Maths @ RSA
C4
Top Ten Topics
Name
Core 4 Teacher
Marking Grid
Set 1
1
Binomials
2
R cos/R sin form
3
Partial Fractions
4
Small Angle Approx
Trapezium Rule
5
Parametric Equations
6
Volumes of
Revolution
7
“Show that…” Trig
8
Hence solve Qu 7
(Trig)
9
Vectors
10 Differential Equations
Set 2
Set 3
Set 4
Set 5
Set 6
Set 7
Set 8
Set 9
Set 10
Revision Set 1
Find the 1st 3 terms of the binomial expansion of
1  2 x 3 in ascending powers of x.
1
1
State the set of values of x for which the expansion
is valid.
Find y in terms of x and find
6
3
4
Using the trapezium rule and 5 strips, find an
approximate value for
8
Solve 2 tan x = sec x for 0˚≤ x ≤ 360˚
9
Given that the points are A (2,3,-1) and B (6,-1,7),
find the equation of the line
a) In vector form
b) In cartesian form
1
5
sin x
Show that
∫0 xe2x dx
Find the volume of revolution generated when the
region bounded by the curve with equation
y = x³ and the x axis between x = 1 and x=2 is
rotated through 360˚ about the x axis.
when
7
solve if equal to 3 when 0 ≤ θ ≤ 360°.
Express the following as partial fractions and then
integrate.
2
(x + 2)(x − 1)
dx
x = 2t and y = t2
Express 3 sin θ + 4 cos θ in the form of R sin (θ + α) and
2
dy
10
If
dy
dx
√1−sin² x
simplifies to tan x.
= x + 2 then find y in terms of x
Revision Set 2
Find the 1st 3 terms of the binomial expansion of
1  3x  2 in ascending powers of x.
1
1
State the set of values of x for which the expansion is
valid.
Find y in terms of x and find
6
dy
dx
when
x = 1 + sin θ and y = cos θ
f(x) = 5 sin x + 4 cos x
Simplify as far as possible
2
3
7
Find the value of the max and state for what x value this
happens in the range of 0 ≤ x ≤ 360°.
sin x cot x
Express the following as partial fractions and then
integrate.
Hence or otherwise, solve sin x cot x =
x
(x + 2)²(x − 1)
8
3
4
0˚≤ x ≤ 360˚
Given A (4,-4,1) and B (0,4,-3) and line ℓ
4
Write down a small angle approximation for
i)
3 sin x
ii)
4 tan x
iii)
sin² x
iv)
tan³ x
9
4 
1 
 
