Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 2 - Number System>EXERCISE - 2.1>Q 5

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IMPORTANT

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What is the smallest positive integer $x$ for which $x^{2} - x + 5$ is not a prime?

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Important Questions on Number System

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IMPORTANT

Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 2 - Number System>EXERCISE - 2.1>Q 6

What is the smallest composite number generated by

p^{2} - p - 1,

where

p

is a prime?

HARD

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IMPORTANT

Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 2 - Number System>EXERCISE - 2.1>Q 7

Some prime numbers are

1

more than a square, as in

4^{2} + 1 = 17

. How many primes

p \leq 100

are of the form

p = n^{2} + 1

?

EASY

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IMPORTANT

Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 2 - Number System>EXERCISE - 2.1>Q 8

How many prime numbers

x

satisfy

\sqrt{300} < x < \sqrt{700}

?

HARD

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IMPORTANT

Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 2 - Number System>EXERCISE - 2.1>Q 9

The number of pairs of prime numbers

(p, q)

satisfying the condition

\frac{51}{100} < \frac{1}{p} + \frac{1}{q} < \frac{5}{6}

will be

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IMPORTANT

Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 2 - Number System>EXERCISE - 2.1>Q 11

The number of positive integers

{}^{'}n^{'}

for which

3 n - 4, 4 n - 5

and

5 n - 3

are all primes is

HARD

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IMPORTANT

Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 2 - Number System>EXERCISE - 2.1>Q 12

m, n

are natural numbers. The number of pairs

(m, n)

for which

m^{2} + n^{2} + 2 m n - 2013 m - 2013 n - 2014 = 0

is

EASY

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IMPORTANT

Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 2 - Number System>EXERCISE - 2.1>Q 13

Suppose that

a, b, c

are distinct numbers such that

{(b - a)}^{2} - 4 (b - c) (c - a) = 0

, find the value of

(\frac{b - c}{c - a})

.

EASY

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IMPORTANT

Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 2 - Number System>EXERCISE - 2.1>Q 14

Suppose that

p

and

q

are prime numbers

(p > q)

and they are roots of the equation

x^{2} - 99 x + m = 0

for some

m .

What is the value of

p - q ?