ze big one Flashcards
What is the product rule ?
Finding the amount of possible outcomes via multiplication.
A spinner that has 18 sections is spun, and a six-sided die is rolled. How many possible outcomes are there ?
108
How do you find the LCM ?
- Find the prime factors of each number and place in a venn diagram
- multiply everything in the venn diagram
How do you find the HCF ?
- Find the prime factors of each number and place in a venn diagram
- multiply everything in the middle section
Write 0.473473473… (recurring) as a fraction.
r = 0.473
10r = 4.734
100r = 47.347
1000r = 473.473
1000r - r = 999r
473.473 - 0.473 = 473
999r = 473
r = 473/999
[473/999]
Express 0.233333… (only the 3 recurring) as a fraction
r = 0.233333…
10r = 2.3333…
100r = 23.3333…
100r - 10r = 90r
23.333… - 2.333… = 21
90r = 21
r = 21/90 -> [7/30]
Estimate the cost of astroturf that is £14.30 +18% for every square metre, for a garden that is 7.7m x 13.2m.
step 1: estimate area of garden
- 7.7 -> 8
- 13.2 -> 13
- 8 x 13 = 104
step 2: estimate pricing:
- £14.30 -> £14
- 18% -> 20%
- 14 x 20% = £16.80 per sqm
step 3: round current estimated values
- 104sqm -> 100sqm
- £16.80 per sqm -> £17 per sqm
step 4: multiply to get final answer
- 17 x 100 = £1700 for the whole garden
How do you estimate the square root of 95 ?
- find square numbers before and after the number
- 81 and 100
- square root is therefore between 9 and 10
- 95 is closer to 100 than 81, so the square root is likely closer to 10 than 9
- estimated value = 9.6, 9.7, 9.8
Work out:
3 x 10^6 x 4.5 x 10^-4
step 1:
3 x 4.5 = 13.5
step 2:
10^6 x 10^-4 = 10^2
step 3:
- 13.5 x 10^2 IS NOT standard form
- 13.5 -> 1.35
10^2 -> 10^3
final answer = 1.35 x 10^3
Work out:
3 x 10^3 / 600 000
step 1:
600 000 -> 6 x 10^5
step 2:
3/6 = 0.5
step 3:
10^3 / 10^5 = 10^-2
step 4:
- 0.5 x 10^-2 IS NOT standard form
- 0.5 -> 5
- 10^-2 -> 10^-3
final answer = 5 x 10^-3
What must be the same before adding or subtracting in standard form ?
the power of 10
Calculate:
7.4 x 10^8 + 9.5 x 10^9
step 1: make powers of 10 the same
- 9.5 x 10^9 + 7.4 x 10^8
- 10^8 -> 10^9 = x10
- 7.4 -> 0.74
- 9.5 x 10^9 + 0.74 x 10^9
step 2: carry out the calculation
- 9.5 + 0.74 = 10.24
- 10.24 x 10^9
step 3: make final answer in standard form
- 10.24 -> 1.024
- 10^9 -> 10^10
final answer = 1.024 x 10^10
Simplify:
4a + 2a^2 + 5ab - 4 - 3a + 7
- 4a - 3a = a
- -4 + 7 = 3
final answer: 2a^2 + 5ab + a + 3
What is the value of (p x p x p)^0 ?
1
Simplify p^5 / p
p^4
Simplify 8b^11 / 2b^5
step 1: numbers
- 8/2 = 4
step 2: indices
- 11 - 5 = 6
final answer = 4b^6
Work out:
(3p^-2)^3
step 1: numbers
3^3 = 27
step 2: indices
p^-2x3 = p^-6
final answer = 27p^-6
Work out:
(x^-2/x^4)^3
step 1: simplify inside the bracket
x^-2/x^4 = x^-6
step 2: raise (multiply) the internal and external powers
-6 x 3 = -18
final answer = x^-18
Evaluate 4^-3
- 4^-3
- 1/4^3
- 1/64
Evaluate 2^-5
- 2^-5
- 1/2^5
- 1/32
Work out (3/2)^-3
- (3/2)^-3
- (2/3)^3
- 2^3/3^3
- 8/27
Evaluate 9^3/2
- 9^3/2
- sqrt 9 = 3
- 3^3 = 27
Evaluate 16^1/2
- 16^1/2
- sqrt 16 = 4
- 4^1 = 4
Evaluate 16^(-3)/2
- 16^(-3)/2
- 1/16^3/2
- sqrt 1/16 = 1/4
- 1/4^3 = 1/64
Evaluate (125/8)^(-4)/3
- (125/8)^(-4)/3
- ## (8/125)^4/3
- 8^4/3
- cube rt 8 = 2
- ## 2^4 = 16
- 125^4/3
- cube rt 125 = 5
- ## 5^4 = 625final answer = 16/625
Evaluate (9/16)^(-3)/2
- (9/16)^(-3)/2
- ## (16/9)^3/2
- 16^3/2
- sqrt 16 = 4
- ## 4^3 = 64
- 9^3/2
- sqrt 9 = 3
- ## 3^3 = 27final answer = 64/27
Evaluate (81/100)^3/2
- 81^3/2
- sqrt 81 = 9
- ## 9^3 = 729
- sqrt 100 = 10
- ## 10^3 = 1000final answer = 729/1000
Expand and simplify:
2x + 3(x - 5) -9
- 3(x - 5)
- 3x - 15
- 2x +3x - 15 - 9
- 5x - 26
What is 3 - (-4) ?
7
What is 8 - 9 - (-7) ?
