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Complex Numbers and Quadratic Equations

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Complex Numbers and Quadratic Equations Class 11th

Let α be root of the equation 1 + x² + x⁴ = 0. Then the value of α¹⁰¹¹ + α²⁰²² - α³⁰³³ is equal to:

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Vishal Baghel

Contributor-Level 10

x⁴ + x² + 1 = 0

x⁴ + 2x² + 1 – x² = 0

$\Rightarrow (x^{2} + 1 + x) (x^{2} - x + 1) = 0$

$x = \pm ω, ? ω^{2}$

Now, = $α^{1011} + α^{2022} - α^{3033}$

= $ω^{1011} + ω^{2022} - ω^{3033}$

= 1 + 1 – 1 = 1

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Complex Numbers and Quadratic Equations Class 11th

Let arg(z) represent the principal argument of the complex number z. Then $| z | = 3$ and art(z – 1) – arg(z + 1) = $\frac{π}{4}$ intersect

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alok kumar singh

Contributor-Level 10

$| z | = 3 \Rightarrow$ circle with radius = 3

arg $(\frac{z - 1}{z + 1}) = \frac{π}{4},$ part of a circle (with radius $\sqrt[]{2}$ ). no common points

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Complex Numbers and Quadratic Equations Class 11th

Let a be root of the equation 1 + x² + x⁴ = 0. Then the value of a¹⁰¹¹ + a²⁰²² - a³⁰³³ is equal to:

0 Follower 2 Views

alok kumar singh

Contributor-Level 10

x⁴ + x² + 1 = 0

x⁴ + 2x² + 1 – x² = 0

$\Rightarrow (x^{2} + 1 + x) (x^{2} - x + 1) = 0$

$x = \pm ω, ? ω^{2}$

Now, = $α^{1011} + α^{2022} - α^{3033}$

= $ω^{1011} + ω^{2022} - ω^{3033}$

= 1 + 1 – 1 = 1

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New answer posted

3 weeks ago

Complex Numbers and Quadratic Equations Class 11th

The number of distinct real roots of the equation x^{5
$(x^{3} - x^{2} - x + 1)$}+ $x (3 x^{3} - 4 x^{2} - 2 x + 4) - 1 = 0$ is ………….

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Raj Pandey

Contributor-Level 9

$x^{8} - x^{7} - x^{6} + x^{5} + 3 x^{4} - 4 x^{3} - 2 x^{2} + 4 x - 1 = 0$

$\Rightarrow x^{7} (x - 1) - x^{5} (x - 1) + 3 x^{3} (x - 1) - x (x^{2} - 1) + 2 x (1 - x) + (x - 1) = 0$

$\Rightarrow (x - 1) (x^{2} - 1) (x^{5} + 3 x - 1) = 0 ∴ x = \pm 1$ are roots of above equation and x⁵ + 3x – 1 is a monotonic term hence vanishes at exactly one value of x other then 1 or -1.

3 real roots.

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Complex Numbers and Quadratic Equations Class 11th

Let O be the origin and A be the point z₁ = 1 + 2i. If B is the point z₂, Re(z₂) < 0, such that OAB is a right angled isosceles triangle with OB as hypotenuses, then which of the following is NOT true?

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Raj Pandey

Contributor-Level 9

$\frac{z_{2} - 0}{(1 + 2 i) - 0} = | \frac{O B}{O A} | e^{\frac{i π}{4}} \Rightarrow z_{2} = (1 + 2 i) (1 + i) = - 1 + 3 i ∴ a r g z_{2} = π - t a n^{- 1} 3 a n d | z_{2} | = \sqrt[]{10}$

$z_{1} - 2 z_{2} = 3 - 4 i ∴ a r g (z_{1} - 2 z_{2}) = - t a n^{- 1} \frac{4}{3} \Rightarrow | z_{1} - 2 z_{2} | = | 2 + 4 i + 1 - 3 i | = \sqrt[]{10}$

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Complex Numbers and Quadratic Equations Class 11th

Let A(z₁), B(z₂) and C(z₃) lie on the circle $| z - i |$ =1 and satisfy the equation 3z₁ + i 2z₂ + 2z₃. If D is the centre of the circle $| z - i | = 1,$ then area of quadrilateral ABCD equals:-

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Vishal Baghel

Contributor-Level 10

$\frac{3 z_{1} + i}{4} = \frac{2 z_{2} + 2 z_{3}}{4}$

P = point of intersection of AD & BC

$A D = 1 \Rightarrow D P = \frac{3}{4}$

$B P = \sqrt[]{1 - \frac{9}{16}} = \frac{\sqrt[]{7}}{4} \Rightarrow B C = \frac{\sqrt[]{7}}{2}$

area of Quad. ABCD = $\frac{1}{2} . A D * B C$

= $\frac{1}{2} * 1 * \frac{\sqrt[]{7}}{2} = \frac{\sqrt[]{7}}{4}$

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Complex Numbers and Quadratic Equations Class 11th

If one of the root of the quadratic equation $λ x^{2} + λ x + (λ + 1)$ = 0 is less than 1 and other is greater than 1, then complete range of values of $λ$ is given by (where $λ \in R$ )