2x^4+5x^3+7x^2-x+41, when x=-2-3i - Vidyadip

Question

$2\text{x}^4+5\text{x}^3+7\text{x}^2-\text{x}+41,$ when $\text{x}=-2-\sqrt{3}\text{i}$

✓

Answer

$\text{x}=-2-\sqrt{3}\text{i}$
$\text{x}^2=(-2-\sqrt{3}\text{i})^2=4+4\sqrt{3}\text{i}+3\text{i}^2=1+4\sqrt{3}\text{i}$
$\text{x}^3=(1+4\sqrt{3}\text{i})(-2-\sqrt{3}\text{i})=-2-8\sqrt{3}\text{i}-\sqrt{3}\text{i}-12\text{i}^2=10+9\sqrt{3}\text{i}$
$\text{x}^4=(1-4\sqrt{3}\text{i})^2=1+8\sqrt{3}\text{i}+48\text{i}^2=-47+8\sqrt{3}\text{i}$
$2\text{x}^4+5\text{x}^3+7\text{x}^2-\text{x}+41$
$=2(-47)+8\sqrt{3}\text{i})+5(10-9\sqrt{3}\text{i})+7(1+4\sqrt{3}\text{i})-(-2-\sqrt{3}\text{i})+41$
$=-94+16\sqrt{3}\text{i}+50-45\sqrt{3}\text{i}+7+28\sqrt{3}\text{i}+2+\sqrt{3}\text{i}+41$
$=(-94+50+7+2+41)+(16\sqrt{3}\text{i}-45\sqrt{3}\text{i}+28\sqrt{3}\text{i}+\sqrt{3}\text{i})$
$=6+0$
$=6$

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "5 Marks Questions" from Complex Numbers→Complex Numbers — all question types→MATHS STD 11 Science question papers→Rajasthan Board STD 11 Science question papers→Rajasthan Board question paper generator→

Similar questions

For any two sets A and B, prove that: $\text{A}\cap\text{B}=\phi\Rightarrow\text{A}\subseteq\text{B}'.$

Solve the following equations:
$\sin2\text{x}-\sin4\text{x}+\sin6\text{x}=0$

Calculate the mean deviation about the mean for the following frequency distribution:

Class interval	0-4	4-8	8-12	12-16	16-20
Frequency	4	6	8	5	2

Differentiate the following from the first principle

$\text{x}^2\text{e}{^\text{x}}$

If the line segment joining the points P(x₁, y₁) and Q(x₂, y₂) subtends an angle $\alpha$ at the origin O, prove that: $\text{O}\text{P}.\text{O}\text{Q}\cos\alpha =\text{x}_1\text{x}_2+\text{y}_1\text{y}_2. $

Solve the following equations:
$3-2\cos\text{x}-4\sin\text{x}-\cos2\text{x}+\sin2\text{x}=0$

Find the equation of the circle which passes through the points (2, 3) and (4,5) and the centre lies on the straight line y - 4x + 3 = 0

Sketch the graphs of the following functions:
$\text{f(x)}=\tan^2\text{x}$

Find the multiplicative inverse of the following complex numbers:
$\sqrt{5}-3\text{i}$

If z₁, z₂ are two complex numbers such that $|\text{z}_1|=|\text{z}_2|$ and $\text{arg(z}_1)+\text{arg(z}_2)=\pi,$ then show that $\text{z}_1=-\bar{\text{z}}_2.$