Higher Algebra>Chapter 9 - The Theory of Quadratic Equations>EXAMPLES. IX. c.>Q 3

MEDIUM

JEE Main

IMPORTANT

Earn 100

Show that the expression $A (x^{2} - y^{2}) - x y (B - C)$ always admits of two real linear factors.

Important Questions on The Theory of Quadratic Equations

EASY

JEE Main

IMPORTANT

Higher Algebra>Chapter 9 - The Theory of Quadratic Equations>EXAMPLES. IX. a.>Q 19

If

α, β

are the roots of the equation

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

then find the value of

α^{4} β^{7} + α^{7} β^{4}

.

EASY

JEE Main

IMPORTANT

Higher Algebra>Chapter 9 - The Theory of Quadratic Equations>EXAMPLES. IX. a.>Q 20

If

α, β

are the roots of the equation

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

, then find the value of

{(\frac{α}{β} - \frac{β}{α})}^{2}

.

MEDIUM

JEE Main

IMPORTANT

Higher Algebra>Chapter 9 - The Theory of Quadratic Equations>EXAMPLES. IX. c.>Q 4

If the equations

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

have a common root, show that it must be equal to

\frac{p q^{'} - p^{'} q}{q - q^{'}}

or

\frac{q - q^{'}}{p^{'} - p}

HARD

JEE Main

IMPORTANT

Higher Algebra>Chapter 9 - The Theory of Quadratic Equations>EXAMPLES. IX. c.>Q 5

Find the condition that the expressions

l x^{2} + m x y + n y^{2}, l^{'} x^{2} + m^{'} x y + n^{'} y^{2}

may have a common linear factor.

MEDIUM

JEE Main

IMPORTANT

Higher Algebra>Chapter 9 - The Theory of Quadratic Equations>EXAMPLES. IX. c.>Q 6

If the expression

3 x^{2} + 2 p x y + 2 y^{2} + 2 a x - 4 y + 1

can be resolved into linear factors, prove that

p

must be one of the roots of the equation

p^{2} + 4 a p + 2 a^{2} + 6 = 0

HARD

JEE Main

IMPORTANT

Higher Algebra>Chapter 9 - The Theory of Quadratic Equations>EXAMPLES. IX. c.>Q 7

Find the condition that the expression

a x^{2} + 2 h x y + b y^{2}, a^{'} x^{2} + 2 h^{'} x y + b^{'} y^{2}

may be respectively divisible by factors of the form

y - m x, m y + x

HARD

JEE Main

IMPORTANT

Higher Algebra>Chapter 9 - The Theory of Quadratic Equations>EXAMPLES. IX. c.>Q 8

Show that the equation

x^{2} - 3 x y + 2 y^{2} - 2 x - 3 y - 35 = 0

for every real value of

x,

there is a real value of

y,

and for every real value of

y

there is a real value of

x

.

HARD

JEE Main

IMPORTANT

Higher Algebra>Chapter 9 - The Theory of Quadratic Equations>EXAMPLES. IX. c.>Q 9

If

x

and

y

are two real quantities connected by the equation

9 x^{2} + 2 x y + y^{2} - 92 x - 20 y + 244 = 0

then

x

will lie between

3

and

6,

and

y

between

1

and

10 .