1. 1
   The population of a country is $9.6\times10^{7}$. The country has $3.2\times10^{2}$ hospitals, spread evenly across a land area of $8\times10^{5}$ km$^2$.

   (a)

   Find the average number of people per hospital, giving your answer in the form $a\times10^k$ where $1\le a<10$.

   [1 mark]

   • A$3\times10^{5}$
   • B$6.4\times10^{5}$
   • C$3\times10^{9}$
   • D$3\times10^{-5}$

   (b)

   Each hospital treats on average $4.5\times10^{2}$ patients per day. Find the total number of patients treated per day across the country, in the form $a\times10^k$.

   [1 mark]

   • A$7.7\times10^{2}$
   • B$1.44\times10^{5}$
   • C$14.4\times10^{4}$
   • D$1.44\times10^{6}$

   (c)

   Find the population density of the country, in people per km$^2$, giving your answer in the form $a\times10^k$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   Let $a = 8^{\frac{2}{3}}$ and $b = 9^{-\frac{1}{2}}$.

   (a)

   Find the value of $a$.

   [1 mark]

   • A$64$
   • B$\frac{1}{4}$
   • C$4$
   • D$\frac{16}{3}$

   (b)

   Find the value of $b$.

   [1 mark]

   • A$3$
   • B$-3$
   • C$\frac{1}{81}$
   • D$\frac{1}{3}$

   (c)

   A calculator may be used. Find the value of $ab$ as a fraction in its simplest form, and hence find $\log_3(ab)$, correct to 3 significant figures.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The annual membership fee charged by a running club to a member depends on the year in which they joined, and the fees for successive joining years form an arithmetic sequence. A member who joined in year 1 paid 120 AED. A member who joined in year 4 paid 180 AED.

   (a)

   Find the common difference $d$ of the sequence, and hence find the fee paid by a member who joins in year 10.

   [3 marks]

   (b)

   Find the first year in which the membership fee exceeds 500 AED, and find the total fees paid by members joining in year 1 up to and including that year.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A company's annual revenue increases geometrically each year. In its first year of trading the company earned 40\,000 AED. In its fourth year it earned 69\,120 AED.

   (a)

   Find the common ratio $r$ of the sequence of annual revenues. Hence find the revenue in the company's 8th year of trading, and the total revenue earned over its first 8 years, each correct to the nearest AED.

   [6 marks]

   (b)

   The company deposits the entire year 8 revenue found in part (a) into a savings account paying a nominal annual interest rate of 5%, compounded quarterly, for 6 years, with no further deposits or withdrawals. Find the value of the deposit after 6 years and the total interest earned, each correct to the nearest AED, and determine whether this compound interest is greater than the simple interest that the same nominal rate of 5% per year would have earned over the same 6 years.

   [6 marks]

   Total for question 4: 12 marks

5. 5
   Consider the expansion of $(2x - 1)^5$ in ascending powers of $x$.

   (a)

   Find the coefficient of $x^2$ in the expansion.

   [1 mark]

   • A$-40$
   • B$40$
   • C$-10$
   • D$-80$

   (b)

   Find the constant term in the expansion.

   [1 mark]

   • A$32$
   • B$-1$
   • C$1$
   • D$-5$

   (c)

   Find the sum of all the coefficients in the expansion of $(2x-1)^5$.

   [2 marks]

   Total for question 5: 4 marks

6. 6
   An infinite geometric series has first term 18 and common ratio $r = \frac{2}{3}$.

   (a)

   Find the sum to infinity of the series.

   [1 mark]

   • A$36$
   • B$10.8$
   • C$54$
   • D$-54$

   (b)

   Find the value of the second term of the sequence.

   [1 mark]

   • A$27$
   • B$8$
   • C$20$
   • D$12$

   (c)

   Find the least value of $n$ for which the sum to infinity exceeds the sum of the first $n$ terms, $S_n$, by less than 0.01.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   Let $p = \log_2 5$ and $q = \log_2 3$.

   (a)

   Write $\log_2 45$ in terms of $p$ and $q$.

   [3 marks]

   (b)

   Solve the equation $2^{x+1} = 5^{x-2}$, giving your answer in terms of $p$, correct to 3 significant figures.

   [4 marks]

   Total for question 7: 7 marks

8. 8
   Let $n$ be a positive integer, and consider the three consecutive integers $n$, $n+1$ and $n+2$.

   (a)

   Prove that the sum of the squares of these three consecutive integers is $3n^2 + 6n + 5$, and hence show that this sum is never a multiple of 3.

   [6 marks]

   (b)

   By using the binomial theorem to expand $(n+1)^4$, find the coefficient of $n$ in this expansion. Hence, given that the coefficient of $n^3$ in the expansion of $(kn+1)^4$ is 32, find the value of the constant $k$.

   [6 marks]

   Total for question 8: 12 marks

End of questions