MEDIUM

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Find all the integral values of $a$ for which the quadratic equation $(x - a) (x - 10) + 1 = 0$ has integral roots.

Important Questions on Theory of Quadratic Equations

HARD

Mathematics>Algebra>Theory of Quadratic Equations>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

HARD

Mathematics>Algebra>Theory of Quadratic Equations>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

HARD

Mathematics>Algebra>Theory of Quadratic Equations>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 Quadratic Equations>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

EASY

Mathematics>Algebra>Theory of Quadratic Equations>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 :

EASY

Mathematics>Algebra>Theory of Quadratic Equations>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

MEDIUM

Mathematics>Algebra>Theory of Quadratic Equations>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 Quadratic Equations>Nature of Roots

The equation

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

, cannot have

HARD

Mathematics>Algebra>Theory of Quadratic Equations>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 Quadratic Equations>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 Quadratic Equations>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 Quadratic Equations>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 Quadratic Equations>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 $x$

MEDIUM

Mathematics>Algebra>Theory of Quadratic Equations>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 Quadratic Equations>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

MEDIUM

Mathematics>Algebra>Theory of Quadratic Equations>Nature of Roots

The sum of all integral values of

k (k \neq 0)

for which the equation

\frac{2}{x - 1} - \frac{1}{x - 2} = \frac{2}{k}

x

has no real roots, is_____.

HARD

Mathematics>Algebra>Theory of Quadratic Equations>Nature of Roots

Suppose

a, b

are integers and

a + b

is a root of

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

What is the maximum possible values of

b^{2} ?

MEDIUM

Mathematics>Algebra>Theory of Quadratic Equations>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

HARD

Mathematics>Algebra>Theory of Quadratic Equations>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.

HARD

Mathematics>Algebra>Theory of Quadratic Equations>Nature of Roots

If the two roots of the equation,

(a - 1) (x^{4} + x^{2} + 1) + (a + 1) {(x^{2} + x + 1)}^{2} = 0

are real and distinct, then the set of all values of

a

is equal to

Find all the integral values of a for which the quadratic equation x-ax-10+1=0 has integral roots.

Important Questions on Theory of Quadratic Equations

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Find all the integral values of $a$ for which the quadratic equation $(x - a) (x - 10) + 1 = 0$ has integral roots.