AlvinAADccaiadfdefAememlryvyeinAnAltAIlvivicGainantdCiAAeomcSncaaydEdaeAemnmlPdyvyiIanAnlsAtvUeictnagndYrietamet1yia5orna
0606 Additional Mathematics Unit 15a Differentiation and integration
2
Mathematical Formulae
cademyQuadratic Equation 1. ALGEBRA
in A AlvinFor the equation ax2 + bx + c = 0, x = −b b2 − 4ac
2a
emy( ) Alv( ) cadem( )y cademyBinomial Theorem
cad ( ) in A in Awhere n is a positive integer and(a + b)n = an +nan–1b+nan–2b2+…+nan–rbr + … + bn,
1 2 r
n n!
r – r)!r!
AlvinAAcadeAmlyvinAlAvcademy AlvIdentities = (n
2. TRIGONOMETRY
sin2 A + cos2 A = 1
sec2 A = 1 + tan2 A
cosec2 A = 1 + cot2 A
Formulae for ∆ABC
a = b = c
sin A sin B sin C
a2 = b2 + c2 – 2bc cos A
∆ = 1 bc sin A
2
www.alvinacademy.com !2 of !59
© UCLES 2015 0606/12/F/M/15
0606 Additional Mathematics Unit 15a Differentiation and integration
0606/11/M/J/14 7
5 (i) Given that y = e x2 , find dy . [2]
dx [2]
[2]
((iAiiii)) lHUvesiencnyeoeuAvrAaalncuscawateaedryd0t2oexAepmeaxmr2ldt x(yvi.y)itonAfAinldAlvyvicxeinaxn2ddxA.AemccaayddeAemmlyvyinAlAvicnademy
www.alvinacademy.com 0606/11/M/J/14 !3 of !59
[Turn over
© UCLES 2014
0606 Additional Mathematics Unit 15a Differentiation and integration
0606/11/M/J/14 9
7 A curve is such that dy = 4x + ^x 1 for x 2 0. The curve passes through the point KJ1 , 5NO.
dx + 1h2 L2 6P
(i) Find the equation of the curve.AlvinAAccaaddeAemmlyvyinAAlAlvvicinandAAemccaayddeAemmlyvyinAlAvicnademy (ii) Find the equation of the normal to the curve at the point where x = 1.[4]
[4]
www.alvinacademy.com 0606/11/M/J/14 !4 of !59
[Turn over
© UCLES 2014
0606 Additional Mathematics Unit 15a Differentiation and integration
0606/21/M/J/14 8
7 Given that a curve has equation y = 1 + 2 x , where x 2 0, find
x
dy
(i) dx , [2]
in AcademyAlvin [2]
cademy Alvin Academyin Academy Hence, or otherwise, find(ii) d2y. [4]
AlvinAAcadeAmlyvinAlAvcademy Alv (iii) the coordinates and nature of the stationary point of the curve.dx2
www.alvinacademy.com 5! of !59
© UCLES 2014 0606/21/M/J/14
0606 Additional Mathematics Unit 15a Differentiation and integration
0606/21/M/J/14 12
10 Find dy when
dx
((Aiii)) lvyy ==inclot+asAnl2nAxxxcs.icnaLKJa3xdNPOd,eAemmlyvyinAAlAlvvicinandAAemccaayddeAemmlyvyinAlAvicnademy [4]
[4]
www.alvinacademy.com !6 of !59
© UCLES 2014 0606/21/M/J/14
0606 Additional Mathematics Unit 15a Differentiation and integration
0606/12/M/J/14 6
4 The region enclosed by the curve y = 2 sin 3x, the x-axis and the line x = a , where
0 1 a 1 1 radian, lies entirely above the x-axis. Given that the area of this region is 1 square unit,
3
AlvinAAccaaddeAemmlyvyinAAlAlvvicinandAAemccaayddeAemmlyvyinAlAvicnademyfind the value of a.
