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

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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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Important Questions on Theory of Equations

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IMPORTANT

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

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Mathematics for Joint Entrance Examination JEE (Advanced) Algebra>Chapter 2 - Theory of Equations>Exercises>Q 66

Let

f (x) = a x^{2} + b x + c, a, b, c \in R

. If

f (x)

takes real values for real values of

x

and non -real values for non-real values of

x,

then

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Mathematics for Joint Entrance Examination JEE (Advanced) Algebra>Chapter 2 - Theory of Equations>Exercises>Q 67

If both roots of the equation

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

are imaginary and

c > - 1,

then

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Mathematics for Joint Entrance Examination JEE (Advanced) Algebra>Chapter 2 - Theory of Equations>Exercises>Q 68

If

{(b^{2} - 4 a c)}^{2} (1 + 4 a^{2}) < 64 a^{2}, a < 0,

then the maximum value of quadratic expression

a x^{2} + b x + c

always less than