# CAT 2023 Question Paper | Quant Slot 2

###### CAT Previous Year Paper | CAT Quant Questions | Question 19

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

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$| x - y | - | x - 5 | = 2$
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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