Advertisements
Advertisements
प्रश्न
Answer the following:
Find the value of x4 + 9x3 + 35x2 − x + 164, if x = −5 + 4i
Advertisements
उत्तर
x = − 5 + 4i
∴ x + 5 = 4i
∴ (x + 5)2 = 16i2
∴ x2 + 10x + 25 = − 16 ...[∵ i2 = − 1]
∴ x2 + 10x + 41 = 0 ...(i)
x2 − x + 4
∵ `x^2 + 10x + 41")"overline(x^4 + 9x^3 + 35x^2 - x + 164`
x4 + 10x3 + 41x2
− − −
− x3 − 6x2 − x + 164
− x3 − 10x2 − 41x
+ + +
4x2 + 40x + 164
4x2 + 40x + 164
− − −
0
∴ x4 + 9x3 + 35x2 − x + 164
= (x2 + 10x + 41) (x2 − x + 4)
= 0(x2 − x + 4) ...[From (i)]
∴ x4 + 9x3 + 35x2 − x + 164 = 0
APPEARS IN
संबंधित प्रश्न
If α and β are different complex numbers with |β| = 1, then find `|(beta - alpha)/(1-baralphabeta)|`
Find the value of: 2x3 – 11x2 + 44x + 27, if x = `25/(3 - 4"i")`
Simplify the following and express in the form a + ib:
`3 + sqrt(-64)`
Simplify the following and express in the form a + ib:
(1 + 3i)2 (3 + i)
Simplify the following and express in the form a + ib:
`(3i^5 + 2i^7 + i^9)/(i^6 + 2i^8 + 3i^18)`
Find the value of : x3 + 2x2 – 3x + 21, if x = 1 + 2i
Write the conjugates of the following complex number:
`-sqrt(5) - sqrt(7)"i"`
Write the conjugates of the following complex number:
cosθ + i sinθ
Simplify:
`(i^592 + i^590 + i^588 + i^586 + i^584)/(i^582 + i^580 + i^578 + i^576 + i^574)`
If a = `(-1 + sqrt(3)"i")/2`, b = `(-1 - sqrt(3)"i")/2` then show that a2 = b and b2 = a
If x + iy = `sqrt(("a" + "ib")/("c" + "id")`, prove that (x2 + y2)2 = `("a"^2 + "b"^2)/("c"^2 + "d"^2)`
Show that `((sqrt(7) + "i"sqrt(3))/(sqrt(7) - "i"sqrt(3)) + (sqrt(7) - "i"sqrt(3))/(sqrt(7) + "i"sqrt(3)))` is real
Answer the following:
Simplify the following and express in the form a + ib:
(2i3)2
Solve the following equation for x, y ∈ R:
2x + i9y (2 + i) = xi7 + 10i16
Answer the following:
Simplify: `("i"^238 + "i"^236 + "i"^234 + "i"^232 + "i"^230)/("i"^228 + "i"^226 + "i"^224 + "i"^222 + "i"^220)`
If z1 = 2 – 4i and z2 = 1 + 2i, then `bar"z"_1 + bar"z"_2` = ______.
State true or false for the following:
The complex number cosθ + isinθ can be zero for some θ.
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.
1 + i2 + i4 + i6 + ... + i2n is ______.
If the complex number z = x + iy satisfies the condition |z + 1| = 1, then z lies on ______.
For a positive integer n, find the value of `(1 - i)^n (1 - 1/i)^"n"`
If |z + 4| ≤ 3, then the greatest and least values of |z + 1| are ______ and ______.
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 inequality |z – 4| < |z – 2| represents the region given by x > 3.
If `((1 + i)/(1 - i))^x` = 1, then ______.
Which of the following is correct for any two complex numbers z1 and z2?
Let x, y ∈ R, then x + iy is a non-real complex number if ______.
The complex number z which satisfies the condition `|(i + z)/(i - z)|` = 1 lies on ______.
Let |z| = |z – 3| = |z – 4i|, then the value |2z| is ______.
Simplify the following and express in the form a+ib.
`(3i^5 + 2i^7 + i^9)/(i^6 + 2i^8 + 3i^18)`
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)`
i2 + i3 + ... + i4000 =
