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: Half each of the components of p

Hint 2: Add these components to q

Hint 3: Make sure your answer has a bracket and two numbers

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

Question 2

Hint 1: BODMAS - add the fractions in the brackets first

Hint 2: You require a common denominator - the lowest common multiple (LCM) of 3 and 7

Hint 3: Write the two equivalent fractions with a common denominator and add the numerators

Hint 4: Multiply this answer by 3/4

Hint 5: Simplify the fractions, or (top × top)/(bottom × bottom) and then simplify

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

Question 3

Hint 1: Write the angle out of 360 to find the fraction of the circle you are finding

Hint 2: Find this fraction of the area of a full circle

Hint 3: Simplify the fraction and calculate your answer in cm²

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

Question 4

Hint 1: State 'c' is the amount of material required for 1 cloak in square metres and 'd' is the amount of material required for 1 dress in square metres

Hint 2: Write an equation for the first day - do not put the units of m² into the equation

Hint 3: Write an equation for the next day - do not put the units of m² into the equation

Hint 4: Multiply both equations to get the quantities of one of your variables (either 'c' or 'd') 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 value of the second variable

Hint 8: State your answers in words, in m² to a few decimal places

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

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

Question 5

5a) Hint 1: Write out the coordinates of D and E

5a) Hint 2: Use these coordinates to calculate the gradient

5a) Hint 3: m = (y₂-y₁)/(x₂-x₁)

5a) Hint 4: Your gradient should be a positive whole number

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

5a) Hint 6: Substitute in your gradient, and the values of A and W

5a) Hint 7: Substitute the coordinates of D into this equation to find c

5a) Hint 8: State your final equation in terms of A and W

5b) Hint 9: Substitute A = 12 into your equation from part (a)

5b) Hint 10: Calculate the weight in kg which should be between 100 and 340kg

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

Question 6

Hint 1: To find the nature, calculate the discriminant

Hint 2: State the values of a, b and c

Hint 3: Discriminant = b² - 4ac

Hint 4: Communicate if b² - 4ac is >0, =0 or <0 and state the nature

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

Question 7

7a) Hint 1: Find the lengths of the sides of the square base

7a) Hint 2: State the coordinates of B in a bracket with commas between the 3 numbers

7b) Hint 3: Find the vector AV by subtracting the components

7b) Hint 4: Find the magnitude of this vector by square rooting the sum of the squares of the components of AV

7b) Hint 5: Your answer should be a positive number

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

Question 8

Hint 1: Multiply each term by the lowest common multiple (LCM) of 3 and 6 to remove the fraction

Hint 2: Move your x terms to one side of the equals sign and your number term to the other

Hint 3: Solve this equation to find the value of x: make sure it is fully simplified

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

Question 9

Hint 1: Substitute x = 5 into the function

Hint 2: To rationalise the denominator, multiply both the numerator and the denominator by √5

Hint 3: Simplify if required: there should be no square root in the denominator

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

Question 10

Hint 1: Completed square form lets us move the turning point from (0,0)

Hint 2: the (x - 3) term moves the graph to the right by 3. The +1 part moves the graph up by 1

Hint 3: Positive coefficient of x² means a 'happy graph', with a minimum turning point

Hint 4: Label the minimum turning point at (3, 1)

Hint 5: y intercept is when x = 0

Hint 6: Substitute in x = 0 to find the y intercept

Hint 7: Make sure the sketch is a smooth, symmetrical curve with the turning point and y intercept labelled

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

Question 11

Hint 1: Use the identity tan(x) = sin(x) / cos(x)

Hint 2: Square each term of the identity

Hint 3: Substitute this in for tan²(x)

Hint 4: Simplify by cancelling the cos²(x) terms

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

Question 12

12a) Hint 1: A = l × b

12a) Hint 2: Substitute in the expressions given, making sure to put brackets around each expression

12b) Hint 3: Find the area of the triangle by substituting in the expressions to A = ½ b h

12b) Hint 4: Make the area of the rectangle and triangle equal to each other

12b) Hint 5: Expand the brackets and simplify

12b) Hint 6: Move all the terms over to one side to make the equation = 0

12c) Hint 7: Factorise the quadratic given in part (b)

12c) Hint 8: Make each bracket = 0

12c) Hint 9: Solve the equation, to obtain two values for x

12c) Hint 10: As x cannot be negative due to working with lengths, one value of x is invalid

12c) Hint 11: Substitute the valid value for x into the expressions for the length and breadth of the rectangle

12c) Hint 12: State your answers clearly in cm

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

Paper 2

Question 1

Hint 1: Get your multiplier ... start at 100% ... decrease by 8% ... turn into a decimal

