Question Papers Marking Instructions

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

Hints offered by N Hopley, with video solutions by 'DLBmaths'

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Paper 1

Question 1

Hint 1: Replace x with -5

Hint 2: Remember to write it as (-5) because of the squared term

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

Question 2

Hint 1: Arrange the 10 numbers in order

Hint 2: Locate the median

Hint 3: Locate the lower quartile and the upper quartile

Hint 4: The inter-quartile range is the range between the lower and upper quartiles

Hint 5: The semi-interquartile range is half of the interquartile range

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

Question 3

Hint 1: Write the first mixed fraction as a single, top-heavy fraction

Hint 2: Change the division into a multiplication

Hint 3: Change the fraction 3/4 into the fraction 4/3

Hint 4: Look to see if you can simply the two fractions, before you multiply them

Hint 5: Mutiply the two fractions

Hint 6: Look to simplify your answer in any way

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

Question 4

Hint 1: The '2x' term in the first bracket will multiply each of the three terms in the second bracket

Hint 2: The '+3' term in the first bracket will multiply each of the three terms in the second bracket

Hint 3: Look at the 6 terms you now have, and gather together any like terms.

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

Question 5

Hint 1: Write on the diagram the coordinates of the other two corners of the square base

Hint 2: After realising the height of the cube, write on the diagram the coordinates of the top four corners of the cube

Hint 3: Read the question carefully to find the height of the pyramid

Hint 4: Realise that point C is directly above the centre of the base of the cube, so it's x and y coordinates will come from the square base

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

Question 6

Hint 1: Work out the gradient of the line

Hint 2: Check that your gradient is negative, as the line is sloping down

Hint 3: Write out the equation of a straight line, and replace m with your value for the gradient

Hint 4: Use the coordinates of either point A or point B to replace letters in your formula with their coordinate values

Hint 5: Work out the value of the y-intercept from your last step

Hint 6: Assemble the equation of the straight line, written in its simplest form (eg. y = mx + c)

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

Question 7

Hint 1: Look at the formula sheet and identify the area formula to use for a triangle when you are given two sides and the angle between them

Hint 2: Substitute the values from the question into the formula and evaluate the area

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

Question 8

Hint 1: Expand the bracket on the right-hand-side of the inequality

Hint 2: Collect like terms on the right-hand-side of the inequality

Hint 3: Subtract x from both sides and/or subtract 9 from both sides

Hint 4: Check that your final answer is either x < …. or ….> x

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

Question 9

Hint 1: Look for where there are right angles in the diagram

Hint 2: Realise that angle OBA is 90° and angle CBD is 90°

Hint 3: Work out angle ABC or angle OBD

Hint 4: Work out angle OCB or angle ODB

Hint 5: Work out angle CAB

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

Question 10

Hint 1: Locate the letter 'b' and determine all the operations that act on the 'b' in the order they happen

Hint 2: Deal with the list of operations, in reverse order

Hint 3: Multiply both sides by the letter 'c'

Hint 4: Subtract r² from both sides

Hint 5: Divide both sides by 4

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

Question 11

Hint 1: Realise that to subtract the fraction, they need to have a common denominator

Hint 2: Once you have the fractions with a common denominator, you can subtract one numerator from the other

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

Question 12

Hint 1: Look at the formula sheet for the standard deviation formula that you normally use

Hint 2: Work out the standard deviation by your preferred method

Hint 3: If done correctly, then you should have the square root of (18/4)

Hint 4: Use your knowledge of simplication of surds to simplify the numerator and denominator of this fraction

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

Question 13

Hint 1: Realise that you have simultaneous equations, in the variables x and y

Hint 2: Decide whether to eliminate the variable x or y

Hint 3: Either mutiply one equation by a number, or rearrange one equation to be either x = …. or y = ….

Hint 4: Combine the two equations so that you end up with just one equation in one variable

Hint 5: Solve for that one variable

Hint 6: Substitute your value back into either of the original equations, to obtain the value of the second variable

Hint 7: State your answer as a set of coordinates

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

Question 14

14a) Hint 1: Realise that the letter 'a' is connected with the location of the symmetry line, on the graph

14b) Hint 2: Using the value of 'a' from part (a), and the coordinates (-3, 8), substitute these values for x, y and a into the equation

14b) Hint 3: Solve the resulting equation for the value of 'b'

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

Question 15

Hint 1: Draw out two separate triangles PTS and PQR

Hint 2: Label all the sides of both triangles with the given information

Hint 3: Determine the scale factor between the diagrams and in which direction you are going in for that scale factor

Hint 4: Create an equation in x, using your scale factor

Hint 5: Solve your equation for x

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

Paper 2

Question 1

Hint 1: Use Pythagoras' Theorem, but with 3 numbers

Hint 2: Remember to write the second term as (-14)² and not -14²

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

Question 2

Hint 1: Remember that 100% + 4.5% = 104.5%

Hint 2: Turn 104.5% into a decimal multiplier of 1.045

Hint 3: Multiply the original value by this multiplier for each of the years

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

Question 3

Hint 1: Recognise you have two sides and an angle between them in a non-right angled triangle

