MyQuestionIcon

1

You visited us 1 times! Enjoying our articles? Unlock Full Access!

Nature of Roots

i If α, β a...

Question

(i) If $α, β$ are the roots of $x^{2} + K x + 2 = 0$ , and $α - β = 1$ then $K = \pm 3$
(ii) The difference of the roots of $x^{2} - b x + c = 0$ is same as the difference of the roots of $x^{2} - c x + b = 0$ . If $b \neq c$ , then $b + c = - 3$ .
Which of the above statements is correct.

A

only (i)

Right on! Give the BNAT exam to get a 100% scholarship for BYJUS courses

B

only (ii)

No worries! We‘ve got your back. Try BYJU‘S free classes today!

C

Both (i) and (ii)

No worries! We‘ve got your back. Try BYJU‘S free classes today!

D

Neither (i) nor (ii)

No worries! We‘ve got your back. Try BYJU‘S free classes today!

Open in App

Solution

The correct option is A only (i)
Given equation is $x^{2} + K x + 2 = 0$
$α + β = - K; α β = 2$
$α - β = \sqrt{(α + β^{2}) - 4 α β}$
$= \sqrt{k^{2} - 8}$
If $α - β = 1 \Rightarrow K = \pm 3$
So, (i) is true.
Also given is $x^{2} - b x + c = 0$
$\Rightarrow α + β = b; α β = c$
$\Rightarrow α - β = \sqrt{(α + β)^{2} - 4 α β}$
$= \sqrt{b^{2} - 4 c}$
For $x^{2} - c x + b; α_{1} - β_{1} = \sqrt{c^{2} - 4 b}$
Given both are equal
$\Rightarrow c^{2} - 4 b = b^{2} - 4 c$
$\Rightarrow c^{2} - b^{2} = 4 (b - c)$
$\Rightarrow c + b = 4$
So, (ii) is false.

Suggest Corrections

thumbs-up

0

Similar questions

Q. Statement I: lf

α, β

are the roots of

x^{2} - a x + b = 0

, then the equation whose roots are

\frac{α + β}{α}, \frac{α + β}{β}

b x^{2} - a^{2} x + a^{2} = 0

Statement II: lf

α, β

are the roots of

x^{2} - b x + c = 0

α + h, β + h

are the roots of

x^{2} + q x + r = 0

h = b - q

.
Which of the above statement(s) is(are) true.

Q. Assertion :If the roots of the equations

x^{2} - b x + c = 0

x^{2} - c x + b = 0

differ by the same quantity, then

b + c

- 4

α, β

are the roots of the equation

A x^{2} + B x + C = 0

α - β = \frac{\sqrt{B^{2} - 4 A C}}{A}

α

β

(α < β)

, are the roots of the equation

x^{2} + b x + c = 0

c < 0 < b

α, β

are the roots of

a x^{2} + b x + c = 0

K

is a real number

[K \neq 0, 1]

then the equation whose roots are

K α, K β

a x^{2} +

+ K^{2} c = 0

.
(ii) 8,2 are roots of

x^{2} + a x + β = 0

and 3, 3 are the roots of

x^{2} + α x + b = 0

then the roots of

x^{2} + a x + b = 0

are 9, 1.
Which of the above statement is correct

α, β

are the roots of

2 x^{2} - 2 x + 3 = 0

α - 1, β - 1

are the roots of

A x^{2} + B x + C = 0,

then the value of

{(\frac{B}{A})}^{2} - 4 (\frac{C}{A})

View More

Join BYJU'S Learning Program

similar_icon

Related Videos

lock

Nature and Location of Roots

MATHEMATICS

Watch in App

explore_more_icon

Explore more

Nature of Roots

Standard XII Mathematics

Join BYJU'S Learning Program

CrossIcon