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

$\left (A \cap B  \right ) \times C$

  1. $\left (A \times B \right ) \cap \left (B \times C \right )$
  2. $\left (A \times C \right ) \cap \left (B \times C \right )$
  3. $\left (A \times B \right ) \cup \left (B \times C \right )$
  4. $\left (A \times B \right ) \cup \left (A \times C \right )$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

The Cartesian product distributes over intersection: (A intersect B) x C = (A x C) intersect (B x C).

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

Which one of the statement is false ?

  1. $\phi \times A = \phi$
  2. A $\times$ B = B $\times$ A
  3. A $\times$ B = {(x $\times$ y) : x A and y B}
  4. $R^{-1}$ = {(y, x) : (x, y) R}
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Statement B 'A × B = B × A' is false when interpreted universally because Cartesian products are not commutative unless A = B. For A = {1, 2} and B = {3}, A × B = {(1,3), (2,3)} but B × A = {(3,1), (3,2)}, which are different sets. Options A, C, and D contain true statements (C has notation issues but the concept is correct).

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

Determine all ordered pairs that satisfy $(x - y)^{2} + x^{2} = 25$, where $x$ and $y$ are integers and $x \geq 0$. Find the number of different values of $y$ that occur

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

${ \left( x-y \right)  }^{ 2 }+{ x }^{ 2 }=25$
Now, ${ 3 }^{ 2 }+{ 4 }^{ 2 }={ 5 }^{ 2 }$
$9+16=25$
$\therefore $There are 2 possibilities:
$I.{ \left( x-y \right)  }^{ 2 }=9$ and ${ x }^{ 2 }=16$
$\therefore x=\pm 4$ and $x-y=\pm 3$
$\left( i \right) .x-y=3\Rightarrow \left( 4,1 \right) $ and $\left( -4,-7 \right) $
$\left( ii \right) .x-y=-3\Rightarrow \left( 4,7 \right) $ and $\left( -4,-1 \right) $
$II.{ \left( x-y \right)  }^{ 2 }=16$ and ${ x }^{ 2 }=9$
$\therefore x=\pm 3$ and $x-y=\pm 4$
$\left( i \right) .x-y=4\Rightarrow \left( 3,-1 \right) $ and $\left( -3,-7 \right) $
$\left( ii \right) .x-y=-4\Rightarrow \left( 3,7 \right) $ and $\left( -3,-1 \right) $
$\therefore $ Different values of y are $1,-1,7,-7$
$\therefore 4$ different values of y occur.

Multiple choice mathematics and statistics hyperbola parametric equation of the hyperbola forms of equations of a hyperbola equations of hyperbola

Find the locus of the point of intersection of the lines $\sqrt 3 x-y-4\sqrt 3\lambda=0$ and $\sqrt 3 \lambda x +\lambda y-4\sqrt{3}=0$ for different values of $\lambda$.

  1. $3x^2-y^2=48$
  2. $y^2-3x^2=24$
  3. $4x^2-3y^2=16$

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

Let $(h,k)$ be the point of intersection of the given lines. Then,

$\sqrt 3 h-k-4\sqrt 3 \lambda=0$ and $\sqrt3 \lambda h +\lambda k-4\sqrt 3=0$
$\sqrt 3 h-k=4\sqrt 3\lambda$ and $\lambda(\sqrt 3h+k)=4\sqrt 3$
$(\sqrt 3 h-k)\lambda (\sqrt 3h +k)=(4\sqrt 3\lambda)(4\sqrt 3)$
$3h^2-k^2=48$
Hence, the locus of $(h,k)$ is $3x^2-y^2=48$.

Multiple choice mathematics and statistics hyperbola parametric equation of the hyperbola forms of equations of a hyperbola equations of hyperbola

If the equation of a hyperbola is $\frac{{{x^2}}}{9} - \frac{{{y^2}}}{{16}} = 1$, then 

  1. traverse axis is along x-axis of length $6$
  2. traverse axis is along y-axis of length $8$
  3. conjugate axis is along y-axis of length $6$

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

For the hyperbola x^2/9 - y^2/16 = 1, a^2 = 9 and b^2 = 16. The transverse axis is along the x-axis with length 2a = 2 * 3 = 6.