Hint 2: Use the cosine rule

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

Question 4

Hint 1: As it says 'to one decimal place' we know to use the quadratic formula

Hint 2: Decide the values of a, b and c, watching for negatives

Hint 3: Be careful with negatives when working out b²-4ac

Hint 4: Write the two answers, unrounded, before rounding to the specified accuracy

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

Question 5

Hint 1: Decide if this is a 'forwards' or 'backwards' percentage question

Hint 2: Note that the number of tickets sold this year represents 115%

Hint 3: Work out 1% and then 100%

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

Question 6

Hint 1: Note that this is a question about spheres, not circles

Hint 2: Look up the formula for the volume of a sphere from the formula sheet

Hint 3: Decide on the radius (not the diameter) of the larger sphere

Hint 4: Decide on the radius of the smaller sphere

Hint 5: Work out the two spheres' volumes

Hint 6: Work out the difference in the volumes

Hint 7: Write your unrounded answer first, then the rounded answer

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

Question 7

Hint 1: Recognise that this is the Converse of Pythagoras' Theorem

Hint 2: Identify the lengths of the larger triangle, formed by putting A and B together

Hint 3: Separately work out the hypothenuse squared, and then work out the sum of the shorter sides, squared

Hint 4: Compare these two numbers to decide whether Pythagoras' Theorem holds, or not

Hint 5: Write a sentence about what you comparison means in terms of the angle inside the larger triangle

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

Question 8

8a) Hint 1: Note that vector PR = vector PQ + vector QR

8a) Hint 2: Note that vector QR = -_c

8b) Hint 3: Note that vector TV = vector TP + vector PV

8b) Hint 4: Note that vector PV = ½ vector PR, which you worked out in part (a)

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

Question 9

9a) Hint 1: Note that 4x² = (2x)²

9a) Hint 2: Note that 25 = 5²

9a) Hint 3: Note that you have a difference of two squares

9b) Hint 4: Note that it would be nice to have in the denominator, one of the brackets that you have in your answer to part (a)

9b) Hint 5: Factorise the denominator

9b) Hint 6: Cancel the common bracket in both the numerator and the denominator

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

Question 10

Hint 1: Note that points D and E are due west/east of each other, so that DE is 'horizontal' and at right angles to the north lines

Hint 2: Work out angle FDE

Hint 3: Work out angle DEF

Hint 4: Work out angle DFE

Hint 5: Focus on triangle DEF and note that you have one side (DE) and two angles in a non-right angled triangle

Hint 6: Decide on whether to use the sine rule or the cosine rule

Hint 7: Use the sine rule with the 'pairs' of length DE and angle DFE, and the length DF and angle DEF

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

Question 11

Hint 1: Aim to rearrange equation into the form of y = mx + c

Hint 2: Add 5y to both sides

Hint 3: Divide both sides by 5

Hint 4: Note that the gradient is the coefficient of x (i.e. the number in front of the x)

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

Question 12

Hint 1: Note that the cube root of x can be written as x to the power of a fraction

Hint 2: Note that 1/… will involve a negative power when the denominator is written as a separate term

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

Question 13

Hint 1: Draw in the radius from C₁ to A

Hint 2: Draw in the radius from C₁ to the point in the middle of the line AB (call this midpoint D)

Hint 3: Draw the triangle with corners C₁, A and D and mark on the length of AD and AC1

Hint 4: Draw vertical line down from C₁ to the middle of AD and recognise that you have a right angled triangle

Hint 5: Use Pythagoras' Theorem on this right angled triangle to work out its height

Hint 6: Work out the total height of the logo by adding together 4 lengths in total

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

Question 14

Hint 1: Note that you have the major arc length

Hint 2: Note that you can work out the circumference of the full circle

Hint 3: Note that you can form a fraction from these two curved lengths

Hint 4: Note that this fraction of 360° is the reflex angle AOB that you want.

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

Question 15

15a) Hint 1: Note that in the given formula, the letter x stands for the angle of turn

15a) Hint 2: Substitute the value of 60 for x in the formula

15b) Hint 3: Note that the minimum height will come from the formula when cos(x°) is at its smallest value

15b) Hint 4: Remember from the graph of cos(x°) that its lowest value is -1

15b) Hint 5: Substitute the 'cos(x°)' part of the formula with the number -1

15c) Hint 6: Note that 61 metres is now the number representing h in the formula

15c) Hint 7: Rearrange the equation to give cos(x°)=…

15c) Hint 8: Calculate the inverse cosine to obtain one value of x°

15c) Hint 9: Use your knowledge of the cosine function to obtain the second value of x° that lies between 0° and 360°

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