Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 2 - Number System>EXERCISE - 2.3>Q 48

EASY

IOQM - PRMO and RMO

IMPORTANT

Earn 100

Let $p$ and $q$ represent two consecutive prime numbers. For some fixed integer $n,$ the set $\{n - 1, 3 n - 19, 38 - 5 n, 7 n - 45\}$ represents $\{p, 2 p, q, 2 q\},$ but not necessarily in that order. Find the value of $n .$

50% studentsanswered this correctly

Important Questions on Number System

MEDIUM

IOQM - PRMO and RMO

IMPORTANT

Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 2 - Number System>EXERCISE - 2.3>Q 49

If

a, b

and

c

are positive real numbers such that

a b + a + b = b c + b + c = c a + c + a = 35

, then the value of

(a + 1) (b + 1) (c + 1) = x^{y} .

Find the product of

x

and

y .

HARD

IOQM - PRMO and RMO

IMPORTANT

Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 2 - Number System>EXERCISE - 2.3>Q 50

Find the largest possible value of

n

such that there exist

n

consecutive positive integers whose sum is equal to

2010

MEDIUM

IOQM - PRMO and RMO

IMPORTANT

Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 2 - Number System>EXERCISE - 2.3>Q 51

Let

S

be the set of all integers

n

such that

\frac{8 n^{3} - 96 n^{2} + 360 n - 400}{2 n - 7}

is an integer. Find the value of

\sum_{n \in S} |n|

.

MEDIUM

IOQM - PRMO and RMO

IMPORTANT

Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 2 - Number System>EXERCISE - 2.3>Q 52

Find the sum of all positive integers

p

such that the expression

(x - p) (x - 13) + 4

can be expressed in the form

(x + q) (x + r)

for distinct integers

q

and

r .

HARD

IOQM - PRMO and RMO

IMPORTANT

Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 2 - Number System>EXERCISE - 2.3>Q 53

Let

P_{k} = 1 + \frac{1}{k} - \frac{1}{k^{2}} - \frac{1}{k^{3}},

where

k

is a positive integer. Find the sum of all the digits of least positive integer

n

such that the product

P_{2} P_{3} \dots P_{n}

exceeds

2010 .

EASY

IOQM - PRMO and RMO

IMPORTANT

Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 2 - Number System>EXERCISE - 2.3>Q 54

What is the sum of the digits of the square of

111, 111, 111 ?

HARD

IOQM - PRMO and RMO

IMPORTANT

Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 2 - Number System>EXERCISE - 2.3>Q 55

Let

a, b, c,

and

d

be real numbers with

|a - b| = 2, |b - c| = 3

and

|c - d| = 4 .

What is the sum of all possible values of

|a - d|

?

HARD

IOQM - PRMO and RMO

IMPORTANT

Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 2 - Number System>EXERCISE - 2.3>Q 56

Let

k = 2008^{2} + 2^{2008} .

What is the units digit of

k^{2} + 2^{k} ?