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 22 of 22

1. Q278 · Original practice · 6 marks
   The supplied direction field represents $dy/dx=xy$. Consider the solution with $y(-1)=1$ on $-2\le x\le2$.
   Rates of change and differential equations
2. Q279 · Original practice · 6 marks
   An ellipse has equation $x^2/9+y^2/4=1$. Point P on its upper half has coordinates $(3\sqrt3/2,1)$.
   Rates of change and differential equations
3. Q280 · Original practice · 6 marks
   A trolley has acceleration $a(t)=6-2t$ m s$^{-2}$ for $0\le t\le6$ s and initial velocity $v(0)=-5$ m s$^{-1}$. The acceleration graph is supplied.
   Modelling motion
4. Q281 · Original practice · 5 marks
   A calibration calculation requires

   $$I=\int_0^{\pi/4}\bigl(\tan^2x+\sin^2(2x)\bigr)\,dx.$$
   Integration techniques
5. Q282 · Original practice · 5 marks
   A curve satisfies $f\prime(x)=e^{-x}/\sqrt{1-e^{-2x}}$ for $x>0$, and $f(\ln2)=\pi/3$. Angles are in radians.
   Integration techniques
6. Q283 · Original practice · 8 marks
   A waiting time T has density $f(t)=kt e^{-2t}$ for $t\ge0$ and zero otherwise, with $k>0$. For this question you may use

   $$E(T)=\int_0^\infty tf(t)\,dt,\qquad E(T^2)=\int_0^\infty t^2f(t)\,dt,\qquad\operatorname{Var}(T)=E(T^2)-[E(T)]^2.$$

   All relevant integrals converge; polynomial factors times $e^{-2t}$ tend to zero at infinity.
   Applications of integral calculus
7. Q284 · Original practice · 4 marks
   A random sample of 12 independent tests records the times shown in the table, in minutes. Assume test times in the population are normal. The sample standard deviation is $s=1.0445$ minutes. Use $z=1.96$ and the QCAA approximate interval $\overline x\pm zs/\sqrt n$.
   Statistical inference
8. Q285 · Original practice · 6 marks
   Let $\Phi$ be the standard normal cumulative distribution function. A normal population X has unknown mean $\mu$ and standard deviation $\sigma>0$. It is known that $P(X<44)=\Phi(1)$ and $P(X<36)=\Phi(-1)$.
   Statistical inference
9. Q286 · Original practice · 6 marks
   Two independent studies each use n=100. Their observed 95% confidence intervals are $(18,22)$ and $(23,29)$. Both use $\overline x\pm1.96s/\sqrt n$. The confidence level is changed using the same two datasets.
   Statistical inference
10. Q287 · Original practice · 6 marks
    A tank is an inverted square-based pyramid, 6 m deep with top side length 4 m. Water enters at 8 m$^3$/min. The water depth h and water-surface side length s satisfy the same similarity ratio as the full tank. A central vertical cross-section is supplied.
    Rates of change and differential equations
11. Q288 · Original practice · 1 mark
    A position vector is $\mathbf a=3\mathbf i+4\mathbf j+5\mathbf k$. Its altitude angle above the xy-plane is
    Vectors in two and three dimensions
12. Q289 · Original practice · 1 mark
    Let $\mathbf a=(2,-3,1)$ and $\mathbf b=(-1,0,0)$. The scalar projection of a onto b is
    Vectors in two and three dimensions
13. Q290 · Original practice · 1 mark
    For $f(x)=\arcsin(3x)$, determine $f\prime(1/6)$.
    Integration techniques
14. Q291 · Original practice · 1 mark
    If $\frac{5x+1}{(x-1)(x+2)}=\frac A{x-1}+\frac B{x+2}$, determine A.
    Integration techniques
15. Q292 · Original practice · 1 mark
    A student verifies a proposition for n=1, then proves that if it holds for n=k, it holds for n=k+2. What can this establish without another base case?
    Mathematical induction and trigonometric proofs
16. Q293 · Original practice · 1 mark
    Independent observations are normal with mean 40 and standard deviation 8. Which describes the distribution of sample means from samples of size 16? The second parameter of N is variance.
    Statistical inference
17. Q294 · Original practice · 1 mark
    A particle moving in the positive x-direction has velocity $v(x)=\sqrt{18-2x^2}$ m s$^{-1}$ while the radicand is positive. Its acceleration at x=2 m is
    Modelling motion
18. Q295 · Original practice · 1 mark
    Which example shows that $|z+w|=|z|+|w|$ is not true for every pair of complex numbers?
    Further complex numbers
19. Q296 · Original practice · 5 marks
    A channel has the depths in the supplied table, measured at equally spaced positions across its 4 m width. The water has uniform downstream speed 0.75 m/s. The flow rate equals cross-sectional area times downstream speed.
    Applications of integral calculus
20. Q297 · Original practice · 6 marks
    A particle has $\mathbf r(t)=4\cos(2t)\mathbf i+4\sin(2t)\mathbf j$ metres. The question diagram shows its circular path and the point at $t=\pi/8$ s.
    Vector calculus