NSAA 2016 Mathematics PART A

A square piece of metal has a semicircular piece cut out of it as shown. The area of the remaining metal is 100 cm².

[diagram not to scale]

Which one of the following is a correct expression for the length of the side of the square in centimetres?

• A.$20 \sqrt{\frac{2}{8 - \pi}}$
• B.$10 \sqrt{\frac{2}{4 - \pi}}$
• C.$20 \sqrt{\frac{2}{8 + \pi}}$
• D.$10 \sqrt{\frac{1}{8 - \pi}}$
• E.$20 \sqrt{\frac{1}{4 - \pi}}$

Answer: A

• A.$8\sqrt{5}$ cm
• B.$10\sqrt{2}$ cm
• C.$2\sqrt{5}$ cm
• D.$5\sqrt{2}$ cm
• E.$4\sqrt{5}$ cm

Answer: E

A cube has sides of unit length. What is the length of a line joining a vertex to the midpoint of one of the opposite faces (the dashed line in the diagram below)?

[diagram not to scale]

• A.$\frac{\sqrt{3}}{2}$
• B.$\sqrt{2}$
• C.$\frac{\sqrt{5}}{2}$
• D.$\sqrt{3}$
• E.$\sqrt{5}$

Answer: A

• A.0°
• B.7.5°
• C.15°
• D.22.5°
• E.30°

Answer: D

The right-angled triangle shown has horizontal and vertical sides measuring $(4+ \sqrt{2})$cm and $(2-\sqrt{2})$cm respectively.

[diagram not to scale]

Calculate the area of the triangle.

• A.$(5+3\sqrt{2})\text{cm}^2$
• B.$(3-\sqrt{2})\text{cm}^2$
• C.$(3+3\sqrt{2})\text{cm}^2$
• D.$(5-\sqrt{2})\text{cm}^2$

Answer: B

• A.$\frac{1}{4}$
• B.$\frac{1}{3}$
• C.$\frac{1}{2}$
• D.$\frac{2}{3}$
• E.$\frac{3}{4}$

Answer: D

• A.$\frac{1}{x}$
• B.$x^2$
• C.$\frac{1}{(1+x)}$
• D.$\frac{1}{\sqrt{x}}$
• E.$\sqrt{x}$

Answer: A

A shape is formed by drawing a triangle ABC inside the triangle ADE.

BC is parallel to DE.

AB = 4 cm BC = x cm DE = x + 3cm DB = x - 4 cm

[diagram not to scale]

Calculate the length of DE.

• A.5 cm
• B.7 cm
• C.9 cm
• D.4+2\sqrt{7}\cm
• E.7+2\sqrt{7}\cm

Answer: C

• A.29% decrease
• B.44% decrease
• C.49% decrease
• D.51% decrease
• E.80% decrease
• F.96% decrease

Answer: C

• A.The square of x is directly proportional to the cube of y.
• B.The square of x is inversely proportional to the cube of y.
• C.The cube of x is directly proportional to the square of y.
• D.The cube of x is inversely proportional to the square of y.
• E.x is directly proportional to $y^6$.

Answer: D

In the triangle PQR shown below:

[diagram not to scale]

X lies on PR

$\angle QXR$ is 90°

$\frac{QX}{PX} = \frac{1}{6}$

$\frac{QX}{XR} = \frac{2}{3}$

M is the midpoint of PR.

What is $\frac{QX}{MX}$?

• A.$\frac{1}{9}$
• B.$\frac{5}{12}$
• C.$\frac{4}{9}$
• D.$\frac{1}{2}$
• E.$\frac{5}{6}$

Answer: C

• A.$x \ge 4$
• B.$x \le 2$ and $x \ge -4$
• C.$x \ge -2$ and $x \le 4$
• D.$x \ge 2$ or $x \le -4$

Answer: D

• A.$h = \frac{2r}{r-2}$
• B.$h = \frac{2r}{r+2}$
• C.$h = r+2$
• D.$h = r-2$
• E.$h = 2r(r-2)$

Answer: A

• A.0
• B.2
• C.4
• D.6
• E.8

Answer: D

• A.A reflection in the x-axis.
• B.A reflection in the y-axis.
• C.A reflection in the line $y = -x$.
• D.A rotation of 90° clockwise about the origin.
• E.A rotation of 90° anticlockwise about the origin.

Answer: B

A design is set up by joining the points which are one third of the way along the sides of a square. This forms a second square as shown.

[diagram not to scale]

This process is repeated.

Calculate the area of the fourth square as a fraction of the original square.

• A.$\frac{1}{27}$
• B.$\frac{5\sqrt{5}}{81}$
• C.$\frac{125}{729}$
• D.$\frac{25}{81}$
• E.$\frac{5\sqrt{5}}{27}$

Answer: C

• A.2.5% decrease
• B.2.5% increase
• C.10% decrease
• D.10% increase
• E.20% decrease
• F.20% increase

Answer: C

• A.$\frac{1}{72}$
• B.$\frac{1}{45}$
• C.$\frac{1}{36}$
• D.$\frac{1}{18}$
• E.$\frac{1}{9}$

Answer: B