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 require a common denominator to subtract fractions

Hint 2: Subtract the whole numbers

Hint 3: Turn the mixed number at the beginning into a top heavy fraction

Hint 4: Find the lowest common multiple of 5 and 3: this should be your common denominator

Hint 5: Subtract the numerators

Hint 6: Simplify and turn the improper fraction into a mixed number

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

Question 2

Hint 1: Expand the single bracket: don't forget you are multiplying by (-2)

Hint 2: Simplify the left hand side of the inequality

Hint 3: Bring the x terms over to the right hand side to make them positive

Hint 4: Bring the number terms over to the left hand side

Hint 5: Get the x term on its own by dividing

Hint 6: Write the x term on the left hand side, swapping the inequality to match

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

Question 3

Hint 1: Highlight the radii in the circle

Hint 2: Fill in the right angles in semicircles and at a tangent and radius

Hint 3: Find angle OBD = angle BDO from the right angle at the tangent and radius

Hint 4: Find EDF from angles in a triangle and right angle in semi circle

Hint 5: Add these together to find angle BDF

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

Question 4

Hint 1: Expand the double brackets using FOIL or the grid method

Hint 2: Be careful of negatives

Hint 3: Collect like terms

Hint 4: Your answer should have 4 terms

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

Question 5

Hint 1: Add all the values

Hint 2: Divide this number by 5: this is your mean

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

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

Hint 5: The second column should all be positive after squaring

Hint 6: Total this second column

Hint 7: Write out the standard deviation formula carefully from the formula sheet

Hint 8: Substitute your total into the numerator and n = 5 into the denominator

Hint 9: Simplify what you have and state your value of a

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

Question 6

Hint 1: the amplitude of the graph is 'a'

Hint 2: the number of full waves between 0 and 360 is 'b'

Hint 3: State your answers for 'a' and 'b'

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

Question 7

Hint 1: In completed square form we move the turning point from (0,0)

Hint 2: the movement left is 'a'

Hint 3: As the turning point has been moved right, the value of 'a' will be negative

Hint 4: the movement up is 'b'

Hint 5: As the turning point has been moved down, the value of 'b' will be negative

Hint 6: Draw a line down the axis of symmetry

Hint 7: What do each of the coordinates on this line have in common?

Hint 8: The equation of the axis of symmetry is x = (this number)

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

Question 8

Hint 1: Use the coordinates to calculate the gradient

Hint 2: m = (y₂-y₁)/(x₂-x₁)

Hint 3: Your gradient should be a positive integer

Hint 4: The equation of a line is y = mx + c

Hint 5: Substitute in your gradient

Hint 6: Substitute the first coordinate into this equation to find c

Hint 7: State your final equation in terms of y and x

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

Question 9

Hint 1: Draw a sketch of the graph y = cos(x)

Hint 2: Find where cos(90), cos(100) and cos(300) would fall on this graph

Hint 3: Write these in order from the smallest to the largest explaining where they fall on the y-axis

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

Question 10

10a) Hint 1: Write the scores in numerical order from smallest to largest

10a) Hint 2: Count into the middle, half way (the mean) of these two values is the median

10a) Hint 3: State the median clearly

10a) Hint 4: Find the median of the lower and upper halves

10a) Hint 5: These will be an actual score given

10a) Hint 6: State the lower quartile (Q1) and upper quartile (Q3) clearly

10a) Hint 7: Semi-interquartile range, SIQR = (Q3 - Q1)/2

10b) Hint 8: Compare the medians by saying which score was less on average

10b) Hint 9: Compare the semi-interquartile ranges by saying which score was less consistent

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

Question 11

Hint 1: You need to multiply both equations to get the number of x or y terms to match

Hint 2: Same sign subtract (to eliminate x or y terms)

Hint 3: Solve the remaining equation to find the unknown

Hint 4: Substitute this value into an original equation to find the other unknown

Hint 5: State your answers for x and y clearly

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

Question 12

Hint 1: Factorise the numerator using a highest common factor

Hint 2: Factorise the denominator using quadratic factorising

Hint 3: You should have two terms being multiplied in the numerator and denominator

Hint 4: Cancel the term on the numerator and denominator that is the same to simplify

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

Question 13

Hint 1: Rationalise means we don’t want a square root in the denominator

Hint 2: Multiply each of the numerator and the denominator by √8

Hint 3: Simplify the numerator and denominator

Hint 4: Remember √8 can be simplified as well: write out the factor pairs for 8 to see

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

Question 14

Hint 1: Fractional power means we first take the cube root 8

Hint 2: Then find this number to the power of 5

Hint 3: and here is a video of the solution:

Paper 2

Question 1

Hint 1: Get your multiplier: start at 100%, increase by 2.8%, turn into a decimal

Hint 2: Predicted value will be 240000 multiplied by the multiplier 2 times for 2 years (to the power of 2)

Hint 3: Type into your calculator and write out the full answer

Hint 4: The answer is already 2 decimal places and put in your units (£)

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

Question 2

Hint 1: Substitute 'a' into the place of all the x's in the function

Hint 2: Substitute f(a) = 23 into the left hand side of the function

Hint 3: Subtract 2 from both sides

Hint 4: Divide both sides by 3

Hint 5: This will give the answer for the value of 'a'

