15a Differentiation and integration

0606 Additional Mathematics Unit 15a Differentiation and integration

0606/12/O/N/15 11

9  You are not allowed to use a calculator in this question.

y  (i)  Find 4 + xdx. [2]
emy Alvin AccaaddeemmyyAlvicnademy  (ii)
y B
A
Acad lvin A lvin Ay = √4+x
lvin y A y A    The diagram shows the graph of y = 4 + x, which meets the y-axis at the point A and the line
O 5x

A em emx = 5 at the point B. Using your answer to part (i), find the area of the region enclosed by the
Acad Alvin Acadcurve and the straight line AB. [5]

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0606 Additional Mathematics Unit 15a Differentiation and integration

0606/22/O/N/15 10
7

Academy lvinrcm 8 cm
Alvin emy A emyhcm
cademy in Acad in Acad6cm
A lv lv  A cone, of height 8 cm and base radius 6 cm, is placed over a cylinder of radius r cm and height h cm
in A Aand is in contact with the cylinder along the cylinder’s upper rim. The arrangement is symmetrical and
lv y ythe diagram shows a vertical cross-section through the vertex of the cone.
A AcadeAmlvin Academ  (i)  Use similar triangles to express h in terms of r. [2]

  (ii)  Hence show that the volume, V cm3, of the cylinder is given by V = 8rr2 - 4 rr3 . [1]
3

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0606 Additional Mathematics Unit 15a Differentiation and integration

0606/22/O/N/15 11

7  (iii)  Given that r can vary, find the value of r which gives a stationary value of V. Find this stationary

value of V in terms of π and determine its nature. [6]

AlvinAAccaaddeAemmlyvyinAAlAlvvicinandAAemccaayddeAemmlyvyinAlAvicnademy

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0606 Additional Mathematics Unit 15a Differentiation and integration

0606/13/O/N/15 7

5  Find the equation of the normal to the curve y = 5 tan x - 3 at the point where x = r . [5]
4

AlvinAAccaaddeAemmlyvyinAAlAlvvicinandAAemccaayddeAemmlyvyinAlAvicnademy

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0606 Additional Mathematics Unit 15a Differentiation and integration

0606/13/O/N/15 9

7  A curve, showing the relationship between two variables x and y, is such that d2y = 6 cos 3x . Given
d x2
3 at the point cr ,
9
AlvinAAccaaddeAemmlyvyinAAlAlvvicinandAAemccaayddeAemmlyvyinAlAvicnademy that the curve has a gradient of 4- 31m, find the equation of the curve. [6]

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0606 Additional Mathematics Unit 15a Differentiation and integration

0606/23/O/N/15 3 at the point where x = 2. [5]
1  Find the equation of the tangent to the curve y = x3 + 3x2 - 5x - 7

AlvinAAccaaddeAemmlyvyinAAlAlvvicinandAAemccaayddeAemmlyvyinAlAvicnademy

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0606 Additional Mathematics Unit 15a Differentiation and integration

0606/23/O/N/15 5

3  (a)  Given that y = x3 , find dy . [3]
2 - x2 dx [3]

  (Ab) lGvivienn tAhaAt cyca=adxde4Aexmm+l6yv,ysihnoAwAtlhAlvatvicddinxyan=dAkAe(4xxmc++ca1a6)ydadnedAemsmtalteyvythienAvalluAve oicfnka. demy

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0606 Additional Mathematics Unit 15a Differentiation and integration

0606/23/O/N/15 12

10   (i)  Given that d ^e2 - x2h = kxe2 - x2 , state the value of k. [1]
dx [2]

lvin AcademyyAlvin yy  (ii)  Using your result from part (i), find 3xe2-x2dx. [2]
AlvinAAccaaddeemmlyvyinAAlAvicnadAemcaydeAmlvin Academ  (iii)  Hence find the area enclosed by the curve y = 3xe2-x2, the x-axis and the lines x = 1 and

Ax = 2.

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0606 Additional Mathematics Unit 15a Differentiation and integration

0606/23/O/N/15 13

1 0 (iv)  Find the coordinates of the stationary points on the curve y = 3xe2-x2. [4]

AlvinAAccaaddeAemmlyvyinAAlAlvvicinandAAemccaayddeAemmlyvyinAlAvicnademy

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