Tag: mathematics and statistics

Questions Related to mathematics and statistics

$\left (A \cap B  \right ) \times C$

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

  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. $\left (A \times B \right ) \cup \left (A \times C \right )$

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Correct Option: B

$\left (A \cap B  \right ) \times C$


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

  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. $\left (A \times B \right ) \cup \left (A \times C \right )$

Show answer
Correct Option: A

Which one of the statement is false ?

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

  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}

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Correct Option: B

A $\times$ (B - C) =

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

  1. $(A \times B) - (A \times C)$

  2. $(A \times C) - (A \times B)$

  3. $(A \times B) \bigcup (A \times C)$

  4. $(A \times B) \bigcap (A \times C)$

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Correct Option: A

If A $=$ {1, 2}, B $=$ {3, 4}, then A$\times$B $=$

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

  1. {(1, 3), (1, 4), (2, 3), (2, 4)}

  2. {(1, 1), (2, 2), (3, 3), (4, 4)}

  3. {(4, 1), (3, 1), (4, 2), (3, 2)}

  4. All the above

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Correct Option: A
Explanation:

Given, $A={1,2}$ and $B={3,4}$.

Then,
$A\times B$
$={(1,3),(1,4),(2,3),(2,4)}$. [ Using (1) and (2)]

If ,$(x-1, y+2)= (7, 5)$ then values of $x$ and $y$ are

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

  1. $5$,$8$

  2. $8$,$3$

  3. $-1$,$5$

  4. $7$,$1$

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Correct Option: B
Explanation:

as the given ordered pairs are equal,

$x-1=7$  and  $y+2=5$
$\therefore x=8$ and $y=3$

Ordered pairs (a, 3) and (5, x) are equal ,the values of $a$ and $x$ are

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

  1. $2$ and $4$

  2. $3$ and $6$

  3. $5$ and $3$

  4. $1$ and $-1$

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Correct Option: C
Explanation:

If two ordered pairs are equal, then both x - coordinates are equal and both y-coordinates are equal.

Since  $ (a, 3)  =  (5,x)  $ then ,$ a = 5 , x = 6 $

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

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

  1. $3$

  2. $4$

  3. $5$

  4. $6$

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Correct Option: B
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.

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 $\lambda$.

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

  1. $3x^2-y^2=48$

  2. $y^2-3x^2=24$

  3. $4x^2-3y^2=16$

  4. None of these

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Correct Option: A
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 $3x^2-y^2=48$.

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

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

  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

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Correct Option: A