Three of the following points lie on the same straight line.
Which point does not lie on this line?
Tick one box.
[1 mark]
\((-2, 14)\)
\((-1, 8)\)
\((1, -1)\)
\((2, -6)\)
A circle has equation \((x - 2)^2 + (y + 3)^2 = 13\)
Find the gradient of the tangent to this circle at the origin.
Circle your answer.
[1 mark]
\(-\frac{3}{2}\) \quad \(-\frac{2}{3}\) \quad \(\frac{2}{3}\) \quad \(\frac{3}{2}\)
Point \(C\) has coordinates \((c, 2)\) and point \(D\) has coordinates \((6, d)\).
The line \(y + 4x = 11\) is the perpendicular bisector of \(CD\).
Find \(c\) and \(d\).
[5 marks]
\(ABC\) is a right-angled triangle.
\includegraphics{figure_6}
\(D\) is the point on hypotenuse \(AC\) such that \(AD = AB\).
The area of \(\triangle ABD\) is equal to half that of \(\triangle ABC\).
Show that \(\tan A = 2 \sin A\)
[4 marks]
Show that the equation given in part (a) has two solutions for \(0° \leq A \leq 90°\)
[2 marks]
State the solution which is appropriate in this context.
[1 mark]
Maxine measures the pressure, \(P\) kilopascals, and the volume, \(V\) litres, in a fixed quantity of gas.
Maxine believes that the pressure and volume are connected by the equation
$$P = cV^d$$
where \(c\) and \(d\) are constants.
Using four experimental results, Maxine plots \(\log_{10} P\) against \(\log_{10} V\), as shown in the graph below.
\includegraphics{figure_8}
Find the value of \(P\) and the value of \(V\) for the data point labelled \(A\) on the graph.
[2 marks]
Calculate the value of each of the constants \(c\) and \(d\).
[4 marks]
Estimate the pressure of the gas when the volume is \(2\) litres.
[2 marks]
Craig is investigating the gradient of chords of the curve with equation \(\mathrm{f}(x) = x - x^2\)
Each chord joins the point \((3, -6)\) to the point \((3 + h, \mathrm{f}(3 + h))\)
The table shows some of Craig's results.
\(x\)
\(\mathrm{f}(x)\)
\(h\)
\(x + h\)
\(\mathrm{f}(x + h)\)
Gradient
\(3\)
\(-6\)
\(1\)
\(4\)
\(-12\)
\(-6\)
\(3\)
\(-6\)
\(0.1\)
\(3.1\)
\(-6.51\)
\(-5.1\)
\(3\)
\(-6\)
\(0.01\)
\(3\)
\(-6\)
\(0.001\)
\(3\)
\(-6\)
\(0.0001\)
Show how the value \(-5.1\) has been calculated.
[1 mark]
Complete the third row of the table above.
[2 marks]
State the limit suggested by Craig's investigation for the gradient of these chords as \(h\) tends to \(0\)
[1 mark]
Using differentiation from first principles, verify that your result in part (c) is correct.
[4 marks]
In this question use \(g = 9.8\,\mathrm{m}\,\mathrm{s}^{-2}\)
A ball, initially at rest, is dropped from a height of \(40\,\mathrm{m}\) above the ground.
Calculate the speed of the ball when it reaches the ground.
Circle your answer.
[1 mark]
\(-28\,\mathrm{m}\,\mathrm{s}^{-1}\) \quad \(28\,\mathrm{m}\,\mathrm{s}^{-1}\) \quad \(-780\,\mathrm{m}\,\mathrm{s}^{-1}\) \quad \(780\,\mathrm{m}\,\mathrm{s}^{-1}\)
An object of mass \(5\,\mathrm{kg}\) is moving in a straight line.
As a result of experiencing a forward force of \(F\) newtons and a resistant force of \(R\) newtons it accelerates at \(0.6\,\mathrm{m}\,\mathrm{s}^{-2}\)
Which one of the following equations is correct?
Circle your answer.
[1 mark]
\(F - R = 0\) \quad \(F - R = 5\) \quad \(F - R = 3\) \quad \(F - R = 0.6\)
A vehicle, which begins at rest at point \(P\), is travelling in a straight line.
For the first \(4\) seconds the vehicle moves with a constant acceleration of \(0.75\,\mathrm{m}\,\mathrm{s}^{-2}\)
For the next \(5\) seconds the vehicle moves with a constant acceleration of \(-1.2\,\mathrm{m}\,\mathrm{s}^{-2}\)
The vehicle then immediately stops accelerating, and travels a further \(33\,\mathrm{m}\) at constant speed.
Draw a velocity-time graph for this journey on the grid below.
[3 marks]
\includegraphics{figure_13}
Find the distance of the car from \(P\) after \(20\) seconds.
[3 marks]
In this question use \(g = 9.81\,\mathrm{m}\,\mathrm{s}^{-2}\)
Two particles, of mass \(1.8\,\mathrm{kg}\) and \(1.2\,\mathrm{kg}\), are connected by a light, inextensible string over a smooth peg.
\includegraphics{figure_14}
Initially the particles are held at rest \(1.5\,\mathrm{m}\) above horizontal ground and the string between them is taut.
The particles are released from rest.
Find the time taken for the \(1.8\,\mathrm{kg}\) particle to reach the ground.
[5 marks]
State one assumption you have made in answering part (a).
[1 mark]
A cyclist, Laura, is travelling in a straight line on a horizontal road at a constant speed of \(25\,\mathrm{km}\,\mathrm{h}^{-1}\)
A second cyclist, Jason, is riding closely and directly behind Laura. He is also moving with a constant speed of \(25\,\mathrm{km}\,\mathrm{h}^{-1}\)
The driving force applied by Jason is likely to be less than the driving force applied by Laura.
Explain why.
[1 mark]
Jason has a problem and stops, but Laura continues at the same constant speed.
Laura sees an accident \(40\,\mathrm{m}\) ahead, so she stops pedalling and applies the brakes.
She experiences a total resistance force of \(40\,\mathrm{N}\)
Laura and her cycle have a combined mass of \(64\,\mathrm{kg}\)
Determine whether Laura stops before reaching the accident.
Fully justify your answer.
[4 marks]
State one assumption you have made that could affect your answer to part (b)(i).
[1 mark]
A remote-controlled toy car is moving over a horizontal surface. It moves in a straight line through a point \(A\).
The toy is initially at the point with displacement \(3\) metres from \(A\). Its velocity, \(v\,\mathrm{m}\,\mathrm{s}^{-1}\), at time \(t\) seconds is defined by
$$v = 0.06(2 + t - t^2)$$
Find an expression for the displacement, \(r\) metres, of the toy from \(A\) at time \(t\) seconds.
[4 marks]
In this question use \(g = 9.8\,\mathrm{m}\,\mathrm{s}^{-2}\)
At time \(t = 2\) seconds, the toy launches a ball which travels directly upwards with initial speed \(3.43\,\mathrm{m}\,\mathrm{s}^{-1}\)
Find the time taken for the ball to reach its highest point.
[3 marks]