09:05 PM
Aug 31, 24 08:51 PM
Aug 31, 24 06:59 PM