[6]
www.alvinacademy.com 7! of !59
© UCLES 2014 0606/12/M/J/14
0606 Additional Mathematics Unit 15a Differentiation and integration
0606/12/M/J/14 11
9 A solid circular cylinder has a base radius of r cm and a volume of 4000 cm3.
(i) Show that the total surface area, A cm2, of the cylinder is given by A = 8000 + 2rr2. [3]
r
emy Alvin AccaaddeemmyyAlvicnademy (ii) Given that r can vary, find the minimum total surface area of the cylinder, justifying that this area
AlvinAAccaaddeAmlyvinAlAvicnadAemy Alvin Ais a minimum. [6]
www.alvinacademy.com 0606/12/M/J/14 !8 of !59
[Turn over
© UCLES 2014
0606 Additional Mathematics Unit 15a Differentiation and integration
0606/22/M/J/14 8
7 Differentiate with respect to x
((i(Aiiiii))) llx1vn4s+^eii2n3nxx+,xcA.oAsxchc,aaddeAemmlyvyinAAlAlvvicinandAAemccaayddeAemmlyvyinAlAvicnademy [2]
[2]
[3]
www.alvinacademy.com 9! of !59
© UCLES 2014 0606/22/M/J/14
0606 Additional Mathematics Unit 15a Differentiation and integration
0606/22/M/J/14 10
9 A curve is such that dy = ^2x + 1 The curve passes through the point (4, 10).
dx
1h2 .
((Aiii)) lFFviinniddntyheAydAexqcuacanatdiaodhndeoenfcAemethmeelvcyvaulyurivanetAe.Ayl0A1lv.5vyicdinxan. dAAemccaayddeAemmlyvyinAlAvicnademy [4]
[5]
www.alvinacademy.com 1! 0 of !59
© UCLES 2014 0606/22/M/J/14
0606 Additional Mathematics Unit 15a Differentiation and integration
0606/22/M/J/14 14
12 A curve has equation y = x3 - 9x2 + 24x.
(i) Find the set of values of x for which dy H 0. [4]
lvin AcademyyAAllvviinnAAccaayddeAemmlyvyinAlAvicnademy The normal to the curve at the point on the curve where x = 3 cuts the y-axis at the point P.dx[5]
A AcadeAmlvin Academ (ii) Find the equation of the normal and the coordinates of P.
www.alvinacademy.com !11 of !59
© UCLES 2014 0606/22/M/J/14
0606 Additional Mathematics Unit 15a Differentiation and integration
0606/13/M/J/14 5
4 The diagram shows a thin square sheet of metal measuring 24 cm by 24 cm. A square of side x cm is
cut off from each corner. The remainder is then folded to form an open box, x cm deep, whose square
ybase is shown shaded in the diagram.
Academ lvin24cm24 cm
Alvin emy A emyxcm
emy cad cadxcm [2]
AlvinAAccaaddeAmlyvinAlAvicnadAemy Alvin A(i) Show that the volume, Vcm3, of the box is given by V = 4x3 - 96x2 + 576x.
(ii) Given that x can vary, find the maximum volume of the box. [4]
www.alvinacademy.com 0606/13/M/J/14 1! 2 of !59
[Turn over
© UCLES 2014
0606 Additional Mathematics Unit 15a Differentiation and integration
0606/13/M/J/14 7
6 Find the equation of the normal to the curve y = x (x2 - 12) 1 at the point on the curve
3
where x = 2. [6]
AlvinAAccaaddeAemmlyvyinAAlAlvvicinandAAemccaayddeAemmlyvyinAlAvicnademy
www.alvinacademy.com 0606/13/M/J/14 !13 of !59
[Turn over
© UCLES 2014
0606 Additional Mathematics Unit 15a Differentiation and integration
0606/13/M/J/14 14
11 The diagram shows the graph of y = cos 3x + 3 sin 3x, which crosses the x-axis at A and has a
maximum point at B.
y y = cos 3x + 3 sin 3x
emyB A
lvin Acad y Alvin yO x
demy A Academ Academ(i) Find the x-coordinate of A.
