12
1
y
y = 3x + 10
D
A
B
y = x3 – 5x2 + 3x + 10
C
x
O
The diagram shows parts of the line y = 3x + 10 and the curve y = x 3 - 5x 2 + 3x + 10 .
The line and the curve both pass through the point A on the y-axis. The curve has a maximum at the
point B and a minimum at the point C. The line through C, parallel to the y-axis, intersects the line
y = 3x + 10 at the point D.
(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]
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(iii) Find the area of the shaded region ABCD, showing all your working.
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[5]
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4
y
B
y = x + 10
y = x2– 6x + 10
A
C
O
x
The graph of y = x 2 - 6x + 10 cuts the y-axis at A. The graphs of y = x 2 - 6x + 10 and y = x + 10
cut one another at A and B. The line BC is perpendicular to the x-axis. Calculate the area of the shaded
region enclosed by the curve and the line AB, showing all your working.
[8]
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5
(i) Find
y ^3x - x h dx .
3
2
[2]
3
The diagram shows part of the curve y = 3x - x 2 and the lines y = 3x and 2y = 27 - 3x . The curve
and the line y = 3x meet the x-axis at O and the curve and the line 2y = 27 - 3x meet the x-axis at A.
y
B
2y = 27 - 3x
y = 3x
3
y = 3x - x 2
O
A
x
(ii) Find the coordinates of A.
[1]
(iii) Verify that the coordinates of B are ^3, 9h.
[1]
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(iv) Find the area of the shaded region.
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[4]
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10
8
y
y = x3 + 4x2 – 5x + 5
A
B
E
O
C
D
y=5
x
The diagram shows part of the curve y = x 3 + 4x 2 - 5x + 5 and the line y = 5. The curve and the
line intersect at the points A, B and C. The points D and E are on the x-axis and the lines AE and
CD are parallel to the y-axis.
(i) Find
y (x3 + 4x2 - 5x + 5) d x .
[2]
(ii) Find the area of each of the rectangles OEAB and OBCD.
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[4]
14
9
y
y=
e 4x + 3
8
A
O
B
x
e 4x + 3
. The curve meets the y-axis at the point A.
8
The normal to the curve at A meets the x-axis at the point B. Find the area of the shaded region
enclosed by the curve, the line AB and the line through B parallel to the y-axis. Give your answer in
e
the form , where a is a constant. You must show all your working.
a
[10]
The diagram shows the graph of the curve y =
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12
12
y
y = 4 + 2 cos 3x
P
Q
O
y=5
x
The diagram shows the curve y = 4 + 2 cos 3x intersecting the line y = 5 at the points P and Q.
(i)
Find, in terms of r, the x-coordinate of P and of Q.
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[3]
13
(ii)
Find the exact area of the shaded region. You must show all your working.
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[6]
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10
10
y
2
y = ( x - 3) 2
y = 4x 3
B (b, 4)
O
A (a, 0)
x
2
The diagram shows part of the graphs of y = 4x 3 and y = (x - 3) 2 . The graph of y = (x - 3) 2
meets the x-axis at the point A(a, 0) and the two graphs intersect at the point B(b, 4).
(a) Find the value of a and of b.
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(b) Find the area of the shaded region.
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8
7
y
A
B
J
rN
y = 2 cos Kx - O
6P
L
C
O
x
J
rN
The diagram shows part of the graph of y = 2 cos Kx - O . The graph intersects the y-axis at the
6P
L
point A, has a maximum point at B and intersects the x-axis at the point C.
(i) Find the coordinates of A.
[1]
(ii) Find the coordinates of B.
[2]
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(iii) Find the coordinates of C.
(iv) Find
J
N
L
P
[2]
y 2 cos Kx - r6Odx .
[1]
(v) Hence find the area of the shaded region.
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[2]
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11
y
y = mx + 8
B
y = 4 + 3x – x2
O
A
x
The diagram shows the curve y = 4 + 3x - x 2 intersecting the positive x-axis at the point A. The
line y = mx + 8 is a tangent to the curve at the point B. Find
(i) the coordinates of A,
[2]
(ii) the value of m,
[3]
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(iii) the coordinates of B,
[2]
(iv) the area of the shaded region, showing all your working.
[5]
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10
9
y
y = 2√x
A (4, 4)
O
The diagram shows part of the curve
x-axis at the point B.
B
x
y = 2 x . The normal to the curve at the point A (4, 4) meets the
(i)
Find the equation of the line AB.
[4]
(ii)
Find the coordinates of B.
[1]
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(iii)
Showing all your working, find the area of the shaded region.
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[4]
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x=2
y
y = x+
x
O
The diagram shows part of the curve y = x +
6
and the line x = 2 .
(3x + 2) 2
(i) Find, correct to 2 decimal places, the coordinates of the stationary point.
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(3x + 2) 2
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[6]
(ii) Find the area of the shaded region, showing all your working.
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[4]
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14
10 (a) Show that
(b)
5x + 12
1
2
+
can be written as
.
2
x + 1 3x + 10
3x + 13x + 10
[1]
y
P
y=
5x + 12
3x + 13x + 10
2
x=2
Q
x
O
5x + 12
, the line x = 2 and a straight line of
3x + 13x + 10
gradient 1. The curve intersects the y-axis at the point P. The line of gradient 1 passes through P
The diagram shows part of the curve y =
2
and intersects the x-axis at the point Q. Find the area of the shaded region, giving your answer in
2
the form a + ln `b 3j , where a and b are constants.
[9]
3
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12
10
y
y=
1
3
2+
(x + 2)
(x + 2)
A
B
-1
0
2
The diagram shows the graph of the curve y =
1
x
3
for x 2- 2 . The points A and B lie
`x + 2j
(x + 2)
on the curve such that the x-coordinates of A and of B are -1 and 2 respectively.
(a) Find the exact y-coordinates of A and of B.
2
+
[2]
(b) Find the area of the shaded region enclosed by the line AB and the curve, giving your answer in the
p
[6]
form - ln r , where p, q and r are integers.
q
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