Question Papers Marking Instructions

The Scottish Qualifications Authority owns the copyright to its exam papers and marking instructions.

Hints offered by B Fleming, with video solutions by 'DLBmaths'

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Paper 1

Question 1

Hint 1: You do not require a common denominator to multiply or divide fractions

Hint 2: You must turn the mixed number into an improper fraction

Hint 3: Simplify by cross cancelling

Hint 4: Multiply the numerators, and multiply denominators

Hint 5: Make sure your final answer is fully simplified

Hint 6: and here is a video of the solution:

Question 2

Hint 1: Multiply out the brackets to obtain four terms

Hint 2: Collect/simplify your like terms

Hint 3: Your answer should have 3 terms

Hint 4: and here is a video of the solution:

Question 3

Hint 1: Half the coefficient of the x term (the number in front of x) - this is the value for 'a'

Hint 2: Subtract this number squared … don't forget the +44

Hint 3: Calculate the value of b

Hint 4: and here is a video of the solution:

Question 4

Hint 1: Calculate 2u by multiplying each component of u by 2

Hint 2: Subtract v from this answer

Hint 3: Your answer should be a column vector with 3 components

Hint 4: and here is a video of the solution:

Question 5

Hint 1: Copy out the formula for the sine rule from the formula sheet

Hint 2: Substitute in what you know

Hint 3: To get LM on its own, multiply both sides by 0.4

Hint 4: Calculate your answer for LM in cm

Hint 5: and here is a video of the solution:

Question 6

6a) Hint 1: Write out the coordinates of A and B

6a) Hint 2: Use these coordinates to calculate the gradient

6a) Hint 3: m = (y₂-y₁)/(x₂-x₁)

6a) Hint 4: Your gradient should be a positive integer

6a) Hint 5: The equation of a line is y = mx + c

6a) Hint 6: Substitute in your gradient, and values for C and F

6a) Hint 7: Substitute your coordinates of A into this equation, to find c

6a) Hint 8: State your final equation in terms of F and C

6b) Hint 9: Substitute F = 40 into your equation from part (a)

6b) Hint 10: Calculate the Calories the sandwich contains, which should be more than 500

Hint 11: and here is a video of the solution:

Question 7

Hint 1: Substitute the coordinates into the equation given

Hint 2: A (negative number)² is a positive number

Hint 3: Calculate the value of 'a' by dividing

Hint 4: and here is a video of the solution:

Question 8

Hint 1: Write out the factor pairs for the numbers 40 and 90

Hint 2: Substitute these factor pairs into the expression, instead of 40 and 90

Hint 3: Simplify the surd terms by calculating the square roots of the square numbers

Hint 4: Collect/simplify your like terms

Hint 5: and here is a video of the solution:

Question 9

Hint 1: We want to find the original amount: 100%

Hint 2: Realise that 480000 ⇔ 80%

Hint 3: Find 10% and then 100%

Hint 4: and here is a video of the solution:

Question 10

Hint 1: What is the amplitude of the graph?

Hint 2: This is your value for 'a'

Hint 3: A standard sine graph starts at (0,0) - this one has been moved right

Hint 4: know that moving a graph right, leads to a negative number inside the bracket

Hint 5: State the values of 'a' (positive number) and 'b' (negative number)

Hint 6: and here is a video of the solution:

Question 11

11a) Hint 1: Rearrange to make y the subject of the formula

11a) Hint 2: The equation of the line is now in the form y = mx + c

11a) Hint 3: State the value of m (this should be a number only, with no 'x' letter)

11b) Hint 4: A line crosses the x axis when y = 0

11b) Hint 5: Substitute y = 0 into the equation of the line

11b) Hint 6: Solve to find x

11b) Hint 7: Write out the coordinates answer using brackets and a comma in between the numbers

Hint 8: and here is a video of the solution:

Question 12

Hint 1: Find the length of AC using the length of AB and the radius

Hint 2: Draw the right angled triangle ACP

Hint 3: Fill in the lengths of AC and CP that you know

Hint 4: Use Pythagoras' Theorem to calculate the length of AP

Hint 5: PQ is twice the length of AP

Hint 6: Write your answer for PQ in cm

Hint 7: and here is a video of the solution:

Question 13

13a) Hint 1: h = 60m

13a) Hint 2: Subsitute h into the given equation

13a) Hint 3: Rearrange to make the quadratic equal to 0

13a) Hint 4: Factorise the quadratic

13a) Hint 5: Solve to find t

13a) Hint 6: Choose the first time and state this answer in seconds

13b) Hint 7: Find how long the rocket was in the air by substituting in h(t) = 0

13b) Hint 8: This will give the initial time of 0 and the time when the rocket lands

13b) Hint 9: The turning point is half way between the roots, so calculate this value which is the time

13b) Hint 10: Calculate the maximum height by substituting this time into the equation

13b) Hint 11: Communicate your answer: will the rocket reach 70m? Say why!