AlvinAAccaadeAmlyvinAlAvicnademy Alvin(ii) [3]
Find dy and hence find the x-coordinate of B. [4]
dx
www.alvinacademy.com !14 of !59
© UCLES 2014 0606/13/M/J/14
0606 Additional Mathematics Unit 15a Differentiation and integration
0606/13/M/J/14 15
11 (iii) Showing all your working, find the area of the shaded region bounded by the curve, the x-axis
and the line through B parallel to the y-axis. [5]
AlvinAAccaaddeAemmlyvyinAAlAlvvicinandAAemccaayddeAemmlyvyinAlAvicnademy
www.alvinacademy.com !15 of !59
© UCLES 2014 0606/13/M/J/14
0606 Additional Mathematics Unit 15a Differentiation and integration
0606/23/M/J/14 10
8 A curve is such that dy = 6x2 - 8x + 3.
dx
(i) Show that the curve has no stationary points. [2]
[2]
emy Alvin AccaaddeemmyyAlvicnademy Given that the curve passes through the point P(2,10), [4]
d A A (ii) find the equation of the tangent to the curve at the point P,
AlvinAAccaadeAmlyvinAlAvicnademy Alvin (iii) find the equation of the curve.
www.alvinacademy.com !16 of !59
© UCLES 2014 0606/23/M/J/14
0606 Additional Mathematics Unit 15a Differentiation and integration
0606/23/M/J/14 12
10 (i) Given that y = 2x , show that dy = k , where k is a constant to be found. [5]
x2 + 21 dx ^x2 + 21h3
AlvinAAccaaddeAemmlyvyinAAlAlvvicinandAAemccaayddeAemmlyvyinAlAvicnademy
www.alvinacademy.com 1! 7 of !59
© UCLES 2014 0606/23/M/J/14
0606 Additional Mathematics Unit 15a Differentiation and integration
0606/23/M/J/14 13
1 0 (Aii) lHveincne fAinAdceddccaa^dxd2e6+Aem2m1lh3yvdyxinaAnAdleAlvvavilcuinaantedddecA21Ae0 mc^caxa2yd6+d2e1Aemh3mdlxyv.yinAlAvicnademy [3]
www.alvinacademy.com 0606/23/M/J/14 !18 of !59
[Turn over
© UCLES 2014
0606 Additional Mathematics Unit 15a Differentiation and integration
0606/11/O/N/14 3
1 Find the coordinates of the stationary point on the curve y = x2 + 16 . [4]
x [3]
2 (Aa) lOvnitnheOaAy642xAesccbaealodwde, Asemkmetlc4yvh5y°tihenAcAulrAvlvevicinayn9=0d°A3Aecmcosca2axyd-de1Aem1f3mo5lr°yv0y°iGnAx GlAv181i0c8°n0a.°dexmy
–2
–4
–6
(b) (i) State the amplitude of 1 - 4 sin 2x. [1]
(ii) State the period of 5 tan 3x + 1. [1]
www.alvinacademy.com 0606/11/O/N/14 1! 9 of !59
[Turn over
© UCLES 2014
0606 Additional Mathematics Unit 15a Differentiation and integration
0606/11/O/N/14 4
3 A curve is such that dy = 2 for x 2-3. The curve passes through the point (6, 10).
dx x+3
(i) Find the equation of the curve. [4]
[1]
AlvinAAccaaddeAemmlyvyinAAlAlvvicinandAAemccaayddeAemmlyvyinAlAvicnademy (ii) Find the x-coordinate of the point on the curve where y = 6.
