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

HARD

IOQM - PRMO and RMO

IMPORTANT

Earn 100

Find the real points $(x, y)$ satisfying $3 x^{2} + 3 y^{2} - 4 x y + 10 x - 10 y + 10 = 0$

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 16

α, β

are the real roots of the equation

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

Find the number of the pairs

(p, q)

such that the quadratic equation with roots

α^{2}, β^{2}

is still

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

.

HARD

IOQM - PRMO and RMO

IMPORTANT

Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 1 - Algebra>EXERCISE - 1.3>Q 17

Given that

α

and

β

are the real roots

x^{2} - 2 x - 1 = 0,

find the value of

\frac{5 α^{4} + 12 β^{3}}{13} .

MEDIUM

IOQM - PRMO and RMO

IMPORTANT

Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 1 - Algebra>EXERCISE - 1.3>Q 18

Given that the real numbers

s, t

satisfy

19 s^{2} + 99 s + 1 = 0, t^{2} + 99 t + 19 = 0

and st

\neq 1 .

Find the value of

|\frac{s t + 4 s + 1}{t}| .

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 .

HARD

IOQM - PRMO and RMO

IMPORTANT

Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 1 - Algebra>EXERCISE - 1.3>Q 20

Given that

α, β

are roots of the equation

x^{2} - 7 x + 8 = 0

, where

α > β

. Find the value of

\frac{2}{α} + 3 β^{2}

without solving the equation.

MEDIUM

IOQM - PRMO and RMO

IMPORTANT

Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 1 - Algebra>EXERCISE - 1.3>Q 21

Both roots of the quadratic equation

x^{2} - 63 x + k = 0

are prime numbers. The number of possible values of

k

is

EASY

IOQM - PRMO and RMO

IMPORTANT

Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 1 - Algebra>EXERCISE - 1.3>Q 22

Find the sum of all possible values of $a$ such that the following equation has real root in $x$ :

${(x - a)}^{2} + {(x^{2} - 3 x + 2)}^{2} = 0$

EASY

IOQM - PRMO and RMO

IMPORTANT

Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 1 - Algebra>EXERCISE - 1.3>Q 23

Let

p

be a real number such that the equation

2 y^{2} - 8 y = p

has only one solution. Then find the value of

p + 64 .