# Past Papers’ Solutions | Edexcel | AS & A level | Mathematics | Core Mathematics 1 (C1-6663/01) | Year 2015 | June | Q#6

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Question

The curve C has equation , a. Find in its simplest form.

b. Find an equation of the tangent to C at the point where x=-1.

Give your answer in the form ax+by+c=0, where a, b and c are integers.

Solution

a.

We are given; We are required to find .

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:    b.

We are required to find the equation of tangent to the curve C at point where x=-1.

To find the equation of the line either we need coordinates of the two points on the line (Two-Point  form of Equation of Line) or coordinates of one point on the line and slope of the line (Point-Slope  form of Equation of Line).

We already have x-coordinate of a point on the tangent ie x=-1.

We can find y-coordinate of this point on the tangent by substituting x=-1 in equation of the curve  found in (a), because the same point also lies on the curve (this is the only point where tangent and  curve intersect).

Therefore, we substitute x=-1 in the following equation of curve C;     Hence, coordinates of a point on the tangent to curve C are (-1,10).

We need to find slope of tangent at in order to write its equation.

The slope of a curve at a particular point is equal to the slope of the tangent to the curve at the  same point;  Therefore, if we can find slope of the curve C at point P(-1,10) then we can find slope of the tangent to the curve at this point.

We need to find the gradient of the curve C at point P(-1,10).

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; We have found in (a); For gradient of the curve at point P(-1,10), substitute in derivative of the equation of the  curve.     Therefore; With coordinates of a point on the tangent P(-1,10) and its slope in hand, we can write equation of the tangent.

Point-Slope form of the equation of the line is;            