Theory question 110
Surds(a) If (√5 + 4)/(3 − 2√5) − (2 + √5)/(4 − 2√5) = a + b√5, find the values of a and b.
(b)(i) Evaluate: |2 −1 2 / 1 3 4 / 1 2 1| (rows separated by /). (ii) Using the result in 10(b)(i), find, correct to two decimal places the value of x in the system of equations: 2x − y + 2z + 5 = 0, x + 3y + 4z − 1 = 0, x + 2y + z + 2 = 0.
Show the model answer
Model answer
(a) (√5 + 4)/(3 − 2√5) = (√5 + 4)(3 + 2√5)/(−11) = −2 − √5.
(2 + √5)/(4 − 2√5) = (2 + √5)(4 + 2√5)/(−4) = −9/2 − 2√5.
Difference = 5/2 + √5: a = 5/2, b = 1.
(b)(i) 2(3 − 8) + 1(1 − 4) + 2(2 − 3) = −15.
(ii) Right-hand sides −5, 1, −2. Δx = |−5 −1 2 / 1 3 4 / −2 2 1| = 25 + 9 + 16 = 50.
x = 50/(−15) ≈ −3.33.
Explanation
Cramer's rule: x = Δx/Δ.