Hint 2: Expected roll will be 35 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 a few decimal places and put in your units (grams)

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

Question 2

Hint 1: Consider setting up a ratio statement like ... (number of pollen grains) : (weight of pollen in grams)

Hint 2: Divide both sides of this ratio by the number of pollen grains to obtain a ratio like ... 1 : (weight of 1 pollen grain in grams)

Hint 3: Write this answer in scientific notation

Hint 4: Your answer in scientific notation should be a (number between 1 and 10) × 10 to the power of (a negative number)

Hint 5: Put in your units (grams)

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

Question 3

Hint 1: Find a route from B to D along known vector paths

Hint 2: Remember negatives change the direction of the vector arrows

Hint 3: State this journey in terms of u and v

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

Question 4

Hint 1: Look for a highest common factor (HCF)

Hint 2: Look for a difference of two squares

Hint 3: Check your answer by expanding the double bracket, multiplying each term by the number in front

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

Question 5

Hint 1: Find the angle EOA from a straight line

Hint 2: Find the angle CAO using alternative angles ('Z angles')

Hint 3: OAB is a right angle between a radius and a tangent

Hint 4: Find BAC = ACB

Hint 5: Find ABC from the angles in a triangle

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

Question 6

6a) Hint 1: Add all the values

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

6a) Hint 3: Create a table with 2 columns: (each value subtract the mean) and (this answer squared)

6a) Hint 4: The first column should have a mix of positive and negative answers

6a) Hint 5: The second column should all be positive after squaring

6a) Hint 6: Total this second column

6a) Hint 7: Write out the standard deviation formula carefully from the formula sheet

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

6a) Hint 9: Type into your calculator and write your answer to several decimal places

6a) Hint 10: Round to 2 decimal places (or more)

6b) Hint 11: Compare the means by saying which waiting time was less on average

6b) Hint 12: Compare the standard deviation by saying which waiting time was less consistent

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

Question 7

Hint 1: Write out the volume of a cone carefully from the formula sheet

Hint 2: State the diameter and radius of each cone

Hint 3: Substitute your values into the formula: once for large cone, once for small cone

Hint 4: Subtract the volume of the small cone from the volume of the large cone, to several decimal places

Hint 5: Round your final answer to 2 significant figures with units

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

Question 8

Hint 1: Matching pairs of sides and angles: sine rule

Hint 2: Invert the sine rule formula from the sheet, so that the it is sin(A)/a = sin(B)/b = sin(C)/c

Hint 3: Substitute in what you know

Hint 4: Rearrange by multiplying both sides by 150

Hint 5: Type into your calculator and write your answer to several decimal places

Hint 6: Round to a few decimal places and put in your units (degrees)

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

Question 9

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

Hint 2: Subtract this number squared … don't forget the -7

Hint 3: Calculate the value of b

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

Question 10

Hint 1: Brackets mean multiply powers

Hint 2: Multiply means add the powers

Hint 3: Now make the negative power positive by turning it into a fraction with a positive power in the denominator

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

Question 11

Hint 1: Find the linear scale factor (LSF): it should be a proper fraction as it is a reduction

Hint 2: Find the area scale factor (ASF) by squaring the LSF

Hint 3: Calculate the cost of the small picture by multiplying the large picture cost by ASF

Hint 4: Remember to write your answer with a £ sign, rounded to 2 decimal places as it is money

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

Question 12

Hint 1: Deal with the square root sign by performing the inverse of square rooting

Hint 2: Deal with the -p by performing the inverse of subtracting

Hint 3: Deal with ×4t by performing the inverse process

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

Question 13

Hint 1: To add fractions, we need 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 brackets in the numerator

Hint 5: Simplify the numerator

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

Question 14

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

Hint 2: Use your calculator to find the angle by using inverse tan

Hint 3: Draw a sketch of y = tan(x) to see where the negative values happen

Hint 4: Your two answers for x should be (180 - acute angle) and (360 - acute angle)

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

Question 15

Hint 1: Draw a right angled triangle, filling in all the values you know

Hint 2: Use Pythagoras' Theorem to find the height of the triangle

Hint 3: To find the height of the label, add the height of the triangle to the radius (6.6)

Hint 4: State the height in cm

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

Question 16

Hint 1: Find the angle BAC, so that we have a 'cosy corner' for the cosine rule

Hint 2: Use right angled trigonometry to find the angle: cos(A) = 3/4

Hint 3: Write down the formula for the cosine rule from the formula sheet

Hint 4: Subsitute in the values that you know

Hint 5: Calculate the answer to several decimal places

Hint 6: Round to one decimal place and put in your units (cm)

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