Fsc Part 2 Mathematics (Complete Solutions)

Q5
Find the values of a and b.

Given f(x) = x3 - ax2 + bx + 1
If f(2) = -3 and f(-1) = 0. Find the values of a and b.

Solutions

f(x) = x3 - ax2 + bx + 1
As f(2) = -3
(2)3 - a(2)2 + b(2) + 1 = -3
(2)3 - a(2)2 + b(2) + 1 = -3
8 - 4a + 2b + 1 = -3
- 4a + 2b + 9 = -3
- 4a + 2b = -3 - 9
- 4a + 2b = -12
-2(2a - b) = -12
2a - b = 6 --------------------------- (i)


As f(-1) = 0
(-1)3 - a(-1)2 + b(-1) + 1 = 0
-1 - a - b + 1 = 0
-a - b = 0
-1(a + b) = 0
a + b = 0 --------------------------- (ii)
Add (i) and (ii)
2a - b = 6
a + b = 0
------------
3a + 0 = 6

3a = 6
a = 2

Put value of 'a' in equation (ii)
a + b = 0
2 + b = 0
b = -2



Answer: a = 2 and b = -2

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