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 17 of 22

1. Q158 · Original practice · 6 marks
   A trolley moves on a straight track for $0\le t\le4$ seconds. Its velocity, in metres per second, is

   $$v(t)=t^2-4t+3.$$

   The velocity-time graph is shown. The trolley starts at position $x(0)=0$.
   Modelling motion: displacement and distance
2. Q159 · Original practice · 6 marks
   The region $R$ is bounded by $y=x$ and $y=x^2$ for $0\le x\le1$, as shown. Two solids are formed by rotating $R$ through one complete revolution: one about the $x$-axis, the other about the $y$-axis.
   Applications of integral calculus: volumes
3. Q160 · Original practice · 7 marks
   A metal block cools according to

   $$\frac{dT}{dt}=-k(T-A),$$

   where $T$ and the surrounding temperature $A$ are in degrees Celsius, $t$ is in minutes and $k>0$ is constant. Initially $T(0)=100$ and $A=20$. At $t=10$, $T=60$. At that instant the block is moved to a room at $A=10$. The block's temperature is continuous at the move and $k$ is unchanged. The d…
   Differential equations: piecewise cooling
4. Q161 · Original practice · 1 mark
   The waiting time $W$, in minutes, for a sensor signal has probability density function

   $$f(w)=\frac18 e^{-w/8},\qquad w\ge0.$$

   No signal has arrived in the first five minutes. A technician chooses a total elapsed time $b>5$ so that

   $$P(W\le b\mid W>5)=0.5.$$

   Which expression gives $b$? The shaded region shows the event $W>5$.
   Applications of integral calculus: exponential probability
5. Q162 · Original practice · 9 marks
   An inverted conical tank has height $6$ m and top radius $3$ m. Water depth above the vertex is $h$ metres, and the radius of its surface is $r$ metres. Water enters at $6\pi$ cubic metres per minute and leaves at $2\pi h$ cubic metres per minute. Initially $h(0)=4$. Assume the model applies for $0<h<6$.
   Rates of change and differential equations: conical tank
6. Q163 · Original practice · 6 marks
   Consider $(\cos\theta+i\sin\theta)^3$.
   Trigonometric proofs using De Moivre
7. Q164 · Original practice · 5 marks
   A monic polynomial $p(z)$ of degree four has real coefficients. Two of its zeros are represented by $A$ and $B$ on the Argand plane:

   $$a=1+2i,\qquad b=-1+i.$$
   Further complex numbers: real polynomials
8. Q165 · Original practice · 6 marks
   A straight line $L$ has vector equation

   $$\mathbf r(t)=t\mathbf i+2t\mathbf j+2\mathbf k,\qquad t\in\mathbb R.$$

   A family of spheres has centre $C=(1,0,0)$ and radius $R>0$. The diagram is schematic.
   Vector equations: line and sphere
9. Q166 · Original practice · 6 marks
   Two planes have Cartesian equations

   $$\Pi_1:x+y+z=6,\qquad\Pi_2:x-y+z=2.$$

   Their line of intersection is $L$. The diagram is schematic.
   Vector equations: intersection of planes
10. Q167 · Original practice · 7 marks
    A projectile is launched from the origin at $t=0$ with velocity $8\mathbf i+12\mathbf j$ metres per second and constant acceleration $-4\mathbf j$ metres per second squared. A target moves with position vector

    $$\mathbf r_T(t)=(30+2t)\mathbf i+10\mathbf j,\qquad t\ge0.$$

    The projectile model applies until it first returns to ground level $y=0$. The diagr…
    Vector calculus: projectile interception
11. Q168 · Original practice · 6 marks
    The region $R$ lies below $y=1/(1+x^2)$ and above the $x$-axis for $0\le x\le2$. The marked points are at $x=0,0.5,1,1.5,2$.
    Applications of integral calculus: Simpson rule
12. Q169 · Original practice · 8 marks
    A function satisfies

    $$\frac{dy}{dx}=x(1-y^2),\qquad y(0)=0.$$

    A slope field for this equation is shown. Consider the solution on $-2\le x\le2$.
    Differential equations: slope field and separation
13. Q170 · Original practice · 8 marks
    A spherical ice shell surrounds a rigid spherical core of radius $3$ cm. The outer radius is initially $5$ cm. Ice volume is lost at a constant rate of $6\pi$ cubic centimetres per minute. Melting occurs only at the outer surface, and the core remains unchanged. Let $r(t)$ be the outer radius, in centimetres. The diagram is a central cross-section.
    Related rates: melting spherical shell
14. Q171 · Original practice · 6 marks
    Measurements $X$ are normally distributed with unknown mean $\mu$ and known standard deviation $12$ mm. Independent measurements are selected at random. A sample of size $n$ has mean $\overline X$.
    Statistical inference: sampling precision
15. Q172 · Original practice · 1 mark
    A random sample consists of 36 independent measurements from a normal population with unknown mean $\mu$ and known standard deviation $12$. The sample mean is $100$. A spreadsheet accidentally records every measurement twice, producing 72 rows. Using $z=1.96$, which statement gives the valid 95% confidence interval for $\mu$ and the reason?
    Statistical inference: duplicated observations
16. Q173 · Original practice · 5 marks
    Four beacons in a light installation have complex coordinates given by the solutions of $(z-1-i)^4=16$. They are represented on the Argand plane. The dashed circle has centre $C=1+i$.
    Further complex numbers
17. Q174 · Original practice · 8 marks
    A nonzero complex number satisfies $|z|=1$. Define $w=z+1/z$.
    Further complex numbers
18. Q175 · Original practice · 1 mark
    The polynomial $p(z)=z^2-2iz-2$ has zero $1+i$. Which is its other zero?
    Further complex numbers
19. Q176 · Original practice · 5 marks
    The polynomial $p(z)=z^3-4z^2+9z-10$ has zero $1+2i$.
    Further complex numbers
20. Q177 · Original practice · 1 mark
    Let $a=4\operatorname{cis}(3\pi/4)$ and $b=2\operatorname{cis}(-3\pi/4)$. Which is a/b?
    Further complex numbers
21. Q178 · Original practice · 6 marks
    Consider $(\cos\theta+i\sin\theta)^4$.
    Mathematical induction and trigonometric proofs
22. Q179 · Original practice · 6 marks
    For positive integers n, let $S_n=\sum_{k=1}^n(2k-1)^3$.
    Mathematical induction and trigonometric proofs
23. Q180 · Original practice · 6 marks
    For positive integers n, consider $8^n-1$.
    Mathematical induction and trigonometric proofs
24. Q181 · Original practice · 6 marks
    A programmable light display contains $k2^k$ lights in row k, for $k=1,\ldots,n$.
    Mathematical induction and trigonometric proofs