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 13 of 26

1. Q57 · Original practice · 1 mark
   In triangle $ABC$, $AC=8$, $AB=11$ and $\angle A=45^\circ$. Determine $BC$ to one decimal place.
   Trigonometry
2. Q58 · Original practice · 1 mark
   A continuous random variable has density $f(x)=c(1+x)$ for $0\le x\le1$, as shown. Determine $c$.
   Continuous random variables and the normal distribution
3. Q59 · Original practice · 1 mark
   For repeated random samples of size $n$ from a population with proportion $p=0.20$, the mean of the sampling distribution of $\hat p$ is
   Sampling and proportions
4. Q60 · Original practice · 1 mark
   An approximate confidence interval for a population proportion has centre $0.63$ and margin of error $0.04$. The interval is
   Interval estimates for proportions
5. Q61 · Original practice · 1 mark
   If $g(x)=\ln(e^x+1)$, then $g'(x)$ is
   Differentiation of exponential and logarithmic functions
6. Q62 · Original practice · 1 mark
   If $h(x)=\cos(3x)\sin x$, then $h'(x)$ is
   Differentiation of trigonometric functions and differentiation rules
7. Q63 · Original practice · 1 mark
   For $f(x)=x^4-4x^2$, the local maximum occurs at
   Further applications of differentiation
8. Q64 · Original practice · 1 mark
   A particle has velocity $v(t)=2t-1$. Its displacement from $t=0$ to $t=3$ is
   Introduction to integration
9. Q65 · Original practice · 1 mark
   If $X\sim\operatorname{Bin}(8,0.25)$, then $E(X)$ is
   Discrete random variables
10. Q66 · Original practice · 1 mark
    An antiderivative of $\dfrac{6x}{3x^2+4}$ is
    Further integration
11. Q67 · Original practice · 1 mark
    Two sides of a triangle have lengths $12$ cm and $9$ cm, with included angle $40^\circ$. Its area is closest to
    Trigonometry
12. Q68 · Original practice · 1 mark
    $X\sim N(100,15^2)$. The shaded region shown represents $P(85<X<115)$. This probability is closest to
    Continuous random variables and the normal distribution
13. Q69 · Original practice · 1 mark
    For random samples of size $900$ from a population with proportion $p=0.55$, the standard deviation of $\hat p$ is closest to
    Sampling and proportions
14. Q70 · Original practice · 1 mark
    A sample of size $400$ has $\hat p=0.30$. Using $z=1.96$, the approximate margin of error of a 95% confidence interval is
    Interval estimates for proportions
15. Q71 · Original practice · 1 mark
    For $f(x)=(x+1)e^{-x}$, the maximum value of $f$ for $x\ge0$ is
    Differentiation of exponential and logarithmic functions
16. Q72 · Original practice · 1 mark
    For $h(x)=\sin x+\cos(2x)$ on $0<x<\dfrac\pi2$, the interior stationary point occurs at
    Differentiation of trigonometric functions and differentiation rules
17. Q73 · Original practice · 1 mark
    A rectangular enclosure beside a straight river requires fencing on only three sides, as shown. If $200$ m of fencing is available, the width $x$ that maximises the enclosed area is
    Further applications of differentiation
18. Q74 · Original practice · 1 mark
    A function satisfies $f'(x)=2x+3$ and $\displaystyle\int_0^2 f(x)\,dx=12$. Determine $f(0)$.
    Introduction to integration
19. Q75 · Original practice · 1 mark
    Let $X\sim\operatorname{Bin}(5,0.3)$. Determine $P(X=2\mid X\ge1)$, to three decimal places.
    Discrete random variables
20. Q76 · Original practice · 2 marks
    Differentiate $y=xe^{2x}$.
    Differentiation of exponential and logarithmic functions
21. Q77 · Original practice · 2 marks
    Differentiate $y=\cos x\sin(2x)$.
    Differentiation of trigonometric functions and differentiation rules
22. Q78 · Original practice · 3 marks
    For $f(x)=-2x^2+12x+5$ on $0\le x\le5$, determine the maximum value of $f$.
    Further applications of differentiation
23. Q79 · Original practice · 2 marks
    Evaluate $\displaystyle\int_1^4(3x+2)\,dx$.
    Introduction to integration
24. Q80 · Original practice · 2 marks
    Let $X\sim\operatorname{Bin}(10,0.20)$. Determine $P(X\le1)$ to four decimal places.
    Discrete random variables