- 8 - 9 = -1
- -1 - (-7) -> -1 + 7 = 6
final answer = 6
Expand and simplify:
(2x + 3)(x + 3a - 2)
- (2x + 3)(x + 3a - 2)
- 2x * x = 2x^2
- 2x * 3a = 6ax
- ## 2x * - 2 = - 4x
- 3 * x = 3x
- 3 * 3a = 9a
- ## 3 * -2 = -62x^2 + 6ax - 4x + 3x + 9a - 6
2x^2 + 6ax - x + 9a - 6
Expand and simplify:
(x + 3)(2x - 1)(x - 2)
step 1: ignore one bracket and multiply the other two
- ignore (x - 2)
- (x + 3)(2x - 1)
- x * 2x = 2x^2
- x * - 1 = -x
—-
- 3 * 2x = 6x
- 3 * - 1 = - 3
—-
- 2x^2 - x + 6x - 3
- 2x^2 + 5x - 3
step 2: multiply this expanded bracket with the formerly ignored bracket
- (2x^2 + 5x - 3)(x - 2)
—-
- 2x^2 * x = 2x^3
- 2x^2 * - 2 = -4x^2
—-
- 5x * x = 5x^2
- 5x * - 2 = -10x
—-
- -3 *x = -3x
- -3 * - 2 = 6
step 3: collect like terms again
2x^3 - 4x^2 + 5x^2 - 10x - 3x + 6
2x^3 + x^2 - 13x + 6
Factorise:
9ab + 15b^2
- 9ab + 15b^2
- 3b is common, so goes on outside
final answer = 3b(3a + 5b)
Factorise:
15xy + 10x + 20(x^2)y
- 15xy + 10x + 20(x^2)y
- 5x is common, so goes on outside
final answer = 5x(3y + 2 + 4xy)
Factorise 49 - p^2
- difference of two squares
- (7 - p)(7 + p)
- 7 * 7 = 49
- 7 * p = 7p
- -p * 7 = -7p
- -p * p = -p^2
- 49 + 7p - 7p - p^2
- 7p - 7p cancel out, so leaves with 49 - p^2
final answer = (7 - p)(7 + p)
Factorise 36 - 4x^2
- difference of two squares
final answer = (6 - 2x)(6 + 2x)
What is sqrt3 x sqrt7 ?
sqrt21
What is sqrt80 / sqrt4 ?
sqrt20
Simplify √60
- √t60 = √5 x √12
- √12 = √4 x √3
- √60 = √5 x √4 x √3
- √4 = 2
- √5 x √3 = √15
√60 simplified = 2√15
Simplify:
√125 - 2√45 + (√5 + 2)^2
√125 - 2√45 + (√5 + 2)^2
step 1: brackets
- (√5 + 2)^2
- (√5 + 2)(√5 + 2)
- √5 * √5 = 5
- √5 * 2 = 2√5
- 2 * √5 = 2√5
- 2 * 2 = 4
- 5 + 2√5 + 2√5 + 4
- 9 + 4√5
√125 - 2√45 + 9 + 4√5
step 2: simplify √125
- √125 = √25 x √5
- √25 = 5
- √125 = 5√5
step 3: simplify 2√45
- 2√45 = 2 x √9 x √5
- √9 = 3
- 2√45 = 2 x 3 x √5
- 2 x 3 = 6
- 6 x √5 = 6√5
5√5 - 6√5 + 9 + 4√5
step 4: collect like terms
3√5 + 9
final answer = 3√5 + 9
Simplify:
√48 + 2√75 + (√3)^2
step 1: brackets
- (√3)^2
- (√3)(√3)
- √3 * √3 = 3
step 2: simplify √48
- √48 = √6 x √8
- √8 = √4 x √2
- √4 = 2
- √48 = √6 x √2 x 2
- √6 x √2 = √12
- √12 = √3 x √4
- √4 = 2
- √48 = √3 x 2 x 2
- 2 x 2 = 4
- √48 = 4√3
step 3: simplify 2√75
- 2√75 = 2 x √3 x √25
- √25 = 5
- 2√75 = 2 x 5 x √3
- 2 x 5 = 10
2√75 = 10√3
step 4: collect like terms
- 10√3 + 4√3 + 3
- 14√3 + 3
final answer = 3 + 14√3
Simplify:
(7 + √5)/(√5 - 1)
Give your answer in the form of a + b√5
step 1: rationalise the denominator
- (7 + √5)/(√5 - 1) x (√5 + 1)/(√5 + 1)
—-
- (7 + √5)(√5 + 1)
- 7 * √5 = 7√5
- 7 * 1 = 7
- √5 * √5 = 5
- √5 * 1 = √5
- (7√5 + 7 + 5 + √5)
- (8√5 + 12)
—-
- (√5 - 1)(√5 + 1)
- √5 * √5 = 5
- √5 * 1 = √5
- -1 * √5 = -√5
- -1 * 1 = -1
- (5 + √5 - √5 - 1)
- (4)
—-
new fraction = (8√5 + 12)/4
step 2: simplify fraction
- 12/4 = 3
- 8√5/4 = 2√5
final answer = 3 + 2√5
Solve 4 + b = 19
4 + b = 19
b = 15
Solve 3 = (14 - x)/4
3 = (14 - x)/4
12 = 14 - x
x + 12 = 14
x = 2
Solve 2x + 3 = 5x - 12
2x + 3 = 5x - 12
2x + 15 = 5x
15 = 3x
5 = x
Solve 4a - 5 = 7 + 6a
4a - 5 = 7 + 6a
4a = 12 + 6a
-2a = 12
2a = -12
a = -6
Rearrange 6a = 3 + b/2 to make b the subject
6a = 3 + b/2
6a - 3 = b/2
2(6a - 3) = b
12a - 6 = b
Make x the subject:
5(x - 3) = 4y(1 - 3x)
5(x - 3) = 4y(1 - 3x)
5x - 15 = 4y - 12xy
5x + 12xy - 15 = 4y
5x + 12xy = 4y + 15
x(5 + 12y) = 4y + 15
x = (4y + 15)/(5 + 12y)
Solve x^2 + 10x + 16 = 0
step 1: find two factors of 16 that add to 10
2 x 8 = 10
step 2: factorise using these factors and double brackets
(x + 2)(x + 8) = 0
step 3: invert the values in the brackets to find the two values for x
x = -2
x = -8
Solve 6x^2 - 23x + 20 = 0
step 1: multiply the coefficient of x^2 and the integer at the end of the equation
6 x 20 = 120
step 2: find two factors of 120 that add to -23
-8 x -15 = 120
step 3: write out the equation with the above values instead of just -23x
6x^2 - 8x - 15x + 20 = 0
step 4: factorise the equation, leaving some terms outside of the brackets and making sure the bracketed expressions are identical
2x (3x - 4) - 5(3x - 4) = 0
step 4: take the terms exterior to the brackets and place them in their own bracket to fully factorise the equation
(3x - 4)(2x - 5) = 0
step 5: invert values in brackets to get values of x (solve where necessary)
3x = 4 -> x = 4/3
2x = 5 -> x = 2.5
Solve 3x^2 + 10x - 8 = 0
step 1: multiply the coefficient and normal number
3 x -8 = -24
step 2: find factors of -24 that add to 10
-2 x 12 = -24
step 3: factorise with these values
(3x + 12)(x - 2) = 0
step 4: invert bracket values to find the value of x (solve where necessary)
x = 2
3x = -12 -> x = -4
Solve 3x^2 + 7x - 13 = 0 using the quadratic formula
quadratic formula = x = (-b +- √b^2 - 4ac)/2a
b = 3
a = 7
c = -13
- input values into calculator
- x = 1.22 (2 d.p)
- x = -3.55 (2 d.p)
What is the quadratic formula ?
