Download App

Quadratic Equations — Maths STD 11 Science — Question

CBSE BoardEnglish MediumSTD 11 ScienceMathsQuadratic Equations3 Marks
Question
Show the following quadratic equation:
$2\text{x}^2+\sqrt{15}\ \text{ix}-\text{i}=0$
✓

Answer

$2\text{x}^2+\sqrt{15}\ \text{ix}-\text{i}=0$
Comparing the given Equation with the general form
$ax^2 + bx + c = 0,$ we get $a = 2, \text{b}=\sqrt{15}\text{i},\text{c}=-\text{i}$
Substituting a and in.
$\alpha=\frac{-\text{b}+\sqrt{\text{b}^2-4\text{ac}}}{2\text{a}}$ and $\beta=\frac{-\text{b}-\sqrt{\text{b}^2-4\text{ac}}}{2\text{a}}$
$\text{a}=\frac{-\sqrt{15}\text{ i}+\sqrt{-15+8}\text{ i}}{4}$ and $\beta=\frac{-\sqrt{15}\text{ i}-\sqrt{-15+8}\text{ i}}{4}$
Let $\sqrt{-15+8\text{i}}=\text{a}+\text{bi}$
$\Rightarrow -15 + 8i = (a + bi)^2$
$\Rightarrow -15 + 8i = a^2 - b^2 + 2abi$
$\Rightarrow a^2 - b^2 = -15$ and $2abi = 8i$
Now $(a^2 + b^2) = (a^2 - b^2) + 4a^2b^2$
$\Rightarrow (a^2 + b^2) = (15)^2 + 64 = 289$
$\Rightarrow a^2 + b^2 = 17$
Solving $a^2 - b^2 = -15$ and $a^2 + b^2 = 17,$ we get
$a^2 = 1$ and $b^2 = 16$
$\Rightarrow\text{a}=\pm1$ and $\text{b}=\pm4$
$\Rightarrow a = 1, b = 4$ or $a = -1, b = -4$
$\therefore\sqrt{-15+8\text{i }}=1+4\text{i},-1-4\text{i}$
when $\sqrt{-15+8}\text{i}=1+4\text{i}$
$\alpha=\frac{-\sqrt{15}\text{i}+1+4\text{i}}{4}=\frac{1+(4-\sqrt{15})\text{i}}{4}$
and $\beta=\frac{-\sqrt{15}\text{i}-(1+4\text{i})}{4}=\frac{-1-(4+\sqrt{15})\text{i}}{4}$
When $\sqrt{-15+8\text{i}}=-1-4\text{i}$
$\alpha=\frac{-\sqrt{15}\text{i}-1-4\text{i}}{4}=\frac{-1-(4+\sqrt{15})}{4}$
and $\beta=\frac{-\sqrt{15}\text{i}-(-1-4\text{i})}{4}=\frac{1+(4-\sqrt{5})\text{i}}{4}$

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 "3 Marks Question" from Quadratic Equations→Quadratic Equations — all question types→Maths STD 11 Science question papers→CBSE Board STD 11 Science question papers→CBSE Board question paper generator→

Similar questions

Check the validity of the statements given below by the method given against it.
q: If n is a real number with $n > 3$, then $n2 > 9$ (by contradiction method).
→
How many automobile license plates can be made if each plate contains two different letters followed by three different digits?
→
Let R be a relation on N × N defined by:
$(\text{a, b})\text{ R }(\text{c, d})\Leftrightarrow\text{a}+\text{d}=\text{b}+\text{c}$ for all $(\text{a, b}),(\text{c, d})\in\text{N}\times\text{N}$
Show that:
$(\text{a},\text{b})\text{ R }(\text{c, d})\text{ and (c, d) R (e, f)}$
$\Rightarrow(\text{a, b})\text{ R (e, f)}$ for all $(\text{a, b}),(\text{c, d}),(\text{e, f})\in\text{N}\times\text{N}$
→
Find the middle terms(s) in the expansion of:
$\Big(\frac{\text{x}}{3}+9\text{y}\Big)^{10}$
→
For any two sets A and B, prove that.
$\text{A}\cap\text{B}\subset\text{A}.$
→
Let $A = {-2, -1, 0, 1, 2}$ and f : $A \rightarrow Z$ be a function defined by $f(x) = x^2 - 2x - 3.$ Find:
Pre-images of $6, −3$ and $5.$
→
Find the general solutions of the following equations:
$\sin9\text{x}=\sin\text{x}$
→
A solution of 8% boric acid is to be diluted by adding a 2% boric acid solution to it. The resulting mixture is to be more than 4% but less than 6% boric acid. If there are 640 litres of the 8% solution, how many litres of 2% solution will have to be added?
→
How many three-digit numbers are there, with distinct digits, with each digit odd?
→
Prove that:
$4(\cos^310^\circ+\sin^320^\circ)=3(\cos10^\circ+\sin20^\circ)$
→