Mathematical Methods questions and solutions

Browse original practice and official past-paper questions. Each question has its own link, with its source and worked solution.

610 questions · Page 17 of 26

1. Q153 · Original practice · 1 mark
   For $x>\dfrac14$, $\dfrac{d}{dx}\ln(4x-1)$ is
   Differentiation of exponential and logarithmic functions
2. Q154 · Original practice · 4 marks
   The temperature of a drink is modelled by $T(t)=18+72e^{-0.2t}$ degrees Celsius, where $t$ is the number of minutes after the drink is poured.

   (a) Determine the instantaneous rate of change of the temperature at $t=4$.

   (b) Determine when the temperature first reaches $30^\circ\mathrm C$.
   Differentiation of exponential and logarithmic functions
3. Q155 · Original practice · 5 marks
   A contaminant concentration is modelled by $C(t)=Ae^{-kt}$, where $A>0$, $k>0$ and $t$ is measured in hours. Measurements give $C(2)=36$ and $C(8)=15$.

   Determine $A$ and $k$, and hence determine the instantaneous rate of change $C'(5)$.
   Differentiation of exponential and logarithmic functions
4. Q156 · Original practice · 1 mark
   Determine $\dfrac{d}{dx}[\cos(5x)]$.
   Differentiation of trigonometric functions and differentiation rules
5. Q157 · Original practice · 1 mark
   For $g(x)=x\sin x$, determine $g'(\pi)$.
   Differentiation of trigonometric functions and differentiation rules
6. Q158 · Original practice · 1 mark
   The graph of $y=\sin(2x)$ is shown with the point $x=\dfrac{\pi}{8}$ marked. The gradient of the tangent at this point is
   Differentiation of trigonometric functions and differentiation rules
7. Q159 · Original practice · 4 marks
   For $h(x)=\dfrac{\ln x}{x+1}$, $x>0$, determine the equation of the tangent to the graph of $h$ at $x=1$.
   Differentiation of trigonometric functions and differentiation rules
8. Q160 · Original practice · 5 marks
   Let $f(x)=\dfrac{\sin x}{2+\cos x}$ for $0\le x\le2\pi$. Determine the global maximum and global minimum values of $f$ on this interval, and the $x$-values at which they occur.
   Differentiation of trigonometric functions and differentiation rules
9. Q161 · Original practice · 1 mark
   A differentiable function has a stationary point at $x=3$ and $f''(3)>0$. The stationary point is
   Further applications of differentiation
10. Q162 · Original practice · 1 mark
    If $f''(x)=6x-12$, the point of inflection occurs at
    Further applications of differentiation
11. Q163 · Original practice · 1 mark
    A particle has displacement $s(t)=t^3-6t^2+9t$ metres. Its acceleration at $t=2$ seconds is
    Further applications of differentiation
12. Q164 · Original practice · 5 marks
    For $f(x)=x^3-6x^2+9x+1$, determine the coordinates and nature of all stationary points and determine the point of inflection.
    Further applications of differentiation
13. Q165 · Original practice · 6 marks
    An open-top box with a square base has volume $500\text{ cm}^3$. Let the base side length be $x$ cm and the height be $h$ cm, as shown. Determine the dimensions that minimise the amount of material required to make the box.
    Further applications of differentiation
14. Q166 · Original practice · 1 mark
    An antiderivative of $4e^x-3\sin x$ is
    Introduction to integration
15. Q167 · Original practice · 1 mark
    A curve has gradient $\dfrac{dy}{dx}=4-3x^2$ and passes through $(1,5)$. Determine its $y$-intercept.
    Introduction to integration
16. Q168 · Original practice · 1 mark
    A particle has acceleration $a(t)=4t$ m/s$^2$ and initial velocity $v(0)=2$ m/s. Determine $v(3)$.
    Introduction to integration
17. Q169 · Original practice · 5 marks
    The velocity–time graph of a particle is shown. The graph is linear between consecutive plotted points, and the initial position is $s(0)=7$ m.

    Determine (a) the displacement from $t=0$ to $t=6$, (b) the total distance travelled, and (c) the position at $t=6$.
    Introduction to integration
18. Q170 · Original practice · 5 marks
    A particle moves on a straight line with acceleration $a(t)=6e^{-t}$ m/s$^2$, initial velocity $v(0)=-1$ m/s and initial position $s(0)=2$ m. Determine the first time after $t=0$ at which the particle is at rest, and determine its position at that time.
    Introduction to integration
19. Q171 · Original practice · 1 mark
    A discrete random variable $X$ takes values $1$, $2$ and $4$ with probabilities $0.2$, $0.5$ and $0.3$ respectively. Determine $E(X)$.
    Discrete random variables
20. Q172 · Original practice · 1 mark
    If $X\sim\operatorname{Bin}(6,0.3)$, then $P(X=2)$ is closest to
    Discrete random variables
21. Q173 · Original practice · 1 mark
    The probability distribution of $X$ is shown. Determine $E(X)$.
    Discrete random variables
22. Q174 · Original practice · 4 marks
    Let $X\sim\operatorname{Bin}(n,0.4)$. Given that $\operatorname{Var}(X)=4.8$, determine $n$ and then calculate $P(X\ge10)$.
    Discrete random variables
23. Q175 · Original practice · 5 marks
    A binomial random variable $X$ has mean $4.8$ and variance $2.88$. Determine the parameters $n$ and $p$, and hence calculate $P(X\ge6)$.
    Discrete random variables
24. Q176 · Original practice · 1 mark
    Given $\displaystyle\int_0^2 f(x)\,dx=5$ and $\displaystyle\int_2^5 f(x)\,dx=-1$, determine $\displaystyle\int_0^5 f(x)\,dx$.
    Further integration