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CAT Geometry Sectional Test 4 – with Solutions

Mixed CAT geometry and mensuration, from a sliced cone and a regular tetrahedron to cyclic quadrilaterals, common tangents and a folded paper sheet, with detailed solutions.

Start Test

Opens the instructions first.

Questions
22
Duration
40 min
Marks
66

Negative marking

Attempts
100
Access
With a course
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Real test conditions

Before you launch the test

This is the computer-based test format, not a quiz page: one clock, a question palette, and no way back once a section closes. Read these first.

  • It runs in full screen. The test takes over the screen when it starts, the way the real computer-based test does.
  • Leaving it is counted. Switching away is recorded and warned about; the third time, the paper is submitted for you.
  • The clock runs for 40 minutes. Set aside that long, uninterrupted. It starts when you begin, keeps running if you close the tab, and submits the paper at zero.

About this test

CAT Geometry Practice Test 4: Triangles, Circles, Coordinates and Solids

Use this set to see whether you can combine several geometry ideas inside one CAT-style problem. CAT Geometry Sectional Test 4 on Aptitude360 covers triangles, circles, polygons, coordinate geometry and solids. A detailed solution accompanies every question, so you can compare each step with your own approach.

What Geometry Sectional Test 4 covers

  • Triangles: an interior point at given distances from the vertices, incentre and exterior bisector angles, orthocentre, integer-sided triangles with an angle condition, area ratios formed by a line
  • Circles: cyclic quadrilaterals, area of a minor segment, circles touching the radii and arc of a sector, equal circles inside a larger circle
  • Polygons and quadrilaterals: hexagon from alternate vertices of a regular dodecagon, area from diagonals and the angle between them, a rectangle split into four parts
  • Coordinate geometry: reflection of a point in a line, common tangents of two circles, a region bounded by a circle and lines, a line through a fixed point meeting both positive axes
  • Solids and applied mensuration: a cone cut into a smaller cone and frustum, volume of a regular tetrahedron, a tilted water tank, folding a paper sheet, heights of towers across a road

Geometry formulas to keep handy for CAT

  • Equilateral triangle area: \( A=\dfrac{\sqrt{3}}{4}a^{2} \)
  • Regular tetrahedron volume: \( V=\dfrac{a^{3}}{6\sqrt{2}} \)
  • Minor segment area (θ in radians): \( \dfrac{r^{2}}{2}(\theta-\sin\theta) \)
  • Quadrilateral from diagonals: \( A=\dfrac{1}{2}d_{1}d_{2}\sin\theta \)
  • Incentre angle: \( \angle BIC=90^{\circ}+\dfrac{\angle A}{2} \)
  • Four common tangents when \( d>r_{1}+r_{2} \)

Study tips for CAT geometry problems

  • Sketch a clear figure and label every given length and angle before calculating
  • For a cone cut parallel to its base, volumes scale as the cube of the height ratio
  • With two circles, compare the distance between centres with the sum and difference of radii first

Who will gain from this sectional

CAT and OMET aspirants who know the geometry basics and want harder mixed problems to test speed and accuracy.

FAQs about CAT Geometry Sectional Test 4 – with Solutions

What is the formula for the volume of a regular tetrahedron?
For a regular tetrahedron with edge a, the volume is a cubed divided by six times the square root of two. Its total surface area is the square root of three times a squared.
How many common tangents can two circles have?
Two separate circles have four common tangents, circles touching externally have three, intersecting circles have two, circles touching internally have one, and a circle lying inside another has none.
What is angle BIC in terms of angle A?
If I is the incentre of triangle ABC, angle BIC equals 90 degrees plus half of angle A.
How do you find the area of a minor segment of a circle?
Subtract the area of the triangle formed by the chord and the two radii from the area of the sector. With the angle in radians, this equals half of r squared times the angle minus its sine.

Test structure

How the paper is split, and how long each part runs. A section you have finished cannot be returned to.

SectionQuestionsTime
CAT Geometry Sectional Test - 42240 min

Marking & rules

Total marks
66 across 22 questions.
Negative marking
A wrong answer costs marks. Leaving a question unanswered costs nothing.
Attempts
100 in total. An attempt is spent the moment you begin one.

What you get when you submit

The analysis is ready the moment you submit the paper.

Score and accuracy

Marks, percentage, right, wrong and skipped, and the time you took.

Section by section

The same figures for each section, so a weak one cannot hide in an average.

The full answer key

Every question with your answer, the right one, the working, and the video where there is one.

Where you stand

Your score against the spread of everyone else who has sat this test.

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