x = (-b +- √b^2 - 4ac)/2a
complete the square:
x^2 - 10x + 18
- (x - 5)^2 - 25 + 18
- (x - 5)^2 - 7
solve x^2 - 10x + 18 = 0 by completing the square
step 1: complete the square
- (x - 5)^2 - 25 + 18 = 0
- (x - 5)^2 - 7 = 0
step 2: solve
- (x - 5)^2 = 7
- x - 5 = +-√7
- x = 5 +- √7
x = 7.65 (2 d.p)
x = 2.35 (2 d.p)
Solve 5x^2 + 30x = - 40
step 1: equate to 0
5x^2 + 30x + 40 = 0
step 2: simplify
5 [x^2 + 6x + 8] = 0
step 3: complete the square
- 5 [x^2 + 6x + 8] = 0
- 5[(x + 3)^2 - 9 + 8] = 0
- 5[(x + 3)^2 - 1] = 0
step 4: expand
- 5 [(x + 3)^2 - 1] = 0
- 5(x + 3)^2 - 5 = 0
step 5: solve
- 5(x + 3)^2 - 5 = 0
- 5(x + 3)^2 = 5
- (x + 3)^2 = 1
- x + 3 = +- √1
- x = - 3 +- √1
x = -2
x = -4
What is an arithmetic sequence ?
a sequence where a value is being added or subtracted between each term (the common difference)
What is a geometric sequence ?
a sequence where a value is multiplied or divided by between each term (the common ratio)
Continue the sequence for 3 terms:
22, 18, 14, 10, 6
2, -2, -6
(4 is the common difference (subtracted each time))
Continue the sequence for 3 terms:
27, 9, 3, 1, 1/3
1/9, 1/27, 1/81
(divided by 3 each time)
Write an expression for the nth term:
1, 2.5, 4, 5.5, 7
step 1: find the common difference
2.5-1 = 1.5
step 2: find the imaginary 1st term
1-1.5 = -0.5
step 3: place the common difference in front of n and the imaginary first term after
1.5n - 0.5
Find the 600th term of a sequence with the nth term 6n - 9
6 x 600 = 3600
3600 - 9 = 3591
600th term = 3591
Write the expression of the nth term:
-5, -9, -13, -17, -21
- common difference = -4
- imaginary first term = -1
- nth term expression = -4n - 1
The first three terms of a sequence are 8, 40 and 200. Work out the next 3 terms.
- geometric sequence
- common ratio is 5
- next three terms = 1000, 5000, 25000
The third and fourth terms of a sequence are 50 and 250. Work out the first term of the sequence.
- geometric sequence
- 250/50 = 5
- 50/5 = 10
- 10/5 = 2
first term is 2
Work out the next two terms of the sequence:
6, 7, 11, 18
step 1: identify the type of sequence and the term to term rule
- quadratic sequence
- first difference = 1, 4, 7
- second difference = 3
step 2: work out the next two terms of the first difference sequence
1, 4, 7, (10), (13)
step 3: apply these two terms to the original sequence
- 18 + 10 = 28
- 28 + 13 = 41
final answer = 28, 41
Give the formula for the nth term of a quadratic sequence.
an^2 + bn + c
Use the formula for the nth term of a quadratic sequence to find the first term and first and second differences of a hypothetical sequence.
step 1: find the first 4 terms of the formula sequence by substitution in an^2 + bn + c
- 1^2 = 1 * a = a
- 1 * b = b
- c = c
first term = a + b + c
—-
- 2^2 = 4 * a = 4a
- 2 * b = 2b
- c = c
second term = 4a + 2b + c
—-
- 3^2 = 9 * a = 9a
- 3 * b = 3b
- c = c
third term = 9a + 3b + c
—-
- 4^2 = 16 * a = 16a
- 4 * b = 4b
- c = c
fourth term = 16a + 4b + c
—-
first four terms = a+b+c, 4a+2b+c, 9a+3b+c, 16a+4b+c
step 2: find the first difference of these four terms
- 4a-a = 3a
- 2b-b = b
- c-c = 0
first difference of first two terms = 3a + b
—-
- 9a-4a = 5a
- 3b-2b = b
- c-c = 0
first difference of next two terms = 5a + b
—-
- 16a-9a = 7a
- 4b-3b = b
- c-c = 0
first difference of next two terms = 7a + b
—-
first difference sequence = 3a+b, 5a+b, 7a+b
step 3: find the second difference of the first difference sequence
- 5a-3a = 2a
- b-b = 0
second difference of first two terms = 2a
—-
- 7a-5a = 2a
- b-b = 0
second difference of next two terms = 2a
—-
second difference = 2a
final answer:
formula = an^2 + bn + c
first term = a + b + c
first difference = 3a + b
second difference = 2a
Work out the nth term:
6, 9, 14, 21, 30
step 1: identify the type of sequence and the formula for the nth term
- quadratic
- an^2 + bn + c
step 2: equate the first term, first difference and second difference of the actual sequence to the hypothetical sequence
- 6 = a+b+c
- 3 = 3a+b
- 2 = 2a
step 3: solve for a, b and c using these
- 2 = 2a
- 1 = a
—-
- 3 = 3a+b
- 3 = (3 x 1) + b
-3 = 3 + b
- 0 = b
—-
- 6 = a+b+c
- 6 = 1 + 0 + c
- 6 = 1 + c
- 5 = c
—-
a = 1
b = 0
c = 5
step 4: substitute these values of a, b and c into the formula for the nth term
- an^2 + bn + c
- 1n^2 + 0n + 5
- n^2 + 5
final answer = n^2 + 5
Work out the 10th term of a sequence with the nth term n^2 + 5
- n = 10
- 10^2 + 5
- 100 + 5 = 105
10th term = 105
Work out the nth term:
1, 3, 9, 19, 33
step 1: find the first 3 first differences
- 3-1 = 2
- 9-3 = 6
- 19-9 = 10
step 2: find the first two second differences
- 6-2 = 4
- 10-6 = 4
step 3: equate the first term, first difference and second difference of the actual sequence to those of the hypothetical sequence
1 = a+b+c
2 = 3a+b
4 = 2a
step 4: use the above to solve for a, b and c
2a = 4
a = 2
—-
3a+b = 2
(3 x 2) + b = 2
6 + b = 2
b = -4
—-
a+b+c = 1
2 - 4 + c = 1
2 + c = 5
c = 3
step 5: substitute the values for a, b and c into the formula an^2 + bn + c
2n^2 - 4n + 3
Which term in the sequence with the nth term 2n^2 - 2n + 3 is equal to 73 ?
