A-Level Mathematics
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Year 12 A-Level Mathematics Summer Prep. Pack 2016/17
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Home academy: CLF Post 16 Contents
Section 1 Number 1
Section 2 Number 2
Section 3 Algebra 1
Section 4 Algebra 2
Section 5 Shape 1 Section 1 Number 1
1. Evaluate: a) b) c)
d)
e) Simplify
2. A shop has a sale of 20% off everything. A shirt costs £32 in the sale. What was the original price of the shirt? 3. Phil wants to increase a number by 35% and then decrease the answer by 28%. He says ‘ a quick way to do this is to multiply by 1·35 and then multiply by 0·28’. Do you agree with Phil’s method? Give reasons for you answer.
4. Calculate (simplify any fractions where possible):
5. Express the following as decimals: i)
ii)
Section 2 Number 2 1. Evaluate:
2. Simplify:
3. Expand and simplify the following:
4. Rationalise the denominator and simplify where possible:
Section 3 Algebra 1
1 Simplify the following:
a) 24k6 c) (2y3)2
6k²
b) 8h4 5h2 d)
4h5
2. Expand and simplify: a) (y +2)(y + 6) b) (x – 1)²
c) (x -3)(x + 4) d) (2m + n)(3m - 2n)
3. Solve the following equations:
d) e)
4. Solve the following pairs of simultaneous equations:
a) 3x + y = 26 b) 3x – 3y = 24 x + y = 10 x – 2y = 9
x = ……………. x =…………….
y = …………… y =…………….
5. The graph of y=5x - 4 is shown below: a) On the same axes, draw the graph of x + y = 5 b) Using the graphs, find the solution of the simultaneous equations:
y = 5x – 4
x + y = 5
6.
Using the equations of the graphs in the four boxes given above:
a) Write down two graphs that are parallel to each other
b) Write down two graphs that cross the y-axis at the same point.
c) Write down any graphs that go through the point (2,10)
d) Write down any graphs that cross the x-axis at x = -3. 7. Find the length between the points A = (-1,4) and B=(3,-2). Express your answer in the form √k where k is an integer.
8. Find the equation of the line parallel to the line y = 4 + 3x and passing through the point (-4,-1)
9. Find the equation of the line that passes through the points (1,-1) and (3,3). 10. Match the equations with the correct graph using the list below Note: You will have to find a missing equation and complete the blank graph.
11. Solve 4x – 5 7 – 2x Show your answer on the number line below:
12. Make t the subject of the equation in the following:
a)
b)
13. Factorise completely:
i) 3pq + 2p² ii) 4st – 14t³
14. Factorise completely:
i) x² + 5x + 6 iv) p² - 9p + 20 x² + 2x – 8 v) t² - 49
15. Simplify:
a) b)
16. Solve x² + 13x + 12 = 0
Section 4 Algebra 2 1. Rearrange the following expressions in descending order if:
i) a = 16
ii) a =
2. Make t the subject of the formula in each case:
i)
ii)
3. Factorise the following:
i) x² - 36 =
ii) 2a² - 50 =
iii) 4s² - 9t² = iv) 2x² + 7x + 3 =
v) 3y² + y - 4 =
4. i) Simplify
ii) Hence, or otherwise, solve
5. Solve the following, giving your answers to 1 decimal place.
i) x² + 5x + 1 = 0
ii) 2x² +13x – 5 = 2 6. Prove that the sum of three consecutive numbers is always a multiple of 3.
Section 5 Shape 1
1. Find the missing side indicated in each the following triangles, giving your answer to a sensible degree of accuracy: a)
b)
c)
2. Find the missing angle indicated in each the following triangles, giving your answer to a sensible degree of accuracy: a) b)
c)
4. Find the radius of a sphere with a volume of 500cm³.
5. Which is greater – the total surface area of a hemisphere, radius 5cm or the total surface area of a cone, diameter 6cm, sloping edge 5cm?
6. The area of this Trapezium is 60m2. Using this information and the diagram, form an equation which could be solved to determine x. 7. Calculate the size of angle BCA (give your answer to 3 s.f)
8. ABCD is a kite with a diagonal DB.
a. Calculate the length of DB b. Calculate the size of the angle BDC c. Calculate the value of x d. Calculate length AC