Past Exam Questions

(a) The first, second and third terms of an arithmetic progression are \(4k\), \(k^2\) and \(8k\) respectively, where \(k\) is a non-zero constant.

1. Find the value of \(k\).
2. Find the sum of the first 20 terms of the progression.

(b) The fourth and sixth terms of a geometric progression are 36 and 6 respectively. The common ratio of the progression is positive.

Find the sum to infinity of the progression. Give your answer in the form \(\frac{a}{\sqrt{b} - c}\), where \(a\), \(b\) and \(c\) are integers.

(a) Express \(x^2 + 4x + 2\) in the form \((x+a)^2 + b\), where \(a\) and \(b\) are integers.

The functions \(f\) and \(g\) are defined as follows.

\(f(x) = x^2 + 4x + 2\) for \(x \leq -2\)

\(g(x) = -x - 4\) for \(x \geq -2\)

(b) (i) Find an expression for \(f^{-1}(x)\).

(ii) Find an expression for \((gf)^{-1}(x)\).

Find the coordinates of the points of intersection of the curve and the line with equations

\(2xy + 5y^2 = 24\) and \(2x + y + 4 = 0\).

The coefficient of \(x^7\) in the expansion of \(\left( px^2 + \frac{4}{p}x \right)^5\) is 1280.

Find the value of the constant \(p\).

A point P is moving along the curve with equation \(y = ax^{\frac{3}{2}} - 12x\) in such a way that the x-coordinate of P is increasing at a constant rate of 5 units per second.

(a) Find the rate at which the y-coordinate of P is changing when \(x = 9\). Give your answer in terms of the constant \(a\).

(b) Given that the curve has a minimum point when \(x = \frac{1}{4}\), find the value of \(a\).

The equation of a curve is \(y = 4 \cos 2x + 3\) for \(0 \leq x \leq 2\pi\).

1. State the greatest and least possible values of \(y\).
2. Sketch the curve.
3. Hence determine the number of solutions of the equation \(4 \cos 2x + 3 = 2x - 1\) for \(0 \leq x \leq 2\pi\).

The diagram shows the curve with equation \(y = \frac{9}{(5x+4)^{\frac{1}{2}}}\) and the line \(y = 6 - 3x\). The line and the curve intersect at the point \(P\) which has y-coordinate 3.

Find the area of the shaded region.

(a) Prove the identity \(\frac{\tan \theta + 7}{\tan^2 \theta - 3} \equiv \frac{\sin \theta \cos \theta + 7 \cos^2 \theta}{1 - 4 \cos^2 \theta}\).

(b) Hence solve the equation \(\frac{\sin \theta \cos \theta + 7 \cos^2 \theta}{1 - 4 \cos^2 \theta} = \frac{5}{\tan \theta}\) for \(0^\circ \leq \theta \leq 180^\circ\).

The diagram shows the circle with equation \(x^2 + y^2 - 14x + 8y + 36 = 0\) and the line \(y = -2\). The line intersects the circle at the points \(A\) and \(B\). The centre of the circle is \(C\).

(a) Find the coordinates of \(A\), \(B\) and \(C\).

(b) Find the angle \(ACB\) in radians. Give your answer correct to 3 significant figures.

(c) The chord \(AB\) divides the circle into two segments. Find the area of the larger segment.

The equation of a curve is such that \(\frac{d^2y}{dx^2} = -\frac{24}{x^3}\). It is given that the curve has a stationary point at \((-2, 19)\).

(a) Find an expression for \(\frac{dy}{dx}\).

(b) Find the \(x\)-coordinate of the other stationary point of the curve, and determine the nature of this stationary point.

(c) Find the equation of the curve.

(d) Find the equation of the normal to the curve at the point where \(\frac{dy}{dx} = -\frac{9}{4}\) and \(x\) is positive. Express your answer in the form \(px + qy + r = 0\), where \(p, q\) and \(r\) are integers.

