Pastpaper Doubts
Pastpaper Doubts
Pastpaper Doubts
June 2010
1. y = 3x + 2x.
(a) Complete the table below, giving the values of y to 2 decimal places.
(b) Use the trapezium rule, with all the values of y from your table, to find an approximate
1
∫0 (3x +2 x) ¿dx ¿
value for .
8.
Figure 2
y = x3 − 10x2 + kx,
where k is a constant.
The point P on C is the maximum turning point.
The line through P parallel to the x-axis cuts the y-axis at the point N.
The region R is bounded by C, the y-axis and PN, as shown shaded in Figure 2.
(c) find the length of PQ, giving your answer in its simplest surd form.
January 2011
5) Given that
(404 ) 40!
= 4!b! ,
In the binomial expansion of (1 + x)40, the coefficients of x4 and x5 are p and q respectively.
q
(b) Find the value of p .
5
2
6. y = 3 x −2
(a) Copy and complete the table below, giving the values of y to 2 decimal places.
Figure 2
5
2
Figure 2 shows a sketch of part of the curve with equation y = 3 x −2 , x > 1.
The region S is bounded by the curve, the straight line through B and (2, 0), and the line
through A parallel to the y-axis. The region S is shown shaded in Figure 2.
(c) Use your answer to part (b) to find an approximate value for the area of S.
(3)
9. The points A and B have coordinates (–2, 11) and (8, 1) respectively.
Find
5.
Figure 1
The shape shown in Figure 1 is a pattern for a pendant. It consists of a sector OAB of a circle
π
centre O, of radius 6 cm, and angle AOB = 3 . The circle C, inside the sector, touches the two
straight edges, OA and OB, and the arc AB as shown.
Find
The region outside the circle C and inside the sector OAB is shown shaded in Figure 1.
7. (a) Solve for 0 x < 360°, giving your answers in degrees to 1 decimal place,
3 sin (x + 45°) = 2.
(4)
2 sin2 x + 2 = 7cos x,
8.
Figure 2
A cuboid has a rectangular cross-section where the length of the rectangle is equal to twice its
width, x cm, as shown in Figure 2.
(a) Show that the total length, L cm, of the twelve edges of the cuboid is given by
162
2
L = 12x + x .
(3)
JAN 2012
3. (a) Find the first 4 terms of the binomial expansion, in ascending powers of x, of
8
x
( )
1+
4 ,
giving each term in its simplest form.
(4)
(b) Use your expansion to estimate the value of (1.025)8, giving your answer to 4 decimal places.
6.
Figure 1
The finite region R, bounded by the lines x = 1, the x-axis and the curve, is shown shaded in
Figure 1. The curve crosses the x-axis at the point (4, 0).
(a) Complete the table with the values of y corresponding to x = 2 and 2.5.
x 1 1.5 2 2.5 3 3.5 4
y 16.5 7.361 1.278 0.556 0
(2)
(b) Use the trapezium rule with all the values in the completed table to find an approximate
value for the area of R, giving your answer to 2 decimal places.
(4)
7.
Figure 2
Figure 2 shows ABC, a sector of a circle of radius 6 cm with centre A. Given that the size of angle
BAC is 0.95 radians, find
The point D lies on the line AC and is such that AD = BD. The region R, shown shaded in
Figure 2, is bounded by the lines CD, DB and the arc BC.
Find
8.
Figure 3
Figure 3 shows a flowerbed. Its shape is a quarter of a circle of radius x metres with two equal
rectangles attached to it along its radii. Each rectangle has length equal to x metres and width
equal to y metres.
8
P = x + 2x.
(3)
(c) Use calculus to find the minimum value of P.
(5)
(d) Find the width of each rectangle when the perimeter is a minimum.
Give your answer to the nearest centimetre.
(2)
1
9. (i) Find the solutions of the equation sin (3x – 15) = 2 , for which 0 x 180.
(6)
(ii)
Figure 4
Figure 1
The line L has equation x = 13 and crosses C at the points P and Q as shown in Figure 1.
5.
