Tag: mathematics and statistics

Quizzes (38) Questions (955)

Questions Related to mathematics and statistics

Multiple choice mathematics and statistics relations cartesian product of sets cartesian product of two sets cartesian product

If $A = {2, 3, 5}$ and $B = {5, 7}$, find the set with highest number of elements:

  1. $A \times B$
  2. $ B \times A$
  3. $A \times A$
  4. $B \times B$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation
$A=\left \{ 2,3,5 \right \}$

$B=\left \{ 5,7 \right \}$

$A\times B=\left \{ (2,5),(2,7),(3,5),(3,7),(5,5),(5,7) \right \}$

$B\times A=\left \{ (5,2),(5,3),(5,5),(7,2),(7,3),(7,5) \right \}$

$A\times A=\left \{ (2,2),(2,3),(2,5),(3,2),(3,3),(3,5),(5,2),(5,3),(5,5) \right \}$

$B\times B=\left \{ (5,5),(5,7),(7,5),(7,7) \right \}$

$\therefore A\times A$ has the highest number of elements
Multiple choice mathematics and statistics relations cartesian product of sets cartesian product of two sets cartesian product

For two sets $A$ and $B$, $A\times B=B\times A$.

  1. True

  2. False
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

The given statement is false.

Example:-
Let us consider $A={1,2}$ and $B={3,4}$.

Now $A\times B={(1,3),(1,4),(2,3),(2,4)}$......(1).

And $B\times A={(3,1),(3,2),(4,1),(4,2)}$..........(2).

Form (1) and (2) it's evident that $A\times B\ne B\times A$.

Multiple choice mathematics and statistics relations cartesian product of sets cartesian product of two sets cartesian product

Let A and B be sets containing 2 and 4 elements respecetively. The number of subsets $A \times B$ having 3 or more elements is 

  1. $219$
  2. $211$
  3. $256$
  4. $220$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation
Let $A=\left\{x,y\right\}$
$B=\left\{a,bc,d\right\}$
$A\times B$ has $2\times 4=8$ elements
Total substance of $A\times B={2}^{8}=256$
$\therefore\,$Total number of subsets of $A\times B$ having $3$ or more elements
$=256-\left(1\,null \,set+8\,single\,ton\,set-^{8}C _{2}\,having\,2\,elements\right)$
$=256-1-8-\dfrac{8!}{6!2!}$
$=256-1-8-\dfrac{8\times 7\times 6!}{6!2!}$
$=256-1-8-28=219$

Multiple choice mathematics and statistics relations cartesian product of sets cartesian product of two sets cartesian product

If $A$ and $B$ are independent event such that $P(A \cap B')=\dfrac {3}{25}$ and $P(A' \cap B)=\dfrac {8}{25}$, then $P(A)=$

  1. $1/5$
  2. $3/8$
  3. $2/5$
  4. $4/5$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation
$A$ & $B$ are independent
$P(A\cap B)=\dfrac {3}{25}$ and $P(A'\cap B)=\dfrac {8}{25}\quad P(A)=(?)$
$\rightarrow \ P(A)+P(B)=1---(i)$ ($A$& $B$ are independent )
$\rightarrow \ P(A\cap B')=P(A)-P(A\cap B)$
$P(A)-P(A\cap B)=\dfrac {3}{25}----(ii)$
$\rightarrow \ P(A' \cap B)=P(B)-P(A\cap B)$
$P(B)-P(A\cap B)=\dfrac {8}{25}-----(iii)$
$\rightarrow \ $ solving equation $(ii)$ and $(iii)$
$\dfrac {\,\,\, P\left( A \right) -P\left( A\cap B \right) =\dfrac { 3 }{ 25 } \\\,\,\, P\left( B \right) -P\left( A\cap B \right) =\dfrac { 8 }{ 25 } \\ -\quad \,\,\,\,\,\,\,+\quad \quad\quad\quad- }{ P\left( A \right) -P\left( B \right) =\dfrac { 3 }{ 25 } -\dfrac { 8 }{ 25 } \\ P\left( A \right) -P\left( B \right) =\dfrac { 3-8 }{ 25 }  } $
                           $=\dfrac {-5}{25}$
$\rightarrow \ P(A)-P(B)=\dfrac {-1}{5}$
$P(A)-(1-P(A))=\dfrac {-1}{5}$
$P(A)-I+P(A)=\dfrac {-1}{5}$
$\therefore \ 2P(A)=\dfrac {-1}{5}+1$
$\therefore \ 2P(A)=\dfrac {4}{5}$
$\therefore \ P(A)=\dfrac {4}{5\times 2}$
$\therefore \ P(A)=\dfrac {2}{5}$


