Mathematics for Joint Entrance Examination JEE (Advanced) Algebra>Chapter 2 - Theory of Equations>Exercises>Q 46

HARD

JEE Main/Advance

IMPORTANT

Earn 100

$P (x)$ is a polynomial with integral coefficients such that for four distinct integers $a, b, c, d : P (a) = P (b) = P (c) = P (d) = 3$ . If $P (e) = 5$ ( $e$ is an integer), then

100% studentsanswered this correctly

Important Questions on Theory of Equations

MEDIUM

JEE Main/Advance

IMPORTANT

Mathematics for Joint Entrance Examination JEE (Advanced) Algebra>Chapter 2 - Theory of Equations>Exercises>Q 47

Let

f (x) = x^{2} + b x + c,

where

b, c \in R

, If

f (x)

is a factor of both

x^{4} + 6 x^{2} + 25

and

3 x^{4} + 4 x^{2} + 28 x + 5

, then the least value of

f (x)

is

MEDIUM

JEE Main/Advance

IMPORTANT

Mathematics for Joint Entrance Examination JEE (Advanced) Algebra>Chapter 2 - Theory of Equations>Exercises>Q 48

Consider the equation

x^{2} + 2 x - n = 0,

where

n \in N a n d n \in [5, 100]

. The total number of different values of

n

so that the given equation has integral roots is

MEDIUM

JEE Main/Advance

IMPORTANT

Mathematics for Joint Entrance Examination JEE (Advanced) Algebra>Chapter 2 - Theory of Equations>Exercises>Q 49

The total number of integral values of

a

so that

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

has integral root is equal to

MEDIUM

JEE Main/Advance

IMPORTANT

Mathematics for Joint Entrance Examination JEE (Advanced) Algebra>Chapter 2 - Theory of Equations>Exercises>Q 50

The number of integral values of

a

for which the quadratic equation

(x + a) (x + 1991) + 1 = 0

has integral roots are

MEDIUM

JEE Main/Advance

IMPORTANT

Mathematics for Joint Entrance Examination JEE (Advanced) Algebra>Chapter 2 - Theory of Equations>Exercises>Q 51

The number of real solutions of the equation

{(\frac{9}{10})}^{x} = - 3 + x - x^{2}

is

HARD

JEE Main/Advance

IMPORTANT

Mathematics for Joint Entrance Examination JEE (Advanced) Algebra>Chapter 2 - Theory of Equations>Exercises>Q 52

The number of real solutions of

|x| + 2 \sqrt{5 - 4 x - x^{2}} = 16

is/are

MEDIUM

JEE Main/Advance

IMPORTANT

Mathematics for Joint Entrance Examination JEE (Advanced) Algebra>Chapter 2 - Theory of Equations>Exercises>Q 53

Let

p (x) = 0

be a polynomial equation of the least possible degree, with rational coefficients, having

\sqrt[3]{7} + \sqrt[3]{49}

as one of its roots. Then, the product of all the roots of

p (x) = 0

is

HARD

JEE Main/Advance

IMPORTANT

Mathematics for Joint Entrance Examination JEE (Advanced) Algebra>Chapter 2 - Theory of Equations>Exercises>Q 54

If

α, β, γ, σ

are the roots of the equation

x^{4} + 4 x^{3} - 6 x^{2} + 7 x - 9 = 0,

then the value of

(1 + α^{2}) (1 + β^{2}) (1 + γ^{2}) (1 + σ^{2})

is