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 add or subtract fractions

Hint 2: Find the lowest common multiple of 3 and 5

Hint 3: Process each fraction so that they have the same denominator

Hint 4: Add the numerators, keep the denominator

Hint 5: Your answer should be a mixed number

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

Question 2

Hint 1: Expand the double bracket and single bracket separately

Hint 2: Write all your terms on one line

Hint 3: Collect your like terms (simplify)

Hint 4: Your answer should be 3 terms

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

Question 3

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 terms), different sign add (to eliminate 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: and here is a video of the solution:

Question 4

Hint 1: Substitute the vector u into the equation given

Hint 2: To find v subtract the vector u from both sides of the equation

Hint 3: Subtract each of the components to find the vector v

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

Question 5

Hint 1: Factorise the equation

Hint 2: You will have two brackets equal to zero

Hint 3: Put each bracket equal to zero

Hint 4: Solve each of the equations

Hint 5: Your answer should be two values for x

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

Question 6

Hint 1: What is the amplitude of the graph?

Hint 2: This is your value for a

Hint 3: Be careful as the x-axis only goes up to 180

Hint 4: How many waves would there be between 0 and 360?

Hint 5: This is your value for b

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

Question 7

7a) Hint 1: Write out the coordinates of A and B

7a) Hint 2: Use these coordinates to calculate the gradient

7a) Hint 3: m = (y₂-y₁)/(x₂-x₁)

7a) Hint 4: Your gradient should be a positive improper fraction fully simplified

7a) Hint 5: The equation of a line is y = mx + c

7a) Hint 6: Substitute in your gradient, P and d

7a) Hint 7: Substitute your coordinate A into this equation to find c

7a) Hint 8: State your final equation in terms of P and d

7b) Hint 9: Substitute d = 5 into your equation from part (a)

7b) Hint 10: Calculate the cost in £ which should be less than £14

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

Question 8

Hint 1: To find the nature of the roots, 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 9

Hint 1: Find one of the centre angles 360/10

Hint 2: Find the other two angles in the triangle

Hint 3: Angle LKJ will be 180 - (two of these angles)

Hint 4: Angle KJL can be found from the other two angles in the triangle

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

Question 10

Hint 1: Non-right angled triangles, so use formula sheet

Hint 2: Cosy corner -> cosine rule

Hint 3: Write out the formula, fill in what you know

Hint 4: Calculate XY in cm

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

Question 11

Hint 1: Multiply both the numeractor and the denominator by √6

Hint 2: You should be left with a square root in the numerator, but not in the denominator

Hint 3: Simplify

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

Question 12

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

Hint 2: Identify where 240 lies on the x-axis of this sketch

Hint 3: Using symmetry find the y coordinate that corresponds to 240

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

Question 13

Hint 1: B is the same distance along the x and z axes as A so will have the same x and z coordinates

Hint 2: B is 8 units along the y axis

Hint 3: State the coordinates of B with a bracket and a comma between each coordinate

Hint 4: D is 2 units further along the x axis than A

Hint 5: C has the same x coordinate as D

Hint 6: C had the same y coordinate as B

Hint 7: C has the same z coordinate as E

Hint 8: State the coordinates of C with a bracket and a comma between each coordinate

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

Question 14

Hint 1: Deal with the h first

Hint 2: Subtract h from both sides

Hint 3: Deal with the g next

Hint 4: Divide both sides by g

Hint 5: Deal with the square root

Hint 6: Square both sides

Hint 7: Make sure x is on the left hand side of the equals sign

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

Question 15

Hint 1: Square 2/3

Hint 2: To do this, square the numerator and denominator

Hint 3: Square p⁴ by multiplying the powers

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

Question 16

Hint 1: Find the points of intersection of the x axis (roots/solutions/zeros)

Hint 2: Cuts the x axis when y = 0

Hint 3: Put each bracket equal to zero and solve

Hint 4: Cuts the y axis when x = 0

Hint 5: Substitute x = 0 into the equation, and solve to find the y intercept

Hint 6: The turning point is half way between the roots

Hint 7: What is the distance between the roots?

