Algebraic expressions and substitutionIB MYP Maths Extended: Subtopic test

1. 1
   A taxi company in Nairobi charges $C=50+30d$ shillings for a journey of $d$ kilometres.

   (a)

   Find the cost of a journey of 6 km.

   [1 mark]

   • A$230$ shillings
   • B$480$ shillings
   • C$180$ shillings
   • D$86$ shillings

   (b)

   A journey costs 410 shillings. Find its length.

   [1 mark]

   • A$360$ km
   • B$13\frac{2}{3}$ km
   • C$12$ km
   • D$15\frac{1}{3}$ km

   (c)

   A second journey is twice as long as one of $d$ km, so it is $2d$ km. Write a simplified expression for the total cost, in shillings, of one journey of $d$ km and one journey of $2d$ km.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   Let $a=3$, $b=-2$ and $c=5$.

   (a)

   Find the value of $a^2+bc$.

   [1 mark]

   • A$19$
   • B$-1$
   • C$-4$
   • D$-19$

   (b)

   Find the value of $(a-b)^2$.

   [1 mark]

   • A$5$
   • B$13$
   • C$1$
   • D$25$

   (c)

   Find the value of $\frac{c-b}{a-b}$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A school hall is a rectangle of length $(2x+3)$ metres and width $(x+4)$ metres. A square stage of side $x$ metres is built inside the hall.

   (a)

   Show that the area of the hall is $2x^2+11x+12$ square metres.

   [3 marks]

   (b)

   Find a simplified expression for the area of the hall not covered by the stage. Hence find this area when $x=5$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   Mei investigates what happens when she picks a whole number $n$, multiplies the numbers $n-1$ and $n+1$ on either side of it, and compares the answer with $n^2$. For $n=6$ she gets $5\times7=35$, which is 1 less than $6^2=36$.

   (a)

   (i) Find $(n-1)(n+1)$ and $n^2-(n-1)(n+1)$ when $n=9$ and when $n=12$.
   (ii) State the pattern as a general rule.
   (iii) Prove that the rule is true for every whole number $n$ by expanding $(n-1)(n+1)$.

   [6 marks]

   (b)

   Mei now compares $(n-2)(n+2)$ with $n^2$.
   (i) Expand and simplify $(n-2)(n+2)$, and state how much less it is than $n^2$.
   (ii) Use your result with $n=100$ to calculate $98\times102$ without a calculator.
   (iii) Write down, in terms of $k$, how much less $(n-k)(n+k)$ is than $n^2$.

   [6 marks]

   Total for question 4: 12 marks

End of questions