Mathematics 9709

Determine the coefficients of $x^4$ and $x^5$ in the expansion of $(1 - 2x)^5$.

Series · Feb/March 2016

The diagram shows a section of the curve $y = \frac{1}{16}(3x - 1)^2$, which is tangent to the $x$-axis at $P$. The point $Q(3, 4)$ lies on the curve, and the tangent at $Q$ meets the $x$-axis at $R$.

Coordinate geometry · Feb/March 2016

A curve whose gradient is given by $\frac{dy}{dx} = 3x^2 - \frac{2}{x^3}$ passes through $(-1, 3)$.

Integration · Feb/March 2016

In an arithmetic progression, the 12th term is $17$, and the total of the first $31$ terms is $1023$.

Series · Feb/March 2016

Solve the equation $\sin^{-1}(3x) = -\frac{1}{3}\pi$, and give the solution exactly.

Trigonometry · Feb/March 2016

The coordinates of two points are $A(5, 7)$ and $B(9, -1)$.

Coordinate geometry · Feb/March 2016

A vacuum flask, used to keep drinks hot, is represented by a closed cylinder whose internal radius is $r\,\text{cm}$ and internal height is $h\,\text{cm}$. The flask has volume $1000\,\text{cm}^3$. It is most efficient when the total internal surface area, $A\,\text{cm}^2$, is as small as possible.

Differentiation · Feb/March 2016

The diagram depicts a pyramid $OABC$ with a flat triangular base $OAB$ and perpendicular height $OC$. Angles $AOB$, $BOC$ and $AOC$ are each right angles. Unit vectors $\mathbf{i}$, $\mathbf{j}$ and $\mathbf{k}$ run parallel to $OA$, $OB$ and $OC$ respectively, and $OA = 4$ units, $OB = 2.4$ units and $OC = 3$ units. Point $P$ lies on $CA$ so that $CP = 3$ units.

Coordinate geometry · Feb/March 2016

The function $f$ is defined by $f(x) = a^2x^2 - ax + 3b$ for $x \leq \frac{1}{2a}$, with $a$ and $b$ as constants.

Functions · Feb/March 2016

In Fig. $1$, $OAB$ is a sector of a circle with centre $O$ and radius $r$. $AX$ is tangent to the arc $AB$ at $A$, and $\angle BAX = \alpha$. Show that $\angle AOB = 2\alpha$.

Circular measure · Feb/March 2016