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

MEDIUM

Earn 100

Let $p$ and $q$ be two natural numbers such that $p > q$ . When $p$ is divided by $q$ , the remainder is $r$ .

$i)$ $r$ cannot be $(p - q)$

$i i)$ $r$ can either be $q$ or $(p - q)$

$i i i)$ $r$ is definitely less than $q$

Which of the above statements is/are true?

100% studentsanswered this correctly

Important Questions on Real Numbers

EASY

Mathematics>Arithmetic>Real Numbers>Euclid's Division Lemma

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. $20$

HARD

Mathematics>Arithmetic>Real Numbers>Euclid's Division Lemma

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?

EASY

Mathematics>Arithmetic>Real Numbers>Euclid's Division Lemma

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

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

EASY

Mathematics>Arithmetic>Real Numbers>Euclid's Division Lemma

The difference of the cubes of two consecutive numbers is

127

. What are the numbers?

MEDIUM

Mathematics>Arithmetic>Real Numbers>Euclid's Division Lemma

The number of distinct primes dividing

12! + 13! + 14!

MEDIUM

Mathematics>Arithmetic>Real Numbers>Euclid's Division Lemma

59

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

900

. What is the number?

EASY

Mathematics>Arithmetic>Real Numbers>Euclid's Division Lemma

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. $19$

EASY

Mathematics>Arithmetic>Real Numbers>Euclid's Division Lemma

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

MEDIUM

Mathematics>Arithmetic>Real Numbers>Euclid's Division Lemma

The largest non-negative integer

k

such that

24^{k}

divides

13!

HARD

Mathematics>Arithmetic>Real Numbers>Euclid's Division Lemma

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:

EASY

Mathematics>Arithmetic>Real Numbers>Euclid's Division Lemma

\frac{1}{4}

of third-part of number is

5

, then the number is:

MEDIUM

Mathematics>Arithmetic>Real Numbers>Euclid's Division Lemma

There are exactly twelve Sundays in the period from January

1

to March

31

in a certain year. Then, the day corresponding to February

15

in that year is

EASY

Mathematics>Arithmetic>Real Numbers>Euclid's Division Lemma

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?

EASY

Mathematics>Arithmetic>Real Numbers>Euclid's Division Lemma

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

MEDIUM

Mathematics>Arithmetic>Real Numbers>Euclid's Division Lemma

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 _____.

HARD

Mathematics>Arithmetic>Real Numbers>Euclid's Division Lemma

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}]

HARD

Mathematics>Arithmetic>Real Numbers>Euclid's Division Lemma

[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>Real Numbers>Euclid's Division Lemma

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?

MEDIUM

Mathematics>Arithmetic>Real Numbers>Euclid's Division Lemma

The number of positive integers

n

in the set

\{2, 3, \dots . ., 200\}

such that

\frac{1}{n}

has a terminating decimal expansion is

MEDIUM

Mathematics>Arithmetic>Real Numbers>Euclid's Division Lemma

How many natural numbers

n

are there such that

n! + 10

is a perfect square ?

Let p and q be two natural numbers such that p>q. When p is divided by q, the remainder is r. i) r cannot be p-q ii) r can either be q or p-q iii) r is definitely less than q Which of the above statements is/are true?

Important Questions on Real Numbers

Learn and Prepare for any exam you want !

Let $p$ and $q$ be two natural numbers such that $p > q$ . When $p$ is divided by $q$ , the remainder is $r$ .

$i)$ $r$ cannot be $(p - q)$

$i i)$ $r$ can either be $q$ or $(p - q)$

$i i i)$ $r$ is definitely less than $q$

Which of the above statements is/are true?