I lf - are the roots of ax2-bx-c-0- lf K is a real number -K- 0-1 - then the equation whose roots are K- K- is ax2- Kbx -K2c-0-ii 8-2 are roots of x2-ax-0 and 3- 3 are the roots of x2-x-b-0 then the roots of x2-ax-b-0 are 9- 1-Which of the above statement is correct

The correct option is D Both (i) and (ii)i) #160;Equation whose roots are (kα ,kβ ) is f(xk)=0 ⇒ a(xk)^2+b(xk)+c=0 ax^2+bkx+ck^2=0 ii) a ...

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Byju's Answer

Standard XII

Mathematics

Common Roots

i lf α,β ar...

Question

(i) lf $α, β$ are the roots of $a x^{2} + b x + c = 0$ . lf $K$ is a real number $[K \neq 0, 1]$ then the equation whose roots are $K α, K β$ is $a x^{2} +$ Kbx $+ K^{2} c = 0$ .
(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

only (i)

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only (ii)

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Both (i) and (ii)

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Neither (i) nor (ii)

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Solution

The correct option is D Both (i) and (ii)
i) Equation whose roots are $(k α, k β) i s f (\frac{x}{k}) = 0$
$\Rightarrow a (\frac{x}{k})^{2} + b (\frac{x}{k}) + c = 0$
$a x^{2} + b k x + c k^{2} = 0$
ii) $a = - (8 + 2) = - 10$
$b = 3 \times 3 = 9$
Given equation transforms to
$x^{2} - 10 x + 9 = 0$
$\Rightarrow x^{2} - 9 x - x + 9 = 0$
$x (x - 9) - 1 (x - 9) = 0$
$\Rightarrow x = 1, 9$

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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

and

α + h, β + h

are the roots of

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

, then

h = b - q

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

Q. If

{8, 2}

are the 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 the equation

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

are:

Q. (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.

Q. lf

α, β

are the roots of the equation

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

, then the equation whose roots are

2 α + \frac{1}{β}, 2 β + \frac{1}{α}

is:

Q. lf the roots of equation

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

are

α

and

β

, the roots of the equation

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

are

α

and

γ

, then the values of

α, β, γ

are respectively

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(i) lf α,β are the roots of ax2+bx+c=0. lf K is a real number [K≠0,1] then the equation whose roots are Kα, Kβ is ax2+ Kbx +K2c=0.(ii) 8,2 are roots of x2+ax+β=0 and 3, 3 are the roots of x2+αx+b=0 then the roots of x2+ax+b=0 are 9, 1.Which of the above statement is correct

The correct option is D Both (i) and (ii)i) Equation whose roots are (kα,kβ)isf(xk)=0⇒a(xk)2+b(xk)+c=0ax2+bkx+ck2=0ii) a=−(8+2)=−10b=3×3=9Given equation transforms tox2−10x+9=0⇒x2−9x−x+9=0x(x−9)−1(x−9)=0⇒x=1,9