CAT 2023 Question Paper | Quant Slot 2

CAT 2023 Quant was dominated by Algebra followed by Arithmetic. In Arithmetic, the questions were dominated by topics like Speed-time-distance, Mixture and Alligations. This year, there was a surprise. The questions from Geometry were relatively on the lower side as compared to the previous years. There were 8 TITA Qs this year. Overall this section was at a medium level of difficulty.

Question 19 : The area of the quadrilateral bounded by the $Y$-axis, the line $x=5$, and the lines $|x-y|-|x-5|=2$, is

Video Explanation

Explanatory Answer

$| x - y | - | x - 5 | = 2$
let $| x - 5 | = \Delta$ (Observe that $\Delta$ is a non-negative number.)
$x = \Delta + 5$ or $x = 5 - \Delta$
$| x - y | - \Delta = 2$
$| x - y | = \Delta + 2$
$y = x + \Delta + 2$ or $y = x - \Delta - 2$
So, four pairs of points satisfy the condition $| x - y | - | x - 5 | = 2$
Case I )
$x = \Delta + 5$ & $y = x + \Delta + 2$
In this case, x is at least 5.
$y = x + \Delta + 2 = x + \Delta + 2 + 3 - 3 = 2 x - 3$
So, every point on the line $y = 2 x - 3$ where $x \geq 5$ satisfies the given condition.
Case II )
$x = \Delta + 5$ & $y = x - \Delta - 2$
In this case, x is at least 5.
$y = x - \Delta - 2 = \Delta + 5 - \Delta - 2 = 3$
So, every point on of the form $( x , 3 )$ where $x \geq 5$ satisfies the given condition.
Case III )
$x = 5 - \Delta$ & $y = x + \Delta + 2$
In this case, the highest value of x Is 5.
$y = x + \Delta + 2 = 5 - \Delta + \Delta + 2 = 7$
So, every point on of the form $( x , 7 )$ where $x \leq 5$ satisfies the given condition.
Case IV )
$x = 5 - \Delta$ & $y = x - \Delta - 2$
In this case, the highest value of x Is 5.
$y = x - \Delta - 2 = x + x - 5 - 2 = 2 x - 7$
So, every point on the line $y = 2 x - 7$ where $x \leq 5$ satisfies the given condition.
We are to find the area enclosed by the y-axis, x = 5 and the lines of $| x - y | - | x - 5 | = 2$.
Because the area we are interested is bounded by x = 0 (y-axis) and x = 5, 0 ≤ x ≤ 5.
So, we’ll only be concerned about Case III and Case IV.
A rough sketch of the bounded region looks like…

The line y = 2x – 7 touches x = 0 and x = 5 at (0, -7) and (5, 3) respectively.
So, the total area enclosed = $5 \times ( 7 + 7 ) - \frac { 1 } { 2 } \times 5 \times ( 3 + 7 ) = 45$.
Some Suggestions:
● Notice that, since the area is bounded by x = 5, we know that x ≤ 5 and therefore the case where $x = \Delta + 5$ will never arise… which leaves us just with Case III and Case IV.
(We did Case I and Case II just for the solution to be thorough and complete.)
● Graphing the equations using graphing tools before visualizing is not advisable.
The graph of $| x - y | - | x - 5 | = 2$looks like…

The question is " The area of the quadrilateral bounded by the $Y$-axis, the line $x=5$, and the lines $|x-y|-|x-5|=2$, is "

Hence, the answer is '45'

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