The equation of a curve is such that \( \frac{dy}{dx}=2x-6x^{\frac12} \). The curve passes through the point \( (4,-9) \).

Find the equation of the curve.

(a) Describe fully a sequence of two transformations which transforms the graph of \( y=f(x) \) to the graph of \( y=f(4-x) \).

(b) The curve with equation \( y=x^3-3x-4 \) is stretched with scale factor \( \frac12 \) in the \(x\)-direction and then translated by \( \begin{pmatrix}0\\-3\end{pmatrix} \).

Find and simplify the equation of the transformed curve.

The equation of a curve is \( y=kx^2-5x-6 \), and the equation of a line is \( y=3x-7k \).

Find the set of values of the constant \(k\) for which the line intersects the curve.

The coefficient of \(x^2\) in the expansion of \( (2-qx)^4-\left(1+\frac8q x\right)^6 \) is \(324\).

Find the possible values of the constant \(q\).

(a) Prove the identity

\( \frac{1-\sin\theta}{\cos\theta}+\frac{\cos\theta}{1-\sin\theta}=\frac2{\cos\theta} \).

(b) Hence, solve the equation

\( \frac{1-\sin\theta}{\cos\theta}+\frac{\cos\theta}{1-\sin\theta}=\frac{\tan^3\theta}{\sin\theta} \)

for \(0^\circ\leq \theta \leq 360^\circ\).

The coordinates of three points, \(P\), \(Q\) and \(R\), are \( (0,p) \), \( (8,6) \) and \( (r,10) \) respectively, where \(p\) and \(r\) are constants. It is given that \(PQ\) is perpendicular to \(QR\).

(a) Show that \(p=2r-10\).

It is further given that the length of \(PR\) is \( \sqrt{85} \).

(b) Find the possible values of \(p\) and \(r\).

A curve has equation

\( y=3x^{-\frac12}-2x^{-\frac32} \).

The curve has a single stationary point when \(x=k\), where \(k>0\).

(a) Find the value of \(k\).

(b) Find \( \frac{d^2y}{dx^2} \), and hence determine whether the stationary point is a maximum or a minimum.

(c) Find the area enclosed by the curve, the \(x\)-axis and the lines \(x=k\) and \(x=4\). Give your answer in the form \(a+b\sqrt c\), where \(a\), \(b\) and \(c\) are integers to be found.

An arithmetic progression has first term \(20\) and common difference \(d\). A geometric progression also has first term \(20\) and common ratio \(r\), where \(r>0\).

The third term of the geometric progression is \(5\) more than the third term of the arithmetic progression. The fifth term of the geometric progression is \(30\) more than the fifth term of the arithmetic progression.

(a) Find the value of \(r\) and the value of \(d\).

(b) Show that the ninth term of the geometric progression is \(4\) times the ninth term of the arithmetic progression.

The diagram shows a triangle \(ACD\) in which \(AD\) is perpendicular to \(CD\). The arc \(BE\) of a circle with centre \(A\) and radius \(5\) cm meets \(AC\) at \(B\) and \(AD\) at \(E\). Angle \(BAE\) is \( \frac16\pi \) radians and the length \(BC=p\) cm.

(a) Given that the value of \(p\) is \(4\), find the exact perimeter of the shaded region. Give your answer in terms of \( \pi \) and \( \sqrt3 \).

(b) Given instead that the area of the shaded region is \( 8\sqrt3-\frac{25}{12}\pi \text{ cm}^2 \), find the value of \(p\).

Functions \(f\) and \(g\) are defined as follows:

\( f(x)=3x-6 \quad \text{for } x>0, \)

\( g(x)=\frac4{(ax-3)^2} \quad \text{for } x>\frac3a, \)

where \(a\) is a positive constant.

(a) State the range of \(f\).

(b) Find \(g^{-1}(x)\).

(c) Given that \(a=2\), solve the equation \(g^{-1}(x)=4\).

(d) Given instead that \(fg(8)=6\), find the value of \(a\), justifying your answer.