13b) Hint 12: Alternate method - substitute 70m into the equation and use the discriminant to conclude answer

Hint 13: and here is a video of the solution:

Paper 2

Question 1

Hint 1: Get your multiplier ... start at 100% ... decrease by 15% ... turn into a decimal

Hint 2: Expected roll will be 964 multiplied by the multiplier 3 times for 3 years …. use ×(decimal multiplier)³

Hint 3: Type into your calculator and write your answer to several decimal places

Hint 4: Round to the nearest ten and put in your units (pupils)

Hint 5: and here is a video of the solution:

Question 2

Hint 1: You should have 2 coordinates labelled B and C, each with brackets, 3 numbers and commas between them

Hint 2: Find the length of the sides of the cube

Hint 3: B is the same distance along the x- and y-axes, and 4 units further up the z-axis than A

Hint 4: C is 4 units closer to the origin along the x-axis than A, 0 units along the y-axis and 4 units further up the z-axis than A

Hint 5: and here is a video of the solution:

Question 3

Hint 1: State 'a' is the cost of and adult ticket in £ and 'c' is the cost of a childs ticket in £

Hint 2: Write an equation for Bill's tickets - do not put the £ sign into the equation

Hint 3: Write an equation for Ben's tickets - do not put the £ sign into the equation

Hint 4: Multiply both equations to get the quantities of one of your variables (either 'a' or 'c') to match

Hint 5: Eliminate this variable by subtracting the equations ('same sign subtract')

Hint 6: Substitute whatever variable you have found into the first equation

Hint 7: Solve this equation to find the second variable

Hint 8: State your answers in words, with a £ sign and 2 decimal places as it is money

Hint 9: You can check your answer by substituting both variables back into the second equation

Hint 10: and here is a video of the solution:

Question 4

4a)i) Hint 1: Add up all the values

4a)i) Hint 2: Divide this number by 6 ... this is your mean

4a)ii) Hint 3: Create a table with 2 columns: (each value subtract the mean) and (this answer squared)

4a)ii) Hint 4: The first column should have a mix of positive and negative answers

4a)ii) Hint 5: The second column should all be positive after squaring

4a)ii) Hint 6: Total this second column

4a)ii) Hint 7: Write out the standard deviation formula carefully from the formula sheet

4a)ii) Hint 8: Substitute your total into the numerator and n = 6 into the denominator

4a)ii) Hint 9: Type into your calculator and write your answer down to several decimal places

4a)ii) Hint 10: Round to 2 decimal places (or more)

4b) Hint 11: Consistency means we need to compare the standard deviations

4b) Hint 12: Answer yes or no, and explain your answer

Hint 13: and here is a video of the solution:

Question 5

Hint 1: Find the linear scale factor (LSF) ... it should be a top heavy fraction as it is an enlargement

Hint 2: Find the volume scale factor (VSF) by cubing the LSF

Hint 3: Calculate the volume by multiplying the smaller volume by VSF

Hint 4: Remember to put in your units (cm³)

Hint 5: and here is a video of the solution:

Question 6

Hint 1: Use the Converse of Pythagoras' Theorem

Hint 2: Calculate the hypotenuse squared and, separately, the sum of the shorter sides squared

Hint 3: Compare your two answers - are they equal or not?

Hint 4: If they are equal then it is right angled so it is directly north. If they are not equal, it is not right angled so it is not directly north

Hint 5: and here is a video of the solution:

Question 7

Hint 1: Write out the formulae for the volume of a cone and volume of a sphere carefully from the formula sheet

Hint 2: State the diameter and the radius of the shape

Hint 3: Substitute your values into the formulae

Hint 4: To find the volume of a hemisphere, half the volume of the sphere

Hint 5: Subtract the volume of the hemisphere from the volume of the cone, giving your answer to several decimal places

Hint 6: Round your final answer to 2 significant figures with units

Hint 7: and here is a video of the solution:

Question 8

Hint 1: Simplify the numerator

Hint 2: Divide the numbers

Hint 3: Cancel the letters - what you do to the numersator, you also do to the denominator

Hint 4: and here is a video of the solution:

Question 9

Hint 1: To subtract fractions, we need them to have a common denominator

Hint 2: Process each fraction so that they have the same denominator

Hint 3: Leave the denominator, as it is already in factorised form

Hint 4: Expand the single bracket in the numerator (be careful of the negative)

Hint 5: Simplify the numerator

Hint 6: and here is a video of the solution:

Question 10

10a) Hint 1: Identify angle ABC at top as the 'cosy corner' to use the cosine rule

10a) Hint 2: From the formula sheet, write down the cosine rule for working out an angle

10a) Hint 3: Substitute your values into the formula ... the 13 is after the subtraction sign in the numerator

10a) Hint 4: Type into your calculator and write your answer to several decimal places

10a) Hint 5: Find the inverse cosine of this answer to get your angle to a few decimal places in degrees

10b) Hint 6: Find the other unshaded angle at point B using the 060° at A

10b) Hint 7: Subtract this angle and the angle found in part (a) from a full turn of 360°

10b) Hint 8: State your answer in degrees

Hint 9: and here is a video of the solution:

Question 11

Hint 1: Deal with the 'ut' term by doing the inverse operation of adding

Hint 2: Deal with the half (or dividing by 2) by doing the opposite

Hint 3: Deal with the t² by doing the opposite of multiplying

Hint 4: and here is a video of the solution:

Question 12

Hint 1: Rearrange to make cos(x) the subject of the formula

Hint 2: Your answer should be a proper fraction

Hint 3: Use your calculator to find the angle using inverse cosine

Hint 4: Draw a sketch of y = cos(x)

Hint 5: Your two answers for x should be the acute angle and (360 - acute angle)

Hint 6: and here is a video of the solution:

Question 13

Hint 1: You have to find the area of a sector and a triangle

Hint 2: From the formula sheet, write down the formula for the area of a triangle

Hint 3: Substitute in your values and calculate the area of a triangle to a few decimal places

Hint 4: Find the angle at the centre of the sector

Hint 5: State the fraction of the full turn (angle/360)

Hint 6: Find the area of the sector by calculating this fraction of the area of a full circle

Hint 7: Add your two areas together and state the units

Hint 8: and here is a video of the solution: