*p46958a0244* - wordpress.com · edexcel "international a level" "c3/4" papers...
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1. f (x) = (3 – 2x)–4, x 32
Find the binomial expansion of f (x), in ascending powers of x, up to and including the term in x2, giving each coefficient as a simplified fraction.
(4)
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Edexcel "International A level" "C3/4" papers from 2016 and 2015 Please use extra loose-leaf sheetsof paper where you run out of spacein this booklet.IAL PAPER JANUARY 2016
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2. (a) Show that
cot2x – cosecx – 11 = 0
may be expressed in the form cosec2x – cosecx + k = 0, where k is a constant.(1)
(b) Hence solve for 0 x 360
cot2x – cosecx – 11 = 0
Give each solution in degrees to one decimal place.
(Solutions based entirely on graphical or numerical methods are not acceptable.)(5)
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3. A curve C has equation
3x + 6y = 32
xy2
Find the exact value of ddyx
at the point on C with coordinates (2, 3). Give your answer in
the form a b+ ln8
, where a and b are integers.(7)
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4.
Figure 1
The curve C with equation y = 2
4 3( )+ x, x – 4
3 is shown in Figure 1
The region bounded by the curve, the x-axis and the lines x = –1 and x = 23
, is shown shaded in Figure 1
This region is rotated through 360 degrees about the x-axis.
(a) Use calculus to find the exact value of the volume of the solid generated.(5)
A
B
Figure 2
Figure 2 shows a candle with axis of symmetry AB where AB = 15 cm. A is a point at the centre of the top surface of the candle and B is a point at the centre of the base of the candle. The candle is geometrically similar to the solid generated in part (a).
(b) Find the volume of this candle.(2)
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23
O x
y
–1
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5.f (x) = –x3 + 4x2 – 6
(a) Show that the equation f(x) = 0 has a root between x = 1 and x = 2(2)
(b) Show that the equation f(x) = 0 can be rewritten as
xx
=−
⎛⎝⎜
⎞⎠⎟
64
(2)
(c) Starting with x1 = 1.5 use the iteration xn+1 = 6
4 −⎛⎝⎜
⎞⎠⎟xn
to calculate the values of x2,
x3 and x4 giving all your answers to 4 decimal places.(3)
(d) Using a suitable interval, show that 1.572 is a root of f (x) = 0 correct to 3 decimal places.
(2)
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6. A hot piece of metal is dropped into a cool liquid. As the metal cools, its temperature T degrees Celsius, t minutes after it enters the liquid, is modelled by
T = 300e–0.04t + 20, t 0
(a) Find the temperature of the piece of metal as it enters the liquid.(1)
(b) Find the value of t for which T = 180, giving your answer to 3 significant figures.
(Solutions based entirely on graphical or numerical methods are not acceptable.)(4)
(c) Show, by differentiation, that the rate, in degrees Celsius per minute, at which the temperature of the metal is changing, is given by the expression
2025− T
(3)
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7.
Figure 3
Figure 3 shows part of the curve C with equation
y = 3 1
1
2
2ln( )( )
xx
++
, x
(a) Find ddyx (2)
(b) Using your answer to (a), find the exact coordinates of the stationary point on the curve C for which x 0. Write each coordinate in its simplest form.
(5)
The finite region R, shown shaded in Figure 3, is bounded by the curve C, the x-axis and the line x = 3
(c) Complete the table below with the value of y corresponding to x = 1
x 0 1 2 3
y 0 35
ln5310
ln10
(1)
(d) Use the trapezium rule with all the y values in the completed table to find an approximate value for the area of R, giving your answer to 4 significant figures.