 
r =   4 + λ 0 
1 
 1
 
 
Show that line AB is perpendicular to ℓ
5
Find the volume of revolution generated when the
region bounded by the curve with equation
y = (x-2)² and the x axis between x = 1 and x=3 is
rotated through 360˚ about the x axis.
10
If (x + 2)
dy
dx
= 1 , then find y in terms of x
Revision Set 3
Find the 1st 3 terms of the binomial expansion of
1 4 x in ascending powers of x.
1
Find y in terms of x and find
6
State the set of values of x for which the expansion
is valid.
x = t2 and y =
7
and solve if equal to 1 when 0 ≤ θ ≤ 360°.
3
Express the following as partial fractions and then
integrate.
4
(x − 1)(x 2 − 4)
1
4
t
sin x
Hence or otherwise, solve
8
1+cot ² x
=
5
4
The points A and B have coordinates (3,1,4) and
(2,−4,7) respectively and the line ℓ has equation
x−2
9
1
=
y+3
−2
=
z−1
6
The line AB makes an acute angle θ with ℓ.
Show that cos θ =
5
sin x
0˚≤ x ≤ 360˚
∫0 √sin x dx with a strip width of 0.2
Find the volume of revolution generated when the
region bounded by the curve with equation
y = 3x² and the y axis between y = 1 and y=2 is
rotated through 360˚ about the y axis.
when
1+cot ² x
Using the trapezium rule, evaluate
4
dx
Simplify as far as possible
Express 2 sin θ - 5 cos θ in the form of R sin ( θ - α)
2
dy
10
If
dy
dx
=
3x²
y+1
27
√35√41
, then find y in terms of x
Revision Set 4
Find the 1st 3 terms of the binomial expansion of
1
(1 2 x ) in ascending powers of x.
1
2
Find y in terms of x and find
6
State the set of values of x for which the expansion
is valid.
x = tan θ and y = sin θ
f(x) = 3 sin x + 5 cos x.
Simplify as far as possible
Find the value of the max and state for what x value
this happens in the range of 0 ≤ x ≤ 360°.
x
Hence or otherwise, solve
8
9
A triangle has vertices A (−2,4,2) , B (1,6,2) and
C (−3,3,4).
a) Show that AB AC=−5.
b) Show that cos BAC=−0 566.
c) Find the area of triangle ABC.
5
10
If
+ x² dx with 5 strips
Find the volume of revolution generated when the
region bounded by the curve with equation
y = 2x³ and the y axis between y = 1 and y=4 is
rotated through 360˚ about the y axis.
cos x √1 + cot²x = 1
0˚≤ x ≤ 360˚
Using the trapezium rule, evaluate
1
∫0 √1
when
cos x √1 + cot²x
(x + 1)(x 2 + 5x + 6)
4
dx
7
Express the following as partial fractions and then
integrate.
3
dy
dy
dx
=
1
x²y
, then find y in terms of x
Revision Set 5
Find the 1st 3 terms of the binomial expansion of
1
( 2  x ) 3 in ascending powers of x.
1
Find y in terms of x and find
6
State the set of values of x for which the expansion
is valid.
solve if equal to
3
1
2
when
2
x = 2t - 1 and y = t 2
7
√
when 0 ≤ θ ≤ 360°.
Express the following as partial fractions and then
integrate.
2
4 − x²
dx
Simplify as far as possible
Express cos θ - sin θ in the form of R cos (θ+α) and
2
dy
1−cos ² x
sec ² x−1
Hence or otherwise, solve
8
3 sin x √
1−cos ² x
sec ² x−1
=2
(0˚≤ x ≤ 360˚)
The line L is given by the equation
Using the trapezium rule, evaluate
4
5
1 2
∫0 √1+x²
x+3
9
dx
Find the volume of the solid that is made when the
curve x² + y² = 36 is rotated about the x axis.
2
=
z+2
−1
and y = 2
Find the coordinates of the point on L which is
nearest to the origin.
10
If
dy
dx
= x²y , then find y in terms of x
Revision Set 6
Find the 1st 3 terms of the binomial expansion of
3  x 2 in ascending powers of x.
1
State the set of values of x for which the expansion
is valid.
Find y in terms of x and find
6
when
Simplify as far as possible
Find the value of the max and state for what x value
this happens in the range of 0 ≤ x ≤ 2π
7
4 – 4 sin² x
Express the following as partial fractions and then
integrate.
3
dx
x = t2 and y = t2 – t
f(x) = 7 sin x + 2 cos x.
2
dy
Hence or otherwise, solve
8
2 ( 4 x 2 1)
( 2 x 1)( 2 x 1)
2 sin x (4 – 4 sin² x) = 1
0˚≤ x ≤ 360˚
Verify that the point (−2,−4,5) lies on both the lines
Assuming very small values of x
4
i)
ii)
Show that cos 2x ≈ 1-2x²
Find an approximate expression in terms
of x for sin 2x
9
  2
0 
6 
4 
 
 
 
 
r1=  0  + λ   2  and r2=  0  + μ  2 
1 
3 
 1
  3
 
 
 
 
Find the acute angle between the lines.
Find the volume of revolution generated when the
1
region bounded by the curve with equation y = 2 + x,
5
1
the x axis and the lines x = and x = 4 is rotated
2
through 360˚ about the x axis. Leave your answer in
exact form.
10
If
dy
dx
=
x²−1
2y−1
, then find y in terms of x
Revision Set 7
Find the 1st 3 terms of the binomial expansion of
1
( 4 3 x ) in ascending powers of x.
2
1
Find y in terms of x and find
6
7
and solve if equal to 1 when 0 ≤ θ ≤ 2π.
3
when
Show that
Express 3 cos θ + 4 sin θ in the form of R sin ( θ - α )
Express the following as partial fractions and then
integrate.
x² + 4
x² − 4
dx
x = cos θ and y = tan2 θ
State the set of values of x for which the expansion
is valid.
2
dy
sin 3θ =
2 tanθ
1+tan² θ
Hence or otherwise, solve
8
2 tanθ
1+tan² θ
= - 0.5
0˚≤ θ ≤ 360˚
Find angle OAC given that A(3,−3,4) and B(4,−5,4)
Using the trapezium rule, evaluate
4
3 ln x
∫1
√x+1
dx with 4 strips
9
3
 3
 
