Past paper questions

Let $a$ be a constant and $-90^\circ < \theta < 90^\circ$. If the figure shows the graph of $y = a\sin(x^\circ + \theta)$, then

The figure shows a right prism $ABCDEF$ with a right-angled triangle as the cross-section. A, B, E and F lie on the horizontal ground. G is a point lying on AB such that $AG:GB=5:3$. If $\angle DAE=a$, $\angle CBF=b$, $\angle CGF=c$ and $\angle DGE=d$, which of the following is true?

In the figure, $A$ is the common centre of the two circles. $BC$ is a chord of the larger circle and touches the smaller circle at $D$. $AD$ produced meets the larger circle at $E$. $F$ is a point lying on the smaller circle such that $E$, $D$, $A$ and $F$ are collinear. If $BC = 24\text{ cm}$ and $DE = 8\text{ cm}$, then $EF =$

If the straight line $x - y = 0$ and the circle $x^{2}+y^{2}+6x+ky-k=0$ do not intersect with each other, find the range of values of $k$.

Let $O$ be the origin. If the coordinates of the points $A$ and $B$ are $(18,-24)$ and $(18,24)$ respectively, then the $x$-coordinate of the orthocentre of $\Delta OAB$ is

Mary, Tom and 8 other students participate in a solo singing contest. If each participant performs once only and the order of performance is randomly arranged, find the probability that Mary performs just after Tom.

The mean, the variance and the inter-quartile range of a set of numbers are $40$, $9$ and $18$ respectively. If $5$ is added to each number of the set and each resulting number is then tripled to form a new set of numbers, find the mean, the variance and the inter-quartile range of the new set of numbers.

Let $A$ be a group of numbers $\{\alpha, \beta, \gamma, \delta\}$ and $B$ be another group of numbers $\{\alpha+2, \beta+2, \mu+2, \gamma+2, \delta+2\}$, where $\alpha < \beta < \mu < \gamma < \delta$. Which of the following must be true?

I. The median of $A$ is smaller than that of $B$.

II. The range of $A$ and the range of $B$ are the same.

III. The standard deviation of $A$ is greater than that of $B$.