(a) (i) Calculate f(2)
(ii) Write f(x) as a product of two algebraic factors - Edexcel - A-Level Maths Pure - Question 8 - 2018 - Paper 2
Question 8
(a) (i) Calculate f(2)
(ii) Write f(x) as a product of two algebraic factors.
Using the answer to (a)(ii),
(b) prove that there are exactly two real solutions to ... show full transcript
Worked Solution & Example Answer:(a) (i) Calculate f(2)
(ii) Write f(x) as a product of two algebraic factors - Edexcel - A-Level Maths Pure - Question 8 - 2018 - Paper 2
Step 1
Calculate f(2)
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Answer
To calculate f(2), substitute x = 2 into the function f(x):
f(2)=−3(2)3+8(2)2−9(2)+10
Calculating this gives:
=−3(8)+8(4)−18+10 =−24+32−18+10 =0
Thus, f(2) = 0.
Step 2
Write f(x) as a product of two algebraic factors
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Answer
We want to express f(x) as a product of factors. Since we found that f(2) = 0, we can factor out (x - 2).
Using polynomial long division to divide f(x) by (x - 2):
Perform the division, finding the quotient:
f(x)=(x−2)(−3x2+2x−5)
Therefore, the two factors are:
f(x)=(x−2)(−3x2+2x−5)
Step 3
prove that there are exactly two real solutions to the equation –3y² + 8y – 9y² + 10 = 0
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Answer
First, simplify the equation:
−3y2+8y−9y2+10=−12y2+8y+10=0
To find the number of real solutions, we will use the discriminant method, calculating the discriminant D:
D=b2−4ac
Where a = -12, b = 8, and c = 10:
D=(8)2−4(−12)(10)=64+480=544>0
Since D > 0, this indicates that there are exactly two real solutions.
Step 4
deduce the number of real solutions, for 7π < θ < 10π, to the equation 3 tan²θ – 8 tanθ + 9 tanθ – 10 = 0
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Answer
First, simplify the trigonometric equation:
3an2(heta)+tan(heta)−10=0
Letting u=an(heta), we factor the quadratic:
3u2+u−10=(3u−5)(u+2)=0
Thus, we find:
u = rac{5}{3}, ext{ or } u = -2
Now, solve for θ:
For u = rac{5}{3}, there are infinitely many solutions.
For u=−2, we find corresponding angles. Over the interval 7π<θ<10π, we can determine how many of each solution fit in that range. Therefore, overall, we can deduce that the equation has three solutions in this interval.
