CAT Mensuration Bible Top 10 Must Solve Mensuration Questions covering 90% of Syllabus – Page 2
All 19 Mensuration questions from CAT papers. Tap Show answer, or open the detailed solution (free login). Page 2 of 7.
In the XY-plane, the area, in sq. units, of the region defined by the inequalities
\( y \geq x + 4 \) and \( -4 \leq x^2 + y^2 + 4(x - y) \leq 0 \) is:
\( y \geq x + 4 \) and \( -4 \leq x^2 + y^2 + 4(x - y) \leq 0 \) is:
Answer: B. \( 2\pi \)
Solution — first steps
Step 1: Simplify the second inequality
\[
x^2 + y^2 + 4x - 4y = (x + 2)^2 - 4 + (y - 2)^2 - 4 = (x + 2)^2 + (y - 2)^2 - 8
\]
So the inequality becomes:
\[
-4 \leq (x + 2)^2 + (y - 2)^2 - 8 \leq 0
\Rightarrow 4 \leq (x + 2)^2 + (y - 2)^2 \leq 8
\]
This represents an annular region (ring) centered at \( (-2, 2) \), with:
Inner radius: \( \sqrt{4} = 2 \)
Outer radius: \( \sqrt{8} = 2\sqrt{2} \)
Step 2: Apply the line constraint \( y \geq x + 4 \)
The line cuts the annular region approximately in half (passing near the center), so the required area is half of the total area of the ring.
\[
x^2 + y^2 + 4x - 4y = (x + 2)^2 - 4 + (y - 2)^2 - 4 = (x + 2)^2 + (y - 2)^2 - 8
\]
So the inequality becomes:
\[
-4 \leq (x + 2)^2 + (y - 2)^2 - 8 \leq 0
\Rightarrow 4 \leq (x + 2)^2 + (y - 2)^2 \leq 8
\]
This represents an annular region (ring) centered at \( (-2, 2) \), with:
Inner radius: \( \sqrt{4} = 2 \)
Outer radius: \( \sqrt{8} = 2\sqrt{2} \)
Step 2: Apply the line constraint \( y \geq x + 4 \)
The line cuts the annular region approximately in half (passing near the center), so the required area is half of the total area of the ring.
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ABCD is a rectangle with sides AB = 56 cm and BC = 45 cm, and E is the midpoint of side CD. Then, the length, in cm, of radius of incircle of \(\triangle A D E\) is
Answer: 10
Solution — first steps
Step 1. Place A=(0,0), D=(0,45), C=(56,45); since E is the midpoint of CD, DE=28.
Step 2. Triangle ADE is right-angled at D with legs 45 and 28.
Step 2. Triangle ADE is right-angled at D with legs 45 and 28.
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A regular octagon ABCDEFGH has sides of length 6 cm each. Then the area, in sq. cm, of the square ACEG is
Answer: A. 36(2 + √2)
Solution — first steps
If AH is 6 than AG visually looks around 10 or 11. 82 \)
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The midpoints of sides AB, BC, and AC in ΔABC are M, N, and P, respectively. The medians drawn from A, B, and C intersect the line segments MP, MN and NP at X, Y, and Z, respectively. If the area of ΔABC is 1440 sq cm, then the area, in sq cm, of ΔXYZ is
Answer: 90
Solution — first steps
Area of Triangle XYZ The points M, N, and P are the midpoints of sides AB, BC, and AC of triangle \( \triangle ABC \). Therefore, triangle \( \triangle MNP \) is the medial triangle.
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A circular plot of land is divided into two regions by a chord of length \(10 \sqrt{3}\) meters such that the chord subtends an angle of \(120^{\circ}\) at the center. Then, the area, in square meters, of the smaller region is
Answer: C. \(25\left(\frac{4 \pi}{3}-\sqrt{3}\right)\)
Solution — first steps
Step 1. For chord length L=10√3 subtending 120°, use L=2r sin60°.
Step 2. Thus 10√3=r√3 and r=10.
Step 2. Thus 10√3=r√3 and r=10.
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Last day solving checklist
- Name the unknown. Include its unit and state whether it is a count, amount, rate or parameter.
- Form the relationships. Translate each given condition into an equation or restriction.
- Choose the method. Use the formula, substitution, elimination, a sign chart or a bound appropriate to this concept.
- Check restrictions. Check integer counts, signs, nonzero denominators, domains and all original conditions.
- Verify the answer. Substitute into the original question; for an extreme, show a feasible example reaches it.
All 12 repeating CAT PYQs and how much each adds
| CAT PYQ | Repeating CAT PYQ | Questions | Representative | Cumulative questions | Cumulative share |
|---|---|---|---|---|---|
| 1 | Common heights midpoints and triangle area ratios | 10 | CAT 2021 S2 Q11 | 10 | 32.26% |
| 2 | Rhombus diagonals side and area | 3 | CAT 2021 S2 Q9 | 13 | 41.94% |
| 3 | Regular polygon decomposition and area | 3 | CAT 2021 S1 Q6 | 16 | 51.61% |
| 4 | Trapezium area after finding height | 3 | CAT 2022 S1 Q11 | 19 | 61.29% |
| 5 | Right triangle incircle and area | 2 | CAT 2021 S1 Q14 | 21 | 67.74% |
| 6 | Rectangle area constraint and optimisation | 2 | CAT 2021 S3 Q19 | 23 | 74.19% |
| 7 | Circle sectors segments and overlapping regions | 2 | CAT 2022 S3 Q18 | 25 | 80.65% |
| 8 | Area of a region defined by coordinate equations | 2 | CAT 2023 S2 Q19 | 27 | 87.10% |
| 9 | Parallelogram area from angle and diagonal | 1 | CAT 2021 S3 Q11 | 28 | 90.32% |
| 10 | Rectangle on a circle from perimeter and diameter | 1 | CAT 2022 S1 Q19 | 29 | 93.55% |
| 11 | Right triangle on a diameter and proportional legs | 1 | CAT 2023 S2 Q18 | 30 | 96.77% |
| 12 | Cuboid surface area and sphere diameter | 1 | CAT 2024 S1 Q16 | 31 | 100% |
Based on CAT Actual Quant Papers.