If [ x ] denotes the greatest integer ≤ x , then the system of linear equations [ s i n θ ] x + [ - c o s θ ] y = 0 , [ c o t θ ] x + y = 0

have infinitely many solutions if θ ∈ π 2 , 2 π 3 and has a unique solution if θ ∈ π , 7 π 6

has a unique solution if θ ∈ π 2 , 2 π 3 ∪ π , 7 π 6

have infinitely many solution if θ ∈ π 2 , 2 π 3 ∪ π , 7 π 6

has a unique if θ ∈ π 2 , 2 π 3 and have infinitely many solutions if θ ∈ π , 7 π 6

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If the number formed by the last two digits of a three digit integer is an integral multiple of $6$ , the original integer Itself will always be divisible by

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Important Questions on Number Systems

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Mathematics>Arithmetic>Number Systems>Basics of Number Systems

1 USD = Rs. 67.89. A money changer adds a margin of Rs. 1.11. What will be the cost of 150 USD ?

1. Rs. 10,183
2. Rs. 10,350
3. Rs. 10,330
4. Rs. 10,450

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Mathematics>Arithmetic>Number Systems>Basics of Number Systems

How many times does the number $3$ occur in unit's place for numbers ranging from $1$ to $100$ ?

1. $20$
2. $11$
3. $10$
4. $19$

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Mathematics>Arithmetic>Number Systems>Basics of Number Systems

Twice the difference between two numbers is equal to their sum. If one number is $15$ , Find the other number.

1. $15$
2. $10$
3. $5$
4. $20$

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Mathematics>Arithmetic>Number Systems>Basics of Number Systems

\frac{1}{4}

of third-part of number is

5

, then the number is:

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Mathematics>Arithmetic>Number Systems>Basics of Number Systems

The largest non-negative integer

k

such that

24^{k}

divides

13!

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Mathematics>Arithmetic>Number Systems>Basics of Number Systems

The difference of the cubes of two consecutive numbers is

127

. What are the numbers?

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Mathematics>Arithmetic>Number Systems>Basics of Number Systems

The sum of three consecutive numbers is $126$ . Find the highest number-

1. $41$
2. $42$
3. $43$
4. $44$

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Mathematics>Arithmetic>Number Systems>Basics of Number Systems

Find the number that is different from other options in the given list of numbers.

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Mathematics>Arithmetic>Number Systems>Basics of Number Systems

The function

f : N \to I

defined by

f (x) = x - 5 [\frac{x}{5}]

, where

N

is the set of natural numbers and

[x]

denotes the greatest integer less than or equal to

x

, is:

HARD

Mathematics>Arithmetic>Number Systems>Basics of Number Systems

For a real number

r

we denote by

[r]

the largest integer less than or equal to

r

. If

x, y

are real numbers with

x, y \geq 1

then which of the following statements is always true?

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Mathematics>Arithmetic>Number Systems>Basics of Number Systems

The sum of two digits of a number is

13

. If

9

is subtracted from the number, the digits are interchanged. What is the number?

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Mathematics>Arithmetic>Number Systems>Basics of Number Systems

How many natural numbers

n

are there such that

n! + 10

is a perfect square ?

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Mathematics>Arithmetic>Number Systems>Basics of Number Systems

The least number, which is to be added to the greatest number of

4

digits so that the sum may be divisible by

345

, is _____.

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Mathematics>Arithmetic>Number Systems>Basics of Number Systems

59

is added to a square of a number, the result is

900

. What is the number?

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Mathematics>Arithmetic>Number Systems>Basics of Number Systems

Which of the following numbers is perfectly divisible by 4?

HARD

Mathematics>Arithmetic>Number Systems>Basics of Number Systems

For

x \in R,

Let

[x]

denotes the greatest integer

\leq x,

then the sum of the series

[- \frac{1}{3}] + [- \frac{1}{3} - \frac{1}{100}] + [- \frac{1}{3} - \frac{2}{100}] + . . . . . + [- \frac{1}{3} - \frac{99}{100}]

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Mathematics>Arithmetic>Number Systems>Basics of Number Systems

[x]

denotes the greatest integer

\leq x,

then the system of linear equations

[s i n θ] x + [- c o s θ] y = 0

[c o t θ] x + y = 0

HARD

Mathematics>Arithmetic>Number Systems>Basics of Number Systems

For a real number

r

let

[r]

denote the largest integer less than or equal to

r .

Let

a > 1

a real number which is not an integer and

k

be the smallest positive integer such that

[a^{k}] > {[a]}^{k}

Then which of the following statement is always true?

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Mathematics>Arithmetic>Number Systems>Basics of Number Systems

The number of distinct primes dividing

12! + 13! + 14!

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Mathematics>Arithmetic>Number Systems>Basics of Number Systems

Three-digit numbers are formed from the digits

1, 2

and

3

in such a way that the digits are not repeated. What is the sum of such three-digit numbers?

If the number formed by the last two digits of a three digit integer is an integral multiple of 6, the original integer Itself will always be divisible by

Important Questions on Number Systems

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If the number formed by the last two digits of a three digit integer is an integral multiple of $6$ , the original integer Itself will always be divisible by