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Let $a x^{2} + b x + c = 0,$ be a quadratic equation, whose at least one coefficient is an imaginary number. Then the roots of the equation are

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Important Questions on Theory of Equation

MEDIUM

Mathematics>Algebra>Theory of Equation>Nature of Roots

Let

A = (\begin{array}{l} a & b \\ c & d \end{array})

be a

2 \times 2

real matrix with

det A = 1 .

If the equation

det (A - λ I_{2}) = 0

has imaginary roots (

I_{2}

be the identity matrix of order

2

), then

MEDIUM

Mathematics>Algebra>Theory of Equation>Nature of Roots

If the roots of the quadratic equation

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

are imaginary, then for all real values of

x

, the minimum value of the expression

3 a^{2} x^{2} + 6 a b x + 2 b^{2}

HARD

Mathematics>Algebra>Theory of Equation>Nature of Roots

Let

a, b

and

c

be three positive real numbers such that the sum of any two of them is greater than the third. All the values of

λ

such that the roots of the equation

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

are real, are given by

MEDIUM

Mathematics>Algebra>Theory of Equation>Nature of Roots

Let

z_{1}

and

z_{2}

be two imaginary roots of

z^{2} + p z + q = 0,

where

p

and

q

are real. The points

z_{1}, z_{2}

and origin form an equilateral triangle if

HARD

Mathematics>Algebra>Theory of Equation>Nature of Roots

The quadratic equation

P (x) = 0

with real coefficients has purely imaginary roots. Then the equation

P (P (x)) = 0

has

EASY

Mathematics>Algebra>Theory of Equation>Nature of Roots

b_{1} b_{2} = 2 (c_{1} + c_{2})

and

b_{1}, b_{2}, c_{1}, c_{2}

are all real numbers, then at least one of the equations

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

and

x^{2} + b_{2} x + c_{2} = 0

has

HARD

Mathematics>Algebra>Theory of Equation>Nature of Roots

Let

θ

be an acute angle such that the equation

x^{3} + 4 x^{2} \cos θ + x \cot θ = 0

has multiple roots. Then the value of

θ

(in radians) is

MEDIUM

Mathematics>Algebra>Theory of Equation>Nature of Roots

If one root of the equation

x^{2} + p x + 12 = 0

4

and the equation

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

has equal roots, the value of

q

EASY

Mathematics>Algebra>Theory of Equation>Nature of Roots

The number of integral values of

m

for which the equation,

(1 + m^{2}) x^{2} - 2 (1 + 3 m) x + (1 + 8 m) = 0

has no real root, is

MEDIUM

Mathematics>Algebra>Theory of Equation>Nature of Roots

Let

p, q \in Q .

2 - \sqrt{3}

is a root of the quadratic equation

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

then

MEDIUM

Mathematics>Algebra>Theory of Equation>Nature of Roots

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

and

Q (x) = - a x^{2} + d x + c

, where

a c \neq 0

(

a, b, c, d

are all real), then

P (x) \cdot Q (x) = 0

has

MEDIUM

Mathematics>Algebra>Theory of Equation>Nature of Roots

The number of all possible positive integral value of

α

for which the roots of the quadratic equation

6 x^{2} - 11 x + α = 0

are rational numbers is:

HARD

Mathematics>Algebra>Theory of Equation>Nature of Roots

Let $p (x) = x^{2} + a x + b$ have two distinct real roots, where $a, b$ are real number. Define $g (x) = p (x^{3})$ for all real number $x$

Then, which of the following statements are true?
I. $g$ has exactly two distinct real roots.
II. $g$ can have more than two distinct real roots.
III. There exists a real number $α$ such that $g (x) \geq α$ for all real $x$

EASY

Mathematics>Algebra>Theory of Equation>Nature of Roots

The expression

a x^{2} + b x + c, (a, b

and

c

are real

)

has the same sign as that of

a

for all

x

MEDIUM

Mathematics>Algebra>Theory of Equation>Nature of Roots

Let

S = {1, 2, 3, \dots, 100} .

Suppose

b & c

are chosen at random from the set

S

. The probability that

4 x^{2} + b x + c

has equal roots is

MEDIUM

Mathematics>Algebra>Theory of Equation>Nature of Roots

Suppose $A, B, C$ are defined as

$A = a^{2} b + a b^{2} - a^{2} c - a c^{2}$

$B = b^{2} c + b c^{2} - a^{2} b - a b^{2}$

$C = a^{2} c + a c^{2} - b^{2} c - b c^{2}$

Where $a > b > c > 0$ and the equation $A x^{2} + B x + C = 0$ has equal roots, then $a, b, c$ are in

HARD

Mathematics>Algebra>Theory of Equation>Nature of Roots

The equation

x^{2} + n x + m = 0; n, m \in Z

, cannot have

HARD

Mathematics>Algebra>Theory of Equation>Nature of Roots

If the roots

α

and

β

of the equation,

x^{2} - \sqrt{2} x + c = 0

are complex for some real number

c \neq 0

and

|\frac{α - β}{1 - αβ}| = 1,

then a value of

c

EASY

Mathematics>Algebra>Theory of Equation>Nature of Roots

If equations

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

and

2 x^{2} + 3 x + 4 = 0

have a common root, then

a : b : c

equals :

HARD

Mathematics>Algebra>Theory of Equation>Nature of Roots

Consider the quadratic equation

n x^{2} + 7 \sqrt{n} x + n = 0

, where

n

is a positive integer. Which of the following statements are necessarily correct?

I. For any

n

, the roots are distinct.

II. There are infinitely many values of

n

for which both roots are real.

III. The product of the roots is necessarily an integer.

Let ax2+bx+c=0, be a quadratic equation, whose at least one coefficient is an imaginary number. Then the roots of the equation are

Important Questions on Theory of Equation

Learn and Prepare for any exam you want !

Let $a x^{2} + b x + c = 0,$ be a quadratic equation, whose at least one coefficient is an imaginary number. Then the roots of the equation are