Mathematical Methods Paper 1 — Technology-free

Queensland Curriculum and Assessment Authority · Mathematical Methods (Year 12 (QCE)) · 2024 AD · External assessment — 32 questions with worked answers (solutions) — a free old question paper on Goodone.club’s exam-paper archive.

Determine $\int x^4\,dx$ (A) $4x^3 + c$ (B) $5x^5 + c$ (C) $\frac{1}{3}x^3 + c$ (D) $\frac{1}{5}x^5 + c$ [1 marks]
Determine $\frac{dy}{dx}$ for the function $y = e^{\sin(x)}$ (A) $\cos(x)e^{\sin(x)}$ (B) $\sin(x)e^{\cos(x)}$ (C) $e^{\sin(x)}$ (D) $e^{\cos(x)}$ [1 marks]
A sample of size $n$ can be used to obtain a sample proportion $\hat{p}$. An approximate margin of error for the population proportion can be obtained using the formula $$E = z\sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$$ If the level of confidence is increased from 95% to 99%, then (A) the associated z-value would decrease, so $E$ would increase. (B) the associated z-value would increase, so $E$ would increase. (C) the associated z-value would decrease, so $E$ would decrease. (D) the associated z-value would increase, so $E$ would decrease. [1 marks]
Simplify $y = 2\ln(e^x)$ (A) $y = 2x$ (B) $y = 2^x$ (C) $y = \frac{2}{x}$ (D) $y = x^2$ [1 marks]
Determine $\int_a^b 2\cos(x)\,dx$, where $a = \frac{\pi}{3}$ and $b = \frac{\pi}{2}$ (A) $1 - \frac{\sqrt{3}}{2}$ (B) $\frac{\sqrt{3}}{2} - 1$ (C) $2 - \sqrt{3}$ (D) $\sqrt{3} - 2$ [1 marks]
Differentiate $y = \ln(x)\cos(x)$ with respect to $x$. (A) $\frac{\cos(x)}{x}$ (B) $-\frac{\sin(x)}{x}$ (C) $\frac{\cos(x)}{x} + \ln(x)\sin(x)$ (D) $\frac{\cos(x)}{x} - \ln(x)\sin(x)$ [1 marks]
Twenty families are selected to participate in a lifestyle study related to family size. The number of children in these families is uniformly distributed as shown. A random sample of five families is chosen from this group, without replacement. A possible mean number of children in the sample is (A) 5.0 (B) 2.0 (C) 1.0 (D) 0.0 [1 marks]
The graph of $f(x)$ is shown. Identify the graph of the second derivative $f''(x)$. [1 marks]
At a certain location, the temperature (°C) can be modelled by the function $T = 5\sin\left(\frac{\pi}{12}x\right) + 23$, where $x$ is the number of hours after sunrise. Determine the rate of change of temperature (°C/hour) when $x = 4$ (A) $\frac{5\pi}{48}$ (B) $\frac{5\pi}{24}$ (C) $\frac{5\pi\sqrt{3}}{24}$ (D) $\frac{5\pi\sqrt{3}}{6}$ [1 marks]
Given that $\log_{10} 6 = 0.778$, determine the value of $\log_{10} 600$ (A) 77.800 (B) 10.778 (C) 2.778 (D) 1.556 [1 marks]
The graph is of the form $y = \log_a(x + b)$. A point on the graph $(4, 3)$ is labelled. The line $x = -4$ is an asymptote. There is a point $P(x_P, y_P)$ on the graph where $y_P$ is twice the value of the y-intercept of the curve. Determine the value of $x_P$. [4 marks]
A community group that uses social media created a new post on the internet on a day when they had 1000 members. The rate of change in their number of members (members/day) is given by $f'(t) = 3e^{0.5t}$, where $t$ represents days after the new post. Determine the time it will take for the community group to achieve seven times the initial number of members. Express your answer in the form $a\ln(b)$. [3 marks]
The diagram shows some dimensions of a large storage container that is a rectangular prism. The angle ABC is 60°. A person requires a container that is at least 4 metres in height. Make a justified decision about whether this storage container meets the person’s requirements. [5 marks]
A permanent ice glacier is in a valley in New Zealand. Due to the temperature changes of the seasons each year, the glacier expands for six months and recedes for six months. The changing distance of a point on the front edge of the glacier to a car park can be modelled by a sine function. During the colder months, when the glacier expands, the front edge of the glacier moves to within 270 m of the car park. However, in the warmer months, when the glacier recedes, the front edge moves to a maximum distance of 280 m away from the car park. The erosion effects of the glacier on the ground are... [6 marks]