Tag: mathematics and statistics

Questions Related to mathematics and statistics

A closed set with respect to some binary operation is called semi- group if

mathematics and statistics binary operations properties of binary operations discrete mathematics sets and relations

  1. $*$ is associative

  2. $*$ is commutative

  3. $*$ is anti-commutative

  4. identity element exists

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Correct Option: A
Explanation:

In mathematics, a semigroup is an algebraic structure consisting of a set together with an associative binary operation.

Hence, $A$ is correct.

$"*"$ is said to be commutative in $A$ for all $a,b  \epsilon  A$

mathematics and statistics binary operations properties of binary operations discrete mathematics sets and relations

  1. $a+b=b+a$

  2. $a*b=b*a$

  3. $a-b=b-a$

  4. $a*b\neq b*a$

Show answer
Correct Option: B
Explanation:

This is the definition of commutativeness for all binary operators.

If $A \ast B = A \cap B$ on $P(X)$, then identify for $\ast$ is ________$(X \neq \phi)$

mathematics and statistics binary operations properties of binary operations discrete mathematics sets and relations

  1. $\phi$

  2. $X$

  3. $U$

  4. $A$

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Correct Option: B
Explanation:
Given: $A \ast B = A \cap B$ on $P(X)$
Now, $P(X)$ is a power set of $X$ which is the set of all subsets of $X$
So, $P(X)$ will have $\phi$ as its smallest set and $X$ will be the largest set.
Since, $A\ast B$ is defined on $P(X)$
So, $A \cap B\in P(X)$
So, $A\subseteq X$ and $B\subseteq X$
$\therefore A\cap X=A$ and $X\cap A=A$
$\therefore A\ast X=A$ and $X\ast A=A$
Hence, $X$ is the identity element for $\ast$.
If the line $(2x+y+1)+\lambda(x-y+1)=0$ is parallel to $y-axis$ then value of $\lambda$ is ?
mathematics and statistics coordinates, points and lines general equation of a line: ax+by+c=0 general equation of a line reducing equation of straight line to standard form

  1. $1$

  2. $-1$

  3. $\dfrac{1}{2}$

  4. $2$

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Correct Option: A
Explanation:
Parallel to y-axis $\Rightarrow y=0$
$2x+y+1+\lambda x-\lambda y+\lambda =0$
$x(2+\lambda )+y(1-\lambda )+(1+\lambda )=0$
coefficient of $y=0$
$1-\lambda =0\Rightarrow \lambda =1$

The equation of the line passing through $(-4, 3)$, parallel to the $3x+7y+6=0$

mathematics and statistics coordinates, points and lines general equation of a line: ax+by+c=0 general equation of a line reducing equation of straight line to standard form

  1. 3x+7y-9=0

  2. 3x+7y+9=0

  3. 3x+7y+3=0

  4. 3x+7y+12=0

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Correct Option: A
Explanation:

The line parallel to $3x+7y+6=0$ is $3x+7y+k=0$

It passes through $(-4,3)$
$\implies 3(-4)+7(3)+k=0\-12+21+k=0\\implies k=-9$
So the required equation is $3x+7y-9=0$

The slope and  the y-intercept  of the given line, $y-3x -6=0$ are respectively,

mathematics and statistics coordinates, points and lines general equation of a line: ax+by+c=0 general equation of a line reducing equation of straight line to standard form

  1. $3, -6$

  2. $-3, -6$

  3. $3, 6$

  4. $-3, 6$

Show answer
Correct Option: C
Explanation:

The given equation is $y-3x-6=0$ ........ $(1)$


To obtain the slope and $y-$intercept of the given equation, we write it in slope-intercept form which is 


$y=mx+c$, where $m$ and $c$ are slope and $y-$intercept

From $(1)$,

$y-3x-6=0\implies y=3x+6$

Comparing it with $y=mx+c$ we get,

slope$=m=3$ and y-intercept$=c=6$

Hence, option C is correct.

