A Level Maths
Bridging Topics:
These topics were taught at GCSE.
You will need a good knowledge of them to succeed at A level:
•	 Expanding Brackets
•	 Surds
•	 Trigonometry
•	 Simultaneous Equations
•	 Algebraic Proof
•	 Co-ordinate Geometry

Refreshing your knowledge of these topics will put you in a strong position to start the A level course.

Try the attached questions and check your answers using the mark scheme.
Q1.
Expand and simplify  (3x2 + 2)(2x + 5) − 6x(x2 − 3)
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Answer _________________________________________
(Total 4 marks)
Q2.
Expand and simplify     (2x + 5)(2x− 5)(3x + 7)
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Answer _________________________________________
(Total 3 mark)

Q3.
(ax + b)(bx + a) ≡ 10x2 + cx + 10          where a and b are positive integers.
Find the two possible values of c.
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Answer _________________ or _________________
(Total 4 marks)
Q4. (a)     Simplify fully        
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Answer _________________________________________
(2)
(b)     
Work out the value of k.
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Answer _________________________________________
Q5.
Write          in the form      where a is an integer.
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Answer _________________________________________
(Total 4 marks)

Q6.
ABC is a triangle.
AB =  cm
Angle ABC = 45°
Not drawn accurately
 
The area of ABC is 12 cm2
You are given that sin 45° =  
Work out the length x.
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Answer ______________________________________ cm
(Total 2 marks)

Q7.
P (−1, 4) is a point on a circle, centre O
 
Work out the equation of the tangent to the circle at P.
Give your answer in the form  y = mx + c
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Answer _________________________________________
(Total 4 marks)

Q8. A and B are points on the circle with equation     x2 + y2 = 25
          A is (3, 4)	          B is a point on the y-axis.	          PA and PB are tangents.
 
(a)     Show that the coordinates of B are (0, −5)
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(1)
(b)     Give a reason why PA = PB
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(1)
(c)     P is the point (a, b)
Work out the values of a and b.
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a = ____________________________________________
b = ____________________________________________
(4)
Q9.
Prove that      5x(x + 6) − (3x + 5)2      is negative for all values of x.
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(Total 4 marks)
Q10.
Prove that     5n − (2n + 3)(n + 1)     is always negative.
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(Total 3 marks)

Q11.
Prove that x2 + x + 1 is always positive.
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(Total 3 marks)
Q12.
Solve          5x − y = 5
2y − x2 = 11
You must show your working.
Do not use trial and improvement.
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Answer _________________________________________
(Total 6 marks)
Q13.
Solve the simultaneous equations
y = x2 − 6x − 20
y = 4 − x
You must show your working.
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Answer _________________________________________
(Total 5 marks)

Q14.  A, B and C are points on the circle  x2 + y2 = 36  as shown.
A is on the y-axis.	  B is on the x-axis.  M is the midpoint of AB. COM is a straight line.
 
(a)  Show that the coordinates of A are (0, 6)
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(b)  Work out the coordinates of B.
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Answer ( ___________ , ___________ )
(1)
(c)  Show that the equation of the straight line passing through C, O and M is  y = x
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(2)
(d)  Work out the coordinates of C. Give your answers in surd form.
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Answer ( ___________ , ___________ )
(3)
Q15.
The diagram shows a vertical tower CD of height, h, metres.
ABC is horizontal.
AB = 40 metres.
 
Work out the height, h, of the tower.
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Answer ....................................................................... metres
(Total 5 marks)

Q16.
A Big Wheel is modelled as a circle with centre O and radius 15 metres.
The wheel turns in an anticlockwise direction.
The lowest point on the wheel is always 2 metres above horizontal ground.

(a)     C is a point on the wheel, h metres above horizontal ground.
Angle COB = x°
Show that h = 17 – 15 cos x°
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(2)

(b)     D is a point on the wheel.
Angle DOB = 120°

Work out the height of D above horizontal ground.
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Answer ______________________ metres
(2)
(c)     Here is a sketch of the graph h = 17 – 15 cos x° for one complete turn of the wheel.
P is the highest point on the graph.

Work out the coordinates of P
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Answer (___________, ___________)
(2)
(Total 6 marks)

Q17.
Work out the value of     −  
Give your answer in the form  
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Answer _________________________________________
(Total 4 marks)

Q18.
Two triangular lawns are shown.
Wire fencing is needed for all five sides.
 
Wire fencing is sold in 50-metre rolls.
Work out the number of rolls needed.
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Answer _________________________________________
(Total 6 marks)