Mathematics for Joint Entrance Examination JEE (Advanced) Algebra>Chapter 2 - Theory of Equations>Exercises>Q 38

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The quadratic equations $x^{2} - 6 x + a = 0$ and $x^{2} - c x + 6 = 0$ have one root $α$ in common. The other roots of the first and second equations are integers in the ratio $4 : 3 .$ Then equation $x^{2} - 3 x - 4 = 0$ has

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Important Questions on Theory of Equations

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Mathematics for Joint Entrance Examination JEE (Advanced) Algebra>Chapter 2 - Theory of Equations>Exercises>Q 39

If

α

and

β, α

and

γ, α

and

δ

are the roots of the equations

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

and

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

respectively, where

a, b

and

c

are positive real numbers, then

α + α^{2} =

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Mathematics for Joint Entrance Examination JEE (Advanced) Algebra>Chapter 2 - Theory of Equations>Exercises>Q 40

If the equations

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

and

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

have two common roots, then

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Mathematics for Joint Entrance Examination JEE (Advanced) Algebra>Chapter 2 - Theory of Equations>Exercises>Q 41

Value of

a

for which equations

x^{3} + a x + 1 = 0

and

x^{4} + a x^{2} + 1 = 0

have a common root

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Mathematics for Joint Entrance Examination JEE (Advanced) Algebra>Chapter 2 - Theory of Equations>Exercises>Q 42

The number of values of

k

for which

[x^{2} - (k - 2) x + k^{2}] \times [x^{2} + k x + (2 k - 1)]

is a perfect square is

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Mathematics for Joint Entrance Examination JEE (Advanced) Algebra>Chapter 2 - Theory of Equations>Exercises>Q 43

The sum of values of

x

satisfying the equation

{(31 + 8 \sqrt{15})}^{x^{2} - 3} + 1 = {(32 + 8 \sqrt{15})}^{x^{2} - 3}

is

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Mathematics for Joint Entrance Examination JEE (Advanced) Algebra>Chapter 2 - Theory of Equations>Exercises>Q 44

The equation

{(x^{2} + x + 1)}^{2} + 1 = (x^{2} + x + 1) (x^{2} - x - 5)

for

x \in (- 2, 3)

will have number of solutions,

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Mathematics for Joint Entrance Examination JEE (Advanced) Algebra>Chapter 2 - Theory of Equations>Exercises>Q 45

If

α, β

are the roots of

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

and

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

and if

(\frac{α}{β}), (\frac{β}{α})

are the roots of

x^{n} + 1 + {(x + 1)}^{n} = 0,

then

n (\in N)

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Mathematics for Joint Entrance Examination JEE (Advanced) Algebra>Chapter 2 - Theory of Equations>Exercises>Q 46

P (x)

is a polynomial with integral coefficients such that for four distinct integers

a, b, c, d : P (a) = P (b) = P (c) = P (d) = 3

. If

P (e) = 5

(

e

is an integer), then