Mathematics for Joint Entrance Examination JEE (Advanced) Algebra>Chapter 2 - Theory of Equations>Exercises>Q 57

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If $\tan θ_{1}, \tan θ_{2}, \tan θ_{3}$ are the real roots of the $x^{3} - (a + 1) x^{2} + (b - a) x - b = 0,$ where $θ_{1} + θ_{2} + θ_{3} \in (0, π),$ then $θ_{1} + θ_{2} + θ_{3}$ is equal to

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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 58

If

α, β

and

γ

are the roots of

x^{3} - x^{2} - 1 = 0,

then

\frac{1 + α}{1 - α} + \frac{1 + β}{1 - β} + \frac{1 + γ}{1 - γ}

is equal to

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Mathematics for Joint Entrance Examination JEE (Advanced) Algebra>Chapter 2 - Theory of Equations>Exercises>Q 59

If

α, β, γ, δ

are the roots of the equation

x^{4} - K x^{3} + K x^{2} + L x + M = 0,

where

K, L

and

M

are real numbers, then the minimum value of

α^{2} + β^{2} + γ^{2} + δ^{2}

is

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Mathematics for Joint Entrance Examination JEE (Advanced) Algebra>Chapter 2 - Theory of Equations>Exercises>Q 60

Set of all real values of

a

such that

f (x) = \frac{(2 a - 1) x^{2} + 2 (a + 1) x + (2 a - 1)}{x^{2} - 2 x + 40}

is always negative is

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Mathematics for Joint Entrance Examination JEE (Advanced) Algebra>Chapter 2 - Theory of Equations>Exercises>Q 61

If

a, b \in R, a \neq 0

and the quadratic equation

a x^{2} - b x + 1 = 0

has imaginary roots, then

(a + b + 1)

is

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Mathematics for Joint Entrance Examination JEE (Advanced) Algebra>Chapter 2 - Theory of Equations>Exercises>Q 62

If the expression

[m x - 1 + (\frac{1}{x})]

is non-negative for all positive real

x

, then the minimum value of

m

must be

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Mathematics for Joint Entrance Examination JEE (Advanced) Algebra>Chapter 2 - Theory of Equations>Exercises>Q 63

Suppose that

f (x)

is a quadratic expression positive for all real

x .

If

g (x) = f (x) + f^{'} (x) + f^{''} (x),

then for any real

x

(where

f^{'} (x)

and

f^{''} (x)

represent

1^{st}

and

2^{nd}

derivative, respectively)

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Mathematics for Joint Entrance Examination JEE (Advanced) Algebra>Chapter 2 - Theory of Equations>Exercises>Q 64

Let

a, b, c \in R

with

a > 0

such that the equation

a x^{2} + b c x + b^{3} + c^{3} - 4 a b c = 0

has non-real roots. If

P (x) = a x^{2} + b x + c

and

Q (x) = a x^{2} + c x + b,

then

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Mathematics for Joint Entrance Examination JEE (Advanced) Algebra>Chapter 2 - Theory of Equations>Exercises>Q 65

Let

f (x) = a x^{2} - b x + c^{2}, b \neq 0

and

f (x) \neq 0

for all

x \in R .

Then