Question

In which of the following case, do the quantities vary directly with each other?

• A.

  $x$	$0.5$	$2$	$8$	$32$
  $y$	$2$	$8$	$32$	$128$

• B.

  $p$	$1^2$	$2^2$	$3^2$	$4^2$
  $q$	$1^3$	$2^3$	$3^3$	$4^3$

• C.

  $r$	$2$	$5$	$10$	$25$	$50$
  $s$	$25$	$10$	$5$	$2$	$0.5$

• D.

  $u$	$2$	$4$	$6$	$9$	$12$
  $v$	$18$	$9$	$6$	$4$	$3$

Answer 1

Answer: (A), (B)

$p$	$1^2$	$2^2$	$3^2$	$4^2$
$q$	$1^3$	$2^3$	$3^3$	$4^3$

Explanation for correct options:

Option (A):

$x$	$0.5$	$2$	$8$	$32$
$y$	$2$	$8$	$32$	$128$

We know that if the quantities vary directly with each other the if $x$ increases then $y$ also increases

Let $y=f\left(x\right)=kx$

For $x=2,y=8$

$$\begin{gathered} \Rightarrow 8=2k \\ \Rightarrow k=4 \end{gathered}$$

For $x=0.5,y=2$

$$\begin{gathered} \Rightarrow 2=0.5k \\ \Rightarrow k=4 \end{gathered}$$

Similarly we can get $k=4$ for other values of $x,y$

Therefore from the above table $y=f\left(x\right)=4x$

$\therefore x\propto y$

Thus, option(A) is correct.

Option (B):

$p$	$1^2$	$2^2$	$3^2$	$4^2$
$q$	$1^3$	$2^3$	$3^3$	$4^3$

Let $q=f\left(p\right)=k{p}^{\frac{3}{2}}$

For $x=1^2,y=1^3$

$$\begin{gathered} \Rightarrow 1^3={\left(1^2\right)}^{\frac{3}{2}}k \\ \Rightarrow k=1 \end{gathered}$$

For $x=2^2,y=2^3$

$$\begin{gathered} \Rightarrow 2^3={\left(2^2\right)}^{\frac{3}{2}}k \\ \Rightarrow k=1 \end{gathered}$$

Similarly we can get $k=1$ for other values of $x,y$

Therefore from the above table $q=f\left(p\right)=p^{\frac{3}{2}}$

$\therefore$$p$ and $q$ varies directly.

Thus option(B) is correct

Explanation for incorrect options:

Option (C):

We know that if the quantities vary directly with each other the if $r$ increases then $s$ also increases

$r$	$2$	$5$	$10$	$25$	$50$
$s$	$25$	$10$	$5$	$2$	$0.5$

Clearly as $r$ increases $s$ decreases.

It doesn't varies directly

Thus, option(C) is incorrect.

Option (D):

We know that if the quantities vary directly with each other the if $u$ increases then $v$ also increases

$u$	$2$	$4$	$6$	$9$	$12$
$v$	$18$	$9$	$6$	$4$	$3$

Clearly as $u$ increases $v$ decreases.

It doesn't varies directly

Thus, option(D) is incorrect.

Hence, options (A) and (B) are correct.