www.alvinacademy.com 2! 0 of !59
© UCLES 2014 0606/11/O/N/14
0606 Additional Mathematics Unit 15a Differentiation and integration
0606/11/O/N/14 6
5 (i) Find the equation of the tangent to the curve y = x3 - ln x at the point on the curve [4]
where x = 1. [2]
(Aii) lSvhoiwnthAaAt tchciasatdandgeeAemntmbliyvseycitsnAtAhelAllvinviecinjaonindiAnAeg mtchceaapydodinetAsem (m-l2yv,y1i6)nA anldAv(1i2c,n2a).demy
www.alvinacademy.com !21 of !59
© UCLES 2014 0606/11/O/N/14
0606 Additional Mathematics Unit 15a Differentiation and integration
0606/21/O/N/14 9
8 (i) Given that y = 2 x2 , show that dy = (2 kx , where k is a constant to be found. [3]
+ x2 dx + x2) 2 [2]
(Aii) lHveincne fAinAdcedcdca(2ad+dxxe2Aem)2mdlxyv.yinAAlAlvvicinandAAemccaayddeAemmlyvyinAlAvicnademy
www.alvinacademy.com 0606/21/O/N/14 !22 of !59
[Turn over
© UCLES 2014
0606 Additional Mathematics Unit 15a Differentiation and integration
0606/21/O/N/14 14
12 (i) Show that x - 2 is a factor of 3x3 - 14x2 + 32. [1]
[4]
(Aii) lHveincne fAaActocrciaseadde3Aexm3m-lyv1y4ix2nA+A3lA2lvviccinaonmdpAlAeetmeclcya.ayddeAemmlyvyinAlAvicnademy
www.alvinacademy.com 2! 3 of !59
© UCLES 2014 0606/21/O/N/14
0606 Additional Mathematics Unit 15a Differentiation and integration
0606/21/O/N/14 15
1 2 The diagram below shows part of the curve y = 3x - 14 + 32 cutting the x-axis at the points P and Q.
x2
y y = 3x – 14 + 32
x2
Academy lvinO P
emy Alvin cademy A cademy (iii) State the x-coordinates of P and Q.
Qx
AlvinAAccaaddeAmlyvinAlAvicnadAemy Alvin Ay (iv) Find [1]
(3x - 14 + 32 ) dx and hence determine the area of the shaded region. [4]
x2
www.alvinacademy.com !24 of !59
© UCLES 2014 0606/21/O/N/14
0606 Additional Mathematics Unit 15a Differentiation and integration
0606/13/O/N/14 10
8 (i) Given that f ^xh = x ln x3, show that f l^xh = 3^1 + ln xh. [3]
[2]
(Aii) lHveincne fAinAdcyca^1ad+delnAemxhmdlxyv.yinAAlAlvvicinandAAemccaayddeAemmlyvyinAlAvicnademy
y (iii) Hence find 2 lnx dx in the form p + ln q, where p and q are integers. [3]
1
www.alvinacademy.com 2! 5 of !59
© UCLES 2014 0606/13/O/N/14
0606 Additional Mathematics Unit 15a Differentiation and integration
0606/13/O/N/14 14
11 The diagram shows part of the curve y = ^x + 5h^x - 1h2.
(Ai) lFvinidntheAAx-ccocaoraddidneatAemesmolfyvtyhienAstAatliAolvnvaicryinanpodOiAnAetsmcyocfatahydedceuAermvmel. yvyinAlAvyic=na(xd+ e5)m(xx –y1)2 [5]
www.alvinacademy.com 2! 6 of !59
© UCLES 2014 0606/13/O/N/14
0606 Additional Mathematics Unit 15a Differentiation and integration
0606/13/O/N/14 15
y1 1 (ii) Find ^x + 5h^x - 1h2 dx. [3]
[2]
(iAii) lHveincne fAinAdcthcaeaadrdeeaAeemnmclloyvsyeidnAbAy ltAhlvevcicuinravnedaAnAedmcthcaeaxyd-daxeiAesm.mlyvyinAlAvicnademy
(iv) Find the set of positive values of k for which the equation ^x + 5h^x - 1h2 = k has only one real
solution. [2]
www.alvinacademy.com !27 of !59
© UCLES 2014 0606/13/O/N/14
0606 Additional Mathematics Unit 15a Differentiation and integration
0606/23/O/N/14 14
9 (i) Determine the coordinates and nature of each of the two turning points on the
A lcvurivne AAy =cca4axd+dexAem-1m2l.yvyinAAlAlvvicinandAAemccaayddeAemmlyvyinAlAvicnademy [6]
www.alvinacademy.com !28 of !59
© UCLES 2014 0606/23/O/N/14
0606 Additional Mathematics Unit 15a Differentiation and integration
0606/23/O/N/14 15
9 (ii) Find the equation of the normal to the curve at the point (3, 13) and find the x-coordinate of the
point where this normal cuts the curve again. [6]
AlvinAAccaaddeAemmlyvyinAAlAlvvicinandAAemccaayddeAemmlyvyinAlAvicnademy
www.alvinacademy.com 0606/23/O/N/14 !29 of !59
[Turn over
© UCLES 2014
0606 Additional Mathematics Unit 15a Differentiation and integration
0606/12/F/M/15 8
6 (i) Given that y = tan 2x , find dy . [3]
x dx
(Aii) lHveincne fAinAdcthcaeaedqdueaAetmiomnlyovfyithnAeAnolArlvmvaiclintaontdhAeAecmucrcvaeayddeyAem=mtlanyvxy2ixnAaltAtvheicpnaoindt wehmere yx = r . [3]
8
www.alvinacademy.com !30 of !59
© UCLES 2015 0606/12/F/M/15
0606 Additional Mathematics Unit 15a Differentiation and integration
0606/12/F/M/15 11
8 (iii) Solve gl(x) = hl(x).
