Download AS to A2 transition MATHS

AS to A2 Transition Review – Review 1
1. Express the equation y  x 2  6 x  8
a)
i)
In completed square form
ii)
As a product of linear factors
b) Using your answers to a) and b) sketch the graph clearly labelling the
key features
2. Without using a calculator and showing all of the working, find a value for the
following
1
a)
1
4
4  23  4
3
4
b)
16
3. Express
1
82  22
1
4
4 2
in the form a 2  b
2 1
4. Differentiate the following functions with respect to x
1
y  x 2  3x  1
a)
2
b)
y  4 x3  5 x
c)
y  24  4 x 2
5. For the function f ( x)  x3  3x 2  4 x  12
a)
Find the value of f (1) , f (2) and f (3) , hence find a factor of
f (x)
b)
Use your answer to a) to fully factorise the function f (x)
c)
Solve the equation f ( x)  0
6. Find the equation of the line that is parallel to y 
the point (2,-1)
1
x  3 and passes through
2
AS to A2 Transition Review – Review 2
1. Simplify the following:
a) 243
3
5
1
2
b)
 1 2
6 
 4
 5 3
15 
 8
c)
b)
3x 2  6 x
c)
12 x 2  7 x  1
c)
27
3
2. Factorise the following:
a)
4 x 2  25
3. Simplify these surds:
a)
4.
b)
12  3 75
Rationalise the denominators:
a)
5.
147
2
3 3
b)
2
5 1
c)
2 3
11  4
a) Sketch the following graphs on the same axes
y  2  3x
y  x2  2
b) Solve the equation:
x 2  2  2  3x
c) Use your answers to a) and b) to solve the inequality
x 2  2  2  3x
6.
Find the value of the gradient of the curve y  3x 4  7 x 3  2 x  3 at the
point x  1
AS to A2 Transition Review – Review 3
1. Expand and simplify
2. Express
5 2
2 3
(2 x  3)( x  2)( x  1)
in the form a  b c
3. Sketch the following graphs y  4  x 2 and y  2 x  1 the same axes
labelling all important features. Hence solve the inequality 4  x 2  2 x  1
4. a) Use the factor theorem to show that x  1 is a factor of
P( x)  3 x 3  3 x 2  5 x  1
b) Find Q (x) such that Px  ( x  1)(Q( x))
c) Solve fully the equation P( x)  0 giving your answers in exact form.
5. The equation y  2 x 2  5 x  c has two distinct real roots.
a) Write down the dicriminant.
b) Hence or otherwise find all of the values of c for which this is true
6. a)Find the gradient of the equation y  3x 2  1 curve at x  1
b) Hence find the equation of line parallel to the tangent through the point
 1,1
AS to A2 Transition Review – Review 4
1. What are the gradients and intercepts of the following straight lines:
2x  1
a) x  2 y  5
b)
5
y
2. A quadratic function has vertex at  2,1 express the function
in the form f x   x 2  bx  c
3. Sketch the following labelling the important features y  ( x  2)2
4. Simplify and hence solve the equation
x  1x  3  x  3x  5  0
5. Two numbers differ by 1 and have a product of 10. If n is the smallest
number.
a) Explain why n 2  n  10  0
b) Find the exact value of the two numbers.
6. x  1 is a factor of the equation f x   2 x3  5x 2  ax  6
a) Use the factor theorem to find a
b) Fully factorise and solve the equation f x   0
AS to A2 Transition Review – Review 5
1.
Find the equation of the straight line that passes through the points (3,1) and (-2,2), giving your answer in the form ax + by +c = 0.
Hence find the coordinates of the point of intersection of the line with
the x-axis.
2.
The points P, Q and R have coordinates (2,4), (7,-2) and (6,2)
respectively.
Find the equation of the straight line which is perpendicular to the line
PQ and which passes through the mid point of PR
3 + 2
2 +3
3.
Simplify this surd
4.
Find the value of y 6 2 x 6 3 = 6 y
64
5.
a) Write x2 + 4x –13 in the form (x + p)2 + q, where p and q are integer
values.
b) Using your answer from a) solve the equation
x2 + 4x –13= 0
leaving your answer in surds
c) Find the minimum value of x2 + 4x –13 and state the value of x for
which this minimum occurs.
d) Sketch the graph of x2 + 4x –13
6.
Using the discriminant, find the range of values for k so that
y = 3x2 + 6x + k
has
a) two distinct real roots,
b) no real roots.
AS to A2 Transition Review – Review 6
a) 9x2y + 6xy
b) 5x2 + 17x + 6
1.
Factorise
2.
Expand and simplify (x + 6y)2 + 5(y + x)2 – 2(3x + 2)
3.
Sketch the curve with equation y = x2
c) 25x2 – 36
On separate axes, sketch the following curves labelling the vertex and
axis of symmetry
a) y = (x – 2)2 + 1
4.
b) y = (x + 1)2 – 3
a) Factorise the quadratic function y = 6x2 – 17x + 7
b) Find the set of values for which y > 0
c) Solve the simultaneous equations
y = 6x2 – 17x + 7
y+x=7
d) Illustrate your answers to b) and c) graphically.
5.
a) Factorise fully
x3 – 2x2 – 5x +6
using the Factor Theorem.
b) Using your answer from a) sketch the curve y =
c) Hence, or otherwise, Find the set of solutions for
x3 – 2x2 – 4x +8 < x + 2
x3 – 2x2 – 5x +6
AS to A2 Transition Review – Review 7
1.
A curve has equation
y = x3 - 4x2 + 13x.
3
a) Show that the tangent at the point where x = 2 has a gradient 1.
b) Find the x - co-ordinate of another point where a tangent has
gradient 1.
c) Show that there is only one tangent with gradient -3.
2.
An open top box is made from a rectangular sheet of card as shown,
where 5cm is cut out from each corner.
5cm
5cm
x
2x
a) Find an expression for the volume of the box in terms of x.
b) Find the value of x which gives the maximum volume of the box?
3.
Complete the following integrals :
a)
x
4
2
dx
b)  ( x 2  2 x)dx
0
4.
Find y as a function of x for these gradient functions:
dy
 x 2  3x  2
dx
where the (x,y) graph passes through (3,16)