Tag: maths

Questions Related to maths

The mathematical statements that are proved are called axioms.

maths axioms, postulates and theorems conditional statements and converse euclid's postulates introduction to euclid's geometry euclid's fifth postulate

  1. True

  2. False

  3. cannot be determined

  4. None of the above

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Correct Option: B
Explanation:

False, statements that are proved are theorems.

A proof is required for :

maths axioms, postulates and theorems conditional statements and converse euclid's postulates introduction to euclid's geometry euclid's fifth postulate

  1. Postulate

  2. Axiom

  3. Theorem

  4. Definition

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Correct Option: C
Explanation:

Axiom/Postulate — a statement that is assumed to be true without proof. These are the basic building blocks from which all theorems are proved. 


Theorem — a mathematical statement that is proved using rigorous mathematical reasoning.  In a mathematical paper, the term theorem is often reserved for the most important results.
  
So, the correct option is $C$ as a theorem needs a proof.

A lemma is a proven statement used for proving another statement.

maths axioms, postulates and theorems conditional statements and converse euclid's postulates introduction to euclid's geometry euclid's fifth postulate

  1. True

  2. False

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Correct Option: A
Explanation:

'Lemma ' is a subsidiary or intermediate theorem in an argument or proof.

A theorem is:

maths axioms, postulates and theorems conditional statements and converse euclid's postulates introduction to euclid's geometry euclid's fifth postulate

  1. an assumption

  2. always true

  3. always false

  4. sometimes true and sometimes false

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Correct Option: B
Explanation:

A theorem is a Statement (logic) that has been proved on the basis of previously established statements, such as other theorems, and generally accepted statements.
A theorem is based on assumption that theorem is true.


Example: In a Euclidean space, the sum of measures of the three angles of any triangles is invariably equal to the straight angle, also as $180^o$

Convert the following fraction into simple decimal recurring form.

$\displaystyle \frac{1}{6}$= ?

maths real number first forms rational numbers as recurring/terminating decimals decimal expansions of real numbers

  1. $0.1\bar 9$

  2. $0.1\bar 6$

  3. $0.1\bar 4$

  4. $0.1\bar 3$

Show answer
Correct Option: B
Explanation:
    Pure recurring decimal is a decimal fraction in which all the figures after the decimal point are repeated.
    $\displaystyle \frac { 1 }{ 6 }= 0.6666666666..$ is $ 0.\overset { \ _ \ _  }{ 6 } $.

Find whether it is a terminating or a non-terminating decimal.

$2.4 \div 0.072$.

maths real number first forms rational numbers as recurring/terminating decimals decimal expansions of real numbers

  1. Terminating

  2. Non-terminating

  3. Ambiguous

  4. Data insufficient

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Correct Option: B
Explanation:

A terminating decimal is a decimal that ends. It's a decimal with a finite number of digits.

$2.4\div 0.072=33.3333333333....$.
The division gives recurring factor.
Hence, it is a non-terminating decimal.

Express $\displaystyle \frac{4}{9}$ as recurring decimal 

maths real number first forms rational numbers as recurring/terminating decimals decimal expansions of real numbers

  1. $0.\bar 5$

  2. $0.\bar 4$

  3. $0.\overline {45}$

  4. $0.\overline {54}$

Show answer
Correct Option: B
Explanation:

On dividing 4 by 9 we get

$\dfrac { 4 }{ 9 } =0.4444......$
So, correct answer is option B.

Find whether it is a terminating or a non-terminating decimal.

$3.2 \div 2.24$.

maths real number first forms rational numbers as recurring/terminating decimals decimal expansions of real numbers

  1. Terminating

  2. Non-terminating

  3. Ambiguous

  4. Data insufficient

Show answer
Correct Option: B
Explanation:

A terminating decimal is a decimal that ends. It's a decimal with a finite number of digits.

$3.2\div 2.24= 1.428571429....$.
The division does not gives end result.
Hence, it is a non-terminating decimal.

Find whether it is a terminating or a non-terminating decimal.

$0.3 \div 0.09$.

maths real number first forms rational numbers as recurring/terminating decimals decimal expansions of real numbers

  1. Terminating

  2. Non-terminating

  3. Ambiguous

  4. Data insufficient

Show answer
Correct Option: B
Explanation:

A terminating decimal is a decimal that ends. It's a decimal with a finite number of digits.

$0.3\div 0.09= 3.33333333333333...$. 
The division results in recurring factor.
Hence it is a non terminating decimal.

The rational number which can be expressed as a terminating decimal is

maths real number first forms rational numbers as recurring/terminating decimals decimal expansions of real numbers

  1. $\displaystyle \frac{1}{6}$

  2. $\displaystyle \frac{1}{12}$

  3. $\displaystyle \frac{1}{15}$

  4. $\displaystyle \frac{1}{20}$

Show answer
Correct Option: D
Explanation:

A terminating decimal is a decimal that ends. It's a decimal with a finite number of digits. 
$\displaystyle \frac {1}{20}= 0.05$ 

In the other options, the decimal does not end with a finite number of digits.