[3]
AlvinAAccaaddeAemmlyvyinAAlAlvvicinandAAemccaayddeAemmlyvyinAlAvicnademy
www.alvinacademy.com 0606/12/F/M/15 !31 of !59
[Turn over
© UCLES 2015
0606 Additional Mathematics Unit 15a Differentiation and integration
0606/12/F/M/15 12
9
x cm
yx cm
cadem inx cm
A lvy cm
lvin y A y The diagram shows an empty container in the form of an open triangular prism. The triangular faces
A em emare equilateral with a side of x cm and the length of each rectangular face is y cm. The container is
y d dmade from thin sheet metal. When full, the container holds 200 3 cm3.
AlvinAAccaaddeAemmlyvinAlAvicnadAemcay Alvin Aca 3x2 1600
(i) Show that A cm2 , the total area of the thin sheet metal used, is given by A = 2 + x . [5]
www.alvinacademy.com !32 of !59
© UCLES 2015 0606/12/F/M/15
0606 Additional Mathematics Unit 15a Differentiation and integration
0606/12/F/M/15 13
9 (ii) Given that x and y can vary, find the stationary value of A and determine its nature. [6]
AlvinAAccaaddeAemmlyvyinAAlAlvvicinandAAemccaayddeAemmlyvyinAlAvicnademy
www.alvinacademy.com 0606/12/F/M/15 3! 3 of !59
[Turn over
© UCLES 2015
0606 Additional Mathematics Unit 15a Differentiation and integration
0606/22/F/M/15 6
4 (i) Differentiate sin x cos x with respect to x, giving your answer in terms of sin x. [3]
[3]
(Aii) lHveincne fAinAdcycasaind2dxedAemx.mlyvyinAAlAlvvicinandAAemccaayddeAemmlyvyinAlAvicnademy
www.alvinacademy.com !34 of !59
© UCLES 2015 0606/22/F/M/15
0606 Additional Mathematics Unit 15a Differentiation and integration
0606/22/F/M/15 12
9
y
y = 4x
in AcademyAlvinA
y Alv demy demyO y= 4 + 2x
(2x + 1)2
Academ lvin Aca lvin Aca x
The diagram shows part of the curve y = 4 + 2x and the line y = 4x.
(2x + 1) 2
AlvinAcadeAmlyvinA Academy A (i) Find the coordinates of A, the stationary point of the curve.
[5]
(ii) Verify that A is also the point of intersection of the curve y = (2x 4 1) 2 + 2x and the line y = 4x.
+ [1]
www.alvinacademy.com 3! 5 of !59
© UCLES 2015 0606/22/F/M/15
0606 Additional Mathematics Unit 15a Differentiation and integration
0606/22/F/M/15 13
9 (iii) Without using a calculator, find the area of the shaded region enclosed by the line y = 4x, the
curve and the y-axis. [6]
AlvinAAccaaddeAemmlyvyinAAlAlvvicinandAAemccaayddeAemmlyvyinAlAvicnademy
www.alvinacademy.com 0606/22/F/M/15 !36 of !59
[Turn over
© UCLES 2015
0606 Additional Mathematics Unit 15a Differentiation and integration
0606/12/M/J/15 12
y8 (i) Find ^10e2x + e-2xhdx. [2]
[2]
lvin AcademyyAlvin yy (ii) Hence find k (10e2x + e-2x)dx in terms of the constant k. [2]
-k
Alvin AcadeemmyyAAlvin AemcaydeAmlvin Academy (iii) Given that k (10e2x + e-2x)dx =-60, show that 11e2k - 11e-2k + 120 = 0.
Acad Alvin Acad-k
www.alvinacademy.com 0606/12/M/J/15 3! 7 of !59
© UCLES 2015
0606 Additional Mathematics Unit 15a Differentiation and integration
0606/12/M/J/15 13
8 (iv) Using a substitution of y = e2k or otherwise, find the value of k in the form a ln b, where [3]
a and b are constants.
AlvinAAccaaddeAemmlyvyinAAlAlvvicinandAAemccaayddeAemmlyvyinAlAvicnademy
www.alvinacademy.com 0606/12/M/J/15 [Turn!38 oovfe!5r9
© UCLES 2015
0606 Additional Mathematics Unit 15a Differentiation and integration
0606/12/M/J/15 14
9 A curve has equation y = 4x + 3 cos 2x. The normal to the curve at the point where x = r meets
4
the x- and y-axes at the points A and B respectively. Find the exact area of the triangle AOB,
wAhelrevOiins thAeAocricgaiand.deAemmlyvyinAAlAlvvicinandAAemccaayddeAemmlyvyinAlAvicnademy [8]
w©wUwC.aLlEvSin20a1c5ademy.com 0606/12/M/J/15 3! 9 of !59
0606 Additional Mathematics Unit 15a Differentiation and integration
0606/22/M/J/15 9
8 y y = x + 10
B
y = x2– 6x + 10
demyA
y Alvin Academy AlvindemyO C x
em ca ca The graph of y = x2 - 6x + 10 cuts the y-axis at A. The graphs of y = x2 - 6x + 10 and y = x + 10
d A Acut one another at A and B. The line BC is perpendicular to the x-axis. Calculate the area of the shaded
AlvinAAccaadeAmlyvinAlAvicnademy Alvinregion enclosed by the curve and the line AB, showing all your working. [8]
www.alvinacademy.com 0606/22/M/J/15 4! 0 of !59
[Turn over
© UCLES 2015
0606 Additional Mathematics Unit 15a Differentiation and integration
0606/22/M/J/15 14
11
O
cademy inB
lvin A y Alv yA C
A em emQ D
8
Rh
in AcademyAlvin Acad Alvin AcadP 4 S
Alv emy emy The diagram shows a cuboid of height h units inside a right pyramid OPQRS of height 8 units and with
d dsquare base of side 4 units. The base of the cuboid sits on the square base PQRS of the pyramid. The
ca capoints A, B, C and D are corners of the cuboid and lie on the edges OP, OQ, OR and OS, respectively,
of the pyramid OPQRS. The pyramids OPQRS and OABCD are similar.
A in A (i) Find an expression for AD in terms of h and hence show that the volume V of the cuboid is given
Alv byV = h3 - 4h2 + 16h units3. [4]
4
www.alvinacademy.com 4! 1 of !59
© UCLES 2015 0606/22/M/J/15
0606 Additional Mathematics Unit 15a Differentiation and integration
0606/22/M/J/15 15
(ii) Given that h can vary, find the value of h for which V is a maximum. [4]
AlvinAAccaaddeAemmlyvyinAAlAlvvicinandAAemccaayddeAemmlyvyinAlAvicnademy
Question 12 is printed on the next page.
www.alvinacademy.com 0606/22/M/J/15 4! 2 of !59
[Turn over
© UCLES 2015
0606 Additional Mathematics Unit 15a Differentiation and integration
0606/13/M/J/15 9
7 The point A, where x = 0, lies on the curve y= ln ^4x 2 + 3h . The normal to the curve at A meets the
x-axis at the point B. x- 1
(i) Find the equation of this normal. [7]
[2]
AlvinAAccaaddeAemmlyvyinAAlAlvvicinandAAemccaayddeAemmlyvyinAlAvicnademy (ii) Find the area of the triangle AOB, where O is the origin.
www.alvinacademy.com 0606/13/M/J/15 !43 of !59
[Turn over
© UCLES 2015
0606 Additional Mathematics Unit 15a Differentiation and integration
0606/13/M/J/15 12
9
y D y = 3x + 10
cademyA
inB
in A Alv y = x3 – 5x2 + 3x + 10
lv y yC
A em emO x
demy Acad Acad The diagram shows parts of the line y = 3x + 10 and the curve y = x3 - 5x2 + 3x + 10.
