Past Papers’ Solutions  Edexcel  AS & A level  Mathematics  Core Mathematics 1 (C16663/01)  Year 2017  June  Q#10
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Question
Figure shows a sketch of part of the curve y = f(x), , where
f(x) = (2x – 5)^{2}(x + 3)
a. Given that
i. the curve with equation y = f(x) – k, , passes through the origin, find the value of the constant k,
ii. the curve with equation y = f(x + c), , has a minimum point at the origin, find the value of the constant c.
b. Show that
Points A and B are distinct points that lie on the curve y = f(x).
The gradient of the curve at A is equal to the gradient of the curve at B.
Given that point A has x coordinate is 3.
c. find the x coordinate of point B.
Solution
a.
i.
We are given that equation of the curve is;
We are also given that passes through origin. Therefore, O(0,0) lies on the curve with equation;
If a point lies on the curve (or the line), the coordinates of that point satisfy the equation of the curve (or the line).
Therefore, we can substitute the coordinates of origin in given equation of the curve and it must satisfy the equation.
ii.
We are given that equation of the curve is;
We are also given that has a minimum point at the origin.
First we find the coordinates of the minimum point for the given curve with equation;
We can see from the diagram that minimum point is also the xintercept of the curve.
The point at which curve (or line) intercepts xaxis, the value of . So we can find the value of coordinate by substituting in the equation of the curve (or line).
Therefore, we substitute in equation of the curve.
Now we have two options.











It is evident from the diagram that does not represent the minimum point of the curve because minimum point of the curve is on positive side of the xaxis. Hence, minimum point has x coordinate .
The given equation represents translation of graph along xaxis.
Translation through vector represents the move, units in the negative xdirection and units in the ydirection.
Translation through vector transforms the function into .
Transformation of the function into results from translation through vector .
Translation through vector transforms the function into which means shift towards left along xaxis.
Original 
Transformed 
Translation Vector 
Movement 

Function 



units in 
Coordinates 


If we need to translate the graph such that minimum point shifts to origin, it must move to the left side (towards negative xaxis) equal to xcoordinate of minimum point ie . Hence;
b.
We are required to find the expression for gradient of the curve.
Gradient (slope) of the curve is the derivative of equation of the curve. Hence gradient of curve with respect to is:
Therefore,
Rule for differentiation is of is:
Rule for differentiation is of is:
Rule for differentiation is of is:
c.
We are given that the gradient of the curve at A is equal to the gradient of the curve at B.
Gradient (slope) of the curve at the particular point is the derivative of equation of the curve at that particular point.
Gradient (slope) of the curve at a particular point can be found by substituting x coordinates of that point in the expression for gradient of the curve;
Therefore;
We are also given that xcoordinates of the point A is 3. Therefore, we can find gradient of the curve at point A.
From (b) we have found that;
Hence;
Therefore, as per given condition;
We can also write as;
We can solve this equation to find the xcoordinates of all such points where gradient of the curve is 25.
Now we have two options.







Therefore, at two points (one with xcoordinate 3 and other with ) the gradient of the curve is 25 and we are given that these two points are A and B.
We are given that xcoordinate of point A is 3, therefore, must be xcoordinate of the point B.
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