Tag: maths

Questions Related to 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 

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

  1. $2a$

  2. $a\sqrt{2}$

  3. $2\sqrt{2}a$

  4. $ \displaystyle \dfrac{a}{\sqrt{2}}$

Show answer
Correct Option: C
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  $\Rightarrow r=\dfrac{1}{2}$

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

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.

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

  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

Show answer
Correct Option: B
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 $B$

A result in which the (usually short) proof relies heavily on a given theorem is known as 

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

  1. Conjecture

  2. Corollary

  3. Lemma

  4. Axioms

Show answer
Correct Option: B
Explanation:

A result derived from a theorem is its corollary.

A proof is required for a :

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

  1. Postulate

  2. Axiom

  3. Theorem

  4. Definition

Show answer
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.

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

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

  1. axioms

  2. conjecture

  3. corollary

  4. theorem

Show answer
Correct Option: B
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 $6$.

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

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

  1. axioms

  2. conjecture

  3. corollary

  4. theorem

Show answer
Correct Option: B
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.

What is a conjecture?

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

  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

Show answer
Correct Option: B
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.

Square numbers can only have ____________ at the end.

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

  1. Odd number of zeros

  2. Even number of zeros

  3. Both (A) and (B)

  4. None of these

Show answer
Correct Option: B
Explanation:

Square numbers can only have EVEN NUMBER OF ZEROS because zero comes in only the square of $10$ and square of $10$ is $100$ that contains even number of zeros.

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

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

  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

Show answer
Correct Option: B
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.

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?

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

  1. $\cfrac{n-1}{2}$

  2. $\cfrac{n}{2}$

  3. $\cfrac{n+1}{2}$

  4. $2n$

Show answer
Correct Option: C
Explanation:

no of digits in a perfect square is $n$

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