EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 2 - Polynomials>IOQM - PRMO and RMO 2019 (11-08-2019)>Q 3

EASY

IOQM - PRMO and RMO

IMPORTANT

Earn 100

How many positive integers $n$ are there such that $3 \leq n \leq 100$ and $x^{2^{n}} + x + 1$ is divisible by $x^{2} + x + 1 ?$

50% studentsanswered this correctly

Important Questions on Polynomials

HARD

IOQM - PRMO and RMO

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 2 - Polynomials>IOQM - PRMO and RMO 2019 (11-08-2019)>Q 4

Each of the numbers

x_{1}, x_{2}, \dots, x_{101}

is

\pm 1 .

What is the smallest positive value of

\sum_{1 \leq i \leq j \leq 101} x_{i} x_{j} ?

HARD

IOQM - PRMO and RMO

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 2 - Polynomials>IOQM - PRMO and RMO 2019 (25-08-2019)>Q 5

If

x = \sqrt{2} + \sqrt{3} + \sqrt{6}

is a root of

x^{4} + a x^{3} + b x^{2} + c x + d = 0

, where

a, b, c, d

are integers, what is the value of

|a + b + c + d|

?

HARD

IOQM - PRMO and RMO

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 2 - Polynomials>IOQM - PRMO and RMO 2019 (25-08-2019)>Q 6

Find the number of positive integers less than

101

that cannot be written as the difference of two squares of integers.

HARD

IOQM - PRMO and RMO

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 2 - Polynomials>IOQM - PRMO and RMO 2019 (25-08-2019)>Q 7

Find the largest value of

a^{b}

such that the positive integers

a, b > 1

satisfy

a^{b} b^{a} + a^{b} + b^{a} = 5329

.

HARD

IOQM - PRMO and RMO

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 2 - Polynomials>IOQM - PRMO and RMO 2019 (25-08-2019)>Q 8

What is the smallest prime number

p

such that

p^{3} + 4 p^{2} + 4 p

has exactly

30

positive divisors?

HARD

IOQM - PRMO and RMO

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 2 - Polynomials>IOQM - PRMO and RMO 2018>Q 9

The equation

166 \times 56 = 8590

is valid in some base

b \geq 10

(that is,

1, 6, 5, 8, 9, 0

are digits in base

b

in the above equation). Find the sum of all possible values of

b \geq 10

satisfying the equation.

MEDIUM

IOQM - PRMO and RMO

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 2 - Polynomials>IOQM - PRMO and RMO 2018>Q 10

Integers

a, b, c

satisfy

a + b - c = 1

and

a^{2} + b^{2} - c^{2} = - 1 .

What is the sum of all possible values of

a^{2} + b^{2}

+ c^{2} ?

HARD

IOQM - PRMO and RMO

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 2 - Polynomials>IOQM - PRMO and RMO 2018>Q 11

If

a, b, c \geq 4

are integers, not all equal and

4 a b c = (a + 3) (b + 3) (c + 3)

, then what is the value of

a + b + c ?