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

Question 3

Hint 1: Cosy corner cosine rule

Hint 2: Write out the cosine rule formula for the length of a side carefully from the formula sheet

Hint 3: Substitute in the values you know

Hint 4: Type the right hand side into your calculator carefully

Hint 5: Find the square root of this answer to find the length of AB in km

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

Question 4

Hint 1: The magnitude of a vector is its length

Hint 2: Use Pythagoras' Theorem to find the magnitude

Hint 3: Find the sum of the squares of each of the components

Hint 4: Make sure that the negative 13 is in a bracket

Hint 5: Square root this answer

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

Question 5

Hint 1: Trace the vector journey p with your finger

Hint 2: At the point where the journey ended, draw in the vector q

Hint 3: vector q should be drawn from top left to bottom right and its arrow drawn

Hint 4: Draw the resultant vector from the beginning of p to the end of the line q and draw and arrow in this direction

Hint 5: Count the number of boxes left and down

Hint 6: Write this in component form in a bracket

Hint 7: Alternate method: state the components of p and q

Hint 8: Add them together

Hint 9: The horizontal component should be negative for the left movement

Hint 10: The vertical component should be negative for the down movement

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

Question 6

6a) Hint 1: Write out the formula for the volume of a sphere carefully from the formula sheet

6a) Hint 2: Substitute in what you know

6a) Hint 3: Type this into your calculator and write out your answer to a few decimal places

6a) Hint 4: Remember E on your calculator display means 10 to the power of the number following it

6a) Hint 5: Round your answer to 2 significant figures and put in units (km³)

6b) Hint 6: Divide the earth volume by the moon volume

6b) Hint 7: Type this into your calculator: your answer should be an integer

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

Question 7

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

Hint 2: To divide fractions: 'keep, change, flip'

Hint 3: Cross cancel anything that can be simplified

Hint 4: Write your final answer as a single simplified fraction

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

Question 8

Hint 1: Original value is 100%, the sale value is 15% less than this

Hint 2: Find what percentage the sale value is and turn this into a decimal

Hint 3: Find the original price by dividing by this decimal

Hint 4: If you multiply by this decimal you will find the same sale price

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

Question 9

Hint 1: Draw out the larger and smaller right angled triangles separately

Hint 2: Fill in the lengths of the sides and the area of the smaller triangle

Hint 3: Find the linear scale factor (LSF) of enlargement: it should be a simplified improper fraction

Hint 4: Find the area scale factor (ASF) by squaring the LSF

Hint 5: Find the area of the larger triangle by multiplying the area of the smaller triangle by the ASF

Hint 6: Subtract the area of the smaller triangle from this to find the area PQTS in cm²

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

Question 10

Hint 1: Arc length = (angle/360) × π × d

Hint 2: Substitute in the values you know

Hint 3: Divide both sides by (angle/360) × π

Hint 4: This will give you the diameter length

Hint 5: Half this to find the length of the pendulum in cm

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

Question 11

Hint 1: There are 6 triangles of equal size, find the area of one and multiply by 6

Hint 2: The angle in the centre of one triangle is 360/6

Hint 3: The cosy corner side lengths are both 40cm

Hint 4: Write out the area of a triangle formula carefully from the formula sheet

Hint 5: Substitute in the values you know

Hint 6: Type this into your calculator and write out your answer to a few decimal places

Hint 7: Multiply this answer by 6 to get the area of the top of the table

Hint 8: Round your answer to a few decimal places and put in units (cm²)

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

Question 12

Hint 1: Draw triangle OLM, with ML the width of the surface and OL the radius

Hint 2: Cut this in half to form a right angled triangle

Hint 3: Find the height of the triangle using Pythagoras' Theorem

Hint 4: Add this value to the radius to find the depth of the milk in metres

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

Question 13

Hint 1: Fill in angle NQR as 128°

Hint 2: Find angle PQR from a straight line

Hint 3: Find angle PRQ from a triangle

Hint 4: Matching sides and angles: sine rule

Hint 5: Write out the formula for the sine rule carefully from the formula sheet

Hint 6: Substitute in the values you know

Hint 7: Multiply both sides of the equation by sin(52)

Hint 8: Type this into your calculator and write out your answer to a few decimal places

Hint 9: Round your answer to a few decimal places and put in units (km)

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

Question 14

14a)i) Hint 1: Length = 13 + 2x

14a)ii) Hint 2: Write an expression for breadth

14a)ii) Hint 3: Area = length × breadth

14a)ii) Hint 4: Substitute in the values for area, length and breadth

14a)ii) Hint 5: Expand the double bracket and rearrange to make the equation equal to zero

14b) Hint 6: The rounding tells us to use the quadratic formula

14b) Hint 7: Write out the quadratic formula carefully from the formula sheet

14b) Hint 8: Substitute in the values you know

14b) Hint 9: Simplify what you can

14b) Hint 10: You should have two equations: one with a plus before the root, one with a minus before the root

14b) Hint 11: Type both of these into your calculator and write your answers to a few decimal places

14b) Hint 12: Round both your answers to one decimal place

14b) Hint 13: As a length cannot be negative, only one of your answers is valid

14b) Hint 14: Communicate this and state your answer for x in cm

Hint 15: and here is a video of the solution: