General solution with parameters

3

1. Find the inverse of the matrix \(\left( \begin{array} { r r r } 4 & 1 & k \\ 3 & 2 & 5 \\ 8 & 5 & 13 \end{array} \right)\), where \(k \neq 5\).
2. Solve the simultaneous equations $$\begin{aligned} & 4 x + y + 7 z = 12 \\ & 3 x + 2 y + 5 z = m \\ & 8 x + 5 y + 13 z = 0 \end{aligned}$$ giving \(x , y\) and \(z\) in terms of \(m\).
3. Find the value of \(p\) for which the simultaneous equations $$\begin{aligned} & 4 x + y + 5 z = 12 \\ & 3 x + 2 y + 5 z = p \\ & 8 x + 5 y + 13 z = 0 \end{aligned}$$ have solutions, and find the general solution in this case.

3

1. Find the inverse of the matrix $$\left( \begin{array} { r r r } 1 & 1 & a \\ 2 & - 1 & 2 \\ 3 & - 2 & 2 \end{array} \right)$$ where \(a \neq 4\).
   Show that when \(a = - 1\) the inverse is $$\frac { 1 } { 5 } \left( \begin{array} { r r r } 2 & 0 & 1 \\ 2 & 5 & - 4 \\ - 1 & 5 & - 3 \end{array} \right)$$
2. Solve, in terms of \(b\), the following system of equations. $$\begin{aligned} x + y - z & = - 2 \\ 2 x - y + 2 z & = b \\ 3 x - 2 y + 2 z & = 1 \end{aligned}$$
3. Find the value of \(b\) for which the equations $$\begin{aligned} x + y + 4 z & = - 2 \\ 2 x - y + 2 z & = b \\ 3 x - 2 y + 2 z & = 1 \end{aligned}$$ have solutions. Give a geometrical interpretation of the solutions in this case. Section B (18 marks)

3

1. Find the value of \(a\) for which the matrix $$\mathbf { M } = \left( \begin{array} { r r r } 1 & 2 & 3 \\ - 1 & a & 4 \\ 3 & - 2 & 2 \end{array} \right)$$ does not have an inverse.
   Assuming that \(a\) does not have this value, find the inverse of \(\mathbf { M }\) in terms of \(a\).
2. Hence solve the following system of equations. $$\begin{aligned} x + 2 y + 3 z & = 1 \\ - x + 4 z & = - 2 \\ 3 x - 2 y + 2 z & = 1 \end{aligned}$$
3. Find the value of \(b\) for which the following system of equations has a solution. $$\begin{aligned} x + 2 y + 3 z & = 1 \\ - x + 6 y + 4 z & = - 2 \\ 3 x - 2 y + 2 z & = b \end{aligned}$$ Find the general solution in this case and describe the solution geometrically.

10 Find the set of values of \(a\) for which the system of equations $$\begin{aligned} x + 4 y + 12 z & = 5 \\ 2 x + a y + 12 z & = a - 1 \\ 3 x + 12 y + 2 a z & = 10 \end{aligned}$$ has a unique solution. Show that the system does not have any solution in the case \(a = 18\). Given that \(a = 8\), show that the number of solutions is infinite and find the solution for which \(x + y + z = 1\).

10 Find the set of values of \(a\) for which the system of equations $$\begin{aligned} x - 2 y - 2 z & = - 7 \\ 2 x + ( a - 9 ) y - 10 z & = - 11 \\ 3 x - 6 y + 2 a z & = - 29 \end{aligned}$$ has a unique solution. Show that the system has no solution in the case \(a = - 3\). Given that \(a = 5\),

1. show that the number of solutions is infinite,
2. find the solution for which \(x + y + z = 2\).

2 Find the value of the constant \(k\) for which the system of equations $$\begin{aligned} 2 x - 3 y + 4 z & = 1 \\ 3 x - y & = 2 \\ x + 2 y + k z & = 1 \end{aligned}$$ does not have a unique solution. For this value of \(k\), solve the system of equations.

has the form \(\left( \begin{array} { r } 1 \\ - 2 \\ 2 \\ - 1 \end{array} \right) + \lambda \mathbf { e } _ { 1 } + \mu \mathbf { e } _ { 2 }\), where \(\lambda\) and \(\mu\) are scalars and \(\left\{ \mathbf { e } _ { 1 } , \mathbf { e } _ { 2 } \right\}\) is a basis for \(K\). Hence obtain a solution \(\mathbf { x } ^ { \prime }\) of ( \(*\) ) such that the sum of the components \(\mathbf { x } ^ { \prime }\) is 6 and the sum of the squares of the components of \(\mathbf { x } ^ { \prime }\) is 26 . {www.cie.org.uk} after the live examination series. }

4

1. Find the value of \(k\) for which the set of linear equations $$\begin{aligned} x + 3 y + k z & = 4 \\ 4 x - 2 y - 10 z & = - 5 \\ x + y + 2 z & = 1 \end{aligned}$$ has no unique solution.
2. For this value of \(k\), find the set of possible solutions, giving your answer in the form $$\left( \begin{array} { c } x \\ y \\ z \end{array} \right) = \mathbf { a } + t \mathbf { b } ,$$ where \(\mathbf { a }\) and \(\mathbf { b }\) are vectors and \(t\) is a scalar.

