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 12 of 22

1. Q38 · Original practice · 5 marks
   A particle has position $$\mathbf r(t)=5(\cos2t,\sin2t).$$
   Vector calculus
2. Q39 · Original practice · 1 mark
   The parametric equations $x=2+3t$, $y=-1+6t$ describe the line
   Vector calculus
3. Q40 · Original practice · 7 marks
   Consider the system $$\begin{aligned}x+y+z&=3,\\2x+3y+4z&=8,\\x+2y+(a^2-5a+9)z&=a+3.\end{aligned}$$ Use Gaussian elimination to classify the system as having a unique solution, no solution or infinitely many solutions for all real values of $a$. Where the solution is unique, determine it in terms of $a$.
   Matrices
4. Q41 · Original practice · 1 mark
   Particle $A$ follows $\mathbf r_A(t)=(t,2t)$ and particle $B$ follows $\mathbf r_B(t)=(3-t,1+t)$. Which statement is correct?
   Vector calculus
5. Q42 · Original practice · 6 marks
   Four competitors $A,B,C,D$ have dominance matrix $$D=\begin{pmatrix}0&1&1&0\\0&0&1&1\\0&0&0&1\\1&0&0&0\end{pmatrix}.$$ A competition uses the score defined as the row sum of $D+D^2$.
   Matrices
6. Q43 · Original practice · 1 mark
   For $\mathbf r(t)=(t^2,e^t,\sin t)$, the acceleration at $t=0$ is
   Vector calculus
7. Q44 · Original practice · 5 marks
   A three-class population is modelled by $$L=\begin{pmatrix}0&0.8&1.5\\0.4&0&0\\0&0.6&0\end{pmatrix},\qquad\mathbf p_0=\begin{pmatrix}150\\100\\50\end{pmatrix}.$$
   Matrices
8. Q45 · Original practice · 1 mark
   A particle starts at the origin with velocity $(1,3)$ and constant acceleration $(2,-1)$. Its position at $t=2$ is
   Vector calculus
9. Q46 · Original practice · 4 marks
   Evaluate exactly $$\int_0^{\pi/2}\sin^2x\cos^2x\,dx.$$
   Integration
10. Q47 · Original practice · 1 mark
    An augmented matrix in row-echelon form has final row $\left[\begin{array}{ccc|c}0&0&0&3\end{array}\right]$. The corresponding system has
    Matrices
11. Q48 · Original practice · 4 marks
    Evaluate exactly $$\int_0^1\frac{2x}{1+x^4}\,dx.$$
    Integration
12. Q49 · Original practice · 1 mark
    For the dominance matrix $$D=\begin{pmatrix}0&1&1&0\\0&0&1&1\\0&0&0&1\\1&0&0&0\end{pmatrix},$$ the entry $(D^2)_{1,4}$ is
    Matrices
13. Q50 · Original practice · 4 marks
    Use partial fractions to evaluate $$\int\frac{4x+7}{(x+1)(x+3)}\,dx.$$
    Integration
14. Q51 · Original practice · 1 mark
    The exact value of $\displaystyle\int_0^{\pi/2}\sin^2x\,dx$ is
    Integration
15. Q52 · Original practice · 5 marks
    Evaluate exactly $$\int_0^1x^2e^x\,dx.$$
    Integration
16. Q53 · Original practice · 1 mark
    An antiderivative of $\dfrac{3x^2}{x^3+4}$ is
    Integration
17. Q54 · Original practice · 4 marks
    Determine the exact area enclosed by $y=x+2$ and $y=x^2$.
    Integration
18. Q55 · Original practice · 1 mark
    The exact value of $\displaystyle\int_0^1\frac1{1+x^2}\,dx$ is
    Integration
19. Q56 · Original practice · 5 marks
    The region under $y=x(2-x)$ for $0\le x\le2$ is rotated about the $x$-axis. Determine the exact volume of the solid formed.
    Integration
20. Q57 · Original practice · 1 mark
    An antiderivative of $\dfrac{5x+1}{(x-1)(x+2)}$ is
    Integration
21. Q58 · Original practice · 4 marks
    The table gives values of $f(x)=e^{-x^2}$ rounded to four decimal places. $$\begin{array}{c|ccccc}x&0&0.25&0.50&0.75&1.00\\\hline f(x)&1.0000&0.9394&0.7788&0.5698&0.3679\end{array}$$ Use Simpson’s rule with four equal intervals to approximate $\displaystyle\int_0^1e^{-x^2}\,dx$.
    Integration
22. Q59 · Original practice · 1 mark
    The exact value of $\displaystyle\int_0^1xe^x\,dx$ is
    Integration
23. Q60 · Original practice · 4 marks
    The lifetime $T$ (hours) of a component is exponentially distributed with mean 12 hours.
    Exponential distributions
24. Q61 · Original practice · 1 mark
    The area between $y=x$ and $y=x^2$ for $0\le x\le1$ is
    Integration