Specialist Maths questions and solutions

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

524 questions · Page 20 of 22

1. Q230 · Original practice · 6 marks
   A well-mixed 100 L aquarium tank receives solution at 10 L/min with concentration 0.4 g/L. Mixture leaves at the same rate. Initially the tank contains pure water. Let S be salt mass in grams.
   Differential equations and related rates
2. Q231 · Original practice · 9 marks
   A bead moves on $x^2+xy+y^2=12$. At (2,2), $dx/dt=3$ and $d^2x/dt^2=0$.
   Differential equations and related rates
3. Q232 · Original practice · 1 mark
   A slope field has $y'=ay+b$, independent of x. Slopes vanish along y=3 and equal 4 along y=1. Which equation fits?
   Differential equations and related rates
4. Q233 · Original practice · 5 marks
   A sample of 100 independent measurements has mean 20. A 95% mean confidence interval, using $\overline x\pm1.96s/\sqrt n$, is (19.216,20.784).
   Statistical inference
5. Q234 · Original practice · 5 marks
   A 90% mean confidence interval uses z=1.645 and n=64. A new study uses 95% confidence, z=1.96, with the same assumed standard deviation.
   Statistical inference
6. Q235 · Original practice · 7 marks
   A sample of 100 independent observations has $\overline x=100,s=10$. For an illustrative fitted model, individual future observations have $N(100,100)$ distribution, using variance as the second parameter. Use z=1.96.
   Statistical inference
7. Q236 · Original practice · 1 mark
   Independent observations have a non-normal population with mean 5 and standard deviation 6. For n=144, which is justified by the central limit theorem? In N(a,b), b is variance.
   Statistical inference
8. Q237 · Original practice · 9 marks
   Independent samples from a normal population with mean $\mu$ and standard deviation 10 have sizes 25 and 100, with means $\overline X_1,\overline X_2$. Consider $U=(\overline X_1+\overline X_2)/2$ and pooled mean $P=(25\overline X_1+100\overline X_2)/125$. For independent A,B, use $\operatorname{Var}(aA+bB)=a^2\operatorname{Var}(A)+b^2\operatorname{Var}(B)$.
   Statistical inference
9. Q238 · Original practice · 5 marks
   A wildlife monitoring budget is $400. Independent measurements with unbiased instrument A have standard deviation 6 and cost $2 each; B has standard deviation 4 and costs $5 each. Use the full budget with one instrument and z=1.96.
   Statistical inference
10. Q239 · Original practice · 8 marks
    A coral nursery uses $N^{\prime}=kN(1-N/200)$ with $N(0)=50$ and constant $k>0$, where time is in days. A target is $N(10)\ge150$. An independent calibration sample of 100 estimates of k has mean 0.24 and sample standard deviation 0.08. Use $\overline x\pm1.96s/\sqrt n$ for an approximate mean confidence interval. Treat the calibration estimates as independe…
    Statistical inference
11. Q240 · Original practice · 7 marks
    An escape-room panel assigns nonnegative real values r, g and b to three rune types. Its calibration totals satisfy

    $$r+g+b=12,\qquad2r+g+3b=25,\qquad r+2g+b=17.$$
    Further matrices and systems of equations
12. Q241 · Original practice · 8 marks
    Three flat sensor screens are modelled by

    $$\Pi_1:x+y+z=6,\qquad\Pi_2:x-y+z=2,\qquad\Pi_3:x+y+az=b,$$

    where $a,b\in\mathbb R$.
    Further matrices and systems of equations
13. Q242 · Original practice · 1 mark
    A researcher reports an observed 95% confidence interval (8,12) for a fixed population mean. Which interpretation is valid?
    Statistical inference
14. Q243 · Original practice · 6 marks
    A lot of miniature game pieces is accepted when its random sample mean mass is at least 49 g. Independent masses are normal with known standard deviation 6 g. A good lot has mean 50 g and a poor lot mean 48 g.
    Statistical inference
15. Q244 · Original practice · 6 marks
    Independent X equals 10 with probability 0.1, and 0 otherwise. For n=100, let K count the observations equal to 10. Then $K\sim\operatorname{Bin}(100,0.1)$ and $\overline X=K/10$.
    Statistical inference
16. Q245 · Original practice · 1 mark
    A mean confidence interval uses fixed confidence and the same standard deviation estimate. If n changes to 4n, what happens to full width?
    Statistical inference
17. Q246 · Original practice · 5 marks
    A random sample of 64 independent measurements has $\overline x=14.8,s=1.2$. The specification limit is 15. Use $\overline x\pm1.96s/\sqrt n$.
    Statistical inference
18. Q247 · Original practice · 10 marks
    Independent waiting times W are exponential, in minutes, with $P(W>3)=e^{-3/2}$. A study of 100 waits reports mean 2.1 and sample standard deviation 1.8.
    Statistical inference
19. Q248 · Original practice · 12 marks
    A scanner spot has complex position $z(t)=1+i+2\frac{1+it}{1-it}$, where $t\ge0$ is time in seconds and the real and imaginary parts give coordinates in metres. A straight detection gate lies on $x+2y=7$.
    Further complex numbers
20. Q249 · Original practice · 12 marks
    A flexible cable runs from $A=(1,1,4)$ to $B=(5,-1,2)$ through a junction $P$ on the floor $z=0$. Coordinates are in metres. Its length is $|AP|+|PB|$. The junction is first free to move anywhere on the floor, then is constrained to the rail $x+y=2$, $z=0$.
    Vectors in two and three dimensions
21. Q250 · Original practice · 12 marks
    A fountain vessel is formed by rotating the profile $r(h)=\sqrt{h/(1-h)}$ about the vertical axis for $0\le h\le3/4$. Here $h$ is height and $r$ is radius, both measured in metres. Water enters at $\pi/2$ m$^3$/min and leaves at $\pi h$ m$^3$/min. Initially $h=3/4$. Assume the flow model holds for $h>0$.
    Differential equations and related rates
22. Q251 · Original practice · 12 marks
    Each round starts two independent unlocking processes. Their completion times $T_1$ and $T_2$ are exponential with means 2 and 3 minutes respectively. The round ends at $T=\min(T_1,T_2)$. All processes are reset independently between rounds. Let $E$ be the event that process 1 finishes first. You may use…
    Exponential random variables
23. Q252 · Original practice · 13 marks
    For nonnegative integers $n$, define $I_n=\int_0^{\pi/2}\cos^{2n}x\,dx$. A solid ceramic bead is formed by rotating the region $0\le y\le\cos^2x$, $0\le x\le\pi/2$, about the $x$-axis. Its density is uniform. All angles are in radians.
    Integration techniques
24. Q253 · Original practice · 13 marks
    A game has centre $\mathbf c=(1,-1)$ and checkpoints $\mathbf p_0=\mathbf c+(8,0)$, $\mathbf p_{n+1}=\mathbf c+A(\mathbf p_n-\mathbf c)$, where $A=\frac14\begin{pmatrix}1&-\sqrt3\\\sqrt3&1\end{pmatrix}$. A player visits the checkpoints in order along straight segments. Coordinates are in metres. You may use $\sum_{n=0}^\infty r^n=1/(1-r)$ for $|r|<1$.
    Mathematical induction and trigonometric proofs