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

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Paper 1

Question 1

Hint 1: know that the magnitude means the length of the vector

Hint 2: use 3D Pythagoras' Theorem

Hint 3: when squaring the -4, remember to write (-4)² and not -4²

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

Question 2

Hint 1: simplify the whole number parts first

Hint 2: know that to subtract fractions, they need a common denominator

Hint 3: convert each fraction to have a denominator of 14

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

Question 3

Hint 1: consider this question as two separate multiplications, then an addition

Hint 2: when multiplying out the first brackets, you should obtain four terms that simplify to three terms

Hint 3: multiply the second bracket by +2x

Hint 4: simplify the three terms from the first brackets with the two terms from the second multiplication

Hint 5: simplify the final answer to given three terms, which is a quadratic expression.

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

Question 4

Hint 1: recognise that angle OMP is a right angle, as tangent MP meets the radius OM at 90 degrees

Hint 2: work out angle POM using the two other angles in triangle OPM

Hint 3: work out angle NOM using angle POM, and the straight line NOP

Hint 4: recognise that triangle OMN is isosceles

Hint 5: calculate angle ONM using angle NOM and knowing that angles ONM and OMN are equal

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

Question 5

Hint 1: identify the location of the median

Hint 2: identify the median of the lower half, which is Quartile 1

Hint 3: identify the median of the upper half, which is Quartile 3

Hint 4: calculate the Interquartile Range from Quartile 3 subtract Quartile 1

Hint 5: calculate the Semi Interquartile Range by halving the Interquartile Range

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

Question 6

Hint 1: write out the formula y = kx²

Hint 2: replace x and y with their coordinate values from (2, -12)

Hint 3: evaluate and solve for the variable k

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

Question 7

Hint 1: recognise that the coefficients of d in the equations are of opposite signs

Hint 2: aim to multiply each equation so that the coefficients of d are equal, but of opposite signs

Hint 3: add the two equations together to eliminate the terms in d

Hint 4: solve for the variable c

Hint 5: substitute the value of c into either of the original equations to calculate the value of d

Hint 6: clearly present the final values you obtained for variables c and d

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

Question 8

Hint 1: know that the discriminant will be used to determine the nature of the roots

Hint 2: identify the values of a, b and c from looking at f(x)

Hint 3: evaluate b²-4ac and identify if it is positive, zero or negative.

Hint 4: clearly write whether the roots are real or not, how many there are, and whether they are the same or different.

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

Question 9

Hint 1: recognise that each term requires to be simplified to see if any of them can be combined

Hint 2: re-write 50 as 25×2, and re-write 45 as 9×5

Hint 3: simplify each term as far as possible to obtain terms involving √2 and √5

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

Question 10

10a) Hint 1: know that the gradient and intercept of line AB are going to be needed

10a) Hint 2: calculate the gradient, m_{AB}

10a) Hint 3: write out y = mx + c, replacing m with the value of m_{AB}

10a) Hint 4: use either the coordinates of point A, or point B, to substitute in values for x and y, to then work out the value of c

10a) Hint 5: write the equation of the line in terms of S and W, and not x and y

10b) Hint 6: recognise that S takes the value of 1000

10b) Hint 7: use the formula from part (a) to calculate the value for W

10b) Hint 8: and here is a video of the solution:

Question 11

Hint 1: carefully expand the first bracket by multiplying by the negative sign

Hint 2: gather all the x terms on the right side of the inequality, so that they are all positive

Hint 3: divide by the coefficient of x to obtain x on its own

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

Question 12

Hint 1: recognise that 2400 tickets represents 75%, and not 100%

Hint 2: work out how many tickets repesent 25%

Hint 3: scale this value up to 100% to obtain the number of tickets for Glasgow

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

Question 13

Hint 1: notice that the provided graph has a 'horizontal centre line' of y = 3, so the graph has been translated upwards by a distance of 3

