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.)*

●

The graph of

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 " **

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