Parameter values for unique solution

1

1. Find the set of values of \(k\) for which the system of equations $$\begin{aligned} x + 2 y + 3 z & = 1 \\ k x + 4 y + 6 z & = 0 \\ 7 x + 8 y + 9 z & = 3 \end{aligned}$$ has a unique solution.
2. Interpret the situation geometrically in the case where the system of equations does not have a unique solution.

1 Find the set of values of \(k\) for which the system of equations $$\begin{array} { r } x + 5 y + 6 z = 1 \\ k x + 2 y + 2 z = 2 \\ - 3 x + 4 y + 8 z = 3 \end{array}$$ has a unique solution and interpret this situation geometrically.

8

1. 1. Fid bet basb le s a for which the system of equations $$\begin{array} { r l } x - 2 y - 2 z + z & 0 \\ 2 x + ( a - 9 y - 0 z + 1 E & 0 \\ 3 x - 6 y + 2 a z + 9 & 0 \end{array}$$ h san q sbtu in
   2. Given that \(a = - 3\), show that the system of equations in part (i) \(\mathbf { b } \mathbf { s } \mathbf { n }\) sb t in In erp et th s situation geometrically.
2. The matrix \(\mathbf { A }\) is given by $$\mathbf { A } = \left( \begin{array} { r r r } 1 & 1 & 2 \\ 0 & 2 & 2 \\ - 1 & 1 & 3 \end{array} \right)$$
   1. Find b eig le so A.
   2. Use th ch racteristic eq tiw \(\mathbf { A }\) tof id \(\mathbf { A } ^ { - 1 }\). If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

8 The matrix \(\mathbf { M }\) is given by \(\mathbf { M } = \left( \begin{array} { l l l } a & 4 & 2 \\ 1 & a & 0 \\ 1 & 2 & 1 \end{array} \right)\).

1. Find, in terms of \(a\), the determinant of \(\mathbf { M }\).
2. Hence find the values of \(a\) for which \(\mathbf { M }\) is singular.
3. State, giving a brief reason in each case, whether the simultaneous equations $$\begin{aligned} a x + 4 y + 2 z & = 3 a \\ x + a y & = 1 \\ x + 2 y + z & = 3 \end{aligned}$$ have any solutions when
   (a) \(a = 3\),
   (b) \(a = 2\).

7 The matrix \(\mathbf { M }\) is given by \(\mathbf { M } = \left( \begin{array} { l l l } a & 4 & 0 \\ 0 & a & 4 \\ 2 & 3 & 1 \end{array} \right)\).

1. Find, in terms of \(a\), the determinant of \(\mathbf { M }\).
2. In the case when \(a = 2\), state whether \(\mathbf { M }\) is singular or non-singular, justifying your answer.
3. In the case when \(a = 4\), determine whether the simultaneous equations $$\begin{aligned} a x + 4 y \quad = & 6 \\ a y + 4 z & = 8 \\ 2 x + 3 y + z & = 1 \end{aligned}$$ have any solutions.

5 By using the determinant of an appropriate matrix, or otherwise, find the value of \(k\) for which the simultaneous equations $$\begin{aligned} 2 x - y + z & = 7 \\ 3 y + z & = 4 \\ x + k y + k z & = 5 \end{aligned}$$ do not have a unique solution for \(x , y\) and \(z\).

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 \(\sum _ { r = n + 1 } ^ { \infty } \frac { 2 } { r ( r + 1 ) ( r + 2 ) } = \frac { 1 } { ( n + 1 ) ( n + 2 ) }\).

5 By using the determinant of an appropriate matrix, find the values of \(\lambda\) for which the simultaneous equations $$\begin{array} { r } 3 x + 2 y + 4 z = 5 \\ \lambda y + z = 1 \\ x + \lambda y + \lambda z = 4 \end{array}$$ do not have a unique solution for \(x , y\) and \(z\). \includegraphics[max width=\textwidth, alt={}, center]{f074de40-08b6-47a6-a0d2-d3cbe628cacc-3_556_759_233_653} The diagram shows the unit square \(O A B C\), and its image \(O A B ^ { \prime } C ^ { \prime }\) after a transformation. The points have the following coordinates: \(A ( 1,0 ) , B ( 1,1 ) , C ( 0,1 ) , B ^ { \prime } ( 3,2 )\) and \(C ^ { \prime } ( 2,2 )\).

