Questions - A-Level Maths

OCR FP3 2016 June Q7

12 marks Challenging +1.2

Use de Moivre's theorem to show that $$\sin 6 \theta \equiv \cos \theta \left( 6 \sin \theta - 32 \sin ^ { 3 } \theta + 32 \sin ^ { 5 } \theta \right)$$
Hence show that, for $\sin 2 \theta \neq 0$, $$- 1 \leqslant \frac { \sin 6 \theta } { \sin 2 \theta } < 3$$

OCR D1 2006 June Q6

16 marks Standard +0.3

Calculate the shortest distance that the mole must travel if it starts and ends at vertex $A$.
The pipe connecting $B$ to $H$ is removed for repairs. By considering every possible pairing of odd vertices, and showing your working clearly, calculate the shortest distance that the mole must travel to pass along each pipe on this reduced network, starting and finishing at $A$.

OCR MEI D1 2006 January Q2

8 marks Easy -1.2

Complete the table in the insert showing the outcome of applying the algorithm to the following two lists: $$\begin{array} { l r l l l l l } \text { List 1: } & 2 , & 34 , & 35 , & 56 & & \\ \text { List 2: } & 13 , & 22 , & 34 , & 81 , & 90 , & 92 \end{array}$$
What does the algorithm achieve?
How many comparisons did you make in applying the algorithm?
If the number of elements in List 1 is $x$, and the number of elements in List 2 is $y$, what is the maximum number of comparisons that will have to be made in applying the algorithm, and what is the minimum number?

Edexcel D1 Q6

13 marks Moderate -0.3

Draw a bipartite graph to model this situation. Initially it is decided to run the Office application on computer $F$, Animation on computer $H$, and Data on computer $I$.
Starting from this matching, use the maximum matching algorithm to find a complete matching. Indicate clearly how the algorithm has been applied.
Computer $H$ is upgraded to allow it to run CAD. Find an alternative matching to that found in part (b).

OCR MEI D2 2014 June Q2

16 marks Easy -1.2

Rachel thinks that the answer given in the newspaper article is not sensible. Give a verbal argument why Rachel might think that the batsman should be given out. Rachel tries to formalise her argument. She defines four simple propositions.
o: "The batsman is given out."
lb: "The batsman is given out (LBW)."
c: "The batsman is given out (caught)."
b: "The ball hit the bat."
An implication of the batsman not being out (LBW) is that the ball has hit the bat. Write this down in terms of Rachel's propositions.
Similarly, write down the implication of the batsman not being out (caught).
Using your answers to parts (ii) and (iii) write down the implication of a batsman being not out, in terms of $b$ and $\sim b$.
[0pt] [You may assume that if $\mathrm { w } \Rightarrow \mathrm { y }$ and $\mathrm { x } \Rightarrow \mathrm { z }$, then $( \mathrm { w } \wedge \mathrm { x } ) \Rightarrow ( \mathrm { y } \wedge \mathrm { z } )$. ]
By writing down the contrapositive of your implication from part (iv), produce an implication which supports Rachel's argument.
(b) A classroom rule has been broken by either Anja, Bobby, Catherine or Dimitria, or by a subset of those four. The teacher knows that Dimitria could not have done it on her own. Let $a$ be the proposition "Anja is guilty", and similarly for $b , c$ and $d$.
Express the teacher's knowledge as a compound proposition. Evidence emerges that Bobby and Catherine were elsewhere at the time, so they cannot be guilty. This can be expressed as the compound proposition $\sim ( b \vee c )$.
Construct a truth table to show the truth values of the compound proposition given by the conjunction of the two compound propositions, one from part (i) and one given above.
What does your truth table tell you about who is guilty? 3 Three products, A, B and C are to be made.
Three supplements are included in each product. Product A has 10 g per kg of supplement $\mathrm { X } , 5 \mathrm {~g}$ per kg of supplement Y and 5 g per kg of supplement Z . Product B has 5 g per kg of supplement $\mathrm { X } , 5 \mathrm {~g}$ per kg of supplement Y and 3 g per kg of supplement Z .
Product C has 12 g per kg of supplement $\mathrm { X } , 7 \mathrm {~g}$ per kg of supplement Y and 5 g per kg of supplement Z .
There are 12 kg of supplement X available, 12 kg of supplement Y , and 9 kg of supplement Z .
Product A will sell at $\pounds 7$ per kg and costs $\pounds 3$ per kg to produce. Product B will sell at $\pounds 5$ per kg and costs $\pounds 2$ per kg to produce. Product C will sell at $\pounds 4$ per kg and costs $\pounds 3$ per kg to produce. The profit is to be maximised.
Explain how the initial feasible tableau shown in Fig. 3 models this problem. \begin{table}[h]
1(v)
1(vi)
1
\begin{center} \begin{tabular}{|l|l|} \hline 2(a)(i) &
\hline &
\hline &
\hline &
\hline &
\hline &
\hline &
\hline &
\hline &
\hline &
\hline

