Prove that, 3 + 5 is an irrational number.

Assume that 3 + 5 is a rational number equal to ' a ' . ⇒ a = 3 + 5 . Squaring both sides, ⇒ a 2 = 3 + 5 2 ⇒ a 2 = 3 + 5 + 2 3 5 ⇒ a 2 = 8 + 2 15 ⇒ 2 15 = a 2 - 8 ⇒ 15 = a 2 - 8 2 Now, a 2 - 8 2 is a rational number, since a is a rational number. ⇒ 15 is a rational number. But, we know that, 15 is an irrational number. Hence, our assumption is wrong. Therefore, 3 + 5 is an irrational number.

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

Prove that, 2 + 7 is irrational number.

Let 2 + 7 is rational number, then 2 + 7 = p q 7 = p q - 2 Squaring on both sides 7 2 = p 2 q 2 - 2 2 p q + 2 p 2 q 2 - 2 2 p q + 2 = 7 p 2 q 2 - 5 = 2 2 p q p 2 - 5 q 2 = 2 2 p q p 2 - 5 q 2 2 p q = 2 Thus, 2 is an irrational number,which contradicts our assumption. Hence, 2 + 7 is irrational number.

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What is the HCF of 867 and 225 ? (Use Euclid’s division algorithm)

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Important Questions on Number Systems

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Numerical and Mental Ability>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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Numerical and Mental Ability>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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Numerical and Mental Ability>Arithmetic>Number Systems>Basics of Number Systems

B has $32$ pens, $24$ pencils and $16$ erasers. How many sets of these three items can B make without any left over?

1. $6$
2. $7$
3. $8$
4. $9$

EASY

Numerical and Mental Ability>Arithmetic>Number Systems>Basics of Number Systems

\frac{1}{4}

of third-part of number is

5

, then the number is:

HARD

Numerical and Mental Ability>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:

EASY

Numerical and Mental Ability>Arithmetic>Number Systems>Basics of Number Systems

The difference of the cubes of two consecutive numbers is

127

. What are the numbers?

MEDIUM

Numerical and Mental Ability>Arithmetic>Number Systems>Basics of Number Systems

P and Q are two examination halls. If

10

examinees are sent from P into Q, the number of examinees becomes equal in both the halls. If

20

examinees are transferred from Q into P, the number of examinees becomes double in P in comparison with Q. What is the number of examinees in each P and Q separately ?

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Numerical and Mental Ability>Arithmetic>Number Systems>Basics of Number Systems

Which of the following numbers is perfectly divisible by 4?

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Numerical and Mental Ability>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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Numerical and Mental Ability>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. $44$

EASY

Numerical and Mental Ability>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?

MEDIUM

Numerical and Mental Ability>Arithmetic>Number Systems>Basics of Number Systems

Prove that,

\sqrt{3} + \sqrt{5}

is an irrational number.

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Numerical and Mental Ability>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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Numerical and Mental Ability>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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Numerical and Mental Ability>Arithmetic>Number Systems>Basics of Number Systems

Identify the missing number below.

$12 36 3$

$5 20 4$

$19 38_$

(A) $3$

(B) $2$

(D) $4$

HARD

Numerical and Mental Ability>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

EASY

Numerical and Mental Ability>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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Numerical and Mental Ability>Arithmetic>Number Systems>Basics of Number Systems

A number of two digits is equal to '

k

' times to the sum of these digits. If the places of these digits are mutually exchanged. the newly formed number is equal to the sum of these digits multiplied by which one of the following options ?

MEDIUM

Numerical and Mental Ability>Arithmetic>Number Systems>Basics of Number Systems

Prove that,

\sqrt{2} + \sqrt{7}

is irrational number.

EASY

Numerical and Mental Ability>Arithmetic>Number Systems>Basics of Number Systems

Which of the following is not irrational?

What is the HCF of 867 and 225 ? (Use Euclid’s division algorithm)

Important Questions on Number Systems

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