Number and algebraIB Maths: Applications and Interpretation HL: Topic test

1. 1
   An arithmetic sequence $u_n$ has first term $5$ and common difference $3$. A geometric sequence $v_n$ has first term $5$ and common ratio $0.4$.

   (a)

   Find $u_{20}$.

   [1 mark]

   • A$65$
   • B$62$
   • C$60$
   • D$59$

   (b)

   Find the sum to infinity of the sequence $v_n$.

   [1 mark]

   • A$\frac{5}{3}$
   • B$\frac{25}{2}$
   • C$\frac{25}{7}$
   • D$\frac{25}{3}$

   (c)

   Use your GDC to find the smallest value of $n$ for which the sum of the first $n$ terms of $u_n$ exceeds $1000$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The magnitude $M$ of an earthquake that releases energy $E$ joules is modelled by $M=\frac{2}{3}\log_{10}E-2.9$.

   (a)

   Find the magnitude of an earthquake that releases $10^{12}$ joules.

   [1 mark]

   • A$5.1$
   • B$8.0$
   • C$6.07$
   • D$9.1$

   (b)

   The energy released is multiplied by $1000$. By how much does the magnitude increase?

   [1 mark]

   • A$\frac{2}{3}$
   • B$3$
   • C$2$
   • D$2000$

   (c)

   An earthquake has magnitude $6.5$. Find the energy released, giving your answer in the form $a\times10^{k}$, where $1\leq a<10$ and $k\in\mathbb{Z}$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   Noor takes out a loan of 18 000 EUR to buy a car. Interest is charged at a nominal annual rate of 6.6%, compounded monthly. The loan is repaid over 4 years by equal payments made at the end of each month.

   (a)

   Use your GDC to find the monthly payment, giving your answer to the nearest cent.

   [3 marks]

   (b)

   (i) Find the total amount of interest Noor pays over the 4 years.
   (ii) Find the amount still owed immediately after the 24th payment.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A workshop makes tables, chairs and stools. The hours needed per item are given by the matrix $A=\begin{pmatrix}3&2&1\\ 1&2&2\\ 2&1&3\end{pmatrix}$, where the rows are cutting, sanding and assembly, and the columns are tables, chairs and stools. In one week the workshop makes $t$ tables, $c$ chairs and $s$ stools, using exactly $38$ hours of cutting, $30$ hours of sanding and $32$ hours of assembly.

   (a)

   (i) Write down a matrix equation that the numbers $t$, $c$ and $s$ must satisfy.
   (ii) Use your GDC to find $A^{-1}$.
   (iii) Hence find $t$, $c$ and $s$.

   [6 marks]

   (b)

   The selling prices are 150 AED for a table, 110 AED for a chair and 80 AED for a stool, given by the row matrix $P=\begin{pmatrix}150&110&80\end{pmatrix}$.
   (i) Use matrix multiplication and your answer to part (a) to find the revenue for the week.
   (ii) The following week the hours used are $39$ for cutting, $29$ for sanding and $33$ for assembly. Use your GDC to find how many tables, chairs and stools are made.
   (iii) Find the change in revenue compared with the first week.

   [6 marks]

   Total for question 4: 12 marks

5. 5
   Let $a=16^{\frac34}$ and $b=125^{-\frac23}$.

   (a)

   Find the value of $a$.

   [1 mark]

   • A$12$
   • B$64$
   • C$2$
   • D$8$

   (b)

   Find the value of $ab$.

   [1 mark]

   • A$\frac{8}{25}$
   • B$200$
   • C$\frac{8}{5}$
   • D$\frac{8}{125}$

   (c)

   Solve $x^{\frac32}=a$.

   [2 marks]

   Total for question 5: 4 marks

6. 6
   Consider the equation $2x^{2}-4x+5=0$, where $x\in\mathbb{C}$.

   (a)

   Find the value of the discriminant $b^{2}-4ac$.

   [1 mark]

   • A$56$
   • B$24$
   • C$-24$
   • D$-4$

   (b)

   Which of the following gives the solutions of the equation?

   [1 mark]

   • A$2\pm\sqrt{6}\,i$
   • B$1\pm\frac{\sqrt6}{2}i$
   • C$1\pm\sqrt{6}\,i$
   • D$-1\pm\frac{\sqrt6}{2}i$

   (c)

   Find the modulus of one of the solutions, giving your answer to 3 significant figures.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   Let $u=1+\sqrt3\,i$ and $v=\sqrt2-\sqrt2\,i$.

   (a)

   Write $u$ and $v$ in the form $re^{i\theta}$, where $r>0$ and $-\pi<\theta\leq\pi$.

   [3 marks]

   (b)

   Find $u^{3}v^{2}$, giving your answer in the form $re^{i\theta}$ and then in the form $a+bi$.

   [4 marks]

   Total for question 7: 7 marks

8. 8
   A colony of beetles has $J_n$ juveniles and $A_n$ adults at the start of year $n$. Each adult produces $3$ juveniles per year, half of the juveniles survive to become adults, and half of the adults survive. Hence $\begin{pmatrix}J_{n+1}\\ A_{n+1}\end{pmatrix}=L\begin{pmatrix}J_n\\ A_n\end{pmatrix}$, where $L=\begin{pmatrix}0&3\\ 0.5&0.5\end{pmatrix}$.

   (a)

   Find the eigenvalues of $L$ and a corresponding eigenvector for each eigenvalue.

   [6 marks]

   (b)

   Initially there are $100$ juveniles and $100$ adults. Given that $L=PDP^{-1}$, where $P=\begin{pmatrix}2&-3\\ 1&1\end{pmatrix}$ and $D=\begin{pmatrix}1.5&0\\ 0&-1\end{pmatrix}$:
   (i) find $P^{-1}$;
   (ii) hence find an expression for $A_n$ in terms of $n$;
   (iii) find the ratio $J_n:A_n$ as $n\to\infty$.

   [6 marks]

   Total for question 8: 12 marks

End of questions