Surds Word Problem

2012 May June Paper 21/23 Q2

A cuboid has a square base of side (2+√3)cm(2+√3)cm and a volume of (16+9√3)cm3(16+9√3)cm3. Without using a calculator, find the height of the cuboid in the form (a+b√3)cm(a+b√3)cm, where aa and bb are integers.

Height =h=h
Volume of cuboid

V=(2+√3)(2+√3)×h=16+9√3V=(2+√3)(2+√3)×h=16+9√3
h=16+9√34+4√3+3h=16+9√34+4√3+3
=16+9√37+4√3×7−4√37−4√3=16+9√37+4√3×7−4√37−4√3
=112+63√3−64√3−10849−(4√3)2=112+63√3−64√3−10849−(4√3)2
=4−√3=4−√3

2015 May June Paper 22 Q3

Do not use a calculator in this question.

The diagram shows the right-angled triangle ABCABC, where AB=(6+3√5)cmAB=(6+3√5)cm and angle B=90∘.B=90∘. The area of this triangle is (36+15√52)cm2(36+15√52)cm2.

  1. Find the length of the side BCBC in the form (a+b√5)cm(a+b√5)cm, where aa and bb are integers.
  2. Find (AC)2(AC)2 in the form (c+d√5)cm2(c+d√5)cm2, where cc and dd are integers.

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2016 Oct Nov Paper 23

In this question all lengths are in centimetres.

In the triangle ABCABC shown above, AC=√3+1,BC=√3−1AC=√3+1,BC=√3−1 and angle ACB=60∘ACB=60∘.

  1. Without using a calculator, show that the length of AB=√6AB=√6.
  2. Show that angle CAB=15∘CAB=15∘.
  3. Without using a calculator, find the area of triangle ABCABC.

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