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1.
IB HL Mock Test 2nd Nov
of 9
IB Math Paper 1
IB Math HL Induction
Search document
Find the area enclosed by the curve y = arctan x, the x-axis and the line x =
3. (Total
(a)
2.
(i) Sketch the graphs of y = sin x and y = sin x, on the sa!e set of axes, $
for " # x # (ii)
.
Find the x-coordinates of the points of intersection of the graphs in the $
do!ain " # x #
.
(iii) (iii) Find Find the the area area eencl nclose osedd by the graph graphs. s.
&
x
"
% − x
(b)
∫ Find th the value of
(c) (c)
'he 'he inc incre reaasing sing fun functio ctionn f satisfies f (") = " and f (a) = b, here a " and b ".
d x
using the substitution x = % sin θ .
a
(i) (i)
*y refe refere renc ncee to to a sket sketch ch,, sho sho tha thatt ∫
f ( x)d x = ab −
"
(ii)
Hence find the value of
∫ "
b
∫ f "
−&
( x)d x
.
x d x % .
arcsin
(Total 2
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(b)
IB HL Mock Test 2nd Nov
of 9
IB Math HL Induction
Search document
0f the ater in the glass evaporates at the rate of 3 c!3 per hour for each c! of exposed surface area of the ater, sho that, dV dt
(c)
IB Math Paper 1
= −3
$V
, here t is !easured in hours.
0f the glass is filled co!pletely, ho long ill it take for all the ater to evaporate1
(Total 1
4.
(a)
e ∫ sing integration by parts, sho that
d y
(b)
=
x
sin xd x =
& 2
e x ( sin x − cos x ) + C
.
& − y e x
d x Reading a Preview Solve the differential euation sin x, given that y = " hen x = ", You're riting your anser in the for! y = f ( x). Unlock full access with a free trial.
Download With Free Trial
(c)
(i)
Sketch the graph of y = f ( x), found in part (b), for " # x # &.2. 4eter!ine the coordinates of the point 5, the first positive intercept on the x-axis, and !ark it on your sketch.
(ii)
'he region bounded by the graph of y = f ( x) and the x-axis, beteen the origin and 5, is rotated 36"7 about the x-axis to for! a solid of revolution. Sign up to vote on this title 8alculate the volu!e of this solid.
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IB HL Mock Test 2nd Nov
1
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IB Math Paper 1
of 9
IB Math HL Induction
Search document
& 6.
(a)
9iven that α &, use the substitution u = x to sho that &
∫ & & + x α
&
(b)
&
Hence sho that arctan α : arctan α
&
∫ & + u du
d x = &
α
=
.
$
. (Total
&
7.
Find the value of
∫ t ln( t + &)dt . "
(Total
You're Reading a Preview $
Unlock full access with a free trial.
8.
'hroughout this uestion x satisfies " # x ; (a)
.
Download With Free Trial d y = − y d x Solve the differential euation sec x , here y = & hen x = ".
9ive your anser in the for! y = f ( x).
(b)
(i)
5rove that & # sec x # & : tan x. $
(ii)
4educe that
%
$
≤
Sign up to vote on this title
$
&
∫ sec xdx ≤ % + ln %
"
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IB HL Mock Test 2nd Nov
1
of 9
'he diagra! belo shos the graphs of y = intersect in the sa!e to points.
IB Math HL Induction
Search document
3 9.
IB Math Paper 1
x − 3
, y = 3 and a uadratic function, that all
You're Reading a Preview
Unlock full access with a free 9iven that the !ini!u! value of the uadratic function is trial. <3, find an expression for the area o t
(ax + bx + c) Free Trial the shaded region in the for! ∫ Download With d x, here the constants a, b, c and t are to be
"
deter!ined. (ote> 'he integral does not need to be evaluated.)
(Total
10.
+ body is !oving through a liuid so that its acceleration can be expressed as Sign up to vote on this title
v Useful Not useful − − 3 < "" ! s ,
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(a)
IB HL Mock Test 2nd Nov
of 9
IB Math Paper 1
Search document
""
&
%"
∫ v V
+ ?" dv.
v
(c)
Sho that the ti!e taken, T seconds, for the body to slo to V ! s <& is given by T =
(b)
IB Math HL Induction
dv
(i)
@xplain hy acceleration can be expressed as !etres, of the body at ti!e t seconds.
(ii)
Hence find a si!ilar integral to that shon in part (a) for the distance, S !etres, travelled as the body slos to V ! s <&.
d s
, here s is displace!ent, in
Hence, using parts (a) and (b), find the distance travelled and the ti!e taken until the
body !o!entarily co!es to rest.
(Total 1
You're Reading a Preview Unlock full access with a free trial.
11.
Download With Free Trial
'he function f is defined by f ( x) =
here D ⊆ (a)
& ( x + 6 x + 3x − &") 3
, for x ∈ D,
is the greatest possible do!ain of f.
Find the roots of f ( x) = ". Sign up to vote on this title
(b)
Aence specify the set D.
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IB HL Mock Test 2nd Nov
of 9
IB Math Paper 1
IB Math HL Induction
Search document
(d)
Solve the euation f ( x) = 3.
(e)
Sketch the graph of B yB= f ( x), for x ∈ D.
(f)
Find the area of the region co!pletely enclosed by the graph of B yB = f ( x).
(Total 1
12.
'he functions f , g and h are defined by , for x ∈ ,
x
f ( x) = & : e &
∈ C D"E, n + & $, n ∈ ∈ h( x) = sec x, forYou're x C Reading a Preview .
g ( x) = x , for x
Unlock full access with a free trial.
(a)
4eter!ine the range of the co!posite function g f . Download With Free Trial
(b)
4eter!ine the inverse of the function g f , clearly stating the do!ain.
(c)
(i)
Sign up vote on this euation title Sho that the function y = ( f g h)( x) satisfies thetodifferential
d y d x = (& < y) sin x
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Winter HW 2015 Math HL G12 Uploaded by MishaZetov
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1
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of 9
IB Math Paper 1
IB Math HL Induction
Search document
(iii) Gou are given that the do!ain of y = ( f g h)( x) can be extended to the hole real axis. 'hat part of the graph of y = ( f g h)( x), beteen its !axi!u! at x and its first !ini!u! for positive x, is rotated by $ about the y-axis. 8alculate th volu!e of the solid generated.
(Total 2
13.
8onsider the curve y = xe x and the line y = kx, k ∈ . (a)
(b)
Het k = ". (i)
Sho that the curve and the line intersect once.
(ii)
Find the angle beteen the tangent to the curve and the line at the point of intersection.
Het k = &. Sho that the line is a tangent to the curve.
You're Reading a Preview
Unlock full access with a free trial.
(c)
(i)
Trial y = x Find the values of k forDownload hich theWith curveFree e x and the line y = kx !eet in to distinct points.
(ii)
/rite don the coordinates of the points of intersection.
(iii) /rite don an integral representing the area of the region A enclosed by the curve and the line. Sign up to vote on this title
(iv)
Hence, given that " ; k ; &, sho that A ; &. Useful
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(Total 2
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1
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of 9
IB Math Paper 1
IB Math HL Induction
Search document
a + be x 14.
Het f ( x) =
ae x
+ b , here " ; b ; a. (b − a )e x ( ae x + b)
(a)
Sho that f I( x) =
.
(b)
Hence Justify that the graph of f has no local !axi!a or !ini!a.
(c)
9iven that the graph of f has a point of inflexion, find its coordinates.
(d)
Sho that the graph of f has exactly to asy!ptotes.
(e)
R enclosed Het a = % and b = &. 8onsiderYou're the region by the graph of y = f ( x), Reading a Preview
&
Unlock full access with a free trial.
the y-axis and the line ith euation y =
.
Download R is Trial Find the volu!e V of the solid obtained With henFree rotated through $ about the x-axis.
(Total 1
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15.
(a)
IB HL Mock Test 2nd Nov
1
of 9
IB Math HL Induction
Search document
x −
Het a ". 4ra the graph of y =
IB Math Paper 1
a
for < a # x # a on the grid belo.
You're Reading a Preview Unlock full access with a free trial.
Download With Free Trial "
(b)
∫ Find k such that
−a
x −
a
a
∫
d x = k
"
x −
a
d x
. (Total
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