(3)
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y
O
R
3 x
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8. f ( ) = 9cos2 + sin2
(a) Show that f ( ) = a + bcos2 , where a and b are integers which should be found.(3)
(b) Using your answer to part (a) and integration by parts, find the exact value of
2
0
π
∫ 2f ( )d
(6)
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9. (a) Express 3 43 2
2
2x
x x−−( )
in partial fractions.(4)
(b) Given that x 23
, find the general solution of the differential equation
x2(3x – 2) ddyx
= y(3x2 – 4)
Give your answer in the form y = f (x).(6)
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10. (a) Express 3sin2x + 5cos2x in the form Rsin(2x + ), where R 0 and 0 2π
Give the exact value of R and give the value of to 3 significant figures.(3)
(b) Solve, for 0 x ,
3sin2x + 5cos2x = 4
(Solutions based entirely on graphical or numerical methods are not acceptable.)(5)
g(x) = 4(3sin2x + 5cos2x)2 + 3
(c) Using your answer to part (a) and showing your working,
(i) find the greatest value of g(x),
(ii) find the least value of g(x).(4)
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11.
Figure 4
Figure 4 shows a sketch of part of the curve with equation y = f (x), x
The curve meets the coordinate axes at the points A(0, –3) and B (– 13
ln4, 0) and the curve has an asymptote with equation y = –4
In separate diagrams, sketch the graph with equation
(a) y = f (x)(4)
(b) y = 2f(x) + 6(3)
On each sketch, give the exact coordinates of the points where the curve crosses or meets the coordinate axes and the equation of any asymptote.
Given that
f (x) = e–3x – 4, x
g(x) = ln12x +
⎛⎝⎜
⎞⎠⎟ , x –2
(c) state the range of f,(1)
(d) find f –1(x),(3)
(e) express fg(x) as a polynomial in x.(3)
y
(0, –3)
(– 13
ln4, 0)
B O
A
y = – 4
x
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12. With respect to a fixed origin O, the lines l1 and l2 are given by the equations
l1 : r = 1245
−⎛
⎝
⎜⎜
⎞
⎠
⎟⎟ +
542
−⎛
⎝
⎜⎜
⎞
⎠
⎟⎟
, l2 : r = 220
⎛
⎝
⎜⎜
⎞
⎠
⎟⎟
+ 063
⎛
⎝
⎜⎜
⎞
⎠
⎟⎟
where and are scalar parameters.
(a) Show that l1 and l2 meet, and find the position vector of their point of intersection A.(6)
(b) Find, to the nearest 0.1 , the acute angle between l1 and l2(3)
The point B has position vector 703
⎛
⎝
⎜⎜
⎞
⎠
⎟⎟
.
(c) Show that B lies on l1(1)
(d) Find the shortest distance from B to the line l2, giving your answer to 3 significant figures.
(4)
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13. A curve C has parametric equations
x = 6cos2t, y = 2sin t, – 2π
t 2π
(a) Show that ddyx = cosec t, giving the exact value of the constant .
(4)
(b) Find an equation of the normal to C at the point where t = 3π
Give your answer in the form y = mx + c, where m and c are simplified surds.(6)
The cartesian equation for the curve C can be written in the form
x = f (y), –k y k
where f(y) is a polynomial in y and k is a constant.
(c) Find f(y).(3)
(d) State the value of k.(1)
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*P44970A0244*
1. A curve has equation
4x2 – y2 + 2xy + 5 = 0
The points P and Q lie on the curve.
Given that ddyx
= 2 at P and at Q,
(a) use implicit differentiation to show that y – 6x = 0 at P and at Q.(6)
(b) Hence find the coordinates of P and Q.(3)
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IAL PAPER JUNE 2015
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2. Given that
4 61 2 2 1 2 2 2
2
2 2( )
( )( ) ( ) ( ) ( )xx x
Ax
Bx
Cx
+− +
≡−
++
++
(a) find the values of the constants A and C and show that B = 0(4)
(b) Hence, or otherwise, find the series expansion of
4 61 2 2
2
2( )
( )( )xx x
+− +
, |x | < 12
in ascending powers of x, up to and including the term in x2, simplifying each term.(5)
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3. y
xO
A
Figure 1
Figure 1 shows a sketch of part of the curve with equation y = f (x), where
f(x) = (2x – 5)ex, x
The curve has a minimum turning point at A.
(a) Use calculus to find the exact coordinates of A.(5)
Given that the equation f(x) = k, where k is a constant, has exactly two roots,
(b) state the range of possible values of k. (2)
(c) Sketch the curve with equation y = |f (x)|.