Edexcel AEA 2002 June Q5

15 marks Challenging +1.8

the possible values of $n _ { 1 }$ and $n _ { 2 }$,
the exact value of the smallest possible area between $C _ { 1 }$ and $C _ { 2 }$, simplifying your answer,
(8)
the largest value of $x$ for which the gradients of the two curves can be the same. Leave your answer in surd form.

Edexcel AEA 2005 June Q6

19 marks Challenging +1.8

Find the coordinates of the points $P , Q$ and $R$.
Sketch, on separate diagrams, the graphs of
1. $y = \mathrm { f } ( 2 x )$,
2. $y = \mathrm { f } ( | x | + 1 )$,
  indicating on each sketch the coordinates of any maximum points and the intersections with the $x$-axis.
  (6) \begin{figure}[h]
  \captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{f9d3e02c-cef2-435b-9cda-76c43fcac575-5_1015_1464_232_337}
  \end{figure} Figure 2 shows a sketch of part of the curve $C$, with equation $y = \mathrm { f } ( x - v ) + w$, where $v$ and $w$ are constants. The $x$-axis is a tangent to $C$ at the minimum point $T$, and $C$ intersects the $y$-axis at $S$. The line joining $S$ to the maximum point $U$ is parallel to the $x$-axis.
Find the value of $v$ and the value of $w$ and hence find the roots of the equation $$f ( x - v ) + w = 0$$

Edexcel AEA 2006 June Q6

15 marks Challenging +1.2

Show that the point $P ( 1,0 )$ lies on $C$ .
Find the coordinates of the point $Q$ .
Find the area of the shaded region between $C$ and the line $P Q$ .

Edexcel AEA 2007 June Q6

17 marks Hard +2.3

Find an expression, in terms of $x$, for the area $A$ of $R$.
Show that $\frac { \mathrm { d } A } { \mathrm {~d} x } = \frac { 1 } { 4 } ( \pi - 2 x - 2 \sin x ) \sec ^ { 2 } \frac { x } { 2 }$.
Prove that the maximum value of $A$ occurs when $\frac { \pi } { 4 } < x < \frac { \pi } { 3 }$.
Prove that $\tan \frac { \pi } { 8 } = \sqrt { } 2 - 1$.
Show that the maximum value of $A > \frac { \pi } { 4 } ( \sqrt { } 2 - 1 )$.

Edexcel AEA 2014 June Q7

23 marks Challenging +1.8

Find the value of $p$, the value of $m$ and the value of $n$.
Show that the equation of $C$ can be written in the form $y = r + \mathrm { f } ( x - h )$ and specify the function f and the constants $r$ and $h$. The region bounded by $C$, the $x$-axis and the lines $x = \frac { \pi } { 6 }$ and $x = \frac { \pi } { 3 }$ is rotated through $2 \pi$ radians about the $x$-axis.
Find the volume of the solid formed.

