Tag: mathematics and statistics

Quizzes (38) Questions (955)

Questions Related to mathematics and statistics

Multiple choice mathematics and statistics introduction to set union and intersections union and intersection of sets basic operations on sets

Let A = {x : x is a square of a natural number and x is less than 100} and B is a set of even natural numbers. What is the cardinality of $ A \cap B$ ?

  1. 4

  2. 5

  3. 9

  4. None of the above
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

$A = {1, 4. 9. 16, 25, 36, 49, 64, 81 }$

$B = {2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 24, 26, 28, 30, 32, 34, 36, 38,...}$
$A \cap B = {4, 16, 36, 64}$
Hence, $n(a \cap B) = 4$.

Multiple choice mathematics and statistics introduction to set union and intersections union and intersection of sets basic operations on sets

A survey shows that $63\%$ of Indians like mangoes whereas $76\%$ like apple. If $x%$ of the Indians like both mangoes and apples, then

  1. $x=39$
  2. $x=63$
  3. $39\le x\le 63$
  4. $None\ of\ these$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation
Let $100$ denote the population of India, then
$n(A)=63$, $n(B)=76$, $n(A\cap B)=x$
$n(A\cup B)=n(A)+n(B)-n(A\cap B)$
$\therefore x=63+76-n(A\cap B)$ ..$(1)$
If should be noted that $n(A\cup B)\neq 100$ but $n(A\cup B)\le 100$ because there are Indians who may like other fruits besides mangoes and apples.
Hence from $(1)$, $x=139-n(A\cup B)$
$\therefore x\ge 39$ or $39\le x$
Again $A\cap B\subset A$, $A\cap B\subset B$
$\therefore n(A\cap B)\le n(A)=63$, $n(A\cap B)\le n(B)=76$
$\therefore x\le 63$.
Hence $39\le x\le 63$.
Multiple choice mathematics and statistics introduction to set union and intersections union and intersection of sets basic operations on sets

Two set A and B are as under 
A = {(a,b) $\epsilon$ R $\times$ R : $\mid a - 5\mid$ < $1$ and  $\mid b - 5\mid$ < $1$};
B = {(a,b) $\epsilon$ R $\times$ R : $4(a-6)^2 + 9(b-5)^2$ $\leq 36$. Then, 

  1. B $\subset$ A
  2. A $\subset$ B
  3. A $\bigcap$ B = $\phi$ (an empty set)
  4. nither A $\subset$ B nor B $\subset$ A
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Set A is a square centered at (5, 5) with side length 2. Set B is an ellipse centered at (6, 5) with semi-axes 3 and 2. By comparing the boundaries, one can show that all points in A satisfy the inequality for B.

Multiple choice mathematics and statistics introduction to set union and intersections union and intersection of sets basic operations on sets

$R$ is the set of all positive odd integers less than $20$; $S$ is the set of all multiples of $3$ that are less than $20$. How many elements are in the set $R$ $\cap$ $S$?

  1. 0

  2. 1

  3. 2

  4. 3

  5. 4
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Given, $R=$ {$1,3,5,7,9,11,13,15,17,19$} , $S=$ {$3,6,9,12,15,18$}
Therefore the intersection of $S$ and $R$ is {$3,9,15$}.
So the number of elements which are common to both is $3$.

Multiple choice mathematics and statistics introduction to set union and intersections union and intersection of sets basic operations on sets

Let $Z$ denotes the set of all integers and $A=\left{ \left( a,b \right) :{ a }^{ 2 }+3{ b }^{ 2 }=28,a,b\in Z \right} $ and $B=\left{ \left( a,b \right) :a < b,a,b\in Z \right} $. Then, the number of elements in $A\cap B$ is

  1. $2$
  2. $4$
  3. $6$
  4. $5$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

$\because A=\left{ \left( a,b \right) :{ a }^{ 2 }+3{ b }^{ 2 }=28,a,b\in Z \right} $
   $=\left{ \left( 5,1 \right) ,\left( -5,-1 \right) ,\left( 5,-1 \right) ,\left( -5,1 \right) ,\left( 4,2 \right) ,\left( -4,-2 \right) ,\left( 4,-2 \right) ,\left( -4,2 \right) ,\left( 1,3 \right) ,\left( -1,-3 \right) ,\left( 1,-3 \right) ,\left( -1,3 \right)  \right}$
and $B=\left{ \left( a,b \right) :a<b,a,b\in Z \right} $
$\therefore A\cap B=\left{ \left( 1,3 \right) ,\left( -1,3 \right) ,\left( -4,-2 \right) ,\left( -4,2 \right) ,\left( -5,-1 \right) ,\left( -5,1 \right)  \right} $
$\therefore $ The number of elements in $A\cap B$ is $6$.

Multiple choice mathematics and statistics introduction to set union and intersections union and intersection of sets basic operations on sets

If $A={\theta :\tan \theta -\tan^2\theta > 0}, B={\theta :|\sin \theta | < 1/2}$ find $A\cap B$.

  1. $\left(0 , \cfrac{7\pi}{6}\right)$
  2. $\left(0 , \cfrac{\pi}{6}\right)$
  3. $\left(0 , -\cfrac{\pi}{6}\right)$

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

$\tan \theta - \tan ^2 \theta > 0$ is only true when 

$0<\tan \theta < 1$
or,$ 0< \theta < \cfrac{\pi}{4}$
$A = (0,\cfrac{\pi}{4})$
$|\sin \theta| < \cfrac{1}{2}$
$-\cfrac{1}{2} < \sin \theta < \cfrac{1}{2}$
$B = (-\cfrac{\pi}{6} , \cfrac{\pi}{6})$
$A \cap B = (0 , \cfrac{\pi}{6})$