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Composite Function

The ratio of ...

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The ratio of the roots of the equation x2 – 5x + a = 0 is same as the ratio of the roots of the equation x2 – 9x + b = 0. If D1 and D2 are the discriminants of the equation x2 – 5x + a = 0 and x2 – 9x + b = 0 respectively, then D1 : D2 is

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Q. The ratio of the roots of the equation

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

is same as the ratio of the roots of equation

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

D_{1}

D_{2}

are the discriminants of

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

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

respectively, then

D_{1} : D_{2} =

Q. The ratio of the roots of the equation

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

is same as the ratio of the roots of the equation

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

D_{1}

D_{2}

are the discriminants of

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

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

respectively. Find the ratio of

D_{1}

D_{2}

D_{1} = ∣ ∣ ∣ ∣ \begin{matrix} x & a & b - 1 & 0 & x x & 2 & 1 \end{matrix} ∣ ∣ ∣ ∣

D_{2} = ∣ ∣ ∣ ∣ \begin{matrix} c x^{2} & 2 a & - b x & 2 & 1 - 1 & 0 & x \end{matrix} ∣ ∣ ∣ ∣ .

If all the roots of the equation

(x^{2} - 4 x - 7) (x^{2} - 2 x - 3) = 0

satisfy the equation

D_{1} + D_{2} = 0,

then the value of

a + 4 b + c

D_{1} = ∣ ∣ ∣ ∣ \begin{matrix} x & a & b - 1 & 0 & x x & 2 & 1 \end{matrix} ∣ ∣ ∣ ∣

D_{2} = ∣ ∣ ∣ ∣ \begin{matrix} c x^{2} & 2 a & - b x & 2 & 1 - 1 & 0 & x \end{matrix} ∣ ∣ ∣ ∣ .

If all the roots of

(x^{2} - 4 x - 7) (x^{2} - 2 x - 3) = 0

satisfies the equation

D_{1} + D_{2} = 0,

then the value of

a + 4 b + c

α, β

are the roots of

x^{2} + 7 x + 5 = 0

, then the equation whose roots are

α - 1, β - 1

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