1. Write down the matrix, \(\mathbf { X }\), for this transformation.
2. The transformation represented by \(\mathbf { X }\) is equivalent to a transformation P followed by a transformation Q. Give geometrical descriptions of a pair of possible transformations P and Q and state the matrices that represent them.
3. Find the matrix that represents transformation Q followed by transformation P .

9 The matrix \(\mathbf { A }\) is given by \(\mathbf { A } = \left( \begin{array} { l l l } a & 1 & 1 \\ 1 & a & 1 \\ 1 & 1 & 2 \end{array} \right)\).

1. Find, in terms of \(a\), the determinant of \(\mathbf { A }\).
2. Hence find the values of \(a\) for which \(\mathbf { A }\) is singular.
3. State, giving a brief reason in each case, whether the simultaneous equations $$\begin{aligned} a x + y + z & = 2 a \\ x + a y + z & = - 1 \\ x + y + 2 z & = - 1 \end{aligned}$$ have any solutions when
   (a) \(a = 0\),
   (b) \(a = 1\).

9 The matrix \(\mathbf { A }\) is given by \(\mathbf { A } = \left( \begin{array} { r r r } a & a & - 1 \\ 0 & a & 2 \\ 1 & 2 & 1 \end{array} \right)\).

1. Find, in terms of \(a\), the determinant of \(\mathbf { A }\).
2. Three simultaneous equations are shown below. $$\begin{aligned} a x + a y - z & = - 1 \\ a y + 2 z & = 2 a \\ x + 2 y + z & = 1 \end{aligned}$$ For each of the following values of \(a\), determine whether the equations are consistent or inconsistent. If the equations are consistent, determine whether or not there is a unique solution.
   (a) \(a = 0\) (b) \(a = 1\) (c) \(a = 2\)

10 The matrix \(\mathbf { D }\) is given by \(\mathbf { D } = \left( \begin{array} { r r r } a & 2 & - 1 \\ 2 & a & 1 \\ 1 & 1 & a \end{array} \right)\).

1. Find the determinant of \(\mathbf { D }\) in terms of \(a\).
2. Three simultaneous equations are shown below. $$\begin{array} { r } a x + 2 y - z = 0 \\ 2 x + a y + z = a \\ x + y + a z = a \end{array}$$ For each of the following values of \(a\), determine whether or not there is a unique solution. If the solution is not unique, determine whether the equations are consistent or inconsistent.
   (a) \(\quad a = 3\) (b) \(a = 2\) (c) \(\quad a = 0\) \section*{THERE ARE NO QUESTIONS WRITTEN ON THIS PAGE}

9 The matrix \(\mathbf { D }\) is given by \(\mathbf { D } = \left( \begin{array} { l l l } 1 & 3 & 4 \\ 2 & a & 3 \\ 0 & 1 & a \end{array} \right)\).

1. Find the values of \(a\) for which \(\mathbf { D }\) is singular.
2. Three simultaneous equations are shown below. $$\begin{array} { r } x + 3 y + 4 z = 3 \\ 2 x + a y + 3 z = 2 \\ y + a z = 0 \end{array}$$ For each of the following values of \(a\), determine whether or not there is a unique solution. If a unique solution does not exist, determine whether the equations are consistent or inconsistent.
   (a) \(a = 3\) (b) \(a = 1\)

