Tag: cartesian product of sets

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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 $24$.

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

Let A and B be two sets such that $A\times B=\left{ \left( a,1 \right) ,\left( b,3 \right) ,\left( a,3 \right) ,\left( b,1 \right) ,\left( a,2 \right) ,\left( b,2 \right)  \right} ,$ then 

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

$A$ is the first  element in  the cartesian product $A\times B=\left{a\,,b\,,a\,,b\,,a\,,b\right}$

and $B$ is the second element in  the cartesian product $A\times B=\left{1,\,3\,,3\,,1\,,2\,,2\right}$
$\therefore$ elements of $A=\left{a,b\right}$ and $B=\left{1\,,2\,,3\right}$