step 1: equate the nth term and the given term
2n^2 - 2n + 3 = 73
step 2: equate the equation to 0
2n^2 - 2n - 70 = 0
step 3: solve for n
- 2n^2 - 2n - 70 = 0
- 2[n^2 - n - 35] = 0
- 2[(n - 1/2)^2 - 0.25 - 35] = 0
- 2[(n - 1/2)^2 - 35.25] = 0
- 2(n - 1/2)^2 - 70.5 = 0
- 2(n - 1/2)^2 = 70.5
- (n - 1/2)^2 = 35.25
- n - 1/2 = +- √35.25
n = 1/2 +- √35.25
n = 6.437171044
n = -5.437171044
step 4: discredit one of the solutions for n
- cannot have a negative term of a sequence, so n cannot be -5.437171044
- n must be 6.437171044
final answer = 73 is the 6.437171044th term
Show that a solution lies between x = 1 and x = 2 for the equation:
x^3 - 2x - 3 = 0
step 1: substitute in x = 1
x^3 - 2x - 3 = 0
1^3 - (2 x 1) - 3 = ?
1 - 2 - 3 = -4
step 2: substitute in x = 2
x^3 - 2x - 3 = 0
2^3 - (2 x 2) - 3 = ?
8 - 4 - 3 = 1
step 3: identify if there is a change from positive to negative or vice versa in the two answers to the equation
- yes
step 4: explain why a solution lies between the two given values for x
The change in sign from a negative to positive value for the answer to the equation when the two values for x are substituted in shows that a solution lies between them, as on a graph, there would be a coordinate on the x axis, aka the solution to the equation.
—-
simpler version = The change in sign shows there must be a solution for x between x = 1 and x = 2
Find the iterative formula for x^3 - 2x - 3 = 0
step 1: make x^3 the subject of the formula
x^3 - 2x - 3 = 0
x^3 - 2x = 3
x^3 = 2x + 3
step 2: get rid of the power on the left
x = ∛(2x + 3)
step 3: write the above equation with the x having the subscript n+1 on the left, and any x on the right with the subscript n to get the iterative formula
x {n+1} = ∛(2x{n} + 3)
Given that x{0} = 2, calculate the values of x{1}, x{2} and x{3} using the iterative formula:
x{n+1} = ∛(2x{n} + 3)
step 1: substitute the value for x{0} into the iterative formula
- x{0+1} = ∛(2x{0} + 3)
- x{1} = ∛((2 x 2) + 3)
step 2: solve for x{1}
- x{1} = ∛(4 + 3)
- x{1} = ∛7
- x{1} = 1.912931183
step 3: repeat above two steps with the value for x{1} instead of x{0}
- x{1+1} = ∛(2x{1} + 3)
- x{2} = ∛(2x{1} + 3)
- x{2} = ∛((2 x ∛7) + 3)
- x{2} = ∛6.825862366
- x{2} = 1.896935259
step 4: repeat again with the value for x{2} OR type the iterative formula into the calculator using the answer button and press = to get the next value
∛((2 x ans) + 3) = 1.893967062 = x{3}
Solve the equation x^3 - 2x - 3 = 0 using the iterative formula x{n+1} = ∛(2x{n} +3), with x{0} = 2, and give your answer to 5 decimal places.
step 1: substitute in x{0} into the equation like normal to find x{1}
- x{0+1} = ∛(2x{0} + 3)
- x{1} = ∛((2 x 2) + 3)
- x{1} = ∛(4 + 3)
- x{1} = ∛7
- x{1} = 1.912931183
step 2: type the iterative formula into the calculator using the ans button
- ∛((2 x ans) + 3)
step 3: keep hitting the = button until the answer does not change anymore
step 4: round this answer to 5 d.p to get the final answer
- 1.893289196 rounded to 5 d.p = 1.89329
final answer = 1.89329
By substituting the solution 1.89329 (5 d.p) into the equation x^3 - 2x - 3 = 0, comment on the accuracy of your solution.
step 1: substitute the solution in
- x^3 - 2x - 3 = 0
- (1.89329)^3 - 2(1.89329) - 3 = 0.00000703525 (11 d.p)
step 2: comment on the accuracy
This is very close to 0, so the solution is a good estimate of the real solution.
What is the rounded value for pi (2 d.p) ?
