(a) The straight line in the $( x , y )$ plane through the points $( - 1,3 )$ and $( 2,1 )$ is defined by the equation\ (i) $3 x + 2 y = 3$,\ (ii) $- x + 3 y = 1$,\ (iii) $2 x + 3 y = 7$,\ (iv) $x + 2 y = 5$.\ (b) There is a solution to the equation $x ^ { 3 } + x ^ { 2 } + 3 = 0$ between\ (i) - 2 and - 1 ,\ (ii) - 1 and 0 ,\ (iii) 0 and 1 ,\ (iv) 1 and 2 .\ (c) Anne, Bert, Clare, Derek and Emily are planning to play a game for which they need to divide themselves into three teams. Each team must have at least one member. The number of different ways they can do this is\ (i) 10 ,\ (ii) 15 ,\ (iii) 25 ,\ (iv) 30 .\ (d) For the following statements $$P : \frac { x ( x - 2 ) } { 1 - x } > 0 , \quad Q : 1 < x < 2$$ about a real number $x$,\ (i) $P$ implies $Q$, but $Q$ does not imply $P$,\ (ii) $Q$ implies $P$, but $P$ does not imply $Q$,\ (iii) $P$ implies $Q$, and $Q$ implies $P$,\ (iv) $P$ and $Q$ contradict each other.\ (e) The least and greatest values of $\cos ( \cos x )$ in the range $0 \leq x \leq \pi$ are\ (i) 0 and 1 ,\ (ii) - $\cos 1$ and 1 ,\ (iii) - 1 and 1 , (iv) $\cos 1$ and 1 .\ (f) As the integer $n$ becomes very large and positive, $$\frac { \sqrt { n } + ( - 1 ) ^ { n } } { \sqrt { n } }$$ (i) approaches (that is, converges to) 0 ,\ (ii) approaches (that is, converges to) 1 ,\ (iii) approaches infinity,\ (iv) oscillates, but does not converge.\ (g) The power of $x$ which has the greatest coefficient in the expansion of $\left( 1 + \frac { 1 } { 2 } x \right) ^ { 10 }$ is\ (i) $x ^ { 2 }$,\ (ii) $x ^ { 3 }$,\ (iii) $x ^ { 5 }$,\ (iv) $x ^ { 10 }$.\ (h) The (shaded) area under the graph of $y = f ( x )$ between $x = 1$ and $x = 2$ is given to be 1 . The area under the graph of $y = 2 f ( 3 - x )$ between $x = 1$ and $x = 2$ is therefore\ (i) 1 ,\ (ii) 2 ,\ (iii) 3 ,\ (iv) 6 .\ (j) In a plane there are given $n$ straight lines, no two of them parallel and no three of them meeting at a point. The number of parts they divide the plane into is\ (i) $n + 1$,\ (ii) $n ^ { 2 } - n + 2$,\ (iii) $\frac { 1 } { 2 } n ( n + 1 ) + 1$,\ (iv) $2 ^ { n }$.\ (k) The simultaneous equations $$\begin{aligned}
& x - 2 y + 3 z = 1 \\
& 2 x + 3 y - z = 4 \\
& 4 x - y + 5 z = 6
\end{aligned}$$ have\ (i) no solutions,\ (ii) exactly one solution,\ (iii) exactly three solutions,\ (iv) infinitely many solutions.
(a) The straight line in the $( x , y )$ plane through the points $( - 1,3 )$ and $( 2,1 )$ is defined by the equation\
(i) $3 x + 2 y = 3$,\
(ii) $- x + 3 y = 1$,\
(iii) $2 x + 3 y = 7$,\
(iv) $x + 2 y = 5$.\
(b) There is a solution to the equation $x ^ { 3 } + x ^ { 2 } + 3 = 0$ between\
(i) - 2 and - 1 ,\
(ii) - 1 and 0 ,\
(iii) 0 and 1 ,\
(iv) 1 and 2 .\
(c) Anne, Bert, Clare, Derek and Emily are planning to play a game for which they need to divide themselves into three teams. Each team must have at least one member. The number of different ways they can do this is\
(i) 10 ,\
(ii) 15 ,\
(iii) 25 ,\
(iv) 30 .\
(d) For the following statements
$$P : \frac { x ( x - 2 ) } { 1 - x } > 0 , \quad Q : 1 < x < 2$$
about a real number $x$,\
(i) $P$ implies $Q$, but $Q$ does not imply $P$,\
(ii) $Q$ implies $P$, but $P$ does not imply $Q$,\
(iii) $P$ implies $Q$, and $Q$ implies $P$,\
(iv) $P$ and $Q$ contradict each other.\
(e) The least and greatest values of $\cos ( \cos x )$ in the range $0 \leq x \leq \pi$ are\
(i) 0 and 1 ,\
(ii) - $\cos 1$ and 1 ,\
(iii) - 1 and 1 , (iv) $\cos 1$ and 1 .\
(f) As the integer $n$ becomes very large and positive,
$$\frac { \sqrt { n } + ( - 1 ) ^ { n } } { \sqrt { n } }$$
(i) approaches (that is, converges to) 0 ,\
(ii) approaches (that is, converges to) 1 ,\
(iii) approaches infinity,\
(iv) oscillates, but does not converge.\
(g) The power of $x$ which has the greatest coefficient in the expansion of $\left( 1 + \frac { 1 } { 2 } x \right) ^ { 10 }$ is\
(i) $x ^ { 2 }$,\
(ii) $x ^ { 3 }$,\
(iii) $x ^ { 5 }$,\
(iv) $x ^ { 10 }$.\
(h) The (shaded) area under the graph of $y = f ( x )$ between $x = 1$ and $x = 2$ is given to be 1 .
The area under the graph of $y = 2 f ( 3 - x )$ between $x = 1$ and $x = 2$ is therefore\
(i) 1 ,\
(ii) 2 ,\
(iii) 3 ,\
(iv) 6 .\
(j) In a plane there are given $n$ straight lines, no two of them parallel and no three of them meeting at a point. The number of parts they divide the plane into is\
(i) $n + 1$,\
(ii) $n ^ { 2 } - n + 2$,\
(iii) $\frac { 1 } { 2 } n ( n + 1 ) + 1$,\
(iv) $2 ^ { n }$.\
(k) The simultaneous equations
$$\begin{aligned}
& x - 2 y + 3 z = 1 \\
& 2 x + 3 y - z = 4 \\
& 4 x - y + 5 z = 6
\end{aligned}$$
have\
(i) no solutions,\
(ii) exactly one solution,\
(iii) exactly three solutions,\
(iv) infinitely many solutions.