Question 1 is a multiple choice question for which marks are given solely for the correct answers. Answer Question 1 on the grid on Page 2. Write your answers to Questions $2,3,4,5$ in the space provided, continuing on the back of this booklet if necessary.
THE USE OF CALCULATORS OR FORMULA SHEETS IS PROHIBITED.
- For each part of the question on pages $3 - 7$ you will be given four possible answers, just one of which is correct. Indicate for each part $\mathrm { A } - \mathrm { J }$ which answer (a), (b), (c), or (d) you think is correct with a tick ( ✓ ) in the corresponding column in the table below. You may use the spaces between the parts for rough working.
A. Depending on the value of the constant $d$, the equation
$$d x ^ { 2 } - ( d - 1 ) x + d = 0$$
may have two real solutions, one real solution or no real solutions. For how many values of $d$ does it have just one real solution?
(a) for one value of $d$;
(b) for two values of $d$;
(c) for three values of $d$;
(d) for infinitely many values of $d$.
B. You are given that $\mathrm { e } ^ { 3 }$ is approximately 20 and that $2 ^ { 10 }$ is approximately 1000 . Using this information, a student can obtain an approximate value for $\ln 2$. Which of the following is it?
(a) $\frac { 7 } { 10 }$
(b) $\frac { 9 } { 13 }$
(c) $\frac { 38 } { 55 }$
(d) $\frac { 41 } { 59 }$
C. How many solutions does the equation
$$\sin 2 x = \cos x$$
have in the range $0 \leq x \leq \pi$ ?
(a) one solution;
(b) two solutions;
(c) three solutions;
(d) four solutions.
D. What is the value of the definite integral
$$\int _ { 1 } ^ { 2 } \frac { \mathrm {~d} x } { x + x ^ { 3 } } ?$$
(a) $\quad \ln 2 - \pi / 6$
(b) $2 \ln 2 - \ln 5$
(c) $\frac { 1 } { 2 } \ln \frac { 8 } { 5 }$
(d) None of the above.
E. For which real numbers $x$ does the inequality
$$\frac { x } { x ^ { 2 } + 1 } \leq \frac { 1 } { 2 }$$
hold?
(a) for all real numbers $x$;
(b) for real numbers $x \leq \frac { 1 } { 2 }$ and no others;
(c) for real numbers $x \leq 1$ and no others;
(d) none of the above. F. Two players take turns to throw a fair six-sided die until one of them scores a six. What is the probability that the first player to throw the die is the first to score a six?
(a) $\frac { 5 } { 9 }$
(b) $\frac { 3 } { 5 }$
(c) $\frac { 6 } { 11 }$
(d) $\frac { 7 } { 12 }$ G. For which of the following do we have
$$\frac { \mathrm { d } y } { \mathrm {~d} x } = 2 y \ln y ?$$
(a) $y = \mathrm { e } ^ { \mathrm { e } ^ { 2 x } }$
(b) $y = e ^ { 2 e ^ { x } }$
(c) $y = e ^ { e ^ { x ^ { 2 } } }$
(d) $y = 2 e ^ { \mathrm { e } ^ { x } }$ H. Into how many regions is the plane divided when the following three parabolas are drawn?
$$\begin{aligned}
& y = x ^ { 2 } \\
& y = x ^ { 2 } - 2 x \\
& y = x ^ { 2 } + 2 x + 2
\end{aligned}$$
(a) 4
(b) 5
(c) 6
(d) 7 I. You go into a supermarket to buy two packets of biscuits, which may or may not be of the same variety. The supermarket has 20 different varieties of biscuits and at least two packets of each variety. In how many ways can you choose your two packets?
(a) 400
(b) 210
(c) 200
(d) 190 J. There are real numbers $x , y$ such that precisely one of the statements (a), (b), (c), (d) is true. Which is the true statement?
(a) $x \geq 0$
(b) $x < y$
(c) $x ^ { 2 } > y ^ { 2 }$
(d) $| x | \leq | y |$