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: Copy onto your page what f(x) is equal to

Hint 2: Now replace all the x terms with -2

Hint 3: Remember to put the -2 in brackets

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

Question 2

Hint 1: Write the second mixed fraction as a top heavy fraction

Hint 2: Multiply the numerators together

Hint 3: Mutiply the denominators together

Hint 4: Simplify your fraction

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

Question 3

Hint 1: Everything in the first bracket multiplies everything in the second bracket

Hint 2: After multiplying, you should have six terms

Hint 3: Gather like terms to obtain an expression with four terms

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

Question 4

Hint 1: realise we don't want a full circumference, but only part of it

Hint 2: we only want the fraction 240/360 of the full circumference

Hint 3: circumference is C=πd or C=2πr

Hint 4: use r = 30 or d = 60

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

Question 5

5a) Hint 1: write the numbers out in order, from smallest to largest

5a) Hint 2: the median is the middle number

5a) Hint 3: identify the lower quartile and the upper quartile

5a) Hint 4: the interquartile range is the distance between the lower quartile and the upper quartile

5a) Hint 5: the semi-interquartile range is half of the interquartile range

5b) Hint 6: comment 1: write about the meaning of the difference in the values of the medians

5b) Hint 7: comment 2: write about the meaning of the difference in the values of the semi-interquartile ranges

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

Question 6

6a) Hint 1: note that the x-axis scale is different from the y-axis scale

6a) Hint 2: identify the two points that are directly on the line, and write down their coordinates

6a) Hint 3: use these two points' coordinates to work out the gradient of the line

6a) Hint 4: quick check - should your gradient be positive or negative?

6a) Hint 5: use a formula to work out the value of c, the y-axis intercept

6a) Hint 6: remember to write the equation of the line using letters F and E, not y and x

6b) Hint 7: decide whether 1.1 is the value of E or F

6b) Hint 8: replace 1.1 in your answer from part(a) to work out the value of the other letter

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

Question 7

Hint 1: locate the letter that we want to make the subject

Hint 2: write down, in order, the operations that happen to that letter

Hint 3: check that we first add y, then multiply by h then divide by 2

Hint 4: now perform the inverse of each operation to both sides of the equation, starting with multiplying by 2

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

Question 8

8a) Hint 1: write down the two letters you will use to represent cement and gravel and define clearly what they start for (e.g. is it the cost of them? the weight of them? the size of them?)

8a) Hint 2: check that your equation does not include any units of weight - it is just numbers and the two letters you chose, and nothing else.

8c) Hint 3: write your two simultaneous equations from parts (a) and (b), one above the other and decide which letter you will try to eliminate first

8c) Hint 4: after solving for one letter, substitute its value back in to either of the original equations to work out the value of the other letter

8c) Hint 5: be sure to write a sentence at the end, clearly stating what you have found out, including units.

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

Question 9

9a) Hint 1: remember that the equation of a vertical line looks like: x = ...

9b) Hint 2: the coordinates of the turning point (4,20) are closely linked to the values of a and b.

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

Question 10

10a) Hint 1: when adding vectors, you add the x components and separately, you add the y components

10b) Hint 2: decide on the journey to go from point M to point Q that can use the information given at the start of the question

10b) Hint 3: notice that because M is the midpoint of PR, then vector MR is half of vector PR

10b) Hint 4: add vector MR to vector RQ to obtain the resultant vector of MQ

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

Question 11

Hint 1: look at triangle OBF. What type of triangle is it?

Hint 2: notice that OB and OF are the same length (why?)

Hint 3: notice that angle BOF is half way between angles BOC and BOD

Hint 4: from the first diagram, angle BOC is one fifth of the full circle

Hint 5: from the first diagram, angle BOD is two fifths of the full circle

Hint 6: angle BOF = (angle BOC + angle BOD)/2

Hint 7: triangle BOF is isosceles (from Hint 1) and you know angle BOF to be 108°, so angles OFB and OBF are equal and all three angles add up to 180°.

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

Question 12

Hint 1: multiply the numerator and the denominator by the same square root

Hint 2: simplifying the denominator should lead to an integer, with no square root

Hint 3: simplify the numerator by working through the product of two surds

Hint 4: simplify the resulting fraction as much as possible

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

Question 13

Hint 1: sketch what the graph of y = cos(x) looks like

Hint 2: write on your graph the coordinates of the minimum turning point

Hint 3: sketch what the graph of y = cos(x + 45) looks like, and what the coordinates of its minimum turning point now are

Hint 4: sketch what the graph of y = 3cos(x + 45) looks like, and what the coordinates of its minimum turning point now are

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

Question 14

Hint 1: multiply the whole equation through by 10, to deal with the two fraction terms

Hint 2: expand the brackets on the right hand side of the equation

Hint 3: gather x terms, and solve for x

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

Question 15

15a) Hint 1: put t = 2 into the expression for h(t), to work out h(2)

15b) Hint 2: realise that -17 is the value of h(t), not the value of t

15b) Hint 3: write out the equation from h(t) = -17

15b) Hint 4: rearrange the equation to make it equal to zero

15b) Hint 5: factorise the quadratic expression

15b) Hint 6: solve for two values of t, and reject one of them, giving your reason for rejecting it.

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

Paper 2

Question 1

Hint 1: an increase of 15% is 100% + 15% = 115% = 1.15

Hint 2: multiply by this decimal for each year of increase

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

Question 2

Hint 1: square each component, remembering to put brackets around the negative value

Hint 2: add up the squares, then square root the total

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

Question 3

Hint 1: identify the formulae to use, from the formulae sheet, to work out the area of a triangle from knowing two sides and the angle between them

Hint 2: correctly match up the letters P, Q and R to the letters in the formula

Hint 3: substitute the values in and give your answer with the correct units

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

Question 4

Hint 1: 8% = 0.08

Hint 2: multiply the sesame seed weight by 0.08

Hint 3: be sure to give your answer back in scientific notation

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

Question 5

Hint 1: read the first sentence and determine the radius of the cone

Hint 2: for point B, think first of the coordinates of the centre of the circulate base

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

Question 6

Hint 1: use the quadratic formula

Hint 2: correctly identify the values of a, b and c

Hint 3: when working out the expression under the sqaure root, be careful of the (-2) value

Hint 4: write the two unrounded decimals first, before writing the rounded decimals.

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

Question 7

Hint 1: realise that the smallest angle is opposite the smallest side length

Hint 2: decide which formulae, from the formula sheet, you can use to work out an angle from knowing all three side lengths

Hint 3: correctly match up the letters X, Y and Z to the letters in the formula

Hint 4: use inverse cosine to work out the value of the angle.

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

Question 8

Hint 1: draw your own diagram with a side-on view and add in the numbers provided in the question

Hint 2: realise that the height of the cylinder is NOT 70cm

Hint 3: know that the fomula for the volume of a cyclinder is V = πr²h/3

Hint 4: realise that we don't want the volume of a full sphere, but only half of a sphere

Hint 5: write the unrounded answer first, before writing the rounded answer

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

Question 9

Hint 1: realise that £977.85 is 2.5% more than the price should would have paid, if she'd paid on time

Hint 2: realise that £977.85 represents 100% + 2.5% = 102.5% = 1.025 of the original price

Hint 3: work backwards to obtain the original price

Hint 4: work out the difference between the two prices, to find out how much would have been saved.

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

Question 10

Hint 1: decide which method to use: completing the square or expanding out the (x+p)²+q and then comparing coefficients

Hint 2: when you have a final answer, check it by multiplying out the breackets and seeing if you get x² + 10x - 15

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

Question 11

Hint 1: realise that there are no angles involved, so this is the converse of Pythagoras' theorem

Hint 2: work out length of BC knowing that the perimeter is 1500 metres

Hint 3: test to see if AB² + BC² is the same numerical value as BC²

Hint 4: write a sentence, clearly explaining your conclusion and the reasons for it

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

Question 12

12a) Hint 1: use lengths BC and EF to determine the length scale factor of reduction, going from sector ABC to sector DEF

12a) Hint 2: know that area scale factor = (length scale factor)²

12a) Hint 3: use the area of sector ABC = 2750 cm² along with the area scale factor of reduction, to obtain the area of sector DEF

12b) Hint 4: realise we can work out the area of a whole circle that has radius 50cm

12b) Hint 5: the angle we want is a fraction of 360. That fraction comes from comparing the area of sector ABC with the area of the full circle of radius 50cm

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

Question 13

Hint 1: use a formula for gradient using the given coordinates

Hint 2: in the resulting fraction, factorise each of the numerator and denominator as much as possible

Hint 3: you should then have a bracket in the numerator that cancels with a bracket in the denominator

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

Question 14

Hint 1: rearrange to give an equation that has cos(x) = ....

Hint 2: generate one solution from using inverse cosine

Hint 3: sketch a graph of y = cos(x) to determine a second solution that's also between 0 and 360

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

Question 15

Hint 1: realise to add fractions, they need a common denominator

Hint 2: scale up each fraction to have (x - 2)(x + 5) as their denominators

Hint 3: be careful with the subtraction and numerator of the second fraction .... use brackets to help

Hint 4: simplify the numerator, leaving the denominator in factorised form

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

Question 16

Hint 1: simplify the numerator into a single term

Hint 2: re-write the square root of a in the denominator as a power of a

Hint 3: use a law of indices to simplify the terms in the letter a

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

Question 17

Hint 1: write out (...)² as (...)(...)

Hint 2: multiply out the brackets and simplify to obtain three terms

Hint 3: use a trigonometric identity to simplify two of the terms.

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

Question 18

Hint 1: draw in right-angled triangle STB and mark on the lengths of ST and BT

Hint 2: use Pythagoras' theorem to work out TB

Hint 3: realise that TB is equal to TD

Hint 4: realise that CD = CT + TD

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

Question 19

Hint 1: work out angle B from knowing angles K and M

Hint 2: use the sine rule to work out BM

Hint 3: draw a vertical line down from B to meet KM, to form a right angled triangle

Hint 4: use SOHCAHTOA on length BM and angle BMK to work out the height of B above the ground

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