Advertisements
Advertisements
Question
Find the value of: x3 – 5x2 + 4x + 8, if x = `10/(3 - "i")`.
Solution
x = `10/(3 - "i")`
∴ x = `(10(3 + "i"))/((3 - "i")(3 + "i")`
= `(10(3 + "i"))/(9 - "i"^2)`
= `(10(3 + "i"))/(9 - (- 1)` ...[∵ i2 = –1]
= `(10(3 + "i"))/10`
∴ x = 3 + i
∴ x – 3 = i
∴ (x – 3)2 = i2
∴ x2 – 6x + 9 = –1 ...[∵ i2 = –1]
∴ x2 – 6x + 10 = 0 ...(i)
x + 1
`x^2 – 6x + 10")"overline(x^3 - 5x^2 + 4x + 8)"`
x3 – 6x2 + 10x
– + –
x2 – 6x + 8
x2 – 6x + 10
– + –
– 2
∴ x3 – 5x2 + 4x + 8
= (x2 – 6x + 10)(x + 1) – 2
= 0 (x + 1) – 2 ...[From (i)]
= 0 – 2
∴ x3 – 5x2 + 4x + 8 = – 2.
APPEARS IN
RELATED QUESTIONS
Simplify the following and express in the form a + ib:
(1 + 3i)2 (3 + i)
Write the conjugates of the following complex number:
3 + i
If (x + iy)3 = y + vi then show that `(y/x + "v"/y)` = 4(x2 – y2)
Answer the following:
Simplify the following and express in the form a + ib:
`(sqrt(5) + sqrt(3)"i")/(sqrt(5) - sqrt(3)"i")`
Answer the following:
Simplify the following and express in the form a + ib:
`(5 + 7"i")/(4 + 3"i") + (5 + 7"i")/(4 - 3"i")`
Answer the following:
Show that `(1/sqrt(2) + "i"/sqrt(2))^10 + (1/sqrt(2) - "i"/sqrt(2))^10` = 0
Answer the following:
Simplify: `("i"^29 + "i"^39 + "i"^49)/("i"^30 + "i"^40 + "i"^50)`
The real value of ‘a’ for which 3i3 – 2ai2 + (1 – a)i + 5 is real is ______.
State true or false for the following:
Multiplication of a non-zero complex number by i rotates it through a right angle in the anti-clockwise direction.
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.
State true or false for the following:
If three complex numbers z1, z2 and z3 are in A.P., then they lie on a circle in the complex plane.
What is the smallest positive integer n, for which (1 + i)2n = (1 – i)2n?
If (1 + i)z = `(1 - i)barz`, then show that z = `-ibarz`.
Find the complex number satisfying the equation `z + sqrt(2) |(z + 1)| + i` = 0.
The sum of the series i + i2 + i3 + ... upto 1000 terms is ______.
If z1 and z2 are complex numbers such that z1 + z2 is a real number, then z2 = ______.
The real value of α for which the expression `(1 - i sin alpha)/(1 + 2i sin alpha)` is purely real is ______.
If α and β are the roots of the equation x2 + 2x + 4 = 0, then `1/α^3 + 1/β^3` is equal to ______.
Simplify the following and express in the form a+ib.
`(3"i"^5 + 2"i"^7 + "i"^9)/("i"^6 + 2"i"^8 + 3"i"^18)`
Simplify the following and express in the form a+ib:
`(3i^5 + 2i^7 + i^9)/(i^6 + 2i^8 + 3i^18)`