OCR H240/01 2020 November Q11

10 marks Challenging +1.2

1. Show that the $x$-coordinate of $A$ satisfies the equation $\left( m ^ { 2 } + 1 \right) x ^ { 2 } - 10 ( m + 1 ) x + 40 = 0$.
2. Hence determine the equation of the tangent to the circle at $A$ which passes through $P$. [4] A second tangent is drawn from $P$ to meet the circle at a second point $B$. The equation of this tangent is of the form $y = n x + 2$, where $n$ is a constant less than 1 .
Determine the exact value of $\tan A P B$.

OCR H240/02 2018 June Q6

13 marks Moderate -0.3

Find the $x$-coordinate of the point where the curve crosses the $x$ axis.
The points $A$ and $B$ lie on the curve and have $x$ coordinates 2 and 4. Show that the line $A B$ is parallel to the $x$-axis.
Find the coordinates of the turning point on the curve.
Determine whether this turning point is a maximum or a minimum.

OCR H240/03 2019 June Q5

9 marks Challenging +1.2

Prove that $( \cot \theta + \operatorname { cosec } \theta ) ^ { 2 } = \frac { 1 + \cos \theta } { 1 - \cos \theta }$.
Hence solve, for $0 < \theta < 2 \pi , 3 ( \cot \theta + \operatorname { cosec } \theta ) ^ { 2 } = 2 \sec \theta$.
\includegraphics[max width=\textwidth, alt={}]{7d1b7598-8f97-43a0-8366-efa8192d549e-06_574_695_306_258}
The diagram shows part of the curve $y = \frac { 2 x - 1 } { ( 2 x + 3 ) ( x + 1 ) ^ { 2 } }$.
Find the exact area of the shaded region, giving your answer in the form $p + q \ln r$, where $p$ and $q$ are positive integers and $r$ is a positive rational number.

OCR H240/03 2020 November Q9

13 marks Standard +0.3

For the motion before $B$ hits the ground, show that the acceleration of $B$ is $0.48 \mathrm {~ms} ^ { - 2 }$.
For the motion before $B$ hits the ground, show that the tension in the string is 23.3 N .
Determine the value of $\mu$. After $B$ hits the ground, $A$ continues to travel up the plane before coming to instantaneous rest before it reaches $P$.
Determine the distance that $A$ travels from the instant that $B$ hits the ground until $A$ comes to instantaneous rest. \includegraphics[max width=\textwidth, alt={}, center]{373fa8e4-9c10-4fcf-9e00-e497161b4c6d-09_917_784_244_242} The diagram shows a wall-mounted light. It consists of a rod $A B$ of mass 0.25 kg and length 0.8 m which is freely hinged to a vertical wall at $A$, and a lamp of mass 0.5 kg fixed at $B$. The system is held in equilibrium by a chain $C D$ whose end $C$ is attached to the midpoint of $A B$. The end $D$ is fixed to the wall a distance 0.4 m vertically above $A$. The rod $A B$ makes an angle of $60 ^ { \circ }$ with the downward vertical. The chain is modelled as a light inextensible string, the rod is modelled as uniform and the lamp is modelled as a particle.
By taking moments about $A$, determine the tension in the chain.
1. Determine the magnitude of the force exerted on the rod at $A$.
2. Calculate the direction of the force exerted on the rod at $A$.
Suggest one improvement that could be made to the model to make it more realistic.

OCR H240/03 2022 June Q5

14 marks Standard +0.8

Show that the $x$-coordinate of $P$ satisfies the equation $$4 x ^ { 3 } + 3 x - 3 = 0 .$$
Show by calculation that the $x$-coordinate of $P$ lies between 0.5 and 1 .
Show that the iteration $$x _ { n + 1 } = \frac { 3 - 4 x _ { n } ^ { 3 } } { 3 }$$ cannot converge to the $x$-coordinate of $P$ whatever starting value is used.
Use the Newton-Raphson method, with initial value 0.5 , to determine the coordinates of $P$ correct to $\mathbf { 5 }$ decimal places.

