Year 11 to 12 AS Mathematics transition work.
Please write your solutions to each problem on A4 lined or squared paper. You must present your solutions to your teacher at the start of your course. Completion of this work will make the transition from GCSE to AS level Mathematics much smoother as it will both consolidate the work you did at GCSE and prepare you for the rigours of AS level Mathematics.
1.Giving your answers in the form a + b2, where a and b are rational numbers, find
(a)(3 – 8)2,
(3)
(b).
(3)
(Total 6 marks)
2.(a)Write 45 in the form a5, where is an integer.
(1)
(b)Express in the form b + c5, where b and c are integers.
(5)
(Total 6 marks)
3.Find the value of
(a),
(1)
(b),
(2)
(c).
(1)
(Total 4 marks)
4.Solve the equation 21 – x = 4x.
(Total 3 marks)
5.Find the set of values of x for which
(a)3(2x + 1) > 5 – 2x,
(2)
(b)2x2 – 7x + 3 > 0,
(4)
(c)both 3(2x + 1) > 5 – 2xand 2x2 – 7x + 3 > 0.
(2)
(Total 8 marks)
6.Solve the simultaneous equations
x – 2y = 1,
x2 + y2 = 29.
(Total 6 marks)
7.(a)Show that eliminating y from the equations
2x + y = 8,
3x2 + xy = 1
produces the equation
x2 + 8x 1 = 0.
(2)
(b)Hence solve the simultaneous equations
2x + y = 8,
3x2 + xy = 1
giving your answers in the form a + b17, where a and b are integers.
(5)
(Total 7 marks)
8.The points A and B have coordinates (4, 6) and (12, 2) respectively.
The straight line l1 passes through A and B.
(a)Find an equation for l1 in the form ax + by = c, where a, b and c are integers.
(4)
The straight line l2 passes through the origin and has gradient –4.
(b)Write down an equation for l2.
(1)
The lines l1 andl2 intercept at the point C.
(c)Find the exact coordinates of the mid-point of AC.
(5)
(Total 10 marks)
9.
The points A(1,7), B(20,7) and C(p, q) form the vertices of a triangle ABC, as shown in the diagram. The point D(8, 2) is the mid-point of AC.
(a)Find the value of p and the value of q.
(2)
The line l, which passes through D and is perpendicular to AC, intersects AB at E.
(b)Find an equation for l, in the form ax + by + c = 0, where a, b and c are integers.
(5)
(c)Find the exact x-coordinate of E.
(2)
(Total 9 marks)
10.The line l1 passes through the point (9, –4) and has gradient .
(a)Find an equation for l1 in the form ax + by + c = 0, where a, b and c are integers.
(3)
The line l2 passes through the origin O and has gradient –2. The lines l1 and l2 intersect at the point P.
(b)Calculate the coordinates of P.
(4)
Given that l1 crosses the y-axis at the point C,
(c)calculate the exact area of OCP.
(3)
(Total 10 marks)
11.On separate diagrams, sketch the graphs of
(a)y= (x+ 3)2,
(3)
(b)y= (x+ 3)2 + k, where k is a positive constant.
(2)
Show on each sketch the coordinates of each point at which the graph meets the axes.
(Total 5 marks)
12.The sequence of positive numbers u1u2, u3, ... is given by:
un + 1 = (un – 3)2, u1 = 1.
(a)Find u2, u3 and u4.
(3)
(b)Write down the value of u20.
(1)
(Total 4 marks)
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