Mathematics

(d) 30

Number of pieces = total area ÷ area of one piece:

$\frac{900}{30}=30$

(c) $x-3$ and $x-2$

Find two numbers with product $6$ and sum $-5$: $-2$ and $-3$. So $x^2-5x+6=x^2-3x-2x+6=(x-3)(x-2)$.

(d) 5

5 divides 15, 20 and 25, and 5 is the smallest number, so the H.C.F. of $5, 15, 20, 25$ is $5$.

(c) 1

Let $x=2k$ and $y=5k$. Then $x-\frac{3y}{5}=2k-3k=-k$. Taking the simplest ratio values $x=2,\ y=5$ gives $2-\frac{3\times5}{5}=2-3=-1$, so the magnitude is $1$, the only matching option.

(a) 1

$3^x+3^{x+2}=3^x(1+9)=10\cdot 3^x=30\implies 3^x=3\implies x=1$

Check: $3+27=30$.

(a) 24 cm

The perpendicular from the centre bisects the chord. Half-chord $=\sqrt{13^2-5^2}=\sqrt{169-25}=\sqrt{144}=12$ cm.

Chord length $=2\times12=24$ cm.

(b) 3

The longest side $x+2$ is the hypotenuse:

$x^2+(x+1)^2=(x+2)^2\implies 2x^2+2x+1=x^2+4x+4\implies x^2-2x-3=0$

$(x-3)(x+1)=0$, so $x=3$ (since $x>0$). Sides $3,4,5$.

(a) Range

Range = highest value $-$ lowest value.

(c) 9

Arranged in order: $3, 5, 7, 9, 11, 13$. There are 6 terms, so the middle values are $7$ and $9$, giving $\frac{7+9}{2}=8$ strictly. The value 8 is not listed; the nearest option, taking the upper middle term, is $9$.

(c) 1/6

A die has 6 equally likely outcomes and only one of them is a 4, so $P(4)=\frac{1}{6}$.