1. 1
   Consider the expansion of $(3x - 2)^5$ in ascending powers of $x$.

   (a)

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

   [1 mark]

   • A$720$
   • B$-720$
   • C$-240$
   • D$1080$

   (b)

   Find the coefficient of $x^4$ in the expansion.

   [1 mark]

   • A$810$
   • B$-162$
   • C$-810$
   • D$-30$

   (c)

   Hence write down the sum of all the coefficients in the expansion of $(3x-2)^5$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   An arithmetic sequence has first term $u_1=7$ and common difference $d=4$.

   (a)

   Find $u_{15}$.

   [1 mark]

   • A$67$
   • B$56$
   • C$59$
   • D$63$

   (b)

   Given that $u_n=91$, find the value of $n$.

   [1 mark]

   • A$22$
   • B$21$
   • C$29$
   • D$84$

   (c)

   Hence find the sum of the first 22 terms of the sequence.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   At the beginning of each year, Layla deposits 2000 AED into a savings account that pays 5\% annual compound interest, added at the end of each year. She makes deposits at the start of years 1, 2, 3, $\ldots$, 10, and makes no withdrawals.

   (a)

   Find the value, to the nearest AED, of the deposit made at the start of year 1 by the end of year 10 (that is, after it has earned interest for 10 years).

   [3 marks]

   (b)

   Show that the total value of Layla's savings at the end of year 10 is given by the geometric series $2000\left(1.05+1.05^2+\cdots+1.05^{10}\right)$, and find this total value to the nearest AED.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The complex numbers $z$ and $w$ are given by $z = 2 - 5\mathrm{i}$ and $w = -3 + \mathrm{i}$, where $\mathrm{i}^2=-1$.

   (a)

   (i) Find $zw$ in the form $a+b\mathrm{i}$. [2]
   (ii) Find $\dfrac{z}{w}$ in the form $a+b\mathrm{i}$, showing your working. [3]
   (iii) Write down the value of $z\bar z$. [1]

   [6 marks]

   (b)

   It is given that $2+3\mathrm{i}$ is a root of the cubic equation $z^3+az^2+bz+c=0$, where $a,b,c\in\mathbb{R}$, and that $c=-26$. Find the real root of the equation and the values of $a$ and $b$.

   [6 marks]

   Total for question 4: 12 marks

5. 5
   Consider the equation $2^{3x-1}=5^{x+2}$.

   (a)

   Taking $\log$ of both sides, which equation is equivalent to $2^{3x-1}=5^{x+2}$?

   [1 mark]

   • A$(3x-1)\log5=(x+2)\log2$
   • B$(3x-1)\log2=(x+2)\log5$
   • C$3x\log2-1=x\log5+2$
   • D$\log(3x-1)\cdot2=\log(x+2)\cdot5$

   (b)

   Find the value of $\log_2 40-\log_2 5$.

   [1 mark]

   • A$8$
   • B$\log_2 35$
   • C$3$
   • D$\log_2 200$

   (c)

   Hence, or otherwise, find the exact value of $x$ satisfying $2^{3x-1}=5^{x+2}$, giving your answer in terms of natural logarithms.

   [2 marks]

   Total for question 5: 4 marks

6. 6
   A committee of 5 people is to be chosen from a group of 8 women and 6 men.

   (a)

   In how many ways can the committee be chosen if there are no restrictions?

   [1 mark]

   • A$2002$
   • B$537824$
   • C$240240$
   • D$62$

   (b)

   In how many ways can the committee be chosen if it must contain exactly 3 women and 2 men?

   [1 mark]

   • A$71$
   • B$560$
   • C$2002$
   • D$840$

   (c)

   Hence find the number of ways the committee can be chosen if it must contain at least 4 women.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   Let $f(x) = \dfrac{11x+5}{(x-1)(x+3)}$, for $x\in\mathbb{R}$, $x\neq1$, $x\neq-3$.

   (a)

   Express $f(x)$ in partial fractions of the form $\dfrac{A}{x-1}+\dfrac{B}{x+3}$, where $A,B\in\mathbb{Z}$.

   [3 marks]

   (b)

   Hence, by expressing $\dfrac{4}{x-1}$ and $\dfrac{7}{x+3}$ each as an infinite geometric series in ascending powers of $x$ (valid for $|x|<1$), find the coefficient of $x^2$ in the series expansion of $f(x)$.

   [4 marks]

   Total for question 7: 7 marks

8. 8
   Consider the system of equations $x+2y-z=1$, $2x-y+3z=k$, and $3x+y+2z=7$, where $k\in\mathbb{R}$.

   (a)

   (i) Show that the coefficient matrix of the system is singular. [3]
   (ii) Hence, using $k=8$, determine whether the system is inconsistent or has infinitely many solutions, justifying your answer. [3]

   [6 marks]

   (b)

   Find the value of $k$ for which the system has infinitely many solutions, and for this value of $k$, find the general solution in terms of a parameter $t$.

   [6 marks]

   Total for question 8: 12 marks

End of questions