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IMPORTANT

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If $\cos^{2} \frac{π}{8}$ is a root of the equation $x^{2} + a x + b = 0$ where $α, b, \in Q,$ then the least value of the expression $x^{2} + 8 b x + a$ is .

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Important Questions on Theory of Equations

HARD

JEE Main/Advance

IMPORTANT

Mathematics for Joint Entrance Examination JEE (Advanced) Algebra>Chapter 2 - Theory of Equations>Exercises>Q 10

Given

α

and

β

are the roots of the quadratic equation

x^{2} - 4 x + k = 0 (k \neq 0)

. If

α β, α β^{2} + α^{3} + β^{3}

are in geometric progression, then the value of

7 k

equal .

HARD

JEE Main/Advance

IMPORTANT

Mathematics for Joint Entrance Examination JEE (Advanced) Algebra>Chapter 2 - Theory of Equations>Exercises>Q 12

Let

α

and

β

be the solutions of the quadratic equation

x^{2} - 1154 x + 1 = 0

then the value of

α^{\frac{1}{4}} + β^{\frac{1}{4}}

is equal to .

HARD

JEE Main/Advance

IMPORTANT

Mathematics for Joint Entrance Examination JEE (Advanced) Algebra>Chapter 2 - Theory of Equations>Exercises>Q 14

Suppose

a, b, c

are the roots of the cubic

x^{3} - x^{2} - 2 = 0

. Then the value of

a^{3} + b^{3} + c^{3}

is .

HARD

JEE Main/Advance

IMPORTANT

Mathematics for Joint Entrance Examination JEE (Advanced) Algebra>Chapter 2 - Theory of Equations>Exercises>Q 15

Polynomial

P (x)

conatins only terms of odd degree. When

P (x)

is divided by

(x - 3),

the remainder is

6

. If

P (x)

is divided by

(x^{2} - 9),

then the remainder is

g (x) .

Then the value of

g (2)

is .

HARD

JEE Main/Advance

IMPORTANT

Mathematics for Joint Entrance Examination JEE (Advanced) Algebra>Chapter 2 - Theory of Equations>Exercises>Q 16

α

and

β

are the roots of the equation

x^{2} - 6 x + 12 = 0

and the value of

{(α - 2)}^{24} - \frac{{(β - 6)}^{8}}{α^{8}} + 1

4^{a}

, then the value of

a

_________

HARD

JEE Main/Advance

IMPORTANT

Mathematics for Joint Entrance Examination JEE (Advanced) Algebra>Chapter 2 - Theory of Equations>Exercises>Q 17

Let

a

and

b

be the roots of the equation

x^{2} - 10 c x - 11 d = 0

and those of

x^{2} - 10 a x - 11 b = 0

are

c

and

d

, then the value of

a + b + c + d,

when

a \neq b \neq c \neq d,

is .

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JEE Main/Advance

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Mathematics for Joint Entrance Examination JEE (Advanced) Algebra>Chapter 2 - Theory of Equations>Exercises>Q 19

If there exists at least one real

x

which satisfies both the equations

x^{2} + 2 x \sin y + 1 = 0

, where

y \in (0, \frac{π}{2})

and

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

, then the value of

a + \sin y

is .

HARD

JEE Main/Advance

IMPORTANT

Mathematics for Joint Entrance Examination JEE (Advanced) Algebra>Chapter 2 - Theory of Equations>Exercises>Q 21

All the values of

k

for which the quadratic polynomial

f (x) = - 2 x^{2} + k x + k^{2} + 5

has two distinct zeros and only one of them satisfying

0 < x < 2

, lie in the interval

(a, b)

. The value of

(a + 10 b)

If cos2π8 is a root of the equation x2+ax+b=0 where α, b, ∈Q, then the least value of the expression x2+8bx+a is .

Important Questions on Theory of Equations

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If $\cos^{2} \frac{π}{8}$ is a root of the equation $x^{2} + a x + b = 0$ where $α, b, \in Q,$ then the least value of the expression $x^{2} + 8 b x + a$ is .