WMA11 June 2021

1.​​ WMA11/01 Edexcel​​ IAL P1 June 2021 Q1​​ (Differentiation & Straight Line​​ Equation)

The curve C has equation​​ 

y=x23+4x+83x-5               x>0

(a)​​ Find​​ dydx, giving your answer in simplest form.​​ 

(4)

The point P(4, 3) lies on C.​​ 

(b)​​ Find the equation of the normal to C at the point P. Write your answer in the form ax + by + c = 0, where a, b and c are integers to be found.​​ 

(4)

SOLUTION

a-​​ 

y=x23+4x+83x-5

y=x23+4x-12+83 x-1-5

dydx=23x1-2x-32-83x-2

dydx=23x-2x-32-83x-2

dydx=23x-2x32-8x2

dydx=23x-2x3-83x2

b-​​ 

To find the equation of the normal, first find the gradient of a tangent of a curve at point P and then find the gradient of normal line. At last, use the point slope formula to get the equation of the normal to C.​​ 

So let’s first find the gradient of tangent of curve.​​ 

mT=dydx=23x-2x3-83x2

mT=234-243

mT=8 3 x 42

mT=83-14-16

mT=6424-624-424

mT=5424

mT=94

Since normal line and tangent to the curve are​​ perpendicular, the product of their gradients is equal to -1.​​ 

mT×mN=-1

94×mN=-1

mN=-49

Now, using point slope formula to find the equation of normal, where​​ the point P(4, 3) and gradient of normal is -4/9.

y-y1=mN(x-x1)

y-3=-49(x-4)

9y-27= -4x-4

y-27=-4x+16

4x+9y-43=0