QCE Vault / Specialist Maths Matrices — Question 40 Original QCE Vault practice · 7 marks
Browse all questions Original practice exam Report an issue Q40 · Practice question Complex unfamiliar 7 marks
QUESTION 40 (7 marks) Consider the system x + y + z = 3 , 2 x + 3 y + 4 z = 8 , x + 2 y + ( a 2 − 5 a + 9 ) z = a + 3. \begin{aligned}x+y+z&=3,\\2x+3y+4z&=8,\\x+2y+(a^2-5a+9)z&=a+3.\end{aligned} x + y + z 2 x + 3 y + 4 z x + 2 y + ( a 2 − 5 a + 9 ) z = 3 , = 8 , = a + 3. Use Gaussian elimination to classify the system as having a unique solution, no solution or infinitely many solutions for all real values of a a a . Where the solution is unique, determine it in terms of a a a . WORKED SOLUTION
Practice marking scheme 7 marks ANSWER a = 2 a=2 a = 2 : infinitely many; a = 3 a=3 a = 3 : none; otherwise ( x , y , z ) = ( 1 + 1 / ( a − 3 ) , 2 − 2 / ( a − 3 ) , 1 / ( a − 3 ) ) (x,y,z)=(1+1/(a-3),\,2-2/(a-3),\,1/(a-3)) ( x , y , z ) = ( 1 + 1/ ( a − 3 ) , 2 − 2/ ( a − 3 ) , 1/ ( a − 3 )) . Worked solution
Apply R 2 ← R 2 − 2 R 1 R_2\leftarrow R_2-2R_1 R 2 ← R 2 − 2 R 1 and R 3 ← R 3 − R 1 R_3\leftarrow R_3-R_1 R 3 ← R 3 − R 1 : y + 2 z = 2 , y + ( a 2 − 5 a + 8 ) z = a . y+2z=2,\qquad y+(a^2-5a+8)z=a. y + 2 z = 2 , y + ( a 2 − 5 a + 8 ) z = a . Subtract the first reduced equation from the second: ( a 2 − 5 a + 6 ) z = a − 2 ⇒ ( a − 2 ) ( a − 3 ) z = a − 2. (a^2-5a+6)z=a-2\quad\Rightarrow\quad(a-2)(a-3)z=a-2. ( a 2 − 5 a + 6 ) z = a − 2 ⇒ ( a − 2 ) ( a − 3 ) z = a − 2. If a = 2 a=2 a = 2 , the last equation is 0 = 0 0=0 0 = 0 . Taking z = s z=s z = s gives ( x , y , z ) = ( 1 + s , 2 − 2 s , s ) (x,y,z)=(1+s,2-2s,s) ( x , y , z ) = ( 1 + s , 2 − 2 s , s ) , so there are infinitely many solutions. If a = 3 a=3 a = 3 , the last equation is 0 = 1 0=1 0 = 1 , so there is no solution. For a ≠ 2 , 3 a\ne2,3 a = 2 , 3 , back-substitution gives the unique solution z = 1 a − 3 , y = 2 − 2 a − 3 , x = 1 + 1 a − 3 . z=\frac1{a-3},\qquad y=2-\frac2{a-3},\qquad x=1+\frac1{a-3}. z = a − 3 1 , y = 2 − a − 3 2 , x = 1 + a − 3 1 . Eliminates to two reduced equations.
[2 marks] Obtains the factorised final equation.
[1 mark] Obtains the unique solution for other values.
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