Questions Further Pure Core AS (138 questions)

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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
8
  1. Solve the equation \(\omega + 2 + 7 \mathrm { i } = 3 \omega ^ { * } - \mathrm { i }\).
  2. Prove algebraically that, for non-zero \(z , z = - z ^ { * }\) if and only if \(z\) is purely imaginary.
  3. 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.
  1. 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 \(b = \frac { \sqrt { 2 } } { 4 } ( \sqrt { 3 } - 1 )\).
  2. 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.
  3. Find \(\mathbf { R } ^ { 6 } , \mathbf { R } ^ { 12 }\) and \(\mathbf { R } ^ { 24 }\).
  4. 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. $$\begin{aligned} & l _ { 1 } : \mathbf { r } = \left( \begin{array} { r } 8
- 11
- 2 \end{array} \right) + \lambda \left( \begin{array} { r } - 2
5
3 \end{array} \right)
& l _ { 2 } : \mathbf { r } = \left( \begin{array} { r } - 6
11
8 \end{array} \right) + \mu \left( \begin{array} { r } - 3
1
- 1 \end{array} \right) \end{aligned}$$
  1. Show that \(l _ { 1 }\) and \(l _ { 2 }\) intersect.
  2. 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
4
  1. 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 \(a x + b y + c = 0\).
  2. 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 \(\mathbf { B } = \left( \begin{array} { c c } \frac { 5 } { 13 } & - \frac { 12 } { 13 }
\frac { 12 } { 13 } & \frac { 5 } { 13 } \end{array} \right)\).
  1. Use \(\mathbf { A }\) and \(\mathbf { B }\) to disprove the proposition: "Matrix multiplication is commutative". Matrix \(\mathbf { B }\) represents the transformation \(\mathrm { T } _ { \mathrm { B } }\).
  2. Describe the transformation \(\mathrm { T } _ { \mathrm { B } }\).
  3. 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 \(N\).
  4. Determine the area of \(N\).
OCR Further Pure Core AS 2021 November Q6
6 In this question you must show detailed reasoning.
  1. Solve the equation \(2 z ^ { 2 } - 10 z + 25 = 0\) giving your answers in the form \(\mathrm { a } + \mathrm { bi }\).
  2. 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 \(\mathbf { A } = \left( \begin{array} { c c c } t - 1 & t - 1 & t - 1
1 - t & 6 & t
2 - 2 t & 2 - 2 t & 1 \end{array} \right)\).
  1. Find, in fully factorised form, an expression for \(\operatorname { det } \mathbf { A }\) in terms of \(t\).
  2. 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. $$\begin{aligned} \left( b ^ { 2 } + 1 \right) x + \left( b ^ { 2 } + 1 \right) y + \left( b ^ { 2 } + 1 \right) z & = 5
    \left( - b ^ { 2 } - 1 \right) x + \quad 6 y + \left( b ^ { 2 } + 2 \right) z & = 10
    \left( - 2 b ^ { 2 } - 2 \right) x + \left( - 2 b ^ { 2 } - 2 \right) y + \quad z & = 15 \end{aligned}$$
  3. 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.
  1. Show that \(\mathrm { S } = 2.5 \mathrm { t } - 2\).
  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.
  3. Using the results in parts (a) and (b), determine the shortest possible length of the ventilation shaft.
  4. 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.