Alpha Question Bank for Engineering: Mathematics>Chapter 4 - Quadratic Equations>Exercise-1>Q 104

HARD

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IMPORTANT

Earn 100

If both roots of the equation $x^{2} - 6 a x + 2 - 2 a + 9 a^{2} = 0$ exceed $3$ , then show that $a > \frac{11}{9}$ .

Important Questions on Quadratic Equations

MEDIUM

JEE Main/Advance

IMPORTANT

Alpha Question Bank for Engineering: Mathematics>Chapter 4 - Quadratic Equations>Exercise-1>Q 105

Find all the values of

k

for which one root of the equation

x^{2} - (k + 1) x + k^{2} + k - 8 = 0,

exceeds

2

and the other root is smaller than

2

.

MEDIUM

JEE Main/Advance

IMPORTANT

Alpha Question Bank for Engineering: Mathematics>Chapter 4 - Quadratic Equations>Exercise-1>Q 106

Find all the real values of

' a'

, so that the roots of the equation

(a^{2} - a + 2) x^{2} + 2 (a - 3) x + 9 (a^{4} - 16) = 0

are of opposite sign.

HARD

JEE Main/Advance

IMPORTANT

Alpha Question Bank for Engineering: Mathematics>Chapter 4 - Quadratic Equations>Exercise-1>Q 107

Find all the values of

' a'

, so that exactly one root of the equation

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

lies between the numbers

(i), (ii)

and

4

, and no root of the equation is either equal to

2

or equal to

4

.

HARD

JEE Main/Advance

IMPORTANT

Alpha Question Bank for Engineering: Mathematics>Chapter 4 - Quadratic Equations>Exercise-1>Q 108

If

α

and

β

are the two distinct roots of

x^{2} + 2 (k - 3) x + 9 = 0

, then find the values of

k

such that

α, β \in (- 6, 1)

.

HARD

JEE Main/Advance

IMPORTANT

Alpha Question Bank for Engineering: Mathematics>Chapter 4 - Quadratic Equations>Exercise-1>Q 110

Find the value of

' a'

so that

x^{2} - 11 x + a = 0

and

x^{2} - 14 x + 2 a = 0

have a common root.

HARD

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IMPORTANT

Alpha Question Bank for Engineering: Mathematics>Chapter 4 - Quadratic Equations>Exercise-1>Q 111

If

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

and

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

have a common root and

a, b, c

are non-zero real numbers, then find the value of

\frac{a^{3} + b^{3} + c^{3}}{a b c} .

HARD

JEE Main/Advance

IMPORTANT

Alpha Question Bank for Engineering: Mathematics>Chapter 4 - Quadratic Equations>Exercise-1>Q 112

If

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

and

x^{2} + q x + p = 0, (p \neq q)

have a common root, show that

1 + p + q = 0

and also show that their other roots are the roots of the equation

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

.

HARD

JEE Main/Advance

IMPORTANT

Alpha Question Bank for Engineering: Mathematics>Chapter 4 - Quadratic Equations>Exercise-1>Q 113

Find values of $' k'$ if equation $x^{3} - 3 x^{2} + 2 = k$ has

$(i)$ $3$ real roots

$(ii)$ $1$ real root