3.14
Explain how to draw a pie chart using these values for the favourite subject of some students:
english = 7
maths = 11
history = 4
geography = 2
step 1: work out the total number of students
- 7 + 11 + 4 + 2 = 24
step 2: divide 360 by 24 to work out how many degrees is for each person in the survey
- 360/24 = 15
step 3: work out how many degrees of the pie chart is for each subject
- 7 x 15 = 105
- 11 x 15 = 165
- 11 x 4 = 44
- 11 x 2 = 22
step 4: use a protractor and a ruler to draw the pie chart on the circle given, and label each section with the appropriate subject name
Explain how to draw a frequency polygon using this set of data:
Time:
0 < t </ 20
20 < t </ 40
40 < t </ 60
60 < t </ 80
80 < t </100
100 < t </ 120
Frequency:
10
15
42
21
13
9
step 1: find the midpoints of the data groups
- 10
- 30
- 50
- 70
- 90
- 110
step 2: plot the midpoints against the frequency (time on the x axis, frequency on the y axis)
step 3: join the plotpoints using straight lines drawn with a ruler
NOTE: do NOT join the first and last plotpoints
Explain how to draw a stem and leaf diagram for the dataset:
12, 14, 17, 17, 19, 18, 29, 28, 25, 25, 25, 45, 48, 59
step 1: draw the basis of the diagram (one vertical line on the left and 3 horizontal lines because there are 4 stems)
step 2: write in the 4 stems on the left hand side of the diagram - in this case, 1, 2, 4 and 5
NOTE: a leaf can only ever be one digit
step 3: write in the leaves for the stem of 1(0), in order of smallest to largest. In this case, 2, 4, 7, 7, 8, 9
step 4: repeat for the stem of 2 - 5, 5, 5, 8, 9
step 5: repeat for the 4 stem - 5, 8
step 6: repeat for the 5 stem - 9
step 7: draw a key, using one of the stems and the leaves and equating it to the value it represents, eg. 4| 8 = 48 [insert value type, such as years of age etc.]
Explain how to draw a stem and leaf diagram for the dataset:
128, 157, 157, 155, 189
step 1: draw the basis of the diagram (one vertical line on the left and 2 horizontal lines because there are 3 stems)
step 2: write in the 3 stems on the left hand side of the diagram - in this case, 12, 15 and 18
NOTE: a leaf can only ever be one digit
step 3: write in the leaves for the stem of 12(0), in order of smallest to largest. In this case, 8
step 4: repeat for the stem of 15 - 5, 7, 7
step 5: repeat for the 18 stem - 9
step 7: draw a key, using one of the stems and the leaves and equating it to the value it represents, eg. 15| 5 = 155 [insert value type, such as years of age etc.]
Explain how to draw a stem and leaf diagram for the dataset:
9.76, 9.55, 9.34, 10.8, 2.773
step 1: draw the basis of the diagram (one vertical line on the left and 4 horizontal lines because there are 5 stems)
step 2: write in the 5 stems on the left hand side of the diagram - in this case, 97, 95, 93, 10 and 277
NOTE: a leaf can only ever be one digit
step 3: write in the leaves for the stem of 93, in order of smallest to largest.
step 4: repeat for the stem of 95
step 5: repeat for the 97 stem
step 6: repeat for the 10 stem
step 7: repeat for the 277 stem
step 7: draw a key, using one of the stems and the leaves and equating it to the value it represents, eg. 93| 4 = 9.34 [insert value type, such as years of age, seconds etc.]
What is the formula for pythagoras’ theorem ?
c^2 = a^2 + b^2
When is pythagoras’ theorem applicable ?
right angled triangles
When are sin, cos and tan applicable ?
right angled trangles
When do you need to use sin to work out a missing length ?
- SOH
- when the opposite and hypotenuse lengths are known
When do you need to use cos to work out a missing length ?
- CAH
- when the adjacent and hypotenuse lengths are known
When do you need to use tan to work out a missing length ?
- TOA
- when the opposite and adjacent lengths are known
Explain how to find the exact values of 0, 30, 45, 60 and 90 degrees for sin, cos and tan using the table method.
step 1: draw a square root symbol
step 2: write 0, 30, 45, 60 and 90 across the top
step 3: inside the sqrt symbol, write 0, 1, 2, 3 and 4 and write sin on the left of this row outside of the symbol
step 4: below the row of values for sin, write 4, 3, 2, 1 and 0, and write cos on the left of this row outside of the symbol
step 5: below both of these rows and the whole symbol, draw a horizontal line and write 2 beneath it
to find, for example, cos45, you would then look at the value for cos under 45, which would be 2, then do the square root of 2 over 2
to find, for example, the value of sin60, you would look at the value for sin under 60, which would be 4, and do the square root of 4 (2) over 2, which is just 1
to find values for tan, divide the same degree of sin by the same degree of tan
for example, to find tan30:
- sin30 = 1/2
- cos30 = sqrt3/2
- tan30 = (1/2)/(sqrt3/2)
- 2 cancels out, so left with 1/sqrt3
- rationalise denominator to get sqrt3/3 as the exact value of tan30
State the formula for the area of a triangle (using sine)
0.5 x a x b x sin(C)
NOTE: the a and b must be either side of the angle labelled C
State the sine rule for both lengths and angles
LENGTH:
a/sinA = b/sinB
ANGLE:
sinA/a = sinB/b
State the cosine rule for a missing side
a^2 = b^2 + c^2 - 2bc*cos(A)
State the cosine rule for a missing angle
A = cos^(-1)((b^2 + c^2 - a^2)/2bc)
State the formula for the volume of a cylinder
pi x r^2 x h
State the formula for the volume of a sphere
4/3 x pi x r^3
State the formula for the volume of a pyramid
1/3 x base area x vertical height
Explain how to work out the volume of a frustum
step 1: find the volume of the larger cone by using 1/3 x pi x r^2 x h
step 2: find the volume of the smaller cone by using the same formula
step 3: subtract the smaller cone’s volume from the larger cone’s volume to get the frustum’s volume
What is the error interval for 5.4 ?
step 1: find the upper and lower bound of 5.4
UB = 5.45
LB = 5.35
step 2: write it as an inequality, where n is the set of numbers that would round to 5.4
final answer = 5.35 </ n < 5.45
A number, x, rounded to 2 s.f. is 1300. Write the error interval for x.
UB = 1350
LB = 1250
final answer = 1250 </ x < 1350
Write the UB and LB for the perimeter of a rectangle with side lengths 87.3cm and 0.518m.
step 1: convert the m into cm
0.518m -> 51.8cm
for the UB:
step 2: find the UB of both measurements
- 87.35cm
- 51.85cm
step 3: multiply both by 2 and add together to get the UB perimeter
174.7 + 103.7 = 278.4cm
for the LB:
do the same as UB but with LB measurements to get 278cm as the perimeter
Write the UB and LB for the area of a rectangle with side lengths 24.3cm and 36.7cm.
step 1: find LB of both measurements and multiply together
- 24.25 x 36.65 = 888.7625
step 2: find UB of both measurements and multiply together
- 24.35 x 36.75 = 894.8625
Round 1.7165892 and 1.7246341 to a suitable degree of accuracy
- 1.72
- the UB and LB both agree to this number of decimal places.
a = 3b - c
if b = 8.7 (1 d.p) and c = 15 (nearest integer), work out the upper and lower bounds for a.
step 1: find the UB and LB of these values.
