PMT
INTEGRATION
1 Find
(x2 + 6 x 3) dx. (3)
2 The curve y = f(x) passes through the point (1, 2).
Given that
6
f (x) = 1 ,
x3
a find an expression for f(x). (4)
The point A on the curve y = f(x) has x-coordinate 2.
b Show that the normal to the curve y = f(x) at A has the equation
16x + 4y 19 = 0. (5)
3 The curve y = f(x) passes through the point (3, 22).
Given that
f (x) = 3x2 + 2x 5,
a find an expression for f(x). (4)
Given also that
g(x) = (x + 3)(x 1)2,
b show that g(x) = f(x) + 2, (3)
c sketch the curves y = f(x) and y = g(x) on the same set of axes. (3)
4 Given that
3
y = x2 ,
x2
find
dy
a , (2)
dx
b y dx. (3)
5 The curve C with equation y = f(x) is such that
dy
= 3x2 4x 1.
dx
Given that the tangent to the curve at the point P with x-coordinate 2 passes through
the origin, find an equation for the curve. (7)
6 A curve with equation y = f(x) is such that
dy
= 3 x 2 , x > 0.
dx x
a Find the gradient of the curve at the point where x = 2, giving your answer in its
simplest form. (2)
Given also that the curve passes through the point (4, 7),
b find the y-coordinate of the point on the curve where x = 3, giving your answer in
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INTEGRATION
1 Find
(x2 + 6 x 3) dx. (3)
2 The curve y = f(x) passes through the point (1, 2).
Given that
6
f (x) = 1 ,
x3
a find an expression for f(x). (4)
The point A on the curve y = f(x) has x-coordinate 2.
b Show that the normal to the curve y = f(x) at A has the equation
16x + 4y 19 = 0. (5)
3 The curve y = f(x) passes through the point (3, 22).
Given that
f (x) = 3x2 + 2x 5,
a find an expression for f(x). (4)
Given also that
g(x) = (x + 3)(x 1)2,
b show that g(x) = f(x) + 2, (3)
c sketch the curves y = f(x) and y = g(x) on the same set of axes. (3)
4 Given that
3
y = x2 ,
x2
find
dy
a , (2)
dx
b y dx. (3)
5 The curve C with equation y = f(x) is such that
dy
= 3x2 4x 1.
dx
Given that the tangent to the curve at the point P with x-coordinate 2 passes through
the origin, find an equation for the curve. (7)
6 A curve with equation y = f(x) is such that
dy
= 3 x 2 , x > 0.
dx x
a Find the gradient of the curve at the point where x = 2, giving your answer in its
simplest form. (2)
Given also that the curve passes through the point (4, 7),
b find the y-coordinate of the point on the curve where x = 3, giving your answer in
Downloaded by harun mburu ()