The line and the curve both pass through the point A on the y-axis. The curve has a maximum at the
ca in inpoint B and a minimum at the point C. The line through C, parallel to the y-axis, intersects the line
A lv lvy = 3x + 10 at the point D.
AlvinAcadeAmlyvinA Academy A (i) Show that the line AD is a tangent to the curve at A. [2]
(ii) Find the x-coordinate of B and of C. [3]
www.alvinacademy.com 4! 4 of !59
© UCLES 2015 0606/13/M/J/15
0606 Additional Mathematics Unit 15a Differentiation and integration
0606/13/M/J/15 13
(iii) Find the area of the shaded region ABCD, showing all your working. [5]
AlvinAAccaaddeAemmlyvyinAAlAlvvicinandAAemccaayddeAemmlyvyinAlAvicnademy
www.alvinacademy.com 0606/13/M/J/15 !45 of !59
[Turn over
© UCLES 2015
0606 Additional Mathematics Unit 15a Differentiation and integration
0606/23/M/J/15 10
y8 (a) (i) Find e4x + 3dx. [2]
[2]
lvin AcademyyAlvin yy (ii) Hence evaluate 3 e4x+3dx. [2]
2.5 [2]
lvin AcademyyAAlvin AcaydeAmlvin Academy (b) (i) Find cosLKJ3xPONdx.
A AcadeAmlvin Academy
(ii) Hence evaluate r cos KJL3xPNOdx .
6
0
www.alvinacademy.com !46 of !59
© UCLES 2015 0606/23/M/J/15
0606 Additional Mathematics Unit 15a Differentiation and integration
0606/23/M/J/15 11
y8 ^x-1 2 dx [4]
(c) Find + .
xh
AlvinAAccaaddeAemmlyvyinAAlAlvvicinandAAemccaayddeAemmlyvyinAlAvicnademy
www.alvinacademy.com 0606/23/M/J/15 4! 7 of !59
[Turn over
© UCLES 2015
0606 Additional Mathematics Unit 15a Differentiation and integration
0606/12/O/N/15 4
2 A curve, showing the relationship between two variables x and y, passes through the point P^-1, 3h.
TAhelcvurvienhaAsAacgcraaaddiednetAemofm2lyvatyiPn.AGAivlAelvnvitchinaant dAddAex2mcy2c=aa-yd5d,eAemmfilnydvytihenAeqluAavtioicnnaofdthee cmurvey. [4]
www.alvinacademy.com 4! 8 of !59
© UCLES 2015 0606/12/O/N/15
0606 Additional Mathematics Unit 15a Differentiation and integration
0606/12/O/N/15 7
5 Variables x and y are such that y = ^x - 3hln^2x2 + 1h.
(i) Find the value of dy when x = 2. [4]
AlvinAAccaaddeAemmlyvyinAAlAlvvicinandAAemccaayddeAemmlyvyinAlAvicnademy (ii) Hence find the approximate change in y when x changes from 2 to 2.03.dx[2]
www.alvinacademy.com 0606/12/O/N/15 !49 of !59
[Turn over
© UCLES 2015
0606 Additional Mathematics Unit 15a Differentiation and integration
0606/12/O/N/15 10
8 Find the equation of the tangent to the curve y = 2x - 1 at the point where x = 2. [7]
x2 + 5
AlvinAAccaaddeAemmlyvyinAAlAlvvicinandAAemccaayddeAemmlyvyinAlAvicnademy
www.alvinacademy.com 5! 0 of !59
© UCLES 2015 0606/12/O/N/15