Mathematics for Joint Entrance Examination JEE (Advanced) Algebra>Chapter 2 - Theory of Equations>Exercises>Q 42

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The number of values of $k$ for which $[x^{2} - (k - 2) x + k^{2}] \times [x^{2} + k x + (2 k - 1)]$ is a perfect square is

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Important Questions on Theory of Equations

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Mathematics for Joint Entrance Examination JEE (Advanced) Algebra>Chapter 2 - Theory of Equations>Exercises>Q 43

The sum of values of

x

satisfying the equation

{(31 + 8 \sqrt{15})}^{x^{2} - 3} + 1 = {(32 + 8 \sqrt{15})}^{x^{2} - 3}

is

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Mathematics for Joint Entrance Examination JEE (Advanced) Algebra>Chapter 2 - Theory of Equations>Exercises>Q 44

The equation

{(x^{2} + x + 1)}^{2} + 1 = (x^{2} + x + 1) (x^{2} - x - 5)

for

x \in (- 2, 3)

will have number of solutions,

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Mathematics for Joint Entrance Examination JEE (Advanced) Algebra>Chapter 2 - Theory of Equations>Exercises>Q 45

If

α, β

are the roots of

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

and

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

and if

(\frac{α}{β}), (\frac{β}{α})

are the roots of

x^{n} + 1 + {(x + 1)}^{n} = 0,

then

n (\in N)

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

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

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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

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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

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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

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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