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Heights and Distances Practice Questions with Detailed Solutions – Level 3

Hard Heights and Distances problems on cloud reflections, broken trees, sliding ladders, kites, aeroplanes and boats nearing a cliff, each with a full worked solution.

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Questions, answer key and solutions.

Questions
50
Duration
75 min
Marks
50

Negative marking

Attempts
100
Access
Free
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Normal practice, or a Simulated test

Both use the same questions. Normal practice is untimed and needs no account: choose an option and the answer and solution appear at once. The Simulated test is the real computer-based format, with one 75-minute clock, a question palette and full screen, and your score, the key and the solutions arrive when you submit.

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Untimed

Every question on one page, loading as you scroll. Choose an option and see straight away whether it was right.

  • No timer. Nothing to start and nothing to submit. Stop whenever you like.
  • Answered as you choose. Tap an option and it is marked at once, with the correct answer shown.
  • Solutions in place. The explanation, and the video where one has been added, sit under the question.
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Real exam conditions

Simulated

Timed CBT

The computer-based test format: one clock, a question palette, and a score the moment you submit.

  • A clock that does not pause. 75 minutes from the moment you begin, and the paper is submitted for you when it reaches zero.
  • Question palette. Answered, not answered and marked for review, section by section, as the real interface has it.
  • Full screen, proctored. The test takes over the screen; leaving it three times submits the paper.
  • Answers hidden until you submit. The key, the explanations and your score all arrive together at the end.

About this test

Heights and Distances Level 3 Practice Quiz with Worked Solutions

This Level 3 set on Aptitude360 is a practice quiz on Heights and Distances. It tests whether you can turn multi-step word problems on angles of elevation and depression into right triangles and solve them. Every question has a worked solution, so you can trace a wrong step back to the exact triangle or ratio that caused it.

Topics covered in this Heights and Distances set

  • Observers moving towards or away from a tower, such as a walking person, a car or a boat nearing a cliff
  • Two observation points on the same side or on opposite sides of a tower or lighthouse
  • Building and tower pairs with an angle of elevation to one top and an angle of depression to the other foot
  • Ladders that slide along a wall, and kites with taut strings at changing angles
  • Cloud and reflection problems over a still lake, plus aeroplanes and balloons in motion
  • Broken trees, flagstaffs and statues on pedestals, and shadows cast by the sun or a lamp post

Key formulas for Heights and Distances problems

  • Basic ratio: \( \tan\theta=\dfrac{h}{d} \)
  • Two points on the same side, \( d \) apart, \( \alpha>\beta \): \( h=\dfrac{d\tan\alpha\tan\beta}{\tan\alpha-\tan\beta} \)
  • Complementary angles from distances \( a \) and \( b \): \( h=\sqrt{ab} \)
  • Points on opposite sides: \( d=h(\cot\alpha+\cot\beta) \)
  • Cloud height, observer \( h \) above lake: \( H=\dfrac{h(\tan\beta+\tan\alpha)}{\tan\beta-\tan\alpha} \)

Revision tips for tougher elevation and depression questions

  • Draw a labelled figure first and show clearly which side is the height and which is the horizontal distance
  • Remember that the angle of depression from a point equals the angle of elevation of that point from below, because the horizontal lines are parallel
  • In reflection problems, the image lies as far below the water surface as the object is above it
  • Keep surds such as root three in the working until the final step to avoid rounding errors

Who should practise this Level 3 set

It suits aptitude aspirants who have already finished the basic and moderate levels of Heights and Distances and now want harder, multi-step trigonometry problems.

FAQs about Heights and Distances Practice Questions with Detailed Solutions – Level 3

What is the formula used in heights and distances?
Most problems use the tangent ratio: the tangent of the angle equals the height divided by the horizontal distance. Sine and cosine are used when a slant length is given, such as a ladder or a kite string.
What is the difference between angle of elevation and angle of depression?
The angle of elevation is measured upward from the horizontal line of sight to an object above the observer. The angle of depression is measured downward from the horizontal to an object below the observer.
Why is the angle of elevation equal to the angle of depression?
The horizontal lines through the two points are parallel, and the line of sight acts as a transversal. The two angles are therefore alternate angles, so they are equal.
How do you solve heights and distances questions with two angles?
Draw one right triangle for each angle, sharing the common height. Write the tangent equation for each triangle and solve the two equations together to find the unknown height or distance.

Test structure

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

SectionQuestionsTime
Heights and Distances Level 35075 min

Marking & rules

Total marks
50 across 50 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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