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1. A curve has the equation 22113 xxy !

(i) Find an expression fordy

dx. [3]

(ii) Find the equation of the tangent to the curve at the point where

the curve crosses they-axis. [3]

2. A curve has equation14

3 2

!

x

xy ,

(i) finddx

dy, [3]

(ii)find also the coordinates of the stationary points. [4]

3. Differentiate the following with respect tox:

(a) 31

3y x

x! [2]

(b)2

2 1

xy

x

!

[3]

(c) y = 47 2 x [2]

(d)2

1!

xy

x[2]

(e) 22 xx [3]

(f) y = 4 3( 2) (2 1)x x [3]

4. Given that 34 ! xxy , expressdx

dyin the form

34

x

bax.

Find the value ofa and b. [3]

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6. Given that 2 2! y x x , finddy

dx. [3]

7. Given thatx

xy2

4 ! find the value ofdx

dywhenx =

4

1. [3]

8. Find the equation of the normal to the curve12

2

!

xxy where the curve cuts thex-axis. [6]

9.

The diagram shows part of the graph of the curve3

22

!

x

xxy .

(i) Given that2)3(

1d

d

!

x

k

x

y. Find the value ofk. [4]

(ii) R,Sand Tare intersections of the graph with the axes. PR is the normal tothe curve at R. Find the equation ofPR. [3]

(iii) Find the ratio of the area of triangle PSR to triangle TPR. [3]

5. A curve has the equationx

xy

21

3

! .

(i) Finddx

dy.

[2]

(ii) Find the equation of the normal to the curve at the pointy = 1. [4]

y

R

S P T x

3

22

!

x

xxy

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10. Given that2

)23( 10!

xy , find the value of

dx

dywhenx =

3

1.

[3]

11. (a) Differentiate with respect to x

(i) 47 2 x [2]

(ii)

x

x

31

52

[3]

12. For the curve 12)4( 2 ! xxy , calculate the coordinates of

(a) the points of intersection with the axis, [3]

(b) the turning points and determine the nature of the turning points. [7]

Hence sketch the curve. [2]

13. A curve has the equation2

42

!

x

xy .

(a)Find the gradient of the curve at the point where the curve meets the x-axis. [4]

(b)Show that the curve has no stationary points. [1]

14. A curve 16204 2 ! xxy cuts they-axis atA, thex-axis atB and C respectively.

(i) Write down the coordinates ofA,B and C. [3]

(ii) Find the coordinates of the point Pon the curve such that the

tangent to the curve at P is parallel to the lineAC. [4]

15. The equation of a curve is given by

2

3 , 11

xy xx

! {

.

i) Show that the gradient of the curve at the point Pwhere1

2x ! is -9. [3]

ii) Find the equation of the normal at point P. [2]

iii) This normal cuts thex andy axes at Q and R respectively. Find the coordinatesof the midpoint ofQR. [3]

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16. Given that 32 ! xxy , show that32

125 2

!

x

xx

dx

dy. Hence find the

equation of the normal to the curve at the point 4!x . [7]

17. (i) Express in partial fractions,)3)(2(

15622

2

xx

xx. [4]

(ii) Using the result of (i), find the first derivative of)3)(2(

15622

2

xx

xx, expressing your answer

in the form 22

2

232

x

cbx

x

a. State the value of the constants a,b and c. [4]

18. Given the equation of a curve is2)2(

12

!

x

xy .

Finddx

dyand the equation of the normal to the curve whenx = -4. [7]