Exam Dates:
Pizzi
ΜΑΤΗ S Name:
Contents
Number:
- Surds
- Algebraic proofs
Algebra: - Transformations of graphs
- Equations of circles
- Quadratic and other sequences
- Completing the square
- Inverse and composite functions
- Expanding more than two binomials
- Nonlinear simultaneous equations
Shape, Space and Measure: - Circle theorems
- Vectors
- Sine and cosine rules
- Area under graphs
Data Handling: - Histograms
- Capture-Recapture
Probability: - Set theory
Ratio and Proportion: - Proportion
- Percentages - reverse
Surds
Things to remember:
- √ means square root;
- To simplify surds, find all its factors;
- To rationalise the denominator, find an equivalent fraction where the denominator is rational.
Questions:
- Work out (5 +√3)(5-√3) / √22
Give your answer in its simplest form.
(a) Rationalise the denominator of
(b) Expand (2+√3)(1+√3)
Give your answer in the form a +b√√3 where a and b are integers.
(2) (Total 3 marks) - (1) (a) Rationalise the denominator of
(b) (i)
(ii) Expand and simplify (√3+ √15)²
Give your answer in the form a +b√3 where a and b are integers.
All measurements on the triangle are in centimetres.
ABC is a right-angled triangle.
k is a positive integer.
√3+√15
A
k
B
Find the value of k.
3+√5
Diagram NOT accurately drawn
k=
(2) (5) (Total 7 marks) - Expand and simplify (√3-√2)(√3-√2)
- Write down the value of 49¹/2
- Write √45 in the form k√5, where k is an integer.
√18 + 10 / √2 in the form a +b√2 where a and b are integers.
5 a = b = (Total 2 marks) (1) (1) (Total 2 marks)
Algebraic Proofs
Things to remember:
- Start by expanding the brackets, then factorise.
- Remember the following:
- Even number: 2n
- 2n + 1 odd number
- a(bn + c) → multiple of a
- Consecutive numbers are numbers that appear one after the other.
Questions:
- (a) Expand and simplify x(x + 1)(x - 1)
(2) In a list of three consecutive positive integers at least one of the numbers is even and one of the numbers is a multiple of 3
n is a positive integer greater than 1
(b) Prove that n³ - n is a multiple of 6 for all possible values of n.
2611 is a prime number.
(c) Explain why 261 + 1 is a multiple of 3
7 (2) (2) (Total for question = 6 marks) - Prove that (2n + 3)² (2n- 3)² is a multiple of 8
for all positive integer values of n.
(a) Expand and simplify (y - - 2)(y – 5)
Prove algebraically that (2n + 1)² (2n + 1) is an even number
for all positive integer values of n.
(Total for Question is 3 marks) (2) (3) (Total for Question is 5 marks) - Prove algebraically that the difference between the squares of any two consecutive integers is equal to the sum of these two integers.
(a) Factorise x² + 7x
(b) Factorise y² 10y + 16
(c) (i) Factorise 2t² + 5t + 2
(ii) t is a positive whole number.
The expression 2t² + 5t + 2 can never have a value that is a prime number.
Explain why. 9 (1) (2) (3) (Total for Question is 6 marks) - (a) Factorise 3t + 12
(b) (i) Expand and simplify 7(2x + 1) + 6(x + 3)
(ii) Show that when x is a whole number 7(2x + 1) + 6(x + 3)
is always a multiple of 5
Prove that (n − 1)² + n² + (n + 1)² = 3n² + 2
10 (1) (3) (Total for Question is 4 marks) (Total for Question is 2 marks)
Transformations of graphs
Things to remember:
- f(x) means the function of x.
- -f(x) is a reflection in the x-axis.
- f(-x) is a reflection in the y-axis.
- f(x-a) is a translation in the x-axis, a units.
- f(x) + b is a translation in the y-axis, b units.
- cf(x) is an enlargement in the y-axis, scale factor c.
- f(dx) is an enlargement in the x-axis, scale factor.
Questions:
- y = f(x)
The graph of y = f(x) is shown on the grid.
On the grid above, sketch the graph of y = -f(x).
(a) The graph of y = f(x) is shown on the grid.
The graph G is a translation of the graph of y = f(x).
(b) Write down the equation of graph G. (2) (2) (Total for Question is 3 marks) - The graph of y = f(x) is shown on both grids below.
On the grid above, sketch the graph of y = f(-x)
On this grid, sketch the graph of y = -f(x) + 3
(1) (1) (Total for question = 2 marks) - The graph of y = f(x) is shown on each of the grids.
On this grid, sketch the graph of y = f(x-3)
This revision booklet is an essential tool for students aiming to achieve a grade 7 or higher in Maths. With detailed solutions and explanations, students can practice and improve their understanding of various Maths topics covered in the Grade 7 syllabus. Topics such as surds, algebraic proofs, and transformations of graphs are thoroughly explained, making it an ideal resource for exam preparation. The booklet is available in PDF format, making it accessible for students to use on their electronic devices. The comprehensive nature of the revision booklet ensures that students can effectively prepare for their Grade 7 Maths exams and achieve success.