Indicate clearly on your sketch the coordinates of the points at which the curve crosses or meets the axes.
(3)
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4. A
BOb
a
Figure 2
Figure 2 shows the points A and B with position vectors a and b respectively, relative to a fixed origin O.
Given that |a| = 5, |b| = 6 and a.b = 20
(a) find the cosine of angle AOB,(2)
(b) find the exact length of AB.(2)
(c) Show that the area of triangle OAB is 5 5(3)
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5. (i) Find the x coordinate of each point on the curve yxx
=+ 1
, x –1, at which the
gradient is 14 (4)
(ii) Given that
tt
ta
a+ =∫ 1 7
2
d ln a > 0
find the exact value of the constant a.(4)
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*P44970A01644*
6. The mass, m grams, of a radioactive substance t years after first being observed, is modelled by the equation
m = 25e1– kt
where k is a positive constant.
(a) State the value of m when the radioactive substance was first observed. (1)
Given that the mass is 50 grams, 10 years after first being observed,
(b) show that k = ⎛⎝⎜
⎞⎠⎟
110
12
ln e(4)
(c) Find the value of t when m = 20, giving your answer to the nearest year.(3)
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*P44970A01844*
7. (a) Use the substitution t = tanx to show that the equation
4tan 2x – 3cot x sec2 x = 0
can be written in the form
3t 4 + 8t2 – 3 = 0(4)
(b) Hence solve, for 0 x < 2 ,
4tan 2x – 3cot x sec2 x = 0
Give each answer in terms of . You must make your method clear.(4)
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*P44970A02244*
8. (a) Prove by differentiation that
ddy
yy
ln tansin
2 44
( ) = , 0 < y < 4π
(4)
(b) Given that y = 6π when x = 0, solve the differential equation
ddyx
x y= 2 4cos sin , 0 < y < 4π
Give your answer in the form tan2y = AeBsinx, where A and B are constants to be determined.
(6)
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9. y
xA
BO
Figure 3
Figure 3 shows a sketch of part of the curve with parametric equations
x = t 2 + 2t, y = t3 – 9t, t
The curve cuts the x-axis at the origin and at the points A and B as shown in Figure 3.
(a) Find the coordinates of point A and show that point B has coordinates (15, 0).(3)
(b) Show that the equation of the tangent to the curve at B is 9x – 4y – 135 = 0 (5)
The tangent to the curve at B cuts the curve again at the point X.
(c) Find the coordinates of X.(5)
(Solutions based entirely on graphical or numerical methods are not acceptable.)
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*P44970A03044*
10.
x
6x
Figure 4
Figure 4 shows a right circular cylindrical rod which is expanding as it is heated.
At time t seconds the radius of the rod is x cm and the length of the rod is 6x cm.
Given that the cross-sectional area of the rod is increasing at a constant rate of 20π cm2 s–1, find the rate of increase of the volume of the rod when x = 2
Write your answer in the form k cm3 s–1 where k is a rational number.(6)
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*P44970A03244*
11. (a) Express 1.5sin – 1.2cos in the form Rsin( – ), where R > 0 and 0 < < 2π
Give the value of R and the value of to 3 decimal places.(3)
The height, H metres, of sea water at the entrance to a harbour on a particular day, is modelled by the equation
H = 3 + 1.5sin 6πt⎛ ⎞
⎜ ⎟⎝ ⎠ – 1.2cos
6πt⎛ ⎞
⎜ ⎟⎝ ⎠ , 0 t < 12
where t is the number of hours after midday.
(b) Using your answer to part (a), calculate the minimum value of H predicted by this model and the value of t, to 2 decimal places, when this minimum occurs.
(4)
(c) Find, to the nearest minute, the times when the height of sea water at the entrance to the harbour is predicted by this model to be 4 metres.
(6)
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*P44970A03644*
12. (i) Relative to a fixed origin O, the line l1 is given by the equation
l1: r = −⎛
⎝
⎜⎜
⎞
⎠
⎟⎟
516
+ 231
−⎛
⎝
⎜⎜
⎞
⎠
⎟⎟
where is a scalar parameter.
