SSCE HSC Mathematics Extension 1 Absolute value functions: Find the domain and range of the

Question

Find the domain and range of the function that is the solution to the differential equation
$$\frac{dy}{dx} = e^{x+y}$$
and whose graph passes through the origin.

(b) For what values of the constant $k$ would the function $f(x) = \frac{kx}{1+x^2} + \arctan x$ have an inverse?

(c) (i) Explain why the equation $	an^{-1}(3x) + 	an^{-1}(10x) = 	heta$, where $-\pi < 	heta < \pi$, has exactly one solution.

(ii) Solve $	an^{-1}(3x) + 	an^{-1}(10x) = \frac{3\pi}{4}$.

(d) A particle is projected from the origin, with initial speed $V$ at an angle of $	heta$ to the horizontal. The position vector of the particle, $\mathbf{r}(t)$, where $t$ is the time after projection and $g$ is the acceleration due to gravity, is given by
$$\mathbf{r}(t) = \left( \frac{V \cos 	heta}{g} ight) t \mathbf{i} + \left( \frac{V \sin 	heta}{g} - \frac{gt^2}{2} ight) \mathbf{j} \quad (	ext{DO NOT prove this})$$

Let $D(t)$ be the distance of the particle from the origin at time $t$, so $D(t) = |\mathbf{r}(t)|$.

Show that for $	heta < \sin^{-1} \left( \frac{8}{9} ight)$ the distance, $D(t)$, is increasing for all $t > 0$.

Find the domain and range of the function that is the solution to the differential equation $$\frac{dy}{dx} = e^{x+y}$$ and whose graph passes through the origin - HSC - SSCE Mathematics Extension 1 - Question 14 - 2024 - Paper 1

Worked Solution & Example Answer:Find the domain and range of the function that is the solution to the differential equation $$\frac{dy}{dx} = e^{x+y}$$ and whose graph passes through the origin - HSC - SSCE Mathematics Extension 1 - Question 14 - 2024 - Paper 1

Show that for \theta < \sin^{-1}(\frac{8}{9}), the distance D(t) is increasing for all t > 0.

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