UB b = 8.75
LB b = 8.65
UB c = 15.5
LB c = 14.5
step 2: find the LB of a
- 3b - c
- 3(8.65) - 15.5 = 10.45
step 3: find the UB of a
- 3b - c
- 3(8.75) - 14.5 = 11.75
Calculate the area of a sector with an angle of 70 degrees and a radius of 8cm
area of a circle = pi x r^2
area of this sector = 70/360 x pi x r^2
70/360 x pi x 8^2 = 39.10 cm^2 (2 d.p)
Find the arc length of a sector with an angle of 87 degrees and a radius of 3cm
circumference of circle = pi x d
d = 2r
d = 2 x 3 = 6cm
circumference of whole circle = 6pi
arc length = 87/360 x 6 x pi = 4.56 cm (2 d.p)
Calculate the missing angle if a sector’s arc length is 3.5cm and the radius is 4cm
arc length = A/360 x 2r x pi
arc length/(2r x pi) = A/360
A = (arc length/(2r x pi)) x 360
A = (3.5/(8 x pi)) x 360 = 50.13380707 degrees
final answer = 50.1 degrees (1 d.p)
Calculate the missing radius if the area of a sector is 66cm^2 and the angle is 136 degrees
area = A/360 x pi x r^2
r = sqrt(area/(A/360 x pi))
r = sqrt(66/((136/360) x pi))
r = 7.457252143cm
r = 7.46cm (2 d.p)
State the formula for the internal angles of a polygon
(n-2) x 180
Solve the linear simultaneous equation:
2x + y = 7
3x - y = 8
step 1: make the number of xs or ys the same if needed
step 2: add or subtract both equations to get the ys or xs in isolation
- 2x + 3x = 5x
- y +(-y) = 0
- 7 + 8 = 15
5x = 15
step 3: solve for the isolated unknown
5x = 15
x = 3
step 4: substitute the value of the known unknown into one of the original equations and solve for the other unknown
2x + y = 7
(2 x 3) + y = 7
6 + y = 7
y = 1
final answer:
x = 3
y = 1
Solve the linear simultaneous equation:
7p - 2q = - 1
21p + q = 25
step 1: make the number of ps or qs the same
(21p + q = 25) x 2 = (42p +2q = 50)
step 2: add or subtract the equations to get the ps or qs in isolation
- 42p + 7p = 49p
- 2p + (- 2p) = 0
- 50 + (- 1) = 49
49p = 49
step 3: solve for p
49p = 49
p = 1
step 4: substitute the value for p into one of the original equations
7p - 2q = -1
7 - 2q = -1
-2q = -8
q = 4
final answer:
p = 1
q = 4
Solve:
y = x^2 + 10x + 6
y = 4x - 3
step 1: substitute the linear equation into the non-linear equation
4x - 3 = x^2 + 10x + 6
step 2: equate it to 0
4x - 3 = x^2 + 10x + 6
0 = x^2 + 6x + 9
step 3: solve for the (two) value(s) of x
- 0 = x^2 + 6x + 9
- (x +3)(x+3)
- x = -3
step 4: substitute the value(s) of x back into the equation like normal
y = 4x - 3
y = -12 - 3
y = -15
final answer:
x = -3
y = -15
Solve:
y = 2x^2 - 17x + 14
y + 4x = 8
step 1: rearrange the linear equation to be in the form ‘y = _’
y + 4x = 8
y = -4x + 8
step 2: substitute the linear equation into the non-linear equation, then solve for x
- 8 - 4x = 2x^2 - 17x + 14
- 0 = 2x^2 - 13x + 6
- 26 = 12 = -12-1
- 0 = 2x^2 - 12x - x + 6
- 0 = -2x(-x + 6) + 1(-x + 6)
- 0 = (-x + 6)(-2x + 1)
-x = -6 -> x = 6
-2x = -1 -> 2x = 1 -> x = 0.5
step 3: substitute the values for x back into the original linear equation
y = 8 - 4x
y = 8 - 24
y = -16
—-
y = 8 - 4x
y = 8 - 2
y = 6
final answer:
x = 0.5, y = 6
x = 6, y = -16
Solve:
x^2 + y^2 = 50
y = 2x - 15
step 1: substitute the linear equation into the circle equation
x^2 + (2x - 15)^2 = 50
step 2: expand the double/squared bracket
(2x - 15)(2x - 15) =
4x^2 - 30x - 30x + 225
step 3: collect like terms
x^2 + 4x^2 - 30x - 30x + 225 = 50
5x^2 - 60x + 225 = 50
step 4: equate to 0
5x^2 - 60x + 175 = 0
step 5: simplify and solve for x
5x^2 - 60x + 175 = 0 ->
x^2 - 12x + 35 = 0
-7*-5 = 35
(x - 7)(x - 5) = 0
x = 7
x = 5
step 6: substitute both values for x back into the original linear equation
y = 2x - 15
y = (27) -15
y = 14 - 15
y = - 1
—-
y = 2x - 15
y = (25) - 15
y = 10 - 15
y = -5
final answer:
x = 7, y = -1
x = 5, y = -5
Solve:
x^2 - 5y^2 = 4
x - 2y = 1
step 1: rearrange the linear equation then substitute it into the non-linear equation
x - 2y = 1
x = 1 + 2y
(1 + 2y)^2 - 5y^2 = 4
step 2: expand the bracket and collect like terms, then equate to 0 and solve for y
(1 + 2y)(1+ 2y) =
1 + 2y + 2y + 4y^2
—-
1 + 4y + 4y^2 - 5y^2 = 4
-3 + 4y - y^2 = 0
3 - 4y + y^2 = 0
(y - 1)(y - 3) = 0
y = 1
y = 3
step 3: substitute the values for y back into the original linear equation
x = 1 + 2y
x = 1 + (23)
x = 1 + 6
x = 7
—-
x = 1 + 2y
x = 1 + (21)
x = 1 + 2
x = 3
final answer:
x = 3, y = 1
x = 7, y = 3
Simplify fully:
(12m + 18)/(2m^2 + 3m)
step 1: factorise the top and bottom of the fraction
- 12m + 18
- 6(2m + 3)
—-
- 2m^2 + 3m
- m(2m + 3)
—-
6(2m+3)/m(2m + 3)
step 2: cancel out the bracket which is the same on either side