The point P lies on l1. Given that OP is perpendicular to l1, calculate the coordinates of P.
(5)
(ii) Relative to a fixed origin O, the line l2 is given by the equation
l2: r = 4312−
⎛
⎝
⎜⎜
⎞
⎠
⎟⎟
+ 534
−⎛
⎝
⎜⎜
⎞
⎠
⎟⎟
where is a scalar parameter.
The point A does not lie on l2. Given that the vector OA is parallel to the line l2
and |OA| = 2 units, calculate the possible position vectors of the point A.(5)
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*P44970A04044*
13. y
O eR
e2 x
Figure 5
Figure 5 shows a sketch of part of the curve with equation y = 2 – ln x, x > 0
The finite region R, shown shaded in Figure 5, is bounded by the curve, the x-axis and the line with equation x = e.
The table below shows corresponding values of x and y for y = 2 – ln x
x e e e+ 2
2e2
y l 0
(a) Complete the table giving the value of y to 4 decimal places.(1)
(b) Use the trapezium rule, with all the values of y in the completed table, to obtain an estimate for the area of R, giving your answer to 3 decimal places.
(3)
(c) Use integration by parts to show that (ln x)2 dx = x (ln x)2 – 2x ln x + 2x + c(4)
The area R is rotated through 360° about the x-axis.
(d) Use calculus to find the exact volume of the solid generated.
Write your answer in the form e( pe + q), where p and q are integers to be found.(6)
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*P45058A0248*
1. The curve C has equation
y xx
= −−
3 2
2 2( ), x 2
The point P on C has x coordinate 3
Find an equation of the normal to C at the point P in the form ax + by + c = 0, where a, b and c are integers.
(6)
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IAL PAPER JANUARY 2015
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4
*P45058A0448*
2. Solve, for 0 2 ,
2cos2 = 5–13sin
Give your answers in radians to 3 decimal places.
(Solutions based entirely on graphical or numerical methods are not acceptable.)(5)
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*P45058A0648*
3. The function g is defined by
g : x |8 – 2x |, x , x 0
(a) Sketch the graph with equation y = g(x), showing the coordinates of the points where the graph cuts or meets the axes.
(3)
(b) Solve the equation
|8 – 2x| = x + 5(3)
The function f is defined by
f : x x2 – 3x + 1, x , 0 x 4
(c) Find fg(5).(2)
(d) Find the range of f. You must make your method clear.(4)
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*P45058A01048*
4. Use the substitution x = 2sin to find the exact value of
1
4 2
3
3
2
0( )− x
xd
(7)
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*P45058A01248*
5. (a) Use the binomial expansion, in ascending powers of x, of 1
1 2√ ( )− x to show that
2 3
1 22 5 6 2+
−≈ + +x
xx x
√ ( ), |x| < 0.5
(4)
(b) Substitute x = 1
20 into
2 3
1 22 5 6 2+
−= + +x
xx x
√ ( )
to obtain an approximation to 10
Give your answer as a fraction in its simplest form.(3)
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*P45058A01648*
6. (i) Given x = tan2 4y, 0 y 8
π , find ddyx
as a function of x.
Write your answer in the form 1
A x xp q( ),
+ where A, p and q are constants to
be found.(5)
(ii) The volume V of a cube is increasing at a constant rate of 2 cm3 s–1. Find the rate at which the length of the edge of the cube is increasing when the volume of the cube is 64 cm3.
(5)
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*P45058A02048*
7. (a) Given that
2cos(x + 30)° = sin(x – 30)°
without using a calculator, show that
tan x° = 3 3 – 4(5)
(b) Hence or otherwise solve, for 0 180,
2cos(2 + 40)° = sin(2 – 20)°
Give your answers to one decimal place.(4)
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*P45058A02448*
8. V
O tFigure 1
The value of Lin’s car is modelled by the formula
V = 18000e–0.2t + 4000e–0.1t + 1000, t 0
where the value of the car is V pounds when the age of the car is t years.
A sketch of t against V is shown in Figure 1.
