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OCR FM1 AS 2021 June Q3

9 marks Standard +0.3

A particle of mass $m$ moves in a straight line with constant acceleration $a$. Its initial and final velocities are $u$ and $v$ respectively and its final displacement from its starting position is $s$. In order to model the motion of the particle it is suggested that the velocity is given by the equation $$v^2 = pu^{\alpha} + qa^{\beta}s^{\gamma}$$ where $p$ and $q$ are dimensionless constants.

Explain why $\alpha$ must equal 2 for the equation to be dimensionally consistent. [2]
By using dimensional analysis, determine the values of $\beta$ and $\gamma$. [4]
By considering the case where $s = 0$, determine the value of $p$. [1]
By multiplying both sides of the equation by $\frac{1}{2}m$, and using the numerical values of $\alpha$, $\beta$ and $\gamma$, determine the value of $q$. [2]

OCR FM1 AS 2021 June Q4

12 marks Standard +0.8

Three particles $A$, $B$ and $C$ are free to move in the same straight line on a large smooth horizontal surface. Their masses are 3.3 kg, 2.2 kg and 1 kg respectively. The coefficient of restitution in collisions between any two of them is $e$. Initially, $B$ and $C$ are at rest and $A$ is moving towards $B$ with speed $u \text{ ms}^{-1}$ (see diagram). $A$ collides directly with $B$ and $B$ then goes on to collide directly with $C$. \includegraphics{figure_4}

The velocities of $A$ and $B$ immediately after the first collision are denoted by $v_A \text{ ms}^{-1}$ and $v_B \text{ ms}^{-1}$ respectively. $\bullet$ Show that $v_A = \frac{u(3-2e)}{5}$. $\bullet$ Find an expression for $v_B$ in terms of $u$ and $e$. [4]
Find an expression in terms of $u$ and $e$ for the velocity of $B$ immediately after its collision with $C$. [4]

After the collision between $B$ and $C$ there is a further collision between $A$ and $B$.

Determine the range of possible values of $e$. [4]

OCR FP1 AS 2021 June Q1

5 marks Moderate -0.8

Find a vector which is perpendicular to both $\begin{pmatrix} 1 \\ 3 \\ -2 \end{pmatrix}$ and $\begin{pmatrix} -3 \\ -6 \\ 4 \end{pmatrix}$. [2]
The cartesian equation of a line is $\frac{x}{2} = y - 3 = \frac{z + 4}{4}$. Express the equation of this line in vector form. [3]

OCR FP1 AS 2021 June Q2

9 marks Standard +0.3

In this question you must show detailed reasoning. The complex numbers $z_1$ and $z_2$ are given by $z_1 = 2 - 3i$ and $z_2 = a + 4i$ where $a$ is a real number.

Express $z_1$ in modulus-argument form, giving the modulus in exact form and the argument correct to 3 significant figures. [3]
Find $z_1z_2$ in terms of $a$, writing your answer in the form $c + id$. [2]
The real and imaginary parts of a complex number on an Argand diagram are $x$ and $y$ respectively. Given that the point representing $z_1z_2$ lies on the line $y = x$, find the value of $a$. [2]
Given instead that $z_1z_2 = (z_1z_2)^*$ find the value of $a$. [2]

OCR FP1 AS 2021 June Q3

10 marks Moderate -0.3

In this question you must show detailed reasoning.

Express $(2 + 3i)^3$ in the form $a + ib$. [3]
Hence verify that $2 + 3i$ is a root of the equation $3z^3 - 8z^2 + 23z + 52 = 0$. [3]
Express $3z^3 - 8z^2 + 23z + 52$ as the product of a linear factor and a quadratic factor with real coefficients. [4]

OCR FP1 AS 2021 June Q4

6 marks Standard +0.3

Prove by induction that $2^{n+1} + 5 \times 9^n$ is divisible by 7 for all integers $n \geq 1$. [6]

OCR FS1 AS 2021 June Q1

8 marks Moderate -0.8

A book reviewer estimates that the probability that he receives a delivery of books to review on any one weekday is $0.1$. The first weekday in September on which he receives a delivery of books to review is the $X$th weekday of September.

State an assumption needed for $X$ to be well modelled by a geometric distribution. [1]
Find $P(X = 11)$. [2]
Find $P(X \leq 8)$. [2]
Find $\text{Var}(X)$. [2]
Give a reason why a geometric distribution might not be an appropriate model for the first weekday in a calendar year on which the reviewer receives a delivery of books to review. [1]

OCR FS1 AS 2021 June Q2

8 marks Standard +0.3

The probability distribution for the discrete random variable $W$ is given in the table.

$w$	1	2	3	4
$P(W = w)$	0.25	0.36	$x$	$x^2$

Show that $\text{Var}(W) = 0.8571$. [7]
Find $\text{Var}(3W + 6)$. [1]

OCR FS1 AS 2021 June Q3

5 marks Standard +0.8

In this question you must show detailed reasoning. The random variable $T$ has a binomial distribution. It is known that $E(T) = 5.625$ and the standard deviation of $T$ is $1.875$. Find the values of the parameters of the distribution. [5]

OCR FS1 AS 2021 June Q4

9 marks Challenging +1.2

The table shows the results of a random sample drawn from a population which is thought to have the distribution $U(20)$.

Range	$1 \leq x \leq 8$	$9 \leq x \leq 12$	$13 \leq x \leq 20$
Observed frequency	12	$y$	$28 - y$

Find the range of values of $y$ for which the data are not consistent with the distribution at the $5\%$ significance level. [9]

OCR Further Pure Core 1 2021 June Q1

9 marks Standard +0.3

The equation of the curve shown on the graph is, in polar coordinates, $r = 3\sin 2\theta$ for $0 \leqslant \theta \leqslant \frac{1}{2}\pi$. \includegraphics{figure_1}

The greatest value of $r$ on the curve occurs at the point $P$.
1. Show that $\theta = \frac{1}{4}\pi$ at the point $P$. [2]
2. Find the value of $r$ at the point $P$. [1]
3. Mark the point $P$ on a copy of the graph. [1]
In this question you must show detailed reasoning. Find the exact area of the region enclosed by the curve. [5]

OCR Further Pure Core 1 2021 June Q2

7 marks Standard +0.3

You are given that $f(x) = \ln(2 + x)$.

Determine the exact value of $f'(0)$. [2]
Show that $f''(0) = -\frac{1}{4}$. [2]
Hence write down the first three terms of the Maclaurin series for $f(x)$. [3]

OCR Further Pure Core 1 2021 June Q3

6 marks Standard +0.3

You are given that $\mathbf{A} = \begin{pmatrix} 1 & 2 & 1 \\ 2 & 5 & 2 \\ 3 & -2 & -1 \end{pmatrix}$ and $\mathbf{B} = \begin{pmatrix} 1 & 0 & 1 \\ -8 & 4 & 0 \\ 19 & -8 & -1 \end{pmatrix}$.

Find $\mathbf{AB}$. [1]
Hence write down $\mathbf{A}^{-1}$. [1]
You are given three simultaneous equations $$x + 2y + z = 0$$ $$2x + 5y + 2z = 1$$ $$3x - 2y - z = 4$$
1. Explain how you can tell, without solving them, that there is a unique solution to these equations. [2]
2. Find this unique solution. [2]

OCR Further Pure Core 1 2021 June Q4

9 marks Standard +0.3

Given that $u = \tanh x$, use the definition of $\tanh x$ in terms of exponentials to show that $$x = \frac{1}{2}\ln\left(\frac{1+u}{1-u}\right).$$ [4]
Solve the equation $4\tanh^2 x + \tanh x - 3 = 0$, giving the solution in the form $a\ln b$ where $a$ and $b$ are rational numbers to be determined. [4]
Explain why the equation in part (b) has only one root. [1]

OCR Further Pure Core 1 2021 June Q5

7 marks Challenging +1.2

In this question you must show detailed reasoning. Find $\int_{-1}^{11} \frac{1}{\sqrt{x^2 + 6x + 13}} dx$ giving your answer in the form $\ln(p + q\sqrt{2})$ where $p$ and $q$ are integers to be determined. [7]

OCR Further Pure Core 2 2021 June Q1

8 marks Moderate -0.8

In this question you must show detailed reasoning. S is the 2-D transformation which is a stretch of scale factor 3 parallel to the x-axis. A is the matrix which represents S.

Write down A. [1]
By considering the transformation represented by $\mathbf{A}^{-1}$, determine the matrix $\mathbf{A}^{-1}$. [2]

Matrix B is given by $\mathbf{B} = \begin{pmatrix} 0 & -1 \\ -1 & 0 \end{pmatrix}$. T is the transformation represented by B.

Describe T. [1]
Determine the matrix which represents the transformation S followed by T. [2]
Demonstrate, by direct calculation, that $(\mathbf{BA})^{-1} = \mathbf{A}^{-1}\mathbf{B}^{-1}$. [2]

OCR Further Pure Core 2 2021 June Q2

7 marks Standard +0.8

Find the shortest distance between the point $(-6, 4)$ and the line $y = -0.75x + 7$. [2]

Two lines, $l_1$ and $l_2$, are given by $$l_1: \mathbf{r} = \begin{pmatrix} 4 \\ 3 \\ -2 \end{pmatrix} + \lambda \begin{pmatrix} 2 \\ 1 \\ -4 \end{pmatrix} \text{ and } l_2: \mathbf{r} = \begin{pmatrix} 11 \\ -1 \\ 5 \end{pmatrix} + \mu \begin{pmatrix} 3 \\ -1 \\ 1 \end{pmatrix}$$

Find the shortest distance between $l_1$ and $l_2$. [3]
Hence determine the geometrical arrangement of $l_1$ and $l_2$. [2]

OCR Further Pure Core 2 2021 June Q3

9 marks Standard +0.3

Three matrices, A, B and C, are given by $\mathbf{A} = \begin{pmatrix} 1 & 2 \\ a & -1 \end{pmatrix}$, $\mathbf{B} = \begin{pmatrix} 2 & -1 \\ 4 & 1 \end{pmatrix}$ and $\mathbf{C} = \begin{pmatrix} 5 & 0 \\ -2 & 2 \end{pmatrix}$ where $a$ is a constant.

Using A, B and C in that order demonstrate explicitly the associativity property of matrix multiplication. [4]
Use A and C to disprove by counterexample the proposition 'Matrix multiplication is commutative'. [2]

For a certain value of $a$, $\mathbf{A}\begin{pmatrix} x \\ y \end{pmatrix} = 3\begin{pmatrix} x \\ y \end{pmatrix}$

Find

A plane $\Pi$ has the equation $\mathbf{r} \cdot \begin{pmatrix} 3 \\ 6 \\ -2 \end{pmatrix} = 15$. $C$ is the point $(4, -5, 1)$. Find the shortest distance between $\Pi$ and $C$. [3]
Lines $l_1$ and $l_2$ have the following equations. $l_1: \mathbf{r} = \begin{pmatrix} 4 \\ 3 \\ 1 \end{pmatrix} + \lambda \begin{pmatrix} -2 \\ 4 \\ -2 \end{pmatrix}$ $l_2: \mathbf{r} = \begin{pmatrix} 5 \\ 2 \\ 4 \end{pmatrix} + \mu \begin{pmatrix} 1 \\ -2 \\ 1 \end{pmatrix}$ Find, in exact form, the distance between $l_1$ and $l_2$. [5]