2 Show that the matrix \(\left( \begin{array} { r r r } 1 & 4 & 2 \\ 3 & 0 & - 2 \\ 3 & - 3 & - 4 \end{array} \right)\) has no inverse. Solve the system of equations $$\begin{array} { r } x + 4 y + 2 z = 0 \\ 3 x - 2 z = 4 \\ 3 x - 3 y - 4 z = 5 \end{array}$$

5 Find the value of \(a\) for which the system of equations $$\begin{aligned} & x - y + 2 z = 4 \\ & x + a y - 3 z = b \\ & x - y + 7 z = 13 \end{aligned}$$ where \(a\) and \(b\) are constants, has no unique solution. Taking \(a\) as the value just found,

1. find the general solution in the case \(b = - 5\),
2. interpret the situation geometrically in the case \(b \neq - 5\).

10 The matrix \(\mathbf { A }\) is defined by $$\mathbf { A } = \left( \begin{array} { r r r } 1 & 5 & 1 \\ 1 & - 2 & - 2 \\ 2 & 3 & \theta \end{array} \right)$$

1. (a) Find the rank of \(\mathbf { A }\) when \(\theta \neq - 1\).
   (b) Find the rank of \(\mathbf { A }\) when \(\theta = - 1\).
   Consider the system of equations $$\begin{aligned} x + 5 y + z & = - 1 \\ x - 2 y - 2 z & = 0 \\ 2 x + 3 y + \theta z & = \theta \end{aligned}$$
2. Solve the system of equations when \(\theta \neq - 1\).
3. Find the general solution when \(\theta = - 1\).
4. Show that if \(\theta = - 1\) and \(\phi \neq - 1\) then \(\mathbf { A } \mathbf { x } = \left( \begin{array} { r } - 1 \\ 0 \\ \phi \end{array} \right)\) has no solution.

8

1. Find the solution to the following simultaneous equations. $$\begin{array} { r r r } x + y + & z = & 3 \\ 2 x + 4 y + 5 z = & 9 \\ 7 x + 11 y + 12 z = & 20 \end{array}$$
2. Determine the values of \(p\) and \(k\) for which there are an infinity of solutions to the following simultaneous equations. $$\begin{array} { r r r l } x + & y + & z = & 3 \\ 2 x + & 4 y + & 5 z = & 9 \\ 7 x + & 11 y + & p z = & k \end{array}$$

The matrix \(\mathbf{A}\) is defined by $$\mathbf{A} = \begin{pmatrix} 1 & 5 & 1 \\ 1 & -2 & -2 \\ 2 & 3 & \theta \end{pmatrix}.$$

1. Find the rank of \(\mathbf{A}\) when \(\theta \neq -1\). [3]
2. Find the rank of \(\mathbf{A}\) when \(\theta = -1\). [1]

Consider the system of equations \begin{align} x + 5y + z &= -1,
x - 2y - 2z &= 0,
2x + 3y + \theta z &= \theta. \end{align}

1. Solve the system of equations when \(\theta \neq -1\). [3]
2. Find the general solution when \(\theta = -1\). [3]
3. Show that if \(\theta = -1\) and \(\phi \neq -1\) then \(\mathbf{A}\mathbf{x} = \begin{pmatrix} -1 \\ 0 \\ \phi \end{pmatrix}\) has no solution. [2]

The matrix \(\mathbf{A}\) is given by \(\mathbf{A} = \begin{pmatrix} a & 2 & 3 \\ 4 & 4 & 6 \\ -2 & 2 & 9 \end{pmatrix}\) where \(a\) is a constant. It is given that if \(\mathbf{A}\) is not singular then $$\mathbf{A}^{-1} = \frac{1}{24a-48} \begin{pmatrix} 24 & -12 & 0 \\ -48 & 9a+6 & 12-6a \\ 16 & -2a-4 & 4a-8 \end{pmatrix}.$$

1. Use \(\mathbf{A}^{-1}\) to solve the simultaneous equations below, giving your answer in terms of \(k\). \begin{align} x + 2y + 3z &= 6
   4x + 4y + 6z &= 8
   -2x + 2y + 9z &= k \end{align} [3]
2. Consider the equations below where \(a\) takes the value which makes \(\mathbf{A}\) singular. \begin{align} ax + 2y + 3z &= b
   4x + 4y + 6z &= 10
   -2x + 2y + 9z &= -13 \end{align} \(b\) takes the value for which the equations have an infinite number of solutions.
3. For the equations in part (ii) with the values of \(a\) and \(b\) found in part (ii) describe fully the geometrical arrangement of the planes represented by the equations. [2]