The slope and y-intercept of the following line are respectively

$2y + 2x - 5 = 0$

mathematics and statistics coordinates, points and lines general equation of a line: ax+by+c=0 general equation of a line reducing equation of straight line to standard form

  1. $ slope=m=1\quad and\quad y-intercept=c=\frac { 5 }{ 2 } . $

  2. $ slope=m=1/5\quad and\quad y-intercept=c=\frac { 2 }{ 5 } . $

  3. $ slope=m=-1\quad and\quad y-intercept=c=\frac { 5 }{ 2 } . $

  4. $ slope=m=-1/5\quad and\quad y-intercept=c=\frac { 2 }{ 5 } . $

Show answer
Correct Option: C
Explanation:

The given equation is $2y+2x-5=0$ ........ $(1)$

To obtain the slope and $y-$intercept of the given equation, we write it in slope-intercept form which is 
$y=mx+c$, where $m$ and $c$ are slope and $y-$intercept

From $(1)$,
$2y+2x-5=0\implies y=-x+\dfrac{5}{2}$
Comparing it with $y=mx+c$ we get,
slope$=m=-1$ and y-intercept$=c=\dfrac{5}{2}$
Hence, option C is correct.

The slope and y-intercept of the following line are respectively

$7x-y + 3 =0$

mathematics and statistics coordinates, points and lines general equation of a line: ax+by+c=0 general equation of a line reducing equation of straight line to standard form

  1. $ slope=m=7/3\quad and\quad y-intercept=1.\ $

  2. $ slope=m=-7\quad and\quad y-intercept=3.\ $

  3. $ slope=m=-7/3\quad and\quad y-intercept=1.\ $

  4. $ slope=m=7\quad and\quad y-intercept=3.\ $

Show answer
Correct Option: D
Explanation:

The given equation is $7x-y+3=0$ ........ $(1)$

To obtain the slope and $y-$intercept of the given equation, we write it in slope-intercept form which is 
$y=mx+c$, where $m$ and $c$ are slope and $y-$intercept

From $(1)$,
$7x-y+3=0\implies y=7x+3$
Comparing it with $y=mx+c$ we get,
slope$=m=7$ and y-intercept$=c=3$
Hence, option D is correct.

The slope and  the y-intercept  of the given line, $2x-3y = 7$ are respectively,

mathematics and statistics coordinates, points and lines general equation of a line: ax+by+c=0 general equation of a line reducing equation of straight line to standard form

  1. $\dfrac{3}{2}, \dfrac{-3}{7}$

  2. $\dfrac{2}{3}, \dfrac{-7}{3}$

  3. $\dfrac{3}{2},  \dfrac{3}{7}$

  4. $\dfrac{2}{3}, \dfrac{7}{3}$

Show answer
Correct Option: B
Explanation:

The given equation is $2x-3y=7$ ........ $(1)$


To obtain the slope and $y-$intercept of the given equation, we write it in slope-intercept form which is, 

$y=mx+c$, where $m$ and $c$ are slope and $y-$intercept

From $(1)$,

$2x-3y=7\implies 3y=2x-7\implies y=\dfrac{2}{3} x+\left( -\dfrac{7}{3}  \right)$

Comparing it with $y=mx+c$ we get,

slope$=m=\dfrac{2}{3}$ and y-intercept$=c=-\dfrac{7}{3}$

Hence, option B is correct.

The slope and y-intercept of the following line are respectively

$4x-y=0$

mathematics and statistics coordinates, points and lines general equation of a line: ax+by+c=0 general equation of a line reducing equation of straight line to standard form

  1. $ slope=m=4\quad and\quad y-intercept=0.\ $

  2. $ slope=m=-4\quad and\quad y-intercept=0.\ $

  3. $ slope=m=1/4\quad and\quad y-intercept=0.\ $

  4. $ slope=m=0\quad and\quad y-intercept=1/4.\ $

Show answer
Correct Option: A
Explanation:

The given equation is $4x-y=0$ ........ $(1)$

To obtain the slope and $y-$intercept of the given equation, we write it in slope-intercept form which is 
$y=mx+c$, where $m$ and $c$ are slope and $y-$intercept

From $(1)$,
$4x-y=0\implies y=4x$
Comparing it with $y=mx+c$ we get,
slope$=m=4$ and y-intercept$=c=0$
Hence, option A is correct.