6/m
final answer = 6/m
simplify fully:
(c^4 + c^3)/(c^2 + 6c + 5)
step 1: factorise the top + bottom
- c^4 + c^3
- c^3(c + 1)
—-
- c^2 + 6c + 5
- 1*5 = 5
- (c + 1)(c + 5)
step 2: cancel out any same brackets
- (c + 1) and (c + 1) cancel out
- c + 5 is left
final answer = c + 5
Simplify fully:
(6x^2 - 54)/(2x^2 + 6x)
step 1: factorise the top + bottom
- 6x^2 - 54
- 6(x^2 - 9)
- 6(x + 3)(x - 3)
—-
- 2x^2 + 6x
- 2x(x + 3)
step 2: cancel out any same brackets and simplify
6(x - 3)/2x -> 3(x - 3)/x
Simplify fully:
(9y^2 - 1)/(3y^2 + 19y + 6)
step 1: factorise fully the top and bottom
- 9y^2 - 1
- (3y - 1)(3y + 1)
—-
- 3y^2 + 19y + 6
- 36 = 18
- 181 = 18
- 3y^2 + 18y + y + 6
- 3y(y + 6) + 1(y + 6)
- (3y + 1)(y + 6)
step 2: cancel out any same brackets
3y - 1/y + 6
Simplify fully:
((20a^2+5a)/(3a+6)) x ((6a^2-24)/(4a^2-7a-2))
step 1: factorise top + bottom of both fractions
(first fraction)
- 20a^2 + 5a
- 5a(4a + 1)
—-
- 3a + 6
- 3(a + 2)
—-
(second fraction)
- 6a^2 - 24
- 6(a^2 - 4)
- 6(a - 2)(a + 2)
—-
- 4a^2 - 7a - 2
- 4-2 = -8
- 1-8 = -8
- 4a^2 - 8a + a - 2
- 4a(a - 2) + 1(a - 2)
- (4a + 1)(a - 2)
—-
5a(4a + 1)/3(a + 2)
6(a- 2)(a + 2)/(4a + 1)(a - 2)
step 2: multiply fractions together (but don’t expand brackets)
(top)
- 5a(4a + 1) x 6(a - 2)(a + 2)
- 5a x 6 = 30a
- 30a(4a + 1)(a - 2)(a + 2)
—-
(bottom)
- 3(a + 2) x (4a + 1)(a - 2)
- 3(4a + 1)(a - 2)(a + 2)
—-
30a(4a + 1)(a - 2)(a + 2)
———————————–
3(4a + 1)(a - 2)(a + 2)
step 3: cancel out any same brackets
30a/3
step 4: simplify
- 30a/3
10a/1
final answer = 10a
Write:
9 + ((2x+10)/(3x^4+x^3)) /
((x^2-25)/(x^5-5x^4)
in the form:
(ax + b)/(cx + d)
step 1: use BIDMAS to figure out which calculation to do first
- division comes first so do that
step 2: factorise both fractions’ top and bottom
(fraction 1 top)
- 2x + 10
- 2(x + 5)
—-
(fraction 1 bottom)
- 3x^4 + x^3
- x^3(3x + 1)
—-
(fraction 2 top)
- x^2 - 25
- (x + 5)(x - 5)
—-
(fraction 2 bottom)
- x^5 - 5x^4
- x^4(x - 5)
step 3: flip fraction 2 so the calculation becomes multiplication instead of division
step 4: multiply both fractions together (don’t expand brackets)
2x^4(x+5)(x-5)
———————————-
x^3(3x + 1)(x + 5)(x - 5)
step 5: cancel out any same brackets
2x^4
—————–
x^3(3x + 1)
step 6: cancel the xs
2x
——–
3x + 1
step 7: add the 9 by giving them common denominator
- 2x/3x+1 + 9/1
- 9/1 x 3x+1/3x+1 = 27x+9/3x+1
—-
2x/3x+1 + 27x+9/3x+1 = 29x+9/3x+1
final answer:
29x + 9
————
3x + 1
Chloe buys a phone for £120, then sells it for £138. Work out her percentage profit.
step 1: work out the profit
profit = £138 - £120 = £18
step 2: use change/og x 100 to find the perfecntage profit
18/120 x 100 = 15% profit
final answer = 15%
The number of people visiting the cinema on a Saturday was 20% more than on Friday.
The number of visitors on Saturday was 9840.
Work out the number of visitors on Friday.
step 1: find the multiplier
- +20% = 100 + 20 = 120
- 120/100 = 1.2
step 2: make an equation
x * 1.2 = 9840
step 3: solve the equation
9840/1.2 = x = 8200
final answer = 8200 visitors
Find the reciprocal of 1.25 and give your answer as a decimal
1/1.25 = 0.8
final answer = 0.8
Here is a dataset:
Time (minutes):
10 < t </ 20
20 < t </ 30
30 < t </ 40
40 < t </ 50
Frequency:
28
13
13
6
a) find the class in which the median lies
b) Work out an estimate for the mean time spent revising
(a)
n/2 = median
n = 6 + 13 + 13 + 28 = 60
60/2 = 30
—-
30th datapoint is in the second class
final answer = 20 < t </ 30
(b)
step 1: find the midpoints of all groups
15
25
35
45
step 2: draw an fx column and multiply the midpoints by the frequencies
15 x 28 = 420
25 x 13 = 325
35 x 13 = 455
6 x 45 = 270
step 3: calculate the total of the fx column
420 + 325 + 455 + 270 = 1470
step 4: divide total of the fx column by the total frequency (no. of students)
1470/60 = 24.5
final answer = on average, 24.5 minutes spent revising
Explain how to draw a histogram from this dataset:
Speed (mph):
0 < s </ 20
20 < s </ 30
30 < s </ 45
45 < s </ 50
50 < s </ 60
Frequency:
30
30
42
37
9
step 1: add one more column and calculate the class widths
20
10
15
5
10
step 2: add another column and calculate the frequency densities using the formula frequency/class width
1.5
3
2.8
7.4
0.9
step 3: plot frequency density against speed (class width (x axis)) and draw bars with no spaces in between
What is the formula for frequency density ?
frequency/class width
A floor with area 10m^2 can be tiled by 3 workers in 8 hours.
