Solution of Past Paper of CAIE Math 9709/12 Feb/Mar 2020 Question # 10
4 views 0 purchase
Course
CAIE AS & A Level Mathematics P1 (9709/1)
Institution
CAIE AS & A Level Mathematics P1 (9709/1)
This document presents most carefully worked out and thoroughly explained complete Solution of Past Paper of Cambridge AS/A Level Mathematics (9709), Pure Mathematics 1, Feb/March 2020
We also provide on-on-one online help for students who still need some help beyond our provided solution.
We al...
solution of past paper 9709 12 february march 2020
Written for
CAIE AS & A Level Mathematics P1 (9709/1)
All documents for this subject (23)
Seller
Follow
hummairamubashar
Content preview
CIE/GCE/AS/Math/P1/20/Mar/12/Q#10
Question
1
𝑑𝑦
The gradient of a curve at the point (x, y) is given by = 2(𝑥 + 3)2 − 𝑥. The curve has
𝑑𝑥
a stationary point at (a, 14), where a is a positive constant.
a) Find the value of 𝑎.
b) Determine the nature of the stationary point.
c) Find the equation of the curve.
Solution
a)
We are given that gradient of the curve at the point (x, y) is;
𝑑𝑦 1
= 2 (𝑥 + 3 )2 − 𝑥
𝑑𝑥
We are also given that point at (a, 14) is a stationary point.
A stationary point 𝑆 (𝑥, 𝑦) on the curve 𝑦 is the point where gradient of the curve is
equal to zero;
𝑑𝑦
𝐺𝑟𝑎𝑑𝑖𝑒𝑛𝑡 𝑜𝑓 𝑡ℎ𝑒 𝐶𝑢𝑟𝑣𝑒𝐴𝑡 𝑆𝑡𝑎𝑡𝑖𝑜𝑛𝑎𝑟𝑦 𝑃𝑜𝑖𝑛𝑡 = | = 0
𝑑𝑥 𝑠
Therefore, at point (a, 14);
𝑑𝑦
=0
𝑑𝑥
Gradient (slope) of the curve at the particular point is the derivative of equation of the
curve at that particular point.
, CIE/GCE/AS/Math/P1/20/Mar/12/Q#10
𝑑𝑦
Gradient (slope) of the curve 𝑦 at a particular point 𝑃(𝑥, 𝑦) can be found by
𝑑𝑥
substituting x-coordinates of that point in the expression for gradient of the curve;
𝑑𝑦
|
𝑑𝑥 𝑥=𝑥𝑃
Hence, gradient at point (a, 14);
𝑑𝑦 1
| = 2(𝑥𝑃 + 3)2 − 𝑥𝑃
𝑑𝑥 𝑥=𝑥𝑃
𝑑𝑦 1
| = 2 (𝑎 + 3 )2 − 𝑎
𝑑𝑥 𝑥=𝑎
Coordinates of stationary point on the curve 𝑦 can be found from the derivative of
equation of the curve by equating it with ZERO. This results in value of x-coordinate of
the stationary point 𝑆(𝑥, 𝑦) on the curve.
Hence;
𝑑𝑦
| =0
𝑑𝑥 𝑥=𝑎
1
2 (𝑎 + 3 )2 − 𝑎 = 0
1
2 (𝑎 + 3 )2 = 𝑎
1 2
(2(𝑎 + 3)2 ) = (𝑎 )2
4 (𝑎 + 3 ) = 𝑎 2
4𝑎 + 12 = 𝑎2
𝑎2 − 4𝑎 − 12 = 0
𝑎2 − 6𝑎 + 2𝑎 − 12 = 0
The benefits of buying summaries with Stuvia:
Guaranteed quality through customer reviews
Stuvia customers have reviewed more than 700,000 summaries. This how you know that you are buying the best documents.
Quick and easy check-out
You can quickly pay through credit card or Stuvia-credit for the summaries. There is no membership needed.
Focus on what matters
Your fellow students write the study notes themselves, which is why the documents are always reliable and up-to-date. This ensures you quickly get to the core!
Frequently asked questions
What do I get when I buy this document?
You get a PDF, available immediately after your purchase. The purchased document is accessible anytime, anywhere and indefinitely through your profile.
Satisfaction guarantee: how does it work?
Our satisfaction guarantee ensures that you always find a study document that suits you well. You fill out a form, and our customer service team takes care of the rest.
Who am I buying these notes from?
Stuvia is a marketplace, so you are not buying this document from us, but from seller hummairamubashar. Stuvia facilitates payment to the seller.
Will I be stuck with a subscription?
No, you only buy these notes for $2.99. You're not tied to anything after your purchase.
