AlgebraIB MYP Maths Extended: Topic test

1. 1
   Noor is working with algebraic expressions on a revision card.

   (a)

   Expand and simplify $(x+4)(x-3)$.

   [1 mark]

   • A$x^{2}-12$
   • B$x^{2}-x-12$
   • C$x^{2}+7x-12$
   • D$x^{2}+x-12$

   (b)

   Factorise $x^{2}-81$.

   [1 mark]

   • A$(x-9)(x+9)$
   • B$(x-9)^{2}$
   • C$(x-81)(x+1)$
   • D$(x+9)^{2}$

   (c)

   Expand and simplify $(n+3)^{2}-n^{2}$. Use your answer to explain why the difference between the squares of two whole numbers that differ by $3$ is always a multiple of $3$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   Yusuf is solving equations from his worksheet.

   (a)

   Solve $\dfrac{x+5}{3}=4$.

   [1 mark]

   • A$x=17$
   • B$x=7$
   • C$x=-\frac{11}{3}$
   • D$x=\frac{7}{3}$

   (b)

   Solve $4(x-2)=2x+10$.

   [1 mark]

   • A$x=1$
   • B$x=6$
   • C$x=9$
   • D$x=18$

   (c)

   Solve the inequality $5-2x\ge11$ and write your answer as an inequality for $x$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A school in Kuala Lumpur is building a rectangular science garden. The length of the garden is $5$ m more than three times its width $w$ metres, and the perimeter of the garden is $74$ m.

   (a)

   Form an equation in $w$ and solve it to find the width of the garden.

   [3 marks]

   (b)

   The school has at most RM $1500$ to spend on fencing at RM $15$ per metre. Form an inequality in $w$ for a garden of this shape and find the greatest whole-number width the school can afford.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A school in Chennai is installing a rooftop solar array. The array is a rectangle of width $x$ metres and length $(x+6)$ metres, and its area is $55$ m$^2$. It will be built from $20$ panels of two types: type P costs $180$ US dollars each and type Q costs $260$ US dollars each, and the school spends $4000$ US dollars on panels in total.

   (a)

   Form a quadratic equation in $x$ and solve it algebraically to find the dimensions of the array. Explain why you reject one solution.

   [6 marks]

   (b)

   Let $p$ be the number of type P panels and $q$ the number of type Q panels. Write two equations and solve them to find $p$ and $q$. Check your answer against the total spent.

   [6 marks]

   Total for question 4: 12 marks

5. 5
   The kinetic energy $E$ (in joules) of an object of mass $m$ kg moving at speed $v$ m/s is given by $E=\frac{1}{2}mv^{2}$.

   (a)

   Make $m$ the subject of the formula $E=\frac{1}{2}mv^{2}$.

   [1 mark]

   • A$m=\dfrac{2E}{v^{2}}$
   • B$m=\dfrac{E}{2v^{2}}$
   • C$m=\dfrac{2E}{v}$
   • D$m=2Ev^{2}$

   (b)

   Make $v$ the subject of the formula $E=\frac{1}{2}mv^{2}$, where $v>0$.

   [1 mark]

   • A$v=\dfrac{2E}{m}$
   • B$v=\sqrt{2Em}$
   • C$v=\dfrac{\sqrt{2E}}{m}$
   • D$v=\sqrt{\dfrac{2E}{m}}$

   (c)

   An object of mass $4$ kg has kinetic energy $50$ J. Find its speed.

   [2 marks]

   Total for question 5: 4 marks

6. 6
   Consider the quadratic equation $2x^{2}-7x+3=0$.

   (a)

   Which of these is the factorised form of $2x^{2}-7x+3$?

   [1 mark]

   • A$(2x+1)(x+3)$
   • B$(2x-1)(x-3)$
   • C$(2x-3)(x-1)$
   • D$(2x+1)(x-3)$

   (b)

   What are the solutions of $2x^{2}-7x+3=0$?

   [1 mark]

   • A$x=-\frac12$ or $x=3$
   • B$x=\frac12$ or $x=-3$
   • C$x=\frac12$ or $x=3$
   • D$x=-\frac12$ or $x=-3$

   (c)

   A different equation $x^{2}-4x+k=0$ has exactly one solution. Find the value of $k$.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   A rectangular mural in Mexico City has area $(3x^{2}+11x+6)$ m$^2$ and width $(x+3)$ m, where $x>0$.

   (a)

   By factorising, find an expression for the length of the mural in terms of $x$.

   [3 marks]

   (b)

   The perimeter of the mural is $50$ m. Find $x$ and the area of the mural.

   [4 marks]

   Total for question 7: 7 marks

8. 8
   On a plan of a farm in Kenya, one unit represents $100$ m. The edge of a field is the curve $y=x^{2}+2x-8$, and a straight track follows the line $y=2x+1$.

   (a)

   The distances are in hundreds of metres. Find the coordinates of the points where the track crosses the edge of the field.

   [6 marks]

   (b)

   A second straight track follows $y=2x+c$. Find the value of $c$ for which this track touches the edge of the field at exactly one point, and give the coordinates of that point. Explain how many times the track meets the edge of the field when $c<-8$.

   [6 marks]

   Total for question 8: 12 marks

End of questions