9

1. The matrix \(\mathbf { X }\) is given by \(\mathbf { X } = \left( \begin{array} { r r r } a & 3 & - 2 \\ 0 & a & 5 \\ 1 & 2 & 1 \end{array} \right)\). Show that the determinant of \(\mathbf { X }\) is \(a ^ { 2 } - 8 a + 15\).
2. Explain briefly why the equations $$\begin{array} { r } 3 x + 3 y - 2 z = 1 \\ 3 y + 5 z = 5 \\ x + 2 y + z = 2 \end{array}$$ do not have a unique solution and determine whether these equations are consistent or inconsistent.
3. Use an algebraic method to find the square roots of the complex number \(9 + 40 \mathrm { i }\).
4. Show that \(9 + 40 \mathrm { i }\) is a root of the quadratic equation \(z ^ { 2 } - 18 z + 1681 = 0\).
5. By using the substitution \(z = \frac { 1 } { u ^ { 2 } }\), find the roots of the equation \(1681 u ^ { 4 } - 18 u ^ { 2 } + 1 = 0\). Give your answers in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real.

2 Find the set of values of \(a\) for which the system of equations $$\begin{aligned} a x + y + 2 z & = 0 \\ 3 x - 2 y & = 4 \\ 3 x - 4 y - 6 a z & = 14 \end{aligned}$$ has a unique solution.

10 You are given the matrix \(\mathbf { A }\) where \(\mathbf { A } = \left( \begin{array} { l l l } a & 2 & 0 \\ 0 & a & 2 \\ 4 & 5 & 1 \end{array} \right)\).

1. Find, in terms of \(a\), the determinant of \(\mathbf { A }\), simplifying your answer.
2. Hence find the values of \(a\) for which \(\mathbf { A }\) is singular. You are given the following equations which are to be solved simultaneously. $$\begin{aligned} a x + 2 y & = 6 \\ a y + 2 z & = 8 \\ 4 x + 5 y + z & = 16 \end{aligned}$$
3. For each of the values of \(a\) found in part (b) determine whether the equations have
   • a unique solution, which should be found, or
   • an infinite set of solutions or
   • no solution.

4

1. Find the coordinates of the point where the following three planes intersect. Give your answers in terms of \(a\). $$\begin{aligned} x - 2 y - z & = 6 \\ 3 x + y + 5 z & = - 4 \\ - 4 x + 2 y - 3 z & = a \end{aligned}$$
2. Determine whether the intersection of the three planes could be on the \(z\)-axis.

14 Three planes have equations $$\begin{aligned} - x + a y & = 2 \\ 2 x + 3 y + z & = - 3 \\ x + b y + z & = c \end{aligned}$$ where \(a\), \(b\) and \(c\) are constants.

1. In the case where the planes do not intersect at a unique point,
   1. find \(b\) in terms of \(a\),
   2. find the value of \(c\) for which the planes form a sheaf.
2. In the case where \(b = a\) and \(c = 1\), find the coordinates of the point of intersection of the planes in terms of \(a\).

14 Three planes have equations $$\begin{aligned} k x - z & = 2 \\ - x + k y + 2 z & = 1 \\ 2 k x + 2 y + 3 z & = 0 \end{aligned}$$ where \(k\) is a constant.

1. By considering a suitable determinant, show that the three planes meet at a point for all values of \(k\).
2. Using a matrix method, find, in terms of \(k\), the coordinates of the point of intersection of the planes.

10 The matrix \(\mathbf { M }\) is defined as $$\mathbf { M } = \left[ \begin{array} { c c c } 2 & - 1 & 1 \\ - 1 & - 1 & - 2 \\ 1 & 2 & c \end{array} \right]$$ where \(c\) is a real number. 10

1. The linear transformation T is represented by the matrix \(\mathbf { M }\) Show that, for one particular value of \(c\), the image under \(T\) of every point lies in the plane $$x + 5 y + 3 z = 0$$ State the value of \(c\) for which this occurs.
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2. It is given that \(\mathbf { M }\) is a non-singular matrix.
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3. 1. State any restrictions on the value of \(c\)

      10 (iii) Using your answer from part (b)(ii), solve \(\begin{array} { r } 2 x - y + z = - 3
      - x - y - 2 z = - 6
      x + 2 y + 4 z = 13 \end{array}\)	\(\_\_\_\_\)

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 \(b\).