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 11 of 26

1. Q9 · Original practice · 1 mark
   In a random sample of 200 voters, 54 support a proposal. The sample proportion is
   Sampling and proportions
2. Q10 · Original practice · 1 mark
   A sample has $\hat p=0.40$ and $n=100$. Using $z=1.96$, the approximate margin of error for a 95% confidence interval is
   Interval estimates for proportions
3. Q11 · Original practice · 1 mark
   If $g(x)=\ln(3x^2+1)$, then $g'(x)$ is
   Differentiation of exponential and logarithmic functions
4. Q12 · Original practice · 1 mark
   If $h(x)=x^2\sin x$, then $h'(x)$ is
   Differentiation of trigonometric functions and differentiation rules
5. Q13 · Original practice · 1 mark
   The stationary points of $f(x)=x^3-3x^2-9x+4$ occur when $x=$
   Further applications of differentiation
6. Q14 · Original practice · 1 mark
   A particle has velocity $v(t)=3t^2-4t+2$. Its displacement from $t=0$ to $t=2$ is
   Introduction to integration
7. Q15 · Original practice · 1 mark
   For a discrete random variable, $P(X=0)=0.2$, $P(X=1)=0.5$ and $P(X=2)=0.3$. Determine $E(X)$.
   Discrete random variables
8. Q16 · Original practice · 1 mark
   An antiderivative of $\dfrac{2x}{x^2+5}$ is
   Further integration
9. Q17 · Original practice · 1 mark
   In a triangle, side $a=5$ is opposite $A=30^\circ$. Side $b$ is opposite $B=45^\circ$. Determine $b$.
   Trigonometry
10. Q18 · Original practice · 1 mark
    The random variable $X$ is normally distributed with mean 80 and standard deviation 5. Determine $P(X<85)$.
    Continuous random variables and the normal distribution
11. Q19 · Original practice · 1 mark
    For repeated samples of size 400 from a population with proportion $p=0.30$, the standard deviation of $\hat p$ is closest to
    Sampling and proportions
12. Q20 · Original practice · 1 mark
    A sample of 625 gives $\hat p=0.52$. A 95% confidence interval using $z=1.96$ is closest to
    Interval estimates for proportions
13. Q21 · Original practice · 1 mark
    For $f(x)=x\ln x$, determine the equation of the tangent at $x=e$.
    Differentiation of exponential and logarithmic functions
14. Q22 · Original practice · 1 mark
    For $f(x)=\dfrac{\sin x}{1+\cos x}$, where defined, $f'(x)$ simplifies to
    Differentiation of trigonometric functions and differentiation rules
15. Q23 · Original practice · 1 mark
    For $f(x)=xe^{-x}$ on $x>0$, the $x$-coordinate of the maximum point is
    Further applications of differentiation
16. Q24 · Original practice · 1 mark
    A differentiable function satisfies $f'(x)=3x^2-2$ and $f(1)=4$. Determine $f(2)$.
    Introduction to integration
17. Q25 · Original practice · 1 mark
    Let $X\sim\mathrm{Bin}(3,0.4)$. Determine $P(X\ge2\mid X\ge1)$, to three decimal places.
    Discrete random variables
18. Q26 · Original practice · 2 marks
    Let $f(x)=\ln(2x+1)$. Determine $f'(x)$ and hence the gradient of the tangent at $x=2$.
    Differentiation of exponential and logarithmic functions
19. Q27 · Original practice · 2 marks
    Differentiate $y=2\sin x+\cos(2x)$.
    Differentiation of trigonometric functions and differentiation rules
20. Q28 · Original practice · 4 marks
    A rectangle is inscribed symmetrically under the parabola $y=12-x^2$ and above the $x$-axis, as shown. The upper right corner is $(x,12-x^2)$, where $0<x<\sqrt{12}$. Determine the maximum possible area of the rectangle.
    Further applications of differentiation
21. Q29 · Original practice · 2 marks
    Evaluate $\displaystyle\int_1^3(2x+1)\,dx$.
    Introduction to integration
22. Q30 · Original practice · 3 marks
    Let $X\sim\mathrm{Bin}(5,0.3)$. Determine $P(X=2)$ to four decimal places.
    Discrete random variables
23. Q31 · Original practice · 3 marks
    Evaluate $\displaystyle\int_0^1\frac{4x}{x^2+1}\,dx$.
    Further integration
24. Q32 · Original practice · 2 marks
    Triangle $ABC$ has $AB=11$, $AC=8$ and $\angle A=30^\circ$. Determine its area.
    Trigonometry