 
A second line is r1=  4  + μ  1  find where these 2
7
 3
 
 
lines intersect.
5
Find the volume of revolution generated when the
2
region bounded by the curve with equation y =
x+1
and the x axis is rotated about the x axis between
the lines x = 1 and x = 5.
10
If
dx
dy
=
x²y
x−1
, then find y in terms of x
Revision Set 8
Find the 1st 3 terms of the binomial expansion
3
of√2 + 3𝑥 √3 − 2𝑥 in ascending powers of x.
1
State the set of values of x for which the expansion
is valid.
Find y in terms of x and find
6
dy
dx
when
x = 1 - t2 and y = 4t
f(x) = 5 cos x – 2 sin x.
2
Find the value of the min and state for what x value
this happens in the range of 0 ≤ x ≤ 2π
7
Show that sin 2θ =
2 tanθ
1+tan²θ
Hence or otherwise, solve
3
Express the following as partial fractions and then
integrate.
x−2
(x² + 5x + 6)(x − 1)
8
3cos θ
2 tanθ
1+tan²θ
=1
0˚≤ x ≤ 360˚
9
Find an equation of the plane through point
−5
(0,3,7), given that (−5) is perpendicular to the
−6
plane.
10
If
Using small angle approximations show that
4
2
(1+cos x) cos x
5
3
≈ 1 + x²
4
Find the volume of revolution when the region
bounded by the curve with equation
y = (x -2) √(x − 2)3 and the x axis is rotated about
the x axis between x = 2 and x = 4.
dy
dx
= 3x²(y + 1) , then find y in terms of x
Revision Set 9
1
Find the 1st 3 terms of the binomial expansion of
(2  3x)
in ascending powers of x.
(3  x)
State the set of values of x for which the expansion
is valid.
Find y in terms of x and find
6
7
Express the following as partial fractions and then
integrate.
x³ + 6x − 9
x² + 6x + 8
cosθ + sin2θ – cos3θ = sin2θ(1 + 2sinθ)
Hence or otherwise, solve
8
5
2
∫0 √3
+
2ex
Find the point of intersection of the line
9
2
4
r =(3) + λ(2) and the plane 2x+3y+z=1
1
1
10
If
dx using 5 strips
Find the volume of revolution generated when the
region bounded by the curve with equation y = ln x
and y axis between x = 0 and x = 1.
cosθ + sin2θ – cos3θ = 0.5
0˚ ≤ θ ≤ 360˚
Using the trapezium rule, evaluate
4
when
Show that
and solve if equal to 5 when 0 ≤ θ ≤ 2π.
3
dx
x = t2 and y = t3
Express 2 sin θ + 3 cos θ in the form of R sin ( θ - α)
2
dy
dy
dx
=
2y−1
x +2
, then find y in terms of x
Revision Set 10
1
Find the 1st 3 terms of the binomial expansion of
1 x2
in ascending powers of x.
(1  3 x 2 )
State the set of values of x for which the expansion
is valid.
Find y in terms of x and find
6
1
x=t + t
and
dy
dx
when
1
y= t -
t
f(x) = 5 sin x - 4 cos x.
2
Find the value of the min and state for what x value
this happens in the range of 0 ≤ x ≤ 2π
7
Show that cos 3x = 4 cos³x – 3 cos x
Hence or otherwise, solve
3
Express the following as partial fractions and then
integrate.
4x3  1
4x 2  1
8
1
cos ³ 2θ
9
10
If
Find the volume of revolution generated when the
1
region bounded by the curve with equation y = 2 + x,
5
the x axis and the lines x = 2 and x = 4 is rotated
through 360˚ about the x axis. Leave your answer in
exact form.
3
Find the cartesian equation of the plane going
through point (-1,-6,-9) and perpendicular to
9
[ 8 ] and then find the angle this plane makes
−5
with line
in terms of cos θ.
1
2
0˚ ≤ x ≤ 360˚
Using small angle approximations find
4
sin 2x(4 cos³x – 3 cos x) =
dy
dx
= sin y , then find y in term of x