Geometric interpretation of systems

8 The equations of three planes are $$\begin{array} { r } 2 x + y + 3 z = 3 \\ 3 x - y - 2 z = 2 \\ - 4 x + 3 y + 7 z = k \end{array}$$ where \(k\) is a constant.

1. By considering a suitable determinant, show that the planes do not meet at a single point.
2. Given that the planes form a sheaf, determine the value of \(k\).

7 Three planes have equations $$\begin{array} { r } x + 2 y - 3 z = 0 \\ - x + 3 y - 2 z = 0 \\ x - 2 y + k z = k \end{array}$$ where \(k\) is a constant.

1. For the case \(k = 0\), the origin lies on all three planes. Use a determinant to explain whether there are any other points that lie on all three planes in this case.
2. You are now given that \(k = 1\).
   1. Show that there are no points that lie on all three planes.
   2. Describe the geometrical arrangement of the three planes.

9 Three planes have equations \(k x + y - 2 z = 0\) \(2 x + 3 y - 6 z = - 5\) \(3 x - 2 y + 5 z = 1\) where \(k\) is a constant. Investigate the arrangement of the planes for each of the following cases. If in either case the planes meet at a unique point, find the coordinates of that point.

1. \(k = - 1\)
2. \(k = \frac { 2 } { 3 }\)

7. $$\mathbf { M } = \left( \begin{array} { r r r } 2 & - 1 & 1 \\ 3 & k & 4 \\ 3 & 2 & - 1 \end{array} \right) \quad \text { where } k \text { is a constant }$$

1. Find the values of \(k\) for which the matrix \(\mathbf { M }\) has an inverse.
2. Find, in terms of \(p\), the coordinates of the point where the following planes intersect $$\begin{aligned} & 2 x - y + z = p \\ & 3 x - 6 y + 4 z = 1 \\ & 3 x + 2 y - z = 0 \end{aligned}$$
3. 1. Find the value of \(q\) for which the set of simultaneous equations $$\begin{aligned} & 2 x - y + z = 1 \\ & 3 x - 5 y + 4 z = q \\ & 3 x + 2 y - z = 0 \end{aligned}$$ can be solved.
   2. For this value of \(q\), interpret the solution of the set of simultaneous equations geometrically.

6. $$\mathbf { M } = \left( \begin{array} { r r r } k & 5 & 7 \\ 1 & 1 & 1 \\ 2 & 1 & - 1 \end{array} \right) \quad \text { where } k \text { is a constant }$$

1. Given that \(k \neq 4\), find, in terms of \(k\), the inverse of the matrix \(\mathbf { M }\).
2. Find, in terms of \(p\), the coordinates of the point where the following planes intersect. $$\begin{array} { r } 2 x + 5 y + 7 z = 1 \\ x + y + z = p \\ 2 x + y - z = 2 \end{array}$$
3. 1. Find the value of \(q\) for which the following planes intersect in a straight line. $$\begin{array} { r } 4 x + 5 y + 7 z = 1 \\ x + y + z = q \\ 2 x + y - z = 2 \end{array}$$
   2. For this value of \(q\), determine a vector equation for the line of intersection.

7 Three planes have equations $$\begin{aligned} ( 4 k + 1 ) x - 3 y + ( k - 5 ) z & = 3 \\ ( k - 1 ) x + ( 3 - k ) y + 2 z & = 1 \\ 7 x - 3 y + 4 z & = 2 \end{aligned}$$ 7

1. The planes do not meet at a unique point.
   Show that \(k = 4.5\) is one possible value of \(k\), and find the other possible value of \(k\).
   

   7

   For each value of \(k\) found in part (a), identify the configuration of the given planes. In each case fully justify your answer, stating whether or not the equations of the planes form a consistent system. [4 marks] \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\)

4 A \(3 \times 3\) system of equations is given by the matrix equation \(\left( \begin{array} { r r r } - 1 & 3 & 1 \\ 5 & - 1 & 2 \\ - 1 & 1 & 0 \end{array} \right) \left( \begin{array} { l } x \\ y \\ z \end{array} \right) = \left( \begin{array} { r } 1 \\ 16 \\ - 2 \end{array} \right)\).

1. Show that this system of equations does not have a unique solution.
2. Solve this system of equations and describe the geometrical significance of the solution.

Show that the system of equations $$14x - 4y + 6z = 5,$$ $$x + y + kz = 3,$$ $$-21x + 6y - 9z = 14,$$ where \(k\) is a constant, does not have a unique solution and interpret this situation geometrically. [4]

Three planes have equations \begin{align} 4x - 5y + z &= 8
3x + 2y - kz &= 6
(k - 2)x + ky - 8z &= 6 \end{align} where \(k\) is a real constant. The planes do not meet at a unique point.

1. Find the possible values of \(k\). [3 marks]
2. For each value of \(k\) found in part (a), identify the configuration of the given planes. Fully justify your answer, stating in each case whether or not the equations of the planes form a consistent system. [5 marks]

Three planes have equations, $$x - y + kz = 3$$ $$kx - 3y + 5z = -1$$ $$x - 2y + 3z = -4$$ Where \(k\) is a real constant. The planes do not meet at a unique point.

