Tag: introduction to euclid's geometry

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

Euclidean geometry is valid only for curved surfaces.

  1. True

  2. False

  3. Sometimes True

  4. Data Insufficient
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Euclid's postulates:

$\rightarrow$A straight line can be drawn joining any two points.
$\rightarrow$A straight line segment can be extended indefinitely in a straight line.
$\rightarrow$A circle can be drawn having segment as radius and one endpoint as center. 
$\rightarrow$All right angles are congruent and equal.
$\rightarrow$Parallel postulate.

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

According to Euclid's axioms, the _____ is greater than the part.

  1. Half

  2. Large

  3. Whole

  4. None of these
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

According to axiom $5$ of Euclid, whole is greater than the part and it is a universal truth.


Proof:
Let's take whole $3$ and part $\dfrac{1}{3}$
Subtracting, we get
$3-\dfrac{1}{3} = \dfrac{8}{3} > 0$

Hence, Proved that whole is greater than part.

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

Using Euclid's Division Lemma, for any positive integer $n, n^3-n$ is always divisible by 

  1. $6$
  2. $4$
  3. $3$
  4. $8$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

$n^{3}-n=n(n^{2}-1)=n(n-1)(n+1)$ is divisible by $3$ then  possible remainder is $0, 1$ and $2$


[$\because$ if $P=ab+r$, Then $0\le r < a$ by Euclid lemma]


$\therefore$ Let $n=3r, 3r+1,3r+2$ where $r$ is an integer

Case $1$: When $n=3r$

Then, $n^{3}-n$ is divisible by $3$  [$\because n^{3}-n=n(n-1)(n+1)=3r(3r-1)(3r+1)$, early shown it is divisible by $3$]

Case $2$: When $n=3r+1$

$n-1=3r+1-1=3r$

Then, $n^{3}-n=(3r+1)(3r)(3r+2)$ it is divisible by $3$

Case: when $n=3r-1$

$m+1=3r-1+1=3r$

Then, $n^{3}-n=(3r-1)(3r-2)(3r)$ it is divisible by $3$

Now out of three $(n-1)^{n}$ and $(n+1)$ are must be even so it is divisible by $2$

$n^{3}-n$ is divisible by $2\times 3=6$

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

Euclid stated that all right angles are equal to one another in the form of a/an ..........

  1. Axiom

  2. Defination

  3. Postulate

  4. Proof
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Postulates


1. A straight line may be drawn from any point to any other point.


2. A terminated line (line segment) can be produced indefinitely.
 3. A circle may be described with any centre and any radius.


4. All right angles are equal to one another.


5. If a straight line falling on two straight lines makes the interior angles on the same

side of it, taken together less than two right angles, then the the two straight lines if

produced indefinitely, meet on that side on which the sum of angles is taken together

less than two right angles.

Euclid used the term postulate for the assumptions that were specific to geometry

and otherwise called axioms. A theorem is a mathematical statement whose truth

has been logically established.
Answer (C) Postulate

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

Euclid's second axiom is

  1. the things which are equal to the same thing are equal to one another

  2. if equals be added to equals, the wholes are equal

  3. if equals be subtracted from equals, the remainders are equals

  4. things which coincide with one another are equal to one another
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Euclid's second axiom can be stated as any terminated straight line can be projected indefinitely or it can be stated as if equals be added to equals, the wholes are equal.