Q3 Cambridge International AS and A Level - Edubirdie

University:
University of Cambridge
Course:
0580 IGCSE | Mathematics
Academic year:

2023
Views:

397

Pages:

2
Author:

Shahbaz A.

Q3 Cambridge International AS and A Level 9709-22 February-March 2023 shahbaz ahmed July 2024 1 −2 −a a b −2a 6a −14a a −3a 7a −14a + b a −a a b 3a 6a 21a 2a 7a 21a + b a −14a + b = 0 b = 14a 3 a 21a + b = 35 Putting b = 14a 21a + 14a = 35 35a = 35 a=1 1 Putting in b = 14a b = 14 a=1 b=14 CHECKING Putting a=1,b=14 in P (x) = ax3 − ax2 + ax + b P (x) = x3 − x2 + x + 14 Putting x=-2 P (−2) = (−2)3 − (−2)2 + (−2) + 14 P (−2) = −8 − 4 − 2 + 14 = 0 =⇒ (x+2) is a factor of P(x) Putting x=3 P (3) = 33 − 32 + 3 + 14 = 27 − 9 + 3 + 14 = 35 = Remainder (b) Hence factorise P(x) and show that the equation P(x) = 0 has exactly one real root Solution P (x) = x3 − x2 + x + 14 Since P (−2) = 0 =⇒ x+2 is a factor P (x) = (x + 2)(x2 − 3x + 7) Now x2 − 3x + 7 = 0 Disc=(−3)2 − 4(1)(7) = 9 − 28 = −19 < 0 Roots of x2 − 3x + 7 = 0 will be complex numbers. Hence only real root x = −2 2

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