Mathematics

(b) $45^\circ$

$\sin Q-\cos Q=0 \Rightarrow \sin Q=\cos Q \Rightarrow \tan Q=1 \Rightarrow Q=45^\circ.$

(d) $N_{24}$

$N_6$ is the set of multiples of 6 and $N_8$ the set of multiples of 8. Their intersection is the set of common multiples, i.e. multiples of $\operatorname{lcm}(6,8)=24$. Hence $N_6\cap N_8=N_{24}$.

(b) 24

P's one-day work $=\tfrac16$, Q's one-day work $=\tfrac1{12}$.

Work done in the first 3 days by both: $3\left(\tfrac16+\tfrac1{12}\right)=3\cdot\tfrac14=\tfrac34.$

Remaining work $=\tfrac14$, done by Q and R together in 2 days, so their daily work is $\tfrac18$.

R's daily work: $\tfrac18-\tfrac1{12}=\tfrac{3-2}{24}=\tfrac1{24}.$

So R alone finishes the work in 24 days.

(b) $m-n$

Dividing a polynomial of degree $m$ by one of degree $n$ ($n<m$) gives quotient degree $m-n$ since $\text{degree of dividend}=\text{degree of divisor}+\text{degree of quotient}$.

(c) $\frac{7}{13}$

Black cards $=26$, kings $=4$, black kings $=2$. $P(\text{black or king})=\frac{26+4-2}{52}=\frac{28}{52}=\frac{7}{13}.$

(d) compound interest at 10% p.a. compounded quarterly

For the same rate, simple interest is smallest. Compound interest grows more the more often it is compounded, so quarterly compounding gives the greatest amount: $\left(1+\tfrac{0.10}{4}\right)^{40}>\left(1+\tfrac{0.10}{2}\right)^{20}>(1.10)^{10}>1+10(0.10).$ Numerically these are about $2.685$, $2.653$, $2.594$ and $2$ times the principal.

(a) 17640

The population two years ago grows for 2 years at 5%: $16000\left(1+\tfrac{5}{100}\right)^2=16000\times1.1025=17640.$

(c) $2:9$

Let the radius and height of $y$ be $r$ and $h_y$; those of $x$ are $3r$ and $h_x$. Since $V_y=\tfrac12V_x$, $V_x=2V_y$: $\tfrac13\pi(3r)^2h_x=2\cdot\tfrac13\pi r^2h_y \Rightarrow 9h_x=2h_y \Rightarrow h_x:h_y=2:9.$

(b) 280

Product of two numbers $=$ LCM $\times$ HCF: $42\times N=840\times14 \Rightarrow N=\frac{11760}{42}=280.$ Check: $\operatorname{HCF}(42,280)=14$ and $\operatorname{LCM}(42,280)=840$.

(b) multiply by $10^4$ (104)

The student computed $\dfrac{x}{100}$ instead of $100x$. The correct value is $100x$, and $100x=\frac{x}{100}\times 10^4,$ so the answer must be multiplied by $10^4$, which is option (b) (written as 104).