Questions - OCR Further Pure Core AS

Show that $L _ { 1 }$ and $L _ { 2 }$ intersect, giving the position vector of the point of intersection.
Find the equation of the line which intersects $L _ { 1 }$ and $L _ { 2 }$ and is perpendicular to both. Give your answer in cartesian form.

OCR Further Pure Core AS 2023 June Q3

3 In this question you must show detailed reasoning. In this question the principal argument of a complex number lies in the interval $[ 0,2 \pi )$.
Complex numbers $z _ { 1 }$ and $z _ { 2 }$ are defined by $z _ { 1 } = 3 + 4 \mathrm { i }$ and $z _ { 2 } = - 5 + 12 \mathrm { i }$.

Determine $z _ { 1 } z _ { 2 }$, giving your answer in the form $a + b \mathrm { i }$.
Express $z _ { 2 }$ in modulus-argument form.
Verify, by direct calculation, that $\arg \left( z _ { 1 } z _ { 2 } \right) = \arg \left( z _ { 1 } \right) + \arg \left( z _ { 2 } \right)$.

OCR Further Pure Core AS 2023 June Q4

4 The vector $\mathbf { p }$, all of whose components are positive, is given by $\mathbf { p } = \left( \begin{array} { c } a ^ { 2 }
a - 5
26 \end{array} \right)$ where $a$ is a constant.
You are given that $\mathbf { p }$ is perpendicular to the vector $\left( \begin{array} { c } 2
6
- 3 \end{array} \right)$.
Determine the value of $a$.

OCR Further Pure Core AS 2023 June Q5

5 In this question you must show detailed reasoning. The roots of the equation $5 x ^ { 2 } - 3 x + 12 = 0$ are $\alpha$ and $\beta$. By considering the symmetric functions of the roots, $\alpha + \beta$ and $\alpha \beta$, determine the exact value of $\frac { 1 } { \alpha ^ { 2 } } + \frac { 1 } { \beta ^ { 2 } }$.

OCR Further Pure Core AS 2023 June Q6

6 Prove by induction that $4 \times 8 ^ { n } + 66$ is divisible by 14 for all integers $n \geqslant 0$.

OCR Further Pure Core AS 2023 June Q7

7 In this question you must show detailed reasoning. Matrix $\mathbf { A }$ is given by $\mathbf { A } = \left( \begin{array} { c c c } a & - 6 & a - 3
a + 9 & a & 4
0 & - 13 & a - 1 \end{array} \right)$ where $a$ is a constant.
Find all possible values of $a$ for which $\operatorname { det } \mathbf { A }$ has the same value as it has when $a = 2$.

OCR Further Pure Core AS 2023 June Q8

Solve the equation $\omega + 2 + 7 \mathrm { i } = 3 \omega ^ { * } - \mathrm { i }$.
Prove algebraically that, for non-zero $z , z = - z ^ { * }$ if and only if $z$ is purely imaginary.
The complex numbers $z$ and $z ^ { * }$ are represented on an Argand diagram by the points $A$ and $B$ respectively.
1. State, for any $z$, the single transformation which transforms $A$ to $B$.
2. Use a geometric argument to prove that $z = z ^ { * }$ if and only if $z$ is purely real.

OCR Further Pure Core AS 2023 June Q9

9 Matrix $\mathbf { R }$ is given by $\mathbf { R } = \left( \begin{array} { c c c } a & 0 & - b
0 & 1 & 0
b & 0 & a \end{array} \right)$ where $a$ and $b$ are constants.

Find $\mathbf { R } ^ { 2 }$ in terms of $a$ and $b$. The constants $a$ and $b$ are given by $a = \frac { \sqrt { 2 } } { 4 } ( \sqrt { 3 } + 1 )$ and $b = \frac { \sqrt { 2 } } { 4 } ( \sqrt { 3 } - 1 )$.
By determining exact expressions for $a b$ and $a ^ { 2 } - b ^ { 2 }$ and using the result from part (a), show that $\mathbf { R } ^ { 2 } = k \left( \begin{array} { c c c } \sqrt { 3 } & 0 & - 1
0 & 2 & 0
1 & 0 & \sqrt { 3 } \end{array} \right)$ where $k$ is a real number whose value is to be determined.
Find $\mathbf { R } ^ { 6 } , \mathbf { R } ^ { 12 }$ and $\mathbf { R } ^ { 24 }$.
Describe fully the transformation represented by $\mathbf { R }$. \section*{END OF QUESTION PAPER}

OCR Further Pure Core AS 2021 November Q1

1 The lines $l _ { 1 }$ and $l _ { 2 }$ have the following equations. $$\begin{aligned} & l _ { 1 } : \mathbf { r } = \left( \begin{array} { r } 8
- 11
- 2 \end{array} \right) + \lambda \left( \begin{array} { r } - 2
5
3 \end{array} \right)
& l _ { 2 } : \mathbf { r } = \left( \begin{array} { r } - 6
11
8 \end{array} \right) + \mu \left( \begin{array} { r } - 3
1
- 1 \end{array} \right) \end{aligned}$$

Show that $l _ { 1 }$ and $l _ { 2 }$ intersect.
Write down the point of intersection of $l _ { 1 }$ and $l _ { 2 }$.

OCR Further Pure Core AS 2021 November Q2

2 The equation $2 x ^ { 3 } + 3 x ^ { 2 } - 2 x + 5 = 0$ has roots $\alpha , \beta$ and $\gamma$.
Use a substitution to find a cubic equation with integer coefficients whose roots are $\alpha + 1 , \beta + 1$ and $\gamma + 1$.

OCR Further Pure Core AS 2021 November Q3

3 In this question you must show detailed reasoning.
The equation $x ^ { 4 } - 7 x ^ { 3 } - 2 x ^ { 2 } + 218 x - 1428 = 0$ has a root $3 - 5 i$.
Find the other three roots of this equation.

