Advertisements
Advertisements
प्रश्न
Find the value of : x3 + 2x2 – 3x + 21, if x = 1 + 2i
Advertisements
उत्तर
x = 1 + 2i
∴ x – 1 = 2i
∴ (x – 1)2 = 4i2
∴ x2 – 2x + 1 = – 4 ...[∵ i2 = – 1]
∴ x2 – 2x + 5 = 0 ...(i)
x + 4
∵ `x^2 – 2x + 5")"overline(x^3 + 2x^2 - 3x + 21)"`
x3 – 2x2 + 5x
– + –
4x2 – 8x + 21
4x2 – 8x + 20
– + –
1
∴ x3 + 2x2 – 3x + 21
= (x2 – 2x + 5)(x + 4) + 1
= 0.(x + 4) + 1 ...[From (i)]
= 0 + 1
∴ x3 + 2x2 – 3x + 21 = 1
APPEARS IN
संबंधित प्रश्न
Simplify the following and express in the form a + ib:
`(4 + 3"i")/(1 - "i")`
Simplify:
`(i^592 + i^590 + i^588 + i^586 + i^584)/(i^582 + i^580 + i^578 + i^576 + i^574)`
Prove that `(1 + "i")^4 xx (1 + 1/"i")^4` = 16
Find the value of x and y which satisfy the following equation (x, y∈R).
(x + 2y) + (2x − 3y)i + 4i = 5
Answer the following:
Simplify the following and express in the form a + ib:
`(sqrt(5) + sqrt(3)"i")/(sqrt(5) - sqrt(3)"i")`
Show that `(1/sqrt(2) + "i"/sqrt(2))^10 + (1/sqrt(2) - "i"/sqrt(2))^10` = 0
Answer the following:
Simplify: `("i"^238 + "i"^236 + "i"^234 + "i"^232 + "i"^230)/("i"^228 + "i"^226 + "i"^224 + "i"^222 + "i"^220)`
Find the value of k if for the complex numbers z1 and z2, `|1 - barz_1z_2|^2 - |z_1 - z_2|^2 = k(1 - |z_1|^2)(1 - |"z"_2|^2)`
The area of the triangle on the complex plane formed by the complex numbers z, –iz and z + iz is ______.
Evaluate `sum_(n = 1)^13 (i^n + i^(n + 1))`, where n ∈ N.
If |z + 1| = z + 2(1 + i), then find z.
Find the complex number satisfying the equation `z + sqrt(2) |(z + 1)| + i` = 0.
The number `(1 - i)^3/(1 - i^2)` is equal to ______.
State True or False for the following:
Multiplication of a non-zero complex number by –i rotates the point about origin through a right angle in the anti-clockwise direction.
State True or False for the following:
For any complex number z the minimum value of |z| + |z – 1| is 1.
State True or False for the following:
The locus represented by |z – 1| = |z – i| is a line perpendicular to the join of (1, 0) and (0, 1).
The real value of α for which the expression `(1 - i sin alpha)/(1 + 2i sin alpha)` is purely real is ______.
If α, β, γ and a, b, c are complex numbers such that `α/a + β/b + γ/c` = 1 + i and `a/α + b/β + c/γ` = 0, then the value of `α^2/a^2 + β^2/b^2 + γ^2/c^2` is equal to ______.
If z = 2 + i, then (z − 1) `(barz − 5) + (barz − 1)` (z − 5) is equal to ______.
