Mensuration Questions and Answers

Answer: A

Same material means the same density, so weight is proportional to volume.

\[ V_{\text{solid}} = \frac{4}{3}\pi (3)^{3} = 36\pi \text{ cm}^{3} \;\longrightarrow\; 6 \text{ kg} \]

The hollow sphere has outer radius 9 cm and inner radius 6 cm:

\[ V_{\text{hollow}} = \frac{4}{3}\pi\left(9^{3} - 6^{3}\right) = \frac{4}{3}\pi (729 - 216) = \frac{4}{3}\pi (513) = 684\pi \text{ cm}^{3} \]

\[ \text{weight} = 6 \times \frac{684\pi}{36\pi} = 6 \times 19 = 114 \text{ kg} \]

The cavity must be subtracted; treating the object as a solid 18 cm sphere would give 162 kg.

Answer: D

Let the side of the square be a cm. Let the length and the breadth of the rectangle be l cm and b cm respectively. 

4a = 2(l + b) 

2a = l + b 

l . b = 480 

We cannot find ( l + b) only with the help of l . b. Therefore a cannot be found . 

Area of the square cannot be found.

Answer: B

\[ A = \pi r^{2} \]

An area must vary as the square of a length, which alone rules out \( \pi r^{2} + 2r \) as dimensionally inconsistent. The other two are real formulas from elsewhere: \( 4\pi r^{2} \) is a sphere's surface area and \( 2\pi r^{2} \) a hemisphere's curved surface.

Answer: A

Check the sides first: \( 10^{2} + 24^{2} = 100 + 576 = 676 = 26^{2} \), so this is a right-angled triangle with legs 10 cm and 24 cm.

\[ A = \frac{1}{2} \times 10 \times 24 = 120 \text{ cm}^{2} \]

Heron's formula agrees: with \( s = 30 \), \( \sqrt{30 \times 4 \times 6 \times 20} = \sqrt{14400} = 120 \). Spotting the 5–12–13 triple doubled saves that work.

Answer: C

Work in centimetres throughout.

\[ V_{\text{wall}} = 2400 \times 800 \times 60 = 115{,}200{,}000 \text{ cm}^{3} \]

Mortar fills 10%, so bricks occupy 90%:

\[ 0.9 \times 115{,}200{,}000 = 103{,}680{,}000 \text{ cm}^{3} \]

\[ V_{\text{brick}} = 24 \times 12 \times 8 = 2304 \text{ cm}^{3} \]

\[ n = \frac{103{,}680{,}000}{2304} = 45{,}000 \]

Ignoring the mortar would give 50,000 - a tenth too many.

Answer: D

Work through the metric chain of volume units.

\[ 1 \text{ litre} = 1 \text{ dm}^{3} \]

\[ 1 \text{ kilolitre} = 1000 \text{ litres} = 1000 \text{ dm}^{3} \]

That is also 1 cubic metre, since \( 1 \text{ m}^{3} = 1000 \text{ dm}^{3} \). The 1000 millilitres on offer is only one litre - a thousandth of the answer.

Answer: C

The two parallel sides of a parallelogram are equal, so a sum of 10 cm makes each one 5 cm, and the 2 cm between them is the height.

\[ A = \text{base} \times \text{height} = 5 \times 2 = 10 \text{ cm}^{2} \]

The trapezium formula gives the same figure here, since

\[ A = \frac{1}{2}(a + b)h = \frac{1}{2} \times 10 \times 2 = 10 \text{ cm}^{2} \]

and for a parallelogram the two parallel sides are equal.

Answer: B

\[ A = \frac{\theta}{360} \times \pi r^{2} = \frac{120}{360} \times \pi \times 18^{2} \]

\[ = \frac{1}{3} \times 324\pi = 108\pi \]

\[ = 108 \times 3.1416 = 339.3 \text{ cm}^{2} \]

\( 120^{\circ} \) is a third of a full turn, so the sector is a third of the circle's 1017.9 cm².

Answer: A

A chord is a straight line joining two points on the circle. It splits the interior into exactly two segments - the major one containing the centre and the minor one on the other side.

A single straight cut through any convex region always produces two pieces, however the chord is placed.

Answer: C

A quadrilateral with all four vertices on a circle is a cyclic quadrilateral.

Its defining property is that opposite angles are supplementary:

\[ \angle A + \angle C = 180^{\circ}, \qquad \angle B + \angle D = 180^{\circ} \]

The other three names are not standard terms in geometry.