Figure 2
Figure 2 shows the line with equation y = 10 – x and the curve with equation y = 10x – x2 – 8.
The line and the curve intersect at the points A and B, and O is the origin.
The shaded area R is bounded by the line and the curve, as shown in Figure 2.
Figure 3
A manufacturer produces pain relieving tablets. Each tablet is in the shape of a solid circular
cylinder with base radius x mm and height h mm, as shown in Figure 3.
(3)
(a) Prove that the sum of the first n terms of this series is given by
a(1−r n )
Sn = 1−r
jan2013
3. A company predicts a yearly profit of £120 000 in the year 2013. The company predicts
that the yearly profit will rise each year by 5%. The predicted yearly profit forms a geometric
sequence with common ratio 1.05
(a) Show that the predicted profit in the year 2016 is £138 915 (1)
(b) Find the first year in which the yearly predicted profit exceeds £200 000 (5)
(c) Find the total predicted profit for the years 2013 to 2023 inclusive, giving your answer to
the nearest pound.
7.
Figure 2
The triangle XYZ in Figure 1 has XY = 6 cm, YZ = 9 cm, ZX = 4 cm and angle ZXY = .
The point W lies on the line XY.
The circular arc ZW, in Figure 1 is a major arc of the circle with centre X and radius 4 cm.
The region enclosed by the major arc ZW of the circle and the lines WY and YZ is shown
shaded in Figure 1.
Calculate
(a) Use calculus to show that the curve has a turning point P when x = 2.
(4)
(b) Find the x-coordinate of the other turning point Q on the curve.
(1)
9.
Figure 2
The finite region R, as shown in Figure 2, is bounded by the x-axis and the curve with
equation
16
2
y = 27 − 2x − 9√x − x , x > 0.
The curve crosses the x-axis at the points (1, 0) and (4, 0).
(a) Copy and complete the table below, by giving your values of y to 3 decimal places.
(b) Use the trapezium rule with all the values in the completed table to find an approximate
value for the area of R, giving your answer to 2 decimal places.
June 2013
18, 12 and p
Find
sin tan = 3 co s + 2
4 cos2 + 2 cos – 1 = 0.
(3)
(b) Hence solve, for 0 < 360°,
y = x2 – 32 x + 20, x > 0,
Use calculus
(a) to find the coordinates of P,
(6)
(b) to determine the nature of the stationary point P.
(3)
10.
Figure 4
The circle C has radius 5 and touches the y-axis at the point (0, 9), as shown in Figure 4.
(a) Write down an equation for the circle C, that is shown in Figure 4.
(3)
A line through the point P(8, –7) is a tangent to the circle C at the point T.
x
y
2. (1 x)
(a) Complete the table below with the value of y corresponding to x = 1.3, giving your
answer to 4 decimal places.
(1)
(b) Use the trapezium rule, with all the values of y in the completed table, to obtain an
approximate value for
1.5
x
1 (1 x) dx
5. The first three terms of a geometric series are 4p, (3p + 15) and (5p + 20) respectively,
where p is a positive constant.
(2)
x5
81
(b) log3
(3)
x5
81
log3 (9x) + log3 = 3
(4)
8.
Figure 2
Figure 2 shows the design for a triangular garden ABC where AB = 7 m, AC = 13 m and
BC = 10 m.
The point D lies on AC such that BD is an arc of the circle centre A, radius 7 m.
The shaded region S is bounded by the arc BD and the lines BC and DC. The shaded region S
will be sown with grass seed, to make a lawned area.
Given that 50 g of grass seed are needed for each square metre of lawn,
**(b) find the amount of grass seed needed, giving your answer to the nearest 10 g.
(7)
9.
(ii) Find all the values of x, in the interval 0 ≤ θ < 360° , for which
4. The first term of a geometric series is 5 and the common ratio is 1.2.
The sum of the first n terms of the series is greater than 3000.
t
H 10 5sin
6 , 0 ≤ t < 24
(a) Show that the height of the water 1 hour after midnight is 12.5 metres.
(1)
(b) Find, to the nearest minute, the times before midday when the height of the water is
9 metres.