1. Find the possible values of \(k\) [3 marks]
2. There are two possible geometric configurations of the given planes. Identify each possible configurations, stating the corresponding value of \(k\) Fully justify your answer. [5 marks]
3. Given further that the equations of the planes form a consistent system, find the solution of the system of equations. [3 marks]

Matrix M is given by \(\mathbf{M} = \begin{pmatrix} k & 1 & -5 \\ 2 & 3 & -3 \\ -1 & 2 & 2 \end{pmatrix}\), where \(k\) is a constant.

1. Show that \(\det \mathbf{M} = 12(k - 3)\). [2]
2. Find a solution of the following simultaneous equations for which \(x \neq z\). $$4x^2 + y^2 - 5z^2 = 6$$ $$2x^2 + 3y^2 - 3z^2 = 6$$ $$-x^2 + 2y^2 + 2z^2 = -6$$ [3]
3. 1. Verify that the point \((2, 0, 1)\) lies on each of the following three planes. $$3x + y - 5z = 1$$ $$2x + 3y - 3z = 1$$ $$-x + 2y + 2z = 0$$ [1]
   2. Describe how the three planes in part (iii) (A) are arranged in 3-D space. Give reasons for your answer. [4]
4. Find the values of \(k\) for which the transformation represented by M has a volume scale factor of 6. [3]

The matrix \(\mathbf{M}\) is defined by $$\mathbf{M} = \begin{pmatrix} 12 & 30 & 8 \\ 18 & 25 & 20 \\ 19 & 50 & 16 \end{pmatrix}.$$

1. Given that \(\det \mathbf{M} = -1040\), give a geometrical interpretation of the solution to the following equation. [2] $$\begin{pmatrix} 12 & 30 & 8 \\ 18 & 25 & 20 \\ 19 & 50 & 16 \end{pmatrix} \begin{pmatrix} x \\ y \\ z \end{pmatrix} = \begin{pmatrix} 2668 \\ 3402 \\ 4581 \end{pmatrix}$$
2. Three hotels A, B, C each have different types of room available to book: single, double and family rooms. For each type of room, the price per night is the same in each of the three hotels. The table below gives, for each hotel, details of the number of each type of room and the total revenue per night when the hotel is full.

   \multirow{2}{*}{Hotel}	Types of room			\multirow{2}{*}{Total revenue}
   \cline{2-4}	Single	Double	Family
   A	12	30	8	£2,668
   B	18	25	20	£3,402
   C	19	50	16	£4,581

   Find the price per night of each type of room. [6]

$$\mathbf{M} = \begin{pmatrix} 2 & -1 & 1 \\ 3 & k & 4 \\ 3 & 2 & -1 \end{pmatrix} \quad \text{where } k \text{ is a constant}$$

1. Find the values of \(k\) for which the matrix \(\mathbf{M}\) has an inverse. [2]
2. Find, in terms of \(p\), the coordinates of the point where the following planes intersect \begin{align} 2x - y + z &= p
   3x - 6y + 4z &= 1
   3x + 2y - z &= 0 \end{align} [5]
3. 1. Find the value of \(q\) for which the set of simultaneous equations \begin{align} 2x - y + z &= 1
      3x - 5y + 4z &= q
      3x + 2y - z &= 0 \end{align} can be solved.
   2. For this value of \(q\), interpret the solution of the set of simultaneous equations geometrically. [4]

Three planes have equations \begin{align} 4x - 5y + z &= 8
3x + 2y - kz &= 6
(k - 2)x + ky - 8z &= 6 \end{align} where \(k\) is a real constant. The planes do not meet at a unique point.

1. Find the possible values of \(k\). [3 marks]
2. For each value of \(k\) found in part (a), identify the configuration of the given planes. Fully justify your answer, stating in each case whether or not the equations of the planes form a consistent system. [5 marks]

In the following set of simultaneous equations, \(a\) and \(b\) are constants. \begin{align} 3x + 2y - z &= 5
2x - 4y + 7z &= 60
ax + 20y - 25z &= b \end{align}

1. In the case where \(a = 10\), solve the simultaneous equations, giving your solution in terms of \(b\). [3]
2. Determine the value of \(a\) for which there is no unique solution for \(x\), \(y\) and \(z\). [3]
3. 1. Find the values of \(\alpha\) and \(\beta\) for which \(\alpha(2y - z) + \beta(-4y + 7z) = 20y - 25z\) for any \(y\) and \(z\). [3]
   2. Hence, for the case where there is no unique solution for \(x\), \(y\) and \(z\), determine the value of \(b\) for which there is an infinite number of solutions. [2]
   3. When \(a\) takes the value in part (ii) and \(b\) takes the value in part (iii)(b) describe the geometrical arrangement of the planes represented by the three equations. [1]