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 19 of 22

1. Q206 · Original practice · 5 marks
   A game designer compares two elevation profiles, $y=\sin x$ and $y=\cos x$, shown on $0\le x\le\pi/2$.
   Integration techniques
2. Q207 · Original practice · 1 mark
   What is $\int_0^{3/2}\frac{dx}{\sqrt{9-x^2}}$ exactly?
   Integration techniques
3. Q208 · Original practice · 8 marks
   A vessel design starts with region R below $y=\sqrt x$, above the x-axis and to the left of x=4. R is rotated about the y-axis.
   Applications of integral calculus
4. Q209 · Original practice · 5 marks
   For k>0, the region below $y=k\sin x$ and above the x-axis on $0\le x\le\pi$ is rotated about the x-axis.
   Applications of integral calculus
5. Q210 · Original practice · 5 marks
   A decorative shade uses the region R between $y=e^x$ and $y=e^{-x}$ on $0\le x\le\ln2$.
   Applications of integral calculus
6. Q211 · Original practice · 5 marks
   For $f(x)=(x-1)^3+2$ on [0,2], use ordinates at x=0,1,2. For comparison, the composite trapezoidal rule here is $\tfrac12[f(0)+2f(1)+f(2)]$.
   Integration techniques
7. Q212 · Original practice · 5 marks
   Two independent sensors have exponential lifetimes, each with mean 5 hours.
   Exponential random variables
8. Q213 · Original practice · 6 marks
   Independent components A and B have exponential lifetimes with rates 0.2 and 0.3 per hour. Let T be the first failure time.
   Exponential random variables
9. Q214 · Original practice · 5 marks
   An exponential waiting time W has 90th percentile exactly 12 minutes.
   Exponential random variables
10. Q215 · Original practice · 8 marks
    A survival model, with t in hours, is

    $$S(t)=P(T>t)=\begin{cases}e^{-0.2t},&0\le t\le3,\\e^{-0.6-0.4(t-3)},&t>3.\end{cases}$$
    Exponential random variables
11. Q216 · Original practice · 5 marks
    A vertical line x=c halves the area under $y=1/(1+x^2)$ on $0\le x\le2$.
    Integration techniques
12. Q217 · Original practice · 5 marks
    Let $I=\int_0^1\tan^{-1}x\,dx$, with angles in radians.
    Integration techniques
13. Q218 · Original practice · 10 marks
    A managed algae population satisfies $N^{\prime}=N(1-N/100)-16$ for $N\ge0$. The model stops at extinction N=0. The graph shows growth rate against population.
    Differential equations and related rates
14. Q219 · Original practice · 6 marks
    A population satisfies $N'=kN(120-N)$ with k>0, N(0)=30 and N(4)=60. Time is in days.
    Differential equations and related rates
15. Q220 · Original practice · 5 marks
    A positive solution satisfies $dy/dx=2x\sqrt y$ and y(0)=1.
    Differential equations and related rates
16. Q221 · Original practice · 6 marks
    Consider $dy/dt=\sqrt y$, y nonnegative, with y(0)=0.
    Differential equations and related rates
17. Q222 · Original practice · 6 marks
    A particle has acceleration $a=-2x$ m/s squared. At x=0 it moves in the positive direction with speed 4 m/s.
    Modelling motion
18. Q223 · Original practice · 5 marks
    A robot antenna vibrates with displacement $x(t)=3\cos2t+4\sin2t$ cm for t nonnegative.
    Modelling motion
19. Q224 · Original practice · 7 marks
    A falling object has downward velocity satisfying $dv/dt=10-2v$, v(0)=0. Its downward displacement starts at zero.
    Modelling motion
20. Q225 · Original practice · 10 marks
    An object is thrown upwards with v(0)=10 m/s. Upwards is positive. Linear air resistance gives $dv/dt=-10-v$ throughout flight; h(0)=0.
    Modelling motion
21. Q226 · Original practice · 5 marks
    A 2 kg block is projected up a smooth plane at 6 m/s. The plane is inclined at 30 degrees to horizontal. Take g=10 m/s squared and up-plane displacement positive.
    Modelling motion
22. Q227 · Original practice · 5 marks
    On a frictionless arcade track, a 2 kg cart moves right at 6 m/s and a 1 kg cart moves left at 3 m/s. They collide and remain joined. External impulse is negligible.
    Modelling motion
23. Q228 · Original practice · 5 marks
    A 2 kg particle moves in positive x under resultant force F(x)=6x N, starting from rest at x=1 m.
    Modelling motion
24. Q229 · Original practice · 6 marks
    Two isotopes have expected counts $A(t)=100e^{-2kt}$ and $B(t)=50e^{-kt}$. B has half-life 6 hours.
    Differential equations and related rates