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Mathematics Extension 1

Stage 6 · HSC questions by topic

About this subject

Mathematics Extension 1

Atlas organises human-reviewed questions against the NSW Stage 6 Mathematics Extension 1syllabus. Every published question retains its exam source and page.

Syllabus: NESA Stage 6 (2017)
Explore the NESA Stage 6 (2017) syllabus topics

Functions

  • Further Work with Functions
  • Polynomials

Trigonometric Functions

  • Inverse Trigonometric Functions
  • Further Trigonometric Identities
  • Trigonometric Equations

Calculus

  • Rates of Change
  • Further Calculus Skills
  • Applications of Calculus

Combinatorics

  • Working with Combinatorics

Proof

  • Proof by Mathematical Induction

Vectors

  • Introduction to Vectors

Statistical Analysis

  • The Binomial Distribution
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4 questions found

List
2023HSC
Q1

The temperature T(t)T(t)T(t)°C of an object at time ttt seconds is modelled using Newton’s Law of Cooling,

T(t)=15+4e−3t.T(t)=15+4e^{-3t}.
T
(
t
)
=
15+
4e−3t.

What is the initial temperature of the object?
A. −3-3−3
B. 444
C. 151515
D. 191919

1 mark
Multiple choice
2023HSC
Q13 (a) (i)

A hemispherical water tank has radius RRR cm. The tank has a hole at the bottom which allows water to drain out.

Initially the tank is empty. Water is poured into the tank at a constant rate of 2kR cm3 s−12kR\ \mathrm{cm^3\,s^{-1}}2kR cm3s−1, where kkk is a positive constant.

After ttt seconds, the height of the water in the tank is hhh cm, as shown in the diagram, and the volume of water in the tank is V cm3V\ \mathrm{cm^3}V cm3.

It is known that

V=π(Rh2−h33).V=\pi\left(Rh^2-\frac{h^3}{3}\right).V=π(Rh2−3h

(Do NOT prove this.)

While water flows into the tank and also drains out of the bottom, the rate of change of the volume of water in the tank is given by

dVdt=k(2R−h).\frac{dV}{dt}=k(2R-h).dtdV​=k(2R−h).

(i) Show that

dhdt=kπh.\frac{dh}{dt}=\frac{k}{\pi h}.dtdh​=πhk​.
2 marks
Short answer
2023HSC
Q13 (a) (ii)

A hemispherical water tank has radius RRR cm. The tank has a hole at the bottom which allows water to drain out.

Initially the tank is empty. Water is poured into the tank at a constant rate of 2kR cm3 s−12kR\ \mathrm{cm^3\,s^{-1}}2kR cm3s−1, where kkk is a positive constant.

After ttt seconds, the height of the water in the tank is hhh cm, as shown in the diagram, and the volume of water in the tank is V cm3V\ \mathrm{cm^3}V cm3.

It is known that

V=π(Rh2−h33).V=\pi\left(Rh^2-\frac{h^3}{3}\right).V=π(Rh2−3h

(Do NOT prove this.)

While water flows into the tank and also drains out of the bottom, the rate of change of the volume of water in the tank is given by

dVdt=k(2R−h).\frac{dV}{dt}=k(2R-h).dtdV​=k(2R−h).

(ii) Show that the tank is full of water after

T=πR22kT=\frac{\pi R^2}{2k}T=2kπR2​

seconds.

2 marks
Short answer
2023HSC
Q13 (a) (iii)

A hemispherical water tank has radius RRR cm. The tank has a hole at the bottom which allows water to drain out.

Initially the tank is empty. Water is poured into the tank at a constant rate of 2kR cm3 s−12kR\ \mathrm{cm^3\,s^{-1}}2kR cm3s−1, where kkk is a positive constant.

After ttt seconds, the height of the water in the tank is hhh cm, as shown in the diagram, and the volume of water in the tank is V cm3V\ \mathrm{cm^3}V cm3.

It is known that

V=π(Rh2−h33).V=\pi\left(Rh^2-\frac{h^3}{3}\right).V=π(Rh2−3h

(Do NOT prove this.)

While water flows into the tank and also drains out of the bottom, the rate of change of the volume of water in the tank is given by

dVdt=k(2R−h).\frac{dV}{dt}=k(2R-h).dtdV​=k(2R−h).

(iii) The instant the tank is full, water stops flowing into the tank, but it continues to drain out of the hole at the bottom as before.

Show that the tank takes 3 times as long to empty as it did to fill.

3 marks
Short answer
3
​
)
.
3
​
)
.
3
​
)
.