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 11 of 22

1. Q14 · Original practice · 5 marks
   The polynomial $$P(z)=z^4-4z^3+6z^2-4z-15$$ has real coefficients. Given that $1+2i$ is a root, factorise $P(z)$ completely over $\mathbb C$.
   Complex numbers
2. Q15 · Original practice · 1 mark
   The distance in the Argand plane between two adjacent sixth roots of unity is
   Complex numbers
3. Q16 · Original practice · 6 marks
   Let $\omega$ be a non-real cube root of unity.
   Complex numbers
4. Q17 · Original practice · 1 mark
   A monic cubic polynomial with real coefficients has roots $2+i$ and $-1$. Its constant term is
   Complex numbers
5. Q18 · Original practice · 5 marks
   Use mathematical induction to prove that $$\sum_{r=1}^n r(r+1)=\frac{n(n+1)(n+2)}3$$ for every positive integer $n$.
   Proof and induction
6. Q19 · Original practice · 1 mark
   Which of the following is a solution of $z^3=-8i$?
   Complex numbers
7. Q20 · Original practice · 5 marks
   Use mathematical induction to prove that $8^n-2^n$ is divisible by 6 for every positive integer $n$.
   Proof and induction
8. Q21 · Original practice · 1 mark
   In proving $1+3+\cdots+(2n-1)=n^2$ by induction, the left side for $n=k+1$ becomes
   Proof and induction
9. Q22 · Original practice · 6 marks
   View question and marking material
   Proof and induction
10. Q23 · Original practice · 1 mark
    Assume $7^k-1$ is divisible by 6. Which rearrangement is most useful for the inductive step?
    Proof and induction
11. Q24 · Original practice · 6 marks
    Let $\mathbf a=(4,-2,4)$ and $\mathbf b=(1,2,2)$.
    Vectors in three dimensions
12. Q25 · Original practice · 1 mark
    Using De Moivre’s theorem, $\sin(3x)=3\sin x-4\sin^3x$. The coefficient of $\sin^3x$ is
    Proof and induction
13. Q26 · Original practice · 6 marks
    The sphere $x^2+y^2+z^2=9$ is intersected by the line $$\mathbf r=(0,0,4)+t(1,0,-1).$$
    Vectors in three dimensions
14. Q27 · Original practice · 1 mark
    A unit vector in the direction of $\mathbf a=(2,-2,1)$ is
    Vectors in three dimensions
15. Q28 · Original practice · 5 marks
    Points $A(1,0,2)$, $B(3,1,0)$ and $C(0,2,1)$ lie in a plane.
    Vectors in three dimensions
16. Q29 · Original practice · 1 mark
    Let $\mathbf a=(2,1,2)$ and $\mathbf b=(1,2,2)$. The scalar projection of $\mathbf a$ on $\mathbf b$ is
    Vectors in three dimensions
17. Q30 · Original practice · 6 marks
    A tetrahedron has vertices with position vectors $\mathbf a,\mathbf b,\mathbf c,\mathbf d$. Let $M$ and $N$ be the midpoints of $AB$ and $CD$, respectively. Prove that the midpoint of $MN$ has position vector $$\frac{\mathbf a+\mathbf b+\mathbf c+\mathbf d}{4},$$ and hence show that the three line segments joining the midpoints of opposite edges of a tetrahe…
    Vectors in three dimensions
18. Q31 · Original practice · 1 mark
    Point $P$ divides the segment from $A(1,2,3)$ to $B(7,-1,6)$ internally in the ratio $AP:PB=2:1$. The coordinates of $P$ are
    Vectors in three dimensions
19. Q32 · Original practice · 7 marks
    Two particles move for $t\ge0$ with positions $$\mathbf r_A(t)=(2t,t),\qquad\mathbf r_B(t)=(6-t,4-2t).$$
    Vector calculus
20. Q33 · Original practice · 1 mark
    The sphere $x^2+y^2+z^2-4x+6y-2z=11$ has centre and radius
    Vectors in three dimensions
21. Q34 · Original practice · 6 marks
    A particle follows the path $$\mathbf r(t)=(3\cos t,2\sin t),\qquad0\le t\le2\pi.$$
    Vector calculus
22. Q35 · Original practice · 1 mark
    The line $\mathbf r=(0,0,1)+t(1,2,-1)$ and the plane $2x-y=3$ are
    Vectors in three dimensions
23. Q36 · Original practice · 6 marks
    A projectile is launched from level ground at $20\ \mathrm{m\,s^{-1}}$ at $45^\circ$ above the horizontal. Take $g=10\ \mathrm{m\,s^{-2}}$ and ignore air resistance.
    Vector calculus
24. Q37 · Original practice · 1 mark
    If $\mathbf a=(1,2,0)$ and $\mathbf b=(2,0,1)$ form adjacent sides of a parallelogram, its area is
    Vectors in three dimensions