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IMPORTANT

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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 .

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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 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

_________

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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 .

HARD

JEE Main/Advance

IMPORTANT

Mathematics for Joint Entrance Examination JEE (Advanced) Algebra>Chapter 2 - Theory of Equations>Exercises>Q 18

Let

a, b \in R

and

a b \neq 1

. If

6 a^{2} + 20 a + 15 = 0

and

15 b^{2} + 20 b + 6 = 0

, then the value of

\frac{4030 b^{3}}{a b^{2} - 9 {(a b + 1)}^{3}}

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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 20

If the equation

x^{2} + 2 (λ + 1) x + λ^{2} + λ + 7 = 0

has only negative roots, then the least value of

λ

equals

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)

HARD

JEE Main/Advance

IMPORTANT

Mathematics for Joint Entrance Examination JEE (Advanced) Algebra>Chapter 2 - Theory of Equations>Exercises>Q 22

If the set of values of

a

for which

f (x) = a x^{2} - (3 + 2 a) x + 6, a \neq 0

is positive for exactly three distinct negative integral values of

x

(c, d],

then the value of

d

Suppose a, b, c are the roots of the cubic x3-x2-2=0. Then the value of a3+b3+c3 is .

Important Questions on Theory of Equations

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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 .