(6)
7.
Figure 1
y = x3 – 6x2 + 9x + 5
x + 9y = 85
(6)
The region R, shown shaded in Figure 1, is bounded by the curve C, the y-axis and the normal
to C at P.
Figure 2
3x – 4y = 25
(4)
(c) Show that, to 3 decimal places, angle POQ is 1.287 radians.
(2)
The tangent to C at P and the tangent to C at Q intersect on the y-axis at the point R.
(d) Find the area of the shaded region PQR shown in Figure 2.
(4)
9. (a) Show that the equation
3 sin2 x + 5 sin x – 2 = 0
(2)
(b) Hence solve, for –180° ≤ θ < 180°, the equation
May 2014
1.
Figure 1
Figure 1 shows a sketch of part of the curve with equation y = √(x2 + 1), x ≥ 0.
The finite region R, shown shaded in Figure 1, is bounded by the curve, the x-axis and the
lines x = 1 and x = 2.
The table below shows corresponding values for x and y for y = √(x2 + 1).
(b) Hence, or otherwise, find the first 3 terms, in ascending powers of x, of the expansion of
x
( )
1+
2 (2 – 3x)6.
(3)
5.
Figure 2
The shape ABCDEA, as shown in Figure 2, consists of a right-angled triangle EAB and a
triangle DBC joined to a sector BDE of a circle with radius 5 cm and centre B.
π
Angle EAB = 2 radians, angle EBD = 1.4 radians and CD = 6.1 cm.
Figure 3
Figure 3 shows a circle C with centre Q and radius 4 and the point T which lies on C. The
tangent to C at the point T passes through the origin O and OT = 6√5.
Given that the coordinates of Q are (11, k), where k is a positive constant,
Figure 4 Figure 5
Figure 4 shows a closed letter box ABFEHGCD, which is made to be attached to a wall of a
house.
The letter box is a right prism of length y cm as shown in Figure 4. The base ABFE of the
prism is a rectangle. The total surface area of the six faces of the prism is S cm2.
The cross section ABCD of the letter box is a trapezium with edges of lengths DA = 9x cm,
AB = 4x cm, BC = 6x cm and CD = 5x cm as shown in Figure 5.
The angle DAB = 90° and the angle ABC = 90°. The volume of the letter box is 9600 cm3.
320
2
(a) Show that y = x .
(2)
7680
(b) Hence show that the surface area of the letter box, S cm2, is given by S = 60x2 + x .
(4)
(c) Use calculus to find the minimum value of S.
(6)
(d) Justify, by further differentiation, that the value of S you have found is a minimum.
(2)
May 2016
4. f(x) = 6x3 + 13x2 – 4
(a) Use the remainder theorem to find the remainder when f(x) is divided by (2x + 3).
(2)
(b) Use the factor theorem to show that (x + 2) is a factor of f(x).
(2)
(c) Factorise f(x) completely.
(4)
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4 cos2 x + 7 sin x – 2 = 0,
9.
Figure 4
Figure 4 shows a plan view of a sheep enclosure.
The points B, F and E lie on a straight line with FE = x metres and 10 ≤ x ≤ 25.
(a) Find, in m2, the exact area of the sector FEA, giving your answer in terms of x, in its
simplest form.
(2)
Given that BC = y metres, where y > 0, and the area of the enclosure is 1000 m2,
P
1000 x
x
12
4 36 3 3
.
(3)
(d) Use calculus to find the minimum value of P, giving your answer to the nearest metre.
(5)
(e) Justify, by further differentiation, that the value of P you have found is a minimum.
(2)
May 2017
3. (a) y = 5x + log2(x + 1), 0⩽x⩽2
Complete the table below, by giving the value of y when x = 1
x 0 0.5 1 1.5 2
(1)
(b) Use the trapezium rule, with all the values of y from the completed table, to find an
approximate value for
2
ò (5
x
+ log 2 (x +1))dx
0
Figure 2
Figure
Figure 2 shows
2 shows a sketch
a sketch ofof
of part part
theofcurve
the curve with equation
with equation
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