OCR H240/03 2022 June Q10

8 marks Challenging +1.2

Find the acceleration of $Q$ while $P$ and $B$ are in contact.
Determine the coefficient of friction between $P$ and $B$.
Given that the coefficient of friction between $B$ and the horizontal surface is $\frac { 5 } { 49 }$, determine the least possible value for the mass of $B$. \includegraphics[max width=\textwidth, alt={}, center]{e69f8d73-764e-4f13-a126-faec02c4ad08-09_634_625_255_246} A uniform rod $A B$ of mass 4 kg and length 3 m rests in a vertical plane with $A$ on rough horizontal ground. A particle of mass 1 kg is attached to the rod at $B$. The rod makes an angle of $60 ^ { \circ }$ with the horizontal and is held in limiting equilibrium by a light inextensible string $C D . D$ is a fixed point vertically above $A$ and $C D$ makes an angle of $60 ^ { \circ }$ with the vertical. The distance $A C$ is $x \mathrm {~m}$ (see diagram).
Find, in terms of $g$ and $x$, the tension in the string. The coefficient of friction between the rod and the ground is $\frac { 9 \sqrt { 3 } } { 35 }$.
Determine the value of $x$.

OCR PURE 2018 May Q7

7 marks Standard +0.3

$\overrightarrow { A C }$
$\overrightarrow { O P }$ (ii) Hence prove that the diagonals of a parallelogram bisect one another.

OCR PURE 2019 May Q6

7 marks Standard +0.3

Show that the equation $6 \cos ^ { 2 } \theta = \tan \theta \cos \theta + 4$ can be expressed in the form $6 \sin ^ { 2 } \theta + \sin \theta - 2 = 0$.
\includegraphics[max width=\textwidth, alt={}, center]{d6430776-0b87-4e5e-8f78-c6228ee163d5-4_446_1150_1119_338} The diagram shows parts of the curves $y = 6 \cos ^ { 2 } \theta$ and $y = \tan \theta \cos \theta + 4$, where $\theta$ is in degrees. Solve the inequality $6 \cos ^ { 2 } \theta > \tan \theta \cos \theta + 4$ for $0 ^ { \circ } < \theta < 360 ^ { \circ }$.

OCR MEI AS Paper 1 2019 June Q8

7 marks Moderate -0.8

The model gives the correct velocity of $25.6 \mathrm {~ms} ^ { - 1 }$ at time 8 s . Show that $k = 0.1$. A second model for the motion uses constant acceleration.
Find the value of the acceleration which gives the correct velocity of $25.6 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ at time 8 s .
Show that these two models give the same value for the displacement in the first 8 s .

OCR MEI AS Paper 2 2024 June Q11

6 marks Moderate -0.8

Verify that the curve cuts the $x$-axis at $x = 4$ and at $x = 9$. The curve does not cut or touch the $x$-axis at any other points.
Determine the exact area bounded by the curve and the $x$-axis.

OCR MEI AS Paper 2 2021 November Q10

6 marks Standard +0.3

Show that PQ is perpendicular to QR . A circle passes through $\mathrm { P } , \mathrm { Q }$ and R .
Determine the coordinates of the centre of the circle.

OCR MEI Paper 2 2024 June Q5

4 marks Easy -1.2

In the Printed Answer Booklet, complete the copy of the two-way table.
Calculate the probability that an A-level student selected at random does not study chemistry given that they do not study mathematics.

OCR MEI Paper 2 2020 November Q12

15 marks Standard +0.3

Given that $q < 2 p$, determine the values of $p$ and $q$.
The spinner is spun 10 times. Calculate the probability that exactly one 5 is obtained. Elaine's teacher believes that the probability that the spinner shows a 1 is greater than 0.2 . The spinner is spun 100 times and gives a score of 1 on 28 occasions.
Conduct a hypothesis test at the $5 \%$ level to determine whether there is any evidence to suggest that the probability of obtaining a score of 1 is greater than 0.2 .

Edexcel CP AS Specimen Q6