(a) State the range of V.(2)
According to this model,
(b) find the rate at which the value of the car is decreasing when t = 10 Give your answer in pounds per year.
(3)
(c) Calculate the exact value of t when V = 15000(4)
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*P45058A02848*
9.
ln5ln3O
y
x
C
R
Figure 2
The curve C has parametric equations
x = ln(t + 2), y =42t
t 0
The finite region R, shown shaded in Figure 2, is bounded by the curve C, the x-axis and the lines with equations x = ln3 and x = ln5
(a) Show that the area of R is given by the integral
4
22
3
1 t tt
( )+d
(3)
(b) Hence find an exact value for the area of R.
Write your answer in the form (a + lnb), where a and b are rational numbers.(7)
(c) Find a cartesian equation of the curve C in the form y = f(x).(2)
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Diagram not drawn to scale
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*P45058A03248*
10.
O
y
x
C
A
Figure 3
Figure 3 shows a sketch of part of the curve C with equation
y = x x2
3
ln – 2x + 4, x 0
Point A is the minimum turning point on the curve.
(a) Show, by using calculus, that the x coordinate of point A is a solution of
x = 6
1 2+ ln( )x(5)
(b) Starting with x0 = 2.27, use the iteration
xn + 1 = 6
1 2+ ln( )xn to calculate the values of x1, x2 and x3, giving your answers to 3 decimal places.
(3)
(c) Use your answer to part (b) to deduce the coordinates of point A to one decimal place.(2)
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*P45058A03648*
11. With respect to a fixed origin O the lines l1 and l2 are given by the equations
l1: r = 14 2
6 1
13 4
−⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟− +⎜ ⎟ ⎜ ⎟
−⎝ ⎠ ⎝ ⎠λ l2: r = 7 2
4 1
p q⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟− +⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
μ
where and are scalar parameters and p and q are constants.
Given that l1 and l2 are perpendicular,
(a) show that q = 3(2)
Given further that l1 and l2 intersect at point X,
find
(b) the value of p,(5)
(c) the coordinates of X.(2)
The point A lies on l1 and has position vector 6
2
3
−⎛
⎝
⎜⎜⎜
⎞
⎠
⎟⎟⎟
Given that point B also lies on l1 and that AB = 2AX
(d) find the two possible position vectors of B.(3)
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*P45058A04048*
12.
1 3
C
SO x
y
Figure 4
Figure 4 shows a sketch of part of the curve C with equation
y = x x2
3
ln – 2x + 4, x
The finite region S, shown shaded in Figure 4, is bounded by the curve C, the x-axis and the lines with equations x = 1 and x = 3
(a) Complete the table below with the value of y corresponding to x = 2. Give your answer to 4 decimal places.
x 1 1.5 2 2.5 3
y 2 1.3041 0.9089 1.2958
(1)
(b) Use the trapezium rule, with all the values of y in the completed table, to obtain an estimate for the area of S, giving your answer to 3 decimal places.
(3)
(c) Use calculus to find the exact area of S.
Give your answer in the form ab
+ lnc, where a, b and c are integers.(6)
(d) Hence calculate the percentage error in using your answer to part (b) to estimate the area of S. Give your answer to one decimal place.
(2)
(e) Explain how the trapezium rule could be used to obtain a more accurate estimate for the area of S.
(1)
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*P45058A04448*
13. (a) Express 10cos – 3sin in the form Rcos ( + ), where R 0 and 0 90°
Give the exact value of R and give the value of to 2 decimal places.(3)
Alana models the height above the ground of a passenger on a Ferris wheel by the equation
H = 12 – 10cos(30t)° + 3sin(30t)°
where the height of the passenger above the ground is H metres at time t minutes after the wheel starts turning.
(b) Calculate
(i) the maximum value of H predicted by this model,
(ii) the value of t when this maximum first occurs.
Give each answer to 2 decimal places.(4)
(c) Calculate the value of t when the passenger is 18m above the ground for the first time.
Give your answer to 2 decimal places.(4)
(d) Determine the time taken for the Ferris wheel to complete two revolutions. (2)
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H