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Greatest Integer Function

lf α, β are...

Question

lf $α, β$ are the roots of $a x^{2} + b x + c = 0$ observe the following lists.
List-I List-II
$A$ : $a, b, c$ have same sign 1. $α β = 1$
$B$ : lf $a = c$ 2. $- 1$
$C$ : lf $a + b + c = 0$ then one root is 3. $α$ and $β$ are negative
$D$ : lf $a + c = b$ then one root is 4. $0$
5. $1$
The correct match for list-l from list-I is A B C D

A

1, 3, 2, 5

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B

1, 3, 4, 5

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C

3, 1, 5, 4

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D

3, 1, 5, 2

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Solution

The correct option is C $3, 1, 5, 2$
If a, b and c have same sign
$\Rightarrow α + β = \frac{- b}{a} < 0$
$α β = \frac{c}{a} > 0$
$∴ α$ and $β$ are negative
If $a = c; α β = \frac{c}{a} = 1$
If $a + b + c = 0; f (1) = 0 \Rightarrow a (1)^{2} + b (1) + c = 0$
$∴$ 1 is a root.
If $a + c = b \Rightarrow a (1)^{2} + c = b (1)$
$a (1)^{2} - b (1) + c = 0$
$- 1$ is a root.

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Similar questions

α, β

are the roots of

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

then match the elements of list I with elements of list II:

List I	List II
A) $\frac{α}{β} + \frac{β}{α} =$	1) $\frac{c^{2}}{a^{2}}$
B) $\frac{α^{2} + β^{2}}{α^{- 2} + β^{- 2}} =$	2) $\frac{c^{5} [3 a b c - b^{3}]}{a^{8}}$
C) $α^{3} + β^{3}$	3) $\frac{b^{2} - 2 a c}{a c}$
D) $α^{5} β^{8} + α^{8} β^{5} =$	4) $\frac{3 a b c - b^{3}}{a^{3}}$

The correct match from List-I to List-II:

A, B, C, D

α, β

are the roots of the equation

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

, then the quadratic equation whose roots are

α + β, α β

α

β,

are the roots of the equation

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

c < 0 < b

f (x) = x^{3} + a x^{2} + b x + c = 0

a, b, c

a, b, c \in R

. If the roots of

x^{3} + a_{1} x^{2} + b_{1} x + c_{1} = 0

(α - β)^{2}, (β - γ)^{2}

(γ - α)^{2}

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f (x) = 0

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.

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