CAT Mensuration Bible Top 10 Must Solve Mensuration Questions covering 90% of Syllabus – Page 3
All 19 Mensuration questions from CAT papers. Tap Show answer, or open the detailed solution (free login). Page 3 of 7.
In a right-angled triangle ABC, the altitude AB is 5 cm, and the base BC is 12 cm. P and Q are two points on BC such that the areas of ΔABP, ΔABQ and ΔABC are in arithmetic progression. If the area of ΔABC is 1.5 times the area of ΔABP, the length of PQ, in cm, is
Answer: 2
Solution — first steps
Step 1. All three triangles ABP,ABQ,ABC share altitude AB=5, so their areas are proportional to bases BP,BQ,BC.
Step 2. Let their AP bases be BP, BQ, 12.
Step 2. Let their AP bases be BP, BQ, 12.
The full step-by-step working opens with a free login.
For students who have joined a Cetking program.
In a rectangle ABCD, AB = 9 cm and BC = 6 cm. P and Q are two points on BC such that the areas of the figures ABP, APQ, and AQCD are in geometric progression. If the area of the figure AQCD is four times the area of triangle ABP, then BP : PQ : QC is
Answer: D. 2 : 4 : 1
Solution — first steps
Option 4
The full step-by-step working opens with a free login.
For students who have joined a Cetking program.
Let \(\triangle A B C\) be an isosceles triangle such that \(A B\) and \(A C\) are of equal length. \(A D\) is the altitude from \(A\) on \(B C\) and \(B E\) is the altitude from \(\mathrm{B}\) on \(\mathrm{AC}\). If \(\mathrm{AD}\) and \(\mathrm{BE}\) intersect at \(\mathrm{O}\) such that \(\angle \mathrm{AOB}=105^{\circ}\), then \(\frac{A D}{B E}\) equals
Answer: C. \(2 \cos 15^{\circ}\)
Solution — first steps
Step 1. Since AB=AC, base angles B and C are equal.
Step 2. AO and BO lie along altitudes, so angle AOB=180°−C. Hence C=75° and A=30°.
Step 2. AO and BO lie along altitudes, so angle AOB=180°−C. Hence C=75° and A=30°.
The full step-by-step working opens with a free login.
For students who have joined a Cetking program.
A rectangle with the largest possible area is drawn inside a semicircle of radius \(2 \mathrm{~cm}\). Then, the ratio of the lengths of the largest to the smallest side of this rectangle is
Answer: A. \(2 : 1\)
Solution — first steps
Step 1. Let half-width of the rectangle be x and height be y. Its top corner lies on x²+y²=4.
Step 2. Area A=2xy=2x√(4−x²).
Step 2. Area A=2xy=2x√(4−x²).
The full step-by-step working opens with a free login.
For students who have joined a Cetking program.
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.