Multiple choice mathematics and statistics relations cartesian product of sets cartesian product of two sets cartesian product

If $\int\dfrac{2\cos x-\sin x+\lambda}{\cos x-\sin x-2}dx=A In\left|\cos x+\sin x-2\right|+Bx+C$. Then the ordered triplet $\left(A,B,\lambda\right)$, is 

  1. $\left(\dfrac{1}{2},\dfrac{3}{2},-1\right)$
  2. $\left(\dfrac{3}{2},\dfrac{1}{2},-1\right)$
  3. $\left(\dfrac{1}{2},-1, \dfrac{3}{2}\right)$
  4. $\left(\dfrac{3}{2},-1, \dfrac{1}{2}\right)$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Perform the integration by expressing the numerator as a linear combination of the denominator and its derivative. Let 2cos(x) - sin(x) + lambda = A(cos(x) - sin(x) - 2) + B(-sin(x) - cos(x)). Solving for coefficients yields A=1/2, B=3/2, lambda=-1.

Multiple choice mathematics and statistics relations cartesian product of sets cartesian product of two sets cartesian product

If $A={1, 2, 3}$ and $B={3, 8}$, then $(A\cup B)\times (A\cap B)$ is

  1. $\{(3, 1), (3, 2), (3, 3), (3, 8)\}$
  2. $\{(1, 3), (2, 3), (3, 3), (8, 3)\}$
  3. $\{(1, 2), (2, 2), (3, 3), (8, 8)\}$
  4. $\{(8, 3), (8, 2), (8, 1), (8, 8)\}$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

$A\cup B={1,2,3,8}$
$A\cap B={3}$
$\therefore (A\cup B)\times (A\cap B)$
$={(1,3),(2,3),(3,3),(8,3)}$

Multiple choice mathematics and statistics relations cartesian product of sets cartesian product of two sets cartesian product

Let $A=\left { 1,2,3 \right }$ and $B=\left { a,b \right }$.Which of the following subsets of $A\times B$ is a mapping from $A$ to $B$

  1. $\left \{ \left ( 1,a \right ),\left ( 3,b \right ),\left ( 2,a \right ),\left ( 2,b \right ) \right \}$
  2. $\left \{ \left ( 1,b \right ),\left ( 2,a \right ),\left ( 3,a \right ) \right \}$
  3. $\left \{ \left ( 1,a \right ),\left ( 2,b \right ) \right \}$

  4. none of these
Reveal answer Fill a bubble to check yourself
A,B,C Correct answer
Explanation

$A=\left{ 1,2,3 \right} \ B=\left{ a,b \right} \ A\times B=\left{ \left( 1,a \right) ,\left( 2,a \right) ,\left( 3,a \right) ,\left( 1,b \right) ,\left( 2,b \right) ,\left( 3,b \right)  \right} .$


$ \left{ \left( 1,a \right) ,\left( 3,b \right) ,\left( 2,a, \right) \left( 2,b \right)  \right} \subset A\times B$

$ \left{ \left( 1,b \right) ,\left( 2,a \right) ,\left( 3,a \right)  \right} \subset A\times B$

$ \left{ \left( 1,a \right) ,\left( 2,b \right)  \right} \subset A\times B$

Multiple choice mathematics and statistics relations cartesian product of sets cartesian product of two sets cartesian product

Let $ A= { 1,2,3,.......50} $ and $B={2,4,6.......100}$ .The number of elements $\left ( x, y \right )\in A\times B$ such that $x+y=50$

  1. $24$
  2. $25$
  3. $50$
  4. $75$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The elements will be
$(2,48),(48,2)$
$(4,46), (46,4)$
:
:
$(2n,50-2n), (50-2n,2n)$
Now we have
$2,4,6,8...$ upto $48$
This forms an A.P
The number terms is $24$.