Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 1 - Algebra>EXERCISE - 1.3>Q 4

MEDIUM

IOQM - PRMO and RMO

IMPORTANT

Earn 100

Mr. Fat is going to pick three non-zero real number and Mr. Taf is going to arrange the three numbers as the coefficients of a quadratic equation.

$□ x^{2} + □ x + □ = 0$

Mr. Fat wins the game if and only if the resulting equation has two distinct rational solutions. Who has a winning strategy?

Important Questions on Algebra

MEDIUM

IOQM - PRMO and RMO

IMPORTANT

Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 1 - Algebra>EXERCISE - 1.3>Q 5

a, b, c

are three distinct non-zero real numbers. Prove that the following three equations

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

and

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

cannot all have two equal real roots.

EASY

IOQM - PRMO and RMO

IMPORTANT

Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 1 - Algebra>EXERCISE - 1.3>Q 6

If

x^{2} + x + 1 = 0

, find the absolute value of

x^{1999} + x^{2000}

.

MEDIUM

IOQM - PRMO and RMO

IMPORTANT

Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 1 - Algebra>EXERCISE - 1.3>Q 9

For how many real values of

a

will

x^{2} + 2 a x + 2008 = 0

has two integer roots?

MEDIUM

IOQM - PRMO and RMO

IMPORTANT

Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 1 - Algebra>EXERCISE - 1.3>Q 10

If

x, y

are positive real numbers satisfying the system of equations

x^{2} + y \sqrt{x y} = 336, y^{2} + x \sqrt{x y} = 112,

then

x + y

equals

MEDIUM

IOQM - PRMO and RMO

IMPORTANT

Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 1 - Algebra>EXERCISE - 1.3>Q 12

When

x

is real, the greatest possible value of

10^{x} - 100^{x}

is

MEDIUM

IOQM - PRMO and RMO

IMPORTANT

Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 1 - Algebra>EXERCISE - 1.3>Q 13

Find integers

' a'

and

' b'

such that

(x^{2} - x - 1)

divides

a x^{17} + b x^{16} + 1

HARD

IOQM - PRMO and RMO

IMPORTANT

Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 1 - Algebra>EXERCISE - 1.3>Q 14

Find the real points

(x, y)

satisfying

3 x^{2} + 3 y^{2} - 4 x y + 10 x - 10 y + 10 = 0

HARD

IOQM - PRMO and RMO

IMPORTANT

Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 1 - Algebra>EXERCISE - 1.3>Q 19

Given that

a = 8 - b

and

c^{2} = a b - 16,

prove that

a = b .