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 15 of 26

1. Q105 · Original practice · 5 marks
   The concentration of a substance is modelled by $C(t)=t^2e^{-t/3}$ for $t\ge0$. Determine the time at which $C$ is maximised. Then determine $C(3)$ as a percentage of this maximum value.
   Differentiation of exponential and logarithmic functions
2. Q106 · Original practice · 1 mark
   For $y=\sin^2(2x)$, determine $\dfrac{dy}{dx}$ at $x=\dfrac{\pi}{8}$.
   Differentiation of trigonometric functions and differentiation rules
3. Q107 · Original practice · 1 mark
   For $x\ne0$, if $y=\dfrac{\cos x}{x}$, then $\dfrac{dy}{dx}$ is
   Differentiation of trigonometric functions and differentiation rules
4. Q108 · Original practice · 1 mark
   For $f(x)=x+2\cos x$ on $0<x<\pi$, the stationary points occur at
   Differentiation of trigonometric functions and differentiation rules
5. Q109 · Original practice · 3 marks
   Determine the equation of the tangent to $y=\cos(2x)$ at $x=\dfrac{\pi}{6}$.
   Differentiation of trigonometric functions and differentiation rules
6. Q110 · Original practice · 5 marks
   A particle has displacement $s(t)=t\sin t$ for $0\le t\le\pi$. Determine, to three decimal places, the time at which the velocity is maximised and the corresponding maximum velocity.
   Differentiation of trigonometric functions and differentiation rules
7. Q111 · Original practice · 1 mark
   The graph shown represents $f'(x)$. On which interval is $f$ decreasing?
   Further applications of differentiation
8. Q112 · Original practice · 1 mark
   If $f''(x)=12x^2-12$, the possible points of inflection occur at
   Further applications of differentiation
9. Q113 · Original practice · 1 mark
   For $f(x)=x^3-3x^2-9x+5$, the local maximum point is
   Further applications of differentiation
10. Q114 · Original practice · 4 marks
    Analyse the turning points of $f(x)=x^4-8x^2$: find their coordinates and state the nature of each turning point.
    Further applications of differentiation
11. Q115 · Original practice · 6 marks
    A rectangular poster must contain a printed area of $600\text{ cm}^2$. The side margins are each $2$ cm wide and the top and bottom margins are each $3$ cm wide. Determine the dimensions of the printed area that minimise the total area of the poster, and state the minimum total area.
    Further applications of differentiation
12. Q116 · Original practice · 1 mark
    An antiderivative of $5e^{2x}-\dfrac{3}{x}$, for $x>0$, is
    Introduction to integration
13. Q117 · Original practice · 1 mark
    A function satisfies $f'(x)=4x^3-2$ and $f(1)=5$. Determine $f(0)$.
    Introduction to integration
14. Q118 · Original practice · 1 mark
    A particle has acceleration $a(t)=6t-4$, with $v(0)=3$ and $s(0)=1$. Determine $s(2)$.
    Introduction to integration
15. Q119 · Original practice · 3 marks
    A particle moves along a straight line with velocity $v(t)=4-2t$ and initial position $s(0)=3$. Determine $s(t)$ and the first time after $t=0$ at which the particle returns to its initial position.
    Introduction to integration
16. Q120 · Original practice · 5 marks
    A twice-differentiable function satisfies $f''(x)=6x-4$, $f'(1)=0$ and $\displaystyle\int_0^2 f(x)\,dx=10$. Determine $f(0)$.
    Introduction to integration
17. Q121 · Original practice · 1 mark
    A discrete random variable has $P(X=1)=k$, $P(X=2)=2k$ and $P(X=3)=3k$. Determine $k$.
    Discrete random variables
18. Q122 · Original practice · 1 mark
    If $X\sim\operatorname{Bin}(4,0.3)$, then $P(X=0)$ is
    Discrete random variables
19. Q123 · Original practice · 1 mark
    A binomial random variable $X$ has mean $3$ and variance $2.1$. Determine the number of trials $n$.
    Discrete random variables
20. Q124 · Original practice · 4 marks
    For the discrete distribution $P(X=0)=0.1$, $P(X=1)=0.2$, $P(X=2)=0.4$ and $P(X=3)=0.3$, calculate the mean and standard deviation of $X$.
    Discrete random variables
21. Q125 · Original practice · 5 marks
    Let $X\sim\operatorname{Bin}(5,p)$, where $0<p<1$. It is known that $P(X=0)=0.32768$. Determine $p$ and then calculate $P(X\ge2)$.
    Discrete random variables
22. Q126 · Original practice · 1 mark
    Evaluate $\displaystyle\int_0^{\pi/2}\cos x\,dx$.
    Further integration
23. Q127 · Original practice · 1 mark
    The values of a function are shown graphically at $x=0,1,2,3$. Using the trapezoidal rule with width $1$, estimate $\displaystyle\int_0^3 f(x)\,dx$.
    Further integration
24. Q128 · Original practice · 3 marks
    Determine the area enclosed by the curve $y=x(4-x)$ and the $x$-axis.
    Further integration