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Express \(\log_3 5 + \log_3 2\) as a single logarithm. Circle your answer.

Which single transformation maps \(y=\cos x\) onto \(y=\cos 2x\)? Circle your answer.

Find the coefficient of \(x^2\) in the binomial expansion of \((2+3x)^5\).

Solve \(2\sin^2 x = 3\cos x\) for \(0^\circ \le x \le 360^\circ\).

A circle has the points \(A(-1,2)\) and \(B(5,10)\) as the ends of a diameter. (a) Write down the centre. (b) Show that the radius is 5. (c) Find the equation of the circle. (d) Determine whether the point \((6,6)\) lies inside, on, or outside the circle.

The curve \(y=9-x^2\) meets the x-axis at two points. (a) Find the x-coordinates of these points. (b) Find the exact area enclosed between the curve and the x-axis.

A curve has equation \(y=2x^3-9x^2+12x\). (a) Find \(\dfrac{dy}{dx}\). (b) Find the coordinates of the stationary points. (c) Determine the nature of each stationary point.

Prove that, for every integer \(n\), \(n^3-n\) is divisible by 6.

A curve has equation \(y=x^2-4x+5\). (a) Find the equation of the tangent to the curve at \(x=3\). (b) Find the equation of the normal at \(x=3\). (c) The normal meets the curve again; find the coordinates of that point.

A population is modelled by \(P=500e^{kt}\), where \(t\) is in years. After 3 years the population is 800. (a) Find \(k\) to 3 significant figures. (b) Estimate the population after 10 years. (c) Find, to the nearest year, when the population reaches 2000.

A particle starts from rest and moves in a straight line with constant acceleration. Which graph best shows its velocity \(v\) against time \(t\)? Circle your answer.

A ball is projected vertically upwards from ground level at \(21\,\text{m s}^{-1}\) (take \(g=9.8\)). (a) Find the maximum height reached. (b) Find the total time before it returns to the ground.

A particle of mass \(2\,\text{kg}\) is acted on by forces \(\mathbf{F_1}=(3\mathbf{i}+4\mathbf{j})\,\text{N}\) and \(\mathbf{F_2}=(5\mathbf{i}-2\mathbf{j})\,\text{N}\). (a) Find the resultant force. (b) Find the magnitude of the acceleration.

Two particles of masses \(3\,\text{kg}\) and \(5\,\text{kg}\) are connected by a light inextensible string passing over a smooth fixed pulley, and released from rest. (a) Find the acceleration of the system. (b) Find the tension in the string.

A particle moves in a straight line with velocity \(v=3t^2-12t+9\;\text{m s}^{-1}\). (a) Find the times when the particle is instantaneously at rest. (b) Find its acceleration when \(t=3\). (c) Find its displacement during the first 4 seconds.

A car of mass \(1200\,\text{kg}\) tows a trailer of mass \(400\,\text{kg}\) by a light rigid tow-bar along a straight horizontal road. The driving force is \(4000\,\text{N}\); the resistances are \(450\,\text{N}\) on the car and \(150\,\text{N}\) on the trailer. (a) Find the acceleration. (b) Find the tension in the tow-bar.

Find \(\displaystyle\int 6x^2\,dx\). Circle your answer.

Given that \(\sin\theta=\cos 50^\circ\), a possible value of \(\theta\) is: circle your answer.

\(f(x)=2x^3+x^2-13x+6\). (a) Show that \((x-2)\) is a factor of \(f(x)\). (b) Factorise \(f(x)\) completely.

The equation \(x^2+(k+2)x+9=0\) has equal roots. Find the possible values of \(k\).

A bottle of water cools according to \(T=20+Ae^{-kt}\), where \(T\,^\circ\text{C}\) is the temperature \(t\) minutes after it leaves a fridge. Initially \(T=80\), and after 5 minutes \(T=50\). (a) Find \(A\). (b) Find \(k\) to 3 significant figures. (c) Find the temperature after 12 minutes. (d) Find, to the nearest minute, the time taken to reach \(30^\circ\text{C}\).

In triangle \(ABC\), \(AB=7\,\text{cm}\), \(BC=8\,\text{cm}\) and angle \(ABC=60^\circ\). (a) Find the length \(AC\). (b) Find the area of the triangle. (c) Find angle \(BAC\) to 1 decimal place.

The line \(y=x+1\) and the curve \(y=x^2-2x-3\) intersect at two points. (a) Find the coordinates of the points of intersection. (b) Write down the inequalities that define the region enclosed between the line and the curve. (c) Find the exact area of the enclosed region.

A curve has equation \(y=2x^3-3x^2-12x\). (a) Find \(\dfrac{dy}{dx}\). (b) Find the coordinates of the stationary points. (c) Determine the nature of each.

A geometric series has second term 6 and sum to infinity 27. (a) Show that the common ratio satisfies \(9r^2-9r+2=0\). (b) Find the two possible values of \(r\). (c) For \(r=\tfrac13\), find the first term and the sum of the first 5 terms.

Express \(\dfrac{5}{2-\sqrt3}\) in the form \(a+b\sqrt3\), where \(a\) and \(b\) are integers.

A school has 40 classes of 30 students. A researcher selects 3 whole classes at random and surveys every student in them. This sampling method is: circle your answer.

On a box plot the median is much closer to the lower quartile than to the upper quartile. The distribution is best described as: circle your answer.

The times (in minutes) for six visits are: \(12, 15, 18, 11, 14, 20\). (a) Find the mean. (b) Find the standard deviation to 2 decimal places.

On each gym visit a member independently chooses an apple, banana or cake, with \(P(A)=0.3\), \(P(B)=0.45\), \(P(C)=0.25\). For four randomly chosen visits, find the probability that the member chose: (a) at least one apple; (b) the same item on all four visits; (c) apple exactly twice and cake exactly twice.

The discrete random variable \(X\) has \(P(X=x)=cx\) for \(x=1,2,3,4\) and 0 otherwise. (a) Find \(c\). (b) Find \(E(X)\). (c) Find \(P(X\ge3)\).

A coin is suspected of being biased towards heads. It is tossed 20 times and 15 heads are obtained. Test, at the 5% significance level, whether there is evidence that the probability of a head exceeds 0.5.