Mathematics Crash Course JEE Main>Chapter 24 - Quadratic Equations>Level 2>Q 27

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If $a^{3} - 3 a^{2} + 5 a - 17 = 0$ and $b^{3} - 3 b^{2} + 5 b + 11 = 0$ are such that $a + b$ is a real number, then the value of $a + b$ is

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Important Questions on Quadratic Equations

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IMPORTANT

Mathematics Crash Course JEE Main>Chapter 24 - Quadratic Equations>Level 3>Q 1

Let

P (x) = a_{0} + a_{1} x + a_{2} x^{2} + \dots + a_{n} x^{n}

be a polynomial in which

a_{i}

is a non-negative integer for each

i \in \{0, 1, 2, 3, \dots, n\}

. If

P (1) = 4

and

P (5) = 136

, what is the value of

P (3)

?

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Mathematics Crash Course JEE Main>Chapter 24 - Quadratic Equations>Level 3>Q 2

If

α

and

β, α

and

γ, α

and

δ

are the roots of the equations

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

and

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

respectively, where

a, b

and

c

are positive real numbers, then

α + α^{2} =

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Mathematics Crash Course JEE Main>Chapter 24 - Quadratic Equations>Level 3>Q 3

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 Crash Course JEE Main>Chapter 24 - Quadratic Equations>Level 3>Q 4

Number of real solutions of

\sqrt{x} + \sqrt{x - \sqrt{1 - x}} = 1

is:

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Mathematics Crash Course JEE Main>Chapter 24 - Quadratic Equations>Level 3>Q 5

If

α, β

are the roots of the equation

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

and

S_{n} = α^{n} + β^{n},

then

a S_{n + 1} + b S_{n} + c S_{n - 1} =_________(n \geq 2)

.

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Mathematics Crash Course JEE Main>Chapter 24 - Quadratic Equations>Level 3>Q 6

Let

a

and

b

be positive real numbers such that

\frac{1}{a} - \frac{1}{b} - \frac{1}{a + b} = 0

. Find the value of

{(\frac{b}{a} + \frac{a}{b})}^{2}

.

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Mathematics Crash Course JEE Main>Chapter 24 - Quadratic Equations>Level 3>Q 7

Let

x, y

and

z

be three real numbers such that

x y + y z + x z = 4

. Find the least possible value of

x^{2} + y^{2} + z^{2}

.

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Mathematics Crash Course JEE Main>Chapter 24 - Quadratic Equations>Level 3>Q 8

Find the sum of all the real numbers $x$ that satisfy the equation ${(3^{x} - 27)}^{2} + {(5^{x} - 625)}^{2} = {(3^{x} + 5^{x} - 652)}^{2}$ .