Hint 2: notice that the amplitude of the provided graph is equal to 2

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

Question 14

14a) Hint 1: determine the coordinates of point C

14a) Hint 2: point B will have the same x and y coordinates as point C, but a different z coordinate

14b) Hint 3: determine the radius of the hemisphere from using the x coordinates of points C and A

14b) Hint 4: remember that the volume of a hemisphere is half that of a full sphere

14b) Hint 5: and here is a video of the solution:

Question 15

Hint 1: know that the index of ³/₂ means cubing and square-rooting

Hint 2: first process the square rooting of 16

Hint 3: then process the cubing of that result

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

Question 16

Hint 1: replace the value of x with 90 in the given expression for f(x)

Hint 2: recognise that you need to know the value of sin(270)

Hint 3: sketch the graph of y = sin(x) to determine the value of sin(170)

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

Question 17

Hint 1: consider how the graph of y = x² has been transformed to give y = 2(x-1)² + 4

Hint 2: the (x - 1) has translated the graph in the horizontal direction, so sketch the graph of y = (x-1)²

Hint 3: the 2(...) has stretched the graph vertically away from the x-axis, so sketch the graph of y = 2(x-1)²

Hint 4: the ...+4 has translated the graph upwards, so sketch the graph of y = 2(x-1)² + 4

Hint 5: make sure that you clearly label the coordinates of the turning point and of the intersection with the y-axis

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

Question 18

Hint 1: label the midpoint of UT and call it D

Hint 2: sketch the right angled triangle CDT

Hint 3: length CT is a radius of the circle and length DT is half of length RS

Hint 4: use Pythagoras' Theorem to calculate length DC

Hint 5: length of memory stick = length RU + length DC + radius of circle

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

Question 19

Hint 1: notice the constant term is -5, which only has factors of ±1 and ±5

Hint 2: consider the factor pairs of 6 which are 1 and 6, or 2 and 3

Hint 3: assemble this information by your chosen method to give a factorised expression

Hint 4: know that the solutions will come from making the expression in each bracket to be equal to zero

Hint 5: you should obtain two values for x, both of them being fractions, but with different signs

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

Paper 2

Question 1

Hint 1: know that increasing by 4% involves multiplying by the decimal 1.04

Hint 2: recognise that £250000 will increase by 4% twice

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

Question 2

Hint 1: recognise that you have been given a speed and a distance

Hint 2: know that to calculate time, you need to divide the distance by the speed

Hint 3: when using your calculator to perform the division, make sure that you
divide by all of 3×10^{8}, and not just divide by 3 followed by multiplying by 10^{8}

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

Question 3

Hint 1: notice that 75 = 3 × 25

Hint 2: factorise 3 out of the given expression

Hint 3: notice that the terms inside the brackets are the difference of two squares

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

Question 4

Hint 1: recognise that this requires use of the sine rule

Hint 2: assemble the information into the sine rule formula

Hint 3: rearrange to make expression sin(Q) = . . .

Hint 4: use inverse sine to calculate angle Q

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

Question 5

Hint 1: obtain the components of vectors __u__ and __v__ from the diagram

Hint 2: calculate vector __u__ - __v__

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

Question 6

6a) Hint 1: calculate the mean of the data

6a) Hint 2: use your own familiar method for calculating a standard deviation

6b) Hint 3: make a comment about the means, being sure to use the phrase 'average number of passengers' and clearly comparing Monday to Saturday

6b) Hint 4: make a comment about the standard deviations, being sure to use the phrase 'number of passengers' and clearly comparing Monday to Saturday

6b) Hint 5: and here is a video of the solution:

Question 7

Hint 1: calculate angle FHY using the given information about bearings

Hint 2: recognise that this problem requires the use of the cosine rule to work out length FY

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

Question 8

Hint 1: know that area of shaded segment = area of sector - area of triangle ABC

Hint 2: know that the area of the sector is a fraction of the area of a full circle, which has a radius of 14

Hint 3: calculate the area of the triangle ABC using the formula given on the formula sheet

Hint 4: subtract one area from the other

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

Question 9

9a) Hint 1: know that to read off the gradient, the equation must be in the form y = mx + c

9a) Hint 2: subtract 3x and add 8 to both sides of the equation

9a) Hint 3: divide all terms by 4 to obtain y = . . .

