Questions - A-Level Maths

OCR M4 2015 June Q5

15 marks Challenging +1.8

Taking $H$ as the reference level for gravitational potential energy, show that the total potential energy $V$ of the system is given by $$V = m g \left( 2 \lambda r \cos \theta - 2 r \cos ^ { 2 } \theta - \lambda a \right)$$
Find the set of possible values of $\lambda$ so that there is more than one position of equilibrium.
For the case $\lambda = \frac { 3 } { 2 }$, determine whether each equilibrium position is stable or unstable.

Edexcel M5 2002 June Q6

17 marks Challenging +1.8

Show that the moment of inertia of the rod about the edge of the table is $\frac { 7 } { 3 } m a ^ { 2 }$. The rod is released from rest and rotates about the edge of the table. When the rod has turned through an angle $\theta$, its angular speed is $\dot { \theta }$. Assuming that the rod has not started to slip,
show that $\dot { \theta } ^ { 2 } = \frac { 6 g \sin \theta } { 7 a }$,
find the angular acceleration of the rod,
find the normal reaction of the table on the rod. The coefficient of friction between the rod and the edge of the table is $\mu$.
Show that the rod starts to slip when $\tan \theta = \frac { 4 } { 13 } \mu$ (6)

Edexcel M5 2005 June Q4

11 marks Challenging +1.8

Show that the moment of inertia of the body about $L$ is $\frac { 77 m a ^ { 2 } } { 4 }$. When $P R$ is vertical, the body has angular speed $\omega$ and the centre of the disc strikes a stationary particle of mass $\frac { 1 } { 2 } \mathrm {~m}$. Given that the particle adheres to the centre of the disc,
find, in terms of $\omega$, the angular speed of the body immediately after the impact.

AQA FP1 2013 January Q7

7 marks Standard +0.8

Show that there is a linear relationship between $Y$ and $X$.
The graph of $Y$ against $X$ is shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{cf9337b9-b766-4ce5-967c-5d7522e2aa42-4_748_858_849_593} Find the value of $n$ and the value of $a$.

AQA FP1 2015 June Q6

5 marks Standard +0.3

Sketch the curve $C _ { 1 }$, stating the values of its intercepts with the coordinate axes.
The curve $C _ { 1 }$ is translated by the vector $\left[ \begin{array} { l } k \\ 0 \end{array} \right]$, where $k < 0$, to give a curve $C _ { 2 }$. Given that $C _ { 2 }$ passes through the origin $( 0,0 )$, find the equations of the asymptotes of $C _ { 2 }$.
[0pt] [3 marks]

OCR FP1 2011 January Q7

9 marks Moderate -0.8

Write down the matrix, $\mathbf { A }$, that represents a shear with $x$-axis invariant in which the image of the point $( 1,1 )$ is $( 4,1 )$.
The matrix $\mathbf { B }$ is given by $\mathbf { B } = \left( \begin{array} { c c } \sqrt { 3 } & 0 \\ 0 & \sqrt { 3 } \end{array} \right)$. Describe fully the geometrical transformation represented by $\mathbf { B }$.
The matrix $\mathbf { C }$ is given by $\mathbf { C } = \left( \begin{array} { l l } 2 & 6 \\ 0 & 2 \end{array} \right)$.
1. Draw a diagram showing the unit square and its image under the transformation represented by $\mathbf { C }$.
2. Write down the determinant of $\mathbf { C }$ and explain briefly how this value relates to the transformation represented by $\mathbf { C }$. 8 The quadratic equation $2 x ^ { 2 } - x + 3 = 0$ has roots $\alpha$ and $\beta$, and the quadratic equation $x ^ { 2 } - p x + q = 0$ has roots $\alpha + \frac { 1 } { \alpha }$ and $\beta + \frac { 1 } { \beta }$.
  1. Show that $p = \frac { 5 } { 6 }$.
  2. Find the value of $q$. 9 The matrix $\mathbf { M }$ is given by $\mathbf { M } = \left( \begin{array} { r r r } a & - a & 1 \\ 3 & a & 1 \\ 4 & 2 & 1 \end{array} \right)$.
    1. Find, in terms of $a$, the determinant of $\mathbf { M }$.
    2. Hence find the values of $a$ for which $\mathbf { M } ^ { - 1 }$ does not exist.
    3. Determine whether the simultaneous equations $$\begin{aligned} & 6 x - 6 y + z = 3 k \\ & 3 x + 6 y + z = 0 \\ & 4 x + 2 y + z = k \end{aligned}$$ where $k$ is a non-zero constant, have a unique solution, no solution or an infinite number of solutions, justifying your answer.
    4. Show that $\frac { 1 } { r } - \frac { 2 } { r + 1 } + \frac { 1 } { r + 2 } \equiv \frac { 2 } { r ( r + 1 ) ( r + 2 ) }$.
    5. Hence find an expression, in terms of $n$, for $$\sum _ { r = 1 } ^ { n } \frac { 2 } { r ( r + 1 ) ( r + 2 ) }$$
    6. Show that $\sum _ { r = n + 1 } ^ { \infty } \frac { 2 } { r ( r + 1 ) ( r + 2 ) } = \frac { 1 } { ( n + 1 ) ( n + 2 ) }$.

OCR FP1 2016 June Q8

10 marks Standard +0.3

Show that $\frac { 1 } { 2 r + 1 } - \frac { 1 } { 2 r + 3 } \equiv \frac { 2 } { ( 2 r + 1 ) ( 2 r + 3 ) }$.
Hence find $\sum _ { r = 1 } ^ { n } \frac { 1 } { ( 2 r + 1 ) ( 2 r + 3 ) }$, giving your answer as a single fraction.
Find $\sum _ { r = n } ^ { \infty } \frac { 1 } { ( 2 r + 1 ) ( 2 r + 3 ) }$, giving your answer as a single fraction.

OCR FP3 2007 June Q7

10 marks Standard +0.3

Show that $\left( z - \mathrm { e } ^ { \mathrm { i } \phi } \right) \left( z - \mathrm { e } ^ { - \mathrm { i } \phi } \right) \equiv z ^ { 2 } - ( 2 \cos \phi ) z + 1$.
Write down the seven roots of the equation $z ^ { 7 } = 1$ in the form $\mathrm { e } ^ { \mathrm { i } \theta }$ and show their positions in an Argand diagram.
Hence express $z ^ { 7 } - 1$ as the product of one real linear factor and three real quadratic factors.

OCR FP3 2013 June Q2

9 marks Challenging +1.2

Write down the operation table and, assuming associativity, show that $G$ is a group.
State the order of each element.
Find all the proper subgroups of $G$. The group $H$ consists of the set $\{ 1,3,7,9 \}$ with the operation of multiplication modulo 10 .
Explaining your reasoning, determine whether $H$ is isomorphic to $G$.

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?

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

2(a)(i)
\end{table}

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 $A > \frac { \pi } { 4 } ( \sqrt { } 2 - 1 )$.

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 PURE 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