Mathematics Crash Course COMEDK UGET>Chapter 3 - Theory of Equation>Exercise 1>Q 100

HARD

COMEDK UGET

IMPORTANT

Earn 100

Let $p (x) = x^{2} + a x + b$ have two distinct real roots, where $a, b$ are real number. Define $g (x) = p (x^{3})$ for all real number $x$

Then, which of the following statements are true?
I. $g$ has exactly two distinct real roots.
II. $g$ can have more than two distinct real roots.
III. There exists a real number $α$ such that $g (x) \geq α$ for all real $x$

50% studentsanswered this correctly

Important Questions on Theory of Equation

EASY

COMEDK UGET

IMPORTANT

Mathematics Crash Course COMEDK UGET>Chapter 3 - Theory of Equation>Exercise 1>Q 1

If

x = 2 + 2^{\frac{2}{3}} + 2^{\frac{1}{3}},

then the value of

x^{3} - 6 x^{2} + 6 x

is

MEDIUM

COMEDK UGET

IMPORTANT

Mathematics Crash Course COMEDK UGET>Chapter 3 - Theory of Equation>Exercise 1>Q 2

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

MEDIUM

COMEDK UGET

IMPORTANT

Mathematics Crash Course COMEDK UGET>Chapter 3 - Theory of Equation>Exercise 1>Q 3

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} =

MEDIUM

COMEDK UGET

IMPORTANT

Mathematics Crash Course COMEDK UGET>Chapter 3 - Theory of Equation>Exercise 1>Q 4

How many real numbers

x

satisfy the equation

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

HARD

COMEDK UGET

IMPORTANT

Mathematics Crash Course COMEDK UGET>Chapter 3 - Theory of Equation>Exercise 1>Q 5

Evaluate $X$ in its simplest form then find sum of all digits of $X .$ Where $X$ is given as $X = \sqrt{2008 + 2007 \sqrt{2008 + 2007 \sqrt{2008 + 2007 \sqrt{\dots .}}}}$

MEDIUM

COMEDK UGET

IMPORTANT

Mathematics Crash Course COMEDK UGET>Chapter 3 - Theory of Equation>Exercise 1>Q 6

The value of

4 + \frac{1}{5 + \frac{1}{4 + \frac{1}{5 + \frac{1}{4 + \dots \dots \infty}}}}

is:

HARD

COMEDK UGET

IMPORTANT

Mathematics Crash Course COMEDK UGET>Chapter 3 - Theory of Equation>Exercise 1>Q 7

Find the value of

λ

such that sum of the squares of the roots of

x^{2} + (4 - λ) x + 3 = λ

has the least value.

MEDIUM

COMEDK UGET

IMPORTANT

Mathematics Crash Course COMEDK UGET>Chapter 3 - Theory of Equation>Exercise 1>Q 8

If the sum of the roots of the quadratic equation

(a + 1) x^{2} + (2 a + 3) x + (3 a + 4) = 0

is

- 1,

then find the product of the roots.