Hint 8: Half this and count along to find the axis of symmetry, x = …

Hint 9: Substitute this value of x into the equation to find the corresponding y coordinate of the turning point

Hint 10: Draw an x and y axis with a ruler, put in arrows with labels

Hint 11: Draw a parobola with a minimum turning point ('happy graph')

Hint 12: This should clearly go through all the points found so far and be labelled

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

Question 17

Hint 1: Volume question so refer to the formula sheet

Hint 2: Copy out the formula for the volume of a pyramid

Hint 3: Substitute in what you know

Hint 4: Rearrange to make h the subject of the formula

Hint 5: Calculate your answer for h, in cm

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

Question 18

Hint 1: Use your trig identities

Hint 2: Substitute tan(x) = sin(x)/cos(x)

Hint 3: Simplify

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

Question 19

19a)i) Hint 1: Completed square form

19a)i) Hint 2: Half the coefficient of the x term ... this is p

19a)i) Hint 3: Subtract this number squared ... don't forget the -81

19a)i) Hint 4: Calculate the value of q

19a)ii) Hint 5: Know that the (x - p)² term means that the normal quadratic graph has moved to the right a distance of p

19a)ii) Hint 6: the equation of the axis of symmetery is x = (value of p)

19b) Hint 7: State the values of a, b and c

19b) Hint 8: Refer to the formula sheet and copy out the quadratic formula

19b) Hint 9: Substitute the values of a, b and c into this formula

19b) Hint 10: Simplify using both fractions and surds

19b) Hint 11: State the values of d and e

Hint 12: and here is a video of the solution:

Paper 2

Question 1

Hint 1: Get your multiplier ... start at 100% ... decrease by 2% ... turn into a decimal

Hint 2: Expected waste will be 125000 multiplied by the multiplier 3 times for the 3 years … use ×(decimal multiplier)³

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

Hint 4: Round to a few decimal places and write correct units (tonnes).

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

Question 2

Hint 1: State your radius and diameter

Hint 2: Write the angle out of 360 to find the fraction of the circle you are finding

Hint 3: Find this fraction of the circumference of a full circle

Hint 4: Calculate your answer in cm

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

Question 3

Hint 1: Magnitude means length so we use Pythagoras' Theorem

Hint 2: Find the sum of the square of the components

Hint 3: Square root this answer

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

Question 4

Hint 1: Expand the single bracket

Hint 2: Bring your letter terms to the right hand side to keep them positive

Hint 3: Bring your number terms to the left hand side

Hint 4: Get x on its own by dividing both sides

Hint 5: Write x on the left hand side and reverse the inequality sign

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

Question 5

5a) Hint 1: Add up all the values

5a) Hint 2: Divide this number by 6 - this is your mean

5a) Hint 3: Create a table with 2 columns: (each value subtract the mean) and (this answer squared)

5a) Hint 4: The first column should have a mix of positive and negative answers

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

5a) Hint 6: Total this second column

5a) Hint 7: Write out the standard deviation formula carefully from the formula sheet

5a) Hint 8: Substitute your total into the numerator and n = 6 into the denominator

5a) Hint 9: Type into your calculator and write your answer to a few decimal places

5a) Hint 10: Round to 2 decimal places (or more)

5b) Hint 11: Compare the means by saying which day had more customers on average

5b) Hint 12: Compare the standard deviation by saying which day had a more consistent number of customers

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

Question 6

Hint 1: Replace all x values for 'a' in the function

Hint 2: Substitute f(a) = 73 into the left hand side of this function

Hint 3: Subtract 5

Hint 4: Divide by 4

Hint 5: This is your value for a

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

Question 7

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

Hint 2: State the diameter and radius

Hint 3: Substitute your values into the formula

Hint 4: Calculate your answer to a few 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: Rearrange to make sin(x) the subject of the formula