OCR Further Pure Core AS 2021 November Q4

A locus $C _ { 1 }$ is defined by $C _ { 1 } = \{ \mathrm { z } : | \mathrm { z } + \mathrm { i } | \leqslant \mid \mathrm { z } - 2 \}$.
1. Indicate by shading on the Argand diagram in the Printed Answer Booklet the region representing $C _ { 1 }$.
2. Find the cartesian equation of the boundary line of the region representing $C _ { 1 }$, giving your answer in the form $a x + b y + c = 0$.
A locus $C _ { 2 }$ is defined by $C _ { 2 } = \{ \mathrm { z } : | \mathrm { z } + 1 | \leqslant 3 \} \cap \{ \mathrm { z } : | \mathrm { z } - 2 \mathrm { i } | \geqslant 2 \}$. Indicate by shading on the Argand diagram in the Printed Answer Booklet the region representing $C _ { 2 }$.

OCR Further Pure Core AS 2021 November Q5

5 Matrices $\mathbf { A }$ and $\mathbf { B }$ are given by $\mathbf { A } = \left( \begin{array} { r l } - 1 & 0
0 & 1 \end{array} \right)$ and $\mathbf { B } = \left( \begin{array} { c c } \frac { 5 } { 13 } & - \frac { 12 } { 13 }
\frac { 12 } { 13 } & \frac { 5 } { 13 } \end{array} \right)$.

Use $\mathbf { A }$ and $\mathbf { B }$ to disprove the proposition: "Matrix multiplication is commutative". Matrix $\mathbf { B }$ represents the transformation $\mathrm { T } _ { \mathrm { B } }$.
Describe the transformation $\mathrm { T } _ { \mathrm { B } }$.
By considering the inverse transformation of $\mathrm { T } _ { \mathrm { B } }$, determine $\mathbf { B } ^ { - 1 }$. Matrix $\mathbf { C }$ is given by $\mathbf { C } = \left( \begin{array} { r r } 1 & 0
0 & - 3 \end{array} \right)$ and represents the transformation $\mathrm { T } _ { \mathrm { C } }$.
The transformation $\mathrm { T } _ { \mathrm { BC } }$ is transformation $\mathrm { T } _ { \mathrm { C } }$ followed by transformation $\mathrm { T } _ { \mathrm { B } }$.
An object shape of area 5 is transformed by $\mathrm { T } _ { \mathrm { BC } }$ to an image shape $N$.
Determine the area of $N$.

OCR Further Pure Core AS 2021 November Q6

6 In this question you must show detailed reasoning.

Solve the equation $2 z ^ { 2 } - 10 z + 25 = 0$ giving your answers in the form $\mathrm { a } + \mathrm { bi }$.
Solve the equation $3 \omega - 2 = \mathrm { i } ( 5 + 2 \omega )$ giving your answer in the form $\mathrm { a } + \mathrm { bi }$.

OCR Further Pure Core AS 2021 November Q7

7 Prove that $2 ^ { 3 n } - 3 ^ { n }$ is divisible by 5 for all integers $n \geqslant 1$.

OCR Further Pure Core AS 2021 November Q8

8 The matrix $\mathbf { A }$ is given by $\mathbf { A } = \left( \begin{array} { c c c } t - 1 & t - 1 & t - 1
1 - t & 6 & t
2 - 2 t & 2 - 2 t & 1 \end{array} \right)$.

Find, in fully factorised form, an expression for $\operatorname { det } \mathbf { A }$ in terms of $t$.
State the values of $t$ for which $\mathbf { A }$ is singular. You are given the following system of equations in $x , y$ and $z$, where $b$ is a real number. $$\begin{aligned} \left( b ^ { 2 } + 1 \right) x + \left( b ^ { 2 } + 1 \right) y + \left( b ^ { 2 } + 1 \right) z & = 5
\left( - b ^ { 2 } - 1 \right) x + \quad 6 y + \left( b ^ { 2 } + 2 \right) z & = 10
\left( - 2 b ^ { 2 } - 2 \right) x + \left( - 2 b ^ { 2 } - 2 \right) y + \quad z & = 15 \end{aligned}$$
Determine which one of the following statements about the solution of the equations is true.
- There is a unique solution for all values of $b$.
- There is a unique solution for some, but not all, values of $b$.
- There is no unique solution for any value of $b$.

OCR Further Pure Core AS 2021 November Q9

9 The points $P ( 3,5 , - 21 )$ and $Q ( - 1,3 , - 16 )$ are on the ceiling of a long straight underground tunnel. A ventilation shaft must be dug from the point $M$ on the ceiling of the tunnel midway between $P$ and $Q$ to horizontal ground level (where the $z$-coordinate is 0 ). The ventilation shaft must be perpendicular to the tunnel. The path of the ventilation shaft is modelled by the vector equation $\mathbf { r } = \mathbf { a } + \lambda \mathbf { b }$, where $\mathbf { a }$ is the position vector of $M$. You are given that $\mathbf { b } = \left( \begin{array} { l } 1
\mathrm {~s}
\mathrm { t } \end{array} \right)$ where $s$ and $t$ are real numbers.

Show that $\mathrm { S } = 2.5 \mathrm { t } - 2$.
Show that at the point where the ventilation shaft reaches the ground $\lambda = \frac { \mathrm { C } } { \mathrm { t } }$, where $c$ is a constant to be determined.
Using the results in parts (a) and (b), determine the shortest possible length of the ventilation shaft.
Explain what the fact that $\mathbf { b } \times \left( \begin{array} { l } 0
0
1 \end{array} \right) \neq \mathbf { O }$ means about the direction of the ventilation shaft.

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