Tag: maths

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Questions Related to maths

Multiple choice sum to infinite terms of a gp sequence, progression and series maths

The sum of the intercepts cut off by the axes on the lines  $ x+y=a,x+y=ar,x+y=ar^{2}\ldots\ldots\ldots$ where $a\neq 0$ and $r=\displaystyle \dfrac{1}{2}$  is 

  1. $2a$
  2. $a\sqrt{2}$
  3. $2\sqrt{2}a$
  4. $ \displaystyle \dfrac{a}{\sqrt{2}}$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

$x+y=a$

intercept cut off by the axes $=\sqrt{a^2+a^2}=\sqrt{2}a$

$\therefore$ sum of all intercepts cut off by the axes on the lines,

$x+y=a, x+y=ar,.... x+y=a^{r^n-1}$

$\sqrt{2}a, \sqrt{2}ar,.... \sqrt{2}ar^{n-1}$

$Sum=\sqrt{2}a+\sqrt{2}ar+....+\sqrt{2}a^{r^n-1}....$as

$=\sqrt{2}(\dfrac{a}{1-r})$

$=\dfrac{\sqrt{2}a}{1-r}$

Given  $\Rightarrow r=\dfrac{1}{2}$

$\therefore Sum=2\sqrt{2}a$

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

Select the correct statement for the following:

$A$: The angles of an equilateral triangle are equal.
$B$: Angles opposite to two congruent sides of a triangle are congruent.

  1. $A$ is a theorem and $B$ is its corollary
  2. $B$ is a theorem and $A$ is its corollary
  3. $A$ and $B$ are both theorems
  4. $A$ and $B$ are both corollaries
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

$B$ is called the  Triangle theorem.  $A$ can be proved using this theorem, as in an equilateral triangle, all sides are equal and the angles opposite to these congruent sides are Equal. So, $A$ is the corollary of $B$

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

A proof is required for a :

  1. Postulate

  2. Axiom

  3. Theorem

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

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

The formation or expression of an opinion or theory without sufficient evidence for proof is known as

  1. axioms

  2. conjecture

  3. corollary

  4. theorem
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation
The formation or expression of an opinion or theory without sufficient evidence for proof is known as conjecture.

For example, make a conjecture about the next number in the pattern $2, 6, 11, 17, ...$. The terms increase by $4$, then $5$, and then $6$.
Multiple choice maths axioms, postulates and theorems euclid's fifth postulate conditional statements and converse euclid's postulates

__________ implies to conclude or suppose from grounds or evidence insufficient to ensure reliability.

  1. axioms

  2. conjecture

  3. corollary

  4. theorem
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation
Conjecture implies to conclude or suppose from grounds or evidence insufficient to ensure reliability.

For example, make a conjecture about the next number in the pattern 2 ; 6 ; 11 ; 17 ; ... The terms increase by 4, then 5, and then 6.
Multiple choice maths axioms, postulates and theorems euclid's fifth postulate conditional statements and converse euclid's postulates

What is a conjecture?

  1. A true statement

  2. A statement that you believe is true based upon your observations

  3. A proven statement about something you believe

  4. A statement used by scientists
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

The formation or expression of an opinion or theory without sufficient evidence for proof is known as conjecture. 


Or a conjecture can be stated as a statement that you believe is true based upon your observations.

Multiple choice maths fun with numbers some special sequences triangular numbers properties and patterns of perfect squares

Let $S$ be the set of all ordered pairs $(x,y) $ of positive integers satisfying the condition $x^{2}-y^{2}=12345678$. Then:

  1. $S$ is an infinite set
  2. $S$ is the empty set
  3. $S$ has exactly one element
  4. $S$ is a finite set and has at least two elements
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation
$x^{2}-y^{2}=12345678 (x,y \, \epsilon \,  1^{+})$

RHS is even, so, x & y should be odd integer but difference square of two odd integer is multiple of 8 but RHS is not multiple of $8$

$\therefore  8 $ is an empty set.
Multiple choice maths fun with numbers some special sequences triangular numbers properties and patterns of perfect squares

If a number of $n$-digits is perfect square and $n$ is an odd number, then which of the following is the number of digits of its square root?

  1. $\cfrac{n-1}{2}$
  2. $\cfrac{n}{2}$
  3. $\cfrac{n+1}{2}$
  4. $2n$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

no of digits in a perfect square is $n$

 If $n$ is odd then no of digits in its square roots is $\dfrac{n+1}{2}$