Hint 2: Use your calculator to give the angle, using inverse sin

Hint 3: Draw a sketch of y = sin(x) to see where the positive values are

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

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

Question 9

Hint 1: Find the size of the angle CBD from a straight line

Hint 2: Matching pairs of sides and angles, so this requires the sine rule

Hint 3: Copy the sine rule formula from the formula sheet

Hint 4: Substitute in what you know

Hint 5: Multiply both sides by 105 to obtain length DC on its own

Hint 6: Calculate your answer to many decimal places and then round to a few decimal places with units (cm)

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

Question 10

Hint 1: Find a route from B to D along vector paths that we know

Hint 2: Remember a negative will change the direction of the vector arrow.

Hint 3: Collect together any like terms of u and w

Hint 4: State your journey in terms of u and w

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

Question 11

Hint 1: Volume of Earth = 100%

Hint 2: Volume of Venus = 85%

Hint 3: Get to 1 % by dividing

Hint 4: Get to 100% by multiplying by 100

Hint 5: Write your answer to many decimal places and then round to a few decimal places with units (km³)

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

Question 12

Hint 1: Draw out a right angled triangle

Hint 2: Fill in the values you know - including the radius (OA or OB)

Hint 3: You will have one unknown side

Hint 4: Calculate the length of this side using Pythagoras' Theorem

Hint 5: Find the width by adding this side to the radius

Hint 6: Write your answer to a few decimal places in cm

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

Question 13

Hint 1: Find the size of angle FTY, using 'cosy corner' cosine rule

Hint 2: Write down the cosine rule for the angle, using the formula sheet

Hint 3: Substitute in what you know make sure the 7.2 comes after the subtraction sign

Hint 4: Calculate your answer for the angle

Hint 5: The bearing will be this angle added to 240 degrees

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

Question 14

Hint 1: Any line that crosses the y-axis at x = 0 has its x coordinate to be zero

Hint 2: Substitute in x = 0

Hint 3: Divide by -5

Hint 4: This is the y coordinate

Hint 5: Write the coordinates with a bracket and a comma between the two numbers

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

Question 15

Hint 1: To divide fractions - keep the first fraction, change the ÷ to ×, and 'flip' the second fraction

Hint 2: (Top × Top) / (Bottom × Bottom)

Hint 3: Make sure to keep brackets round each expression

Hint 4: The numerator is already factorised

Hint 5: The denominator can be factorised as it's the difference of two squares

Hint 6: Cancel the brackets that are the same

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

Question 16

Hint 1: Draw a right angled triangle from the square top of the locker

Hint 2: Find the diagonal length (hypotenuse) using Pythagoras' Theorem

Hint 3: Draw a right angled triangle with PM as the hypotenuse

Hint 4: Fill in the values that you know

Hint 5: Calculate the hypotenuse PM using Pythagoras' Theorem

Hint 6: Communicate your answer with a yes or no and given a reason why

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

Question 17

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

Hint 2: Write out area of triangle formula from the formula sheet

Hint 3: Substitute in what you know and calculate your answer to a few decimal places

Hint 4: Find the fraction of the full circle that we have: angle/360

Hint 5: Find the area of the sector: the fraction of the circle area

Hint 6: Subtract the sector area from the triangle area giving the answer in cm²

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

Question 18

18a) Hint 1: Find the linear scale factor (LSF) and simplify - it should be an improper fraction as it is an enlargement

18a) Hint 2: Find the volume scale factor (VSF) - and simplify - it should be an improper fraction as it is an enlargement

18a) Hint 3: If similar shapes, then the VSF = LSF³

18a) Hint 4: Cube the LSF

18a) Hint 5: Communicate your answer - the LSF³ does not equal the VSF as it should if mathematically similar

18b) Hint 6: Find the new VSF - it should be an improper fraction as it is an enlargement

18b) Hint 7: To find the corresponding LSF, cube root the VSF

18b) Hint 8: To find the depth, multiply this LSF by 16

18b) Hint 9: State your answer in cm

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