Work out how long it would take 4 workers to tile a floor that is 25m^2 in area.
Assume that all workers can work at the same rate.
step 1: find k, the constant
y (inversely prop.) 1/x OR y = k/x
x = amt of workers, y = hours taken
—-
if x = 3, and y = 8:
8 = k/3
k = 24
step 2: substitute the second set of values into y = 24/x
when x = 4, y = ?
y = 24/4
y = 6
NOTE: THIS IS FOR 10m^2
step 3: calculate how many times bigger 25m^2 is than 10m^2
- 25/10 = 2.5
- 2.5x bigger
step 4: multiply y = 6 by 2.5
6 x 2.5 = 15h
final answer = 15h
The ingredients for 12 pancakes are:
300g flour
400ml milk
2 eggs
Raul has:
1500g flour
1800ml milk
11 eggs
What is the maximum amount of pancakes he can make ?
step 1: divide all of Raul’s measurements by the original measurements
1500/300 = 5
1800/400 = 4.5
11/2 = 5.5
step 2: identify the smallest of these answers
4.5
step 3: multiply the original amount of pancakes (12) by 4.5
12 x 4.5 = 54
final answer = 54
Nadia buys her favourite Coke in the UK.
In Spain, she sees the same drink.
The UK bottle is 2l and costs £2.99, whilst the Spanish bottle is 1250ml and costs 2.40 euros.
If £1 = 1.17 euros, which of the two bottles has better value for money ?
step 1: convert all liquid measurements to litres
1250ml -> 1.25l
step 2: convert all monetary values to euros
£2.99 x 1.17 = 3.4983
step 3: compare
- 2l = 3.4983 euros
- 1l = 1.74915 euros
—-
- 1.25l = 2.40 euros
- 1l = 1.92 euros
—-
1.75 (2 d.p) < 1.92, so the UK bottle is better value for money
Tia has £5000 to invest for 3 years. She compares the details of two banks.
Bank A:
- 2.5% compound interest
Bank B:
- First year = 4% compound interest
- All other years = 1% compound interest
How much more money will Tia make going to Bank A than Bank B ?
step 1: work out the multiplier for Bank A’s compound interest
- 100 + 2.5 = 102.5
- 102.5/100 = 1.025
- is for 3 years, so will be 1.025^3
step 2: work out how much money Tia will have at the end of 3 years in total with Bank A
1.025^3 x 5000 = 5384.453125
step 3: work out how much money Tia earned with Bank A
5384.453125 - 5000 = £384.453125
step 4: work out the first multiplier for Bank B and multiply by 5000
- 100 + 4 = 104
- 104/100 = 1.04
- one year, so 1.04^1 = 1.04
—-
1.04 x 5000 = 5200
step 5: work out the second multiplier for bank B and multiply by 5200
- 100 + 1 = 101
- 101/100 = 1.01
- is for two years, so 1.01^2
—-
1.01^2 x 5200 = 5304.52
step 6: work out how much money Tia earned with Bank B at the end of 3 years
5304.52 - 5000 = £304.52
step 7: work out how much more money she earned with Bank A
£384.453125 - £304.52 = £79.933125
final answer = £79.93 (nearest penny)
Given that f(x) = 3x - 1, calculate f(2)
step 1: substitute 2 into the equation (2 = x)
f(2) = 3*2 - 1
step 2: simplify to get the answer
f(2) = 5
What does the x in f(x) represent ?
the input of the ‘function machine’ (f)
Given that g(x) = x^3 + 5x, calculate g(5)
g(x) = x^3 + 5x
g(5) = 5^3 + 25
g(5) = 150
(a) Given that f(x) = x + 1, and g(x) = x^2 - 5, calculate gf(x)
(b) Calculate fg(x)
(a)
step 1: write gf(x) with double brackets
g(f(x))
step 2: substitute f(x) into g(x) wherever there is an x
g(f(x)) = (x + 1)^2 - 5
step 3: expand the brackets and collect like terms
- (x + 1)(x + 1) = x^2 + x + x + 1
- x^2 + 2x + 1
- x^2 + 2x + 1 - 5
- x^2 + 2x - 4
final answer = gf(x) = x^2 + 2x - 4
(b)
step 1: write fg(x) with double brackets
f(g(x))
step 2: substitute g(x) into f(x) wherever there’s an x in the equation and simplify
- f(g(x)) = x^2 - 5 + 1
- f(g(x)) = x^2 - 4
final answer = fg(x) = x^2 - 4
Given f(x) = 8x - 5, calculate f^-1(x)
step 1: replace ‘f(x)’ with y
y = 8x - 5
step 2: replace any x with a y and any y with an x
x = 8y - 5
step 3: make y the subject
x = 8y - 5
x + 5 = 8y
(x + 5)/8 = y
step 4: replace y with f^-1(x)
f^-1(x) = (x + 5)/8
Given f(x) = (x/5) + 1, calculate f^-1(x)
step 1: change f(x) into y
y = (x/5) + 1
step 2: change any y into an x and any x into a y
x = (y/5) + 1
step 3: make y the subject
x = (y/5) + 1
x - 1 = y/5
5(x - 1) = y
step 4: replace y with f^-1(x)
f^-1(x) = 5(x - 1)
Given that f(x) = x+5, and g(x) = x^2 - 2, work out (gf)^-1(x)
step 1: find the function of g(f(x)) as normal
- g(x) = x^2 - 2
- g(f(x)) = (x + 5)^2 - 2
step 2: find the inverse function of gf(x)
- gf(x) = (x + 5)^2 - 2
- y = (x + 5)^2 - 2
- x = (y + 5)^2 - 2
- x + 2 = (y + 5)^2
- sqrt(x + 2) = y + 5
- y = sqrt(x + 2) - 5
- (gf)^-1(x) = sqrt(x + 2) - 5
final answer:
(gf)^-1(x) = sqrt(x + 2) - 5