9b) Hint 4: know that a line crosses the y-axis when the x coordinate equals 0

9b) Hint 5: in the given equation of 3x + 4y - 8 = 0, replace x with 0, and calculate the value of y

9b) Hint 6: write the final answers as coordinates, with brackets and two numbers separated by a comma

9b) Hint 7: and here is a video of the solution:

Question 10

Hint 1: notice that h is being multiplied by 3, divided by 2 and then square rooted, to give d

Hint 2: perform the inverse of these operations, in reverse order

Hint 3: square both sides of the equation

Hint 4: multiply both sides of the equation by 2

Hint 5: divide both sides of the equation by 3

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

Question 11

Hint 1: know that the volume of a cone requires a radius and a height

Hint 2: draw the right angled triangle OAB, marking in lengths OA and AB

Hint 3: use Pythagoras' Theorem to calculate length OB

Hint 4: use the formula for the volume of a cone from the formula sheet

Hint 5: after writing a full decimal answer, write it to 2 significant figures accuracy

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

Question 12

Hint 1: know that dividing fractions requires leaving the first fraction alone, change the ÷ to a ×, and write down the reciprocal of the second fraction (i.e. swap the numerator with the denominator)

Hint 2: notice that 6x in the numerator of the first fraction can be simplified using the 2x² in the denominator of the second fraction

Hint 3: combine the terms into a single fraction

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

Question 13

Hint 1: recognise that we have information for an area scale factor, but we need a length scale factor

Hint 2: calculate the area scale factor that takes you from the smaller to the larger photograph

Hint 3: length scale factor = √(area scale factor)

Hint 4: use the length scale factor and 12cm to calculate the width of the larger photograph

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

Question 14

14a) Hint 1: recognise that h = 115

14a) Hint 2: replace h with 115 in the given formula

14a) Hint 3: rearrange equation to say cos(x) = . . .

14a) Hint 4: use inverse cosine to calculate the value of x

14b) Hint 5: use your knowledge of trigonometric equations and/or graphs to obtain a second solution for x that is between 180° and 360°

14b) Hint 6: and here is a video of the solution:

Question 15

15a) Hint 1: know that length = breadth + 5

15a) Hint 2: replace 'breadth' with the variable, 'x'

15b) Hint 3: know that area = length × breadth

15b) Hint 4: in that formula, replace area with 20, length with '(x+5)' and breadth with 'x'

15b) Hint 5: expand out the brackets and gather terms on one side to give the stated quadratic equation

15c) Hint 6: recognise the clue that 'correct to one decimal place' means that the quadratic formula will be required to solve the quadratic equation

15c) Hint 7: use the quadratic formula to obtain two values for x

15c) Hint 8: recognise that x represents the breadth of the rectangle, and so it cannot be negative

15c) Hint 9: reject one of your values of x, giving a reason, and present the single remaining value for x

15c) Hint 10: and here is a video of the solution:

Question 16

Hint 1: expand out the brackets to obtain two terms

Hint 2: know that tan(x) = sin(x)/cos(x)

Hint 3: replace tan(x) with sin(x)/cos(x)

Hint 4: simplify the term that involves cos(x) in the numerator and cos(x) in the denominator

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

Question 17

Hint 1: consider a route from A to G that goes via C

Hint 2: notice that vector AC is __t__

Hint 3: notice that vector CG is ^{1}/_{3} of vector CB

Hint 4: notice that vector CB is vector CA + vector AB

Hint 5: know that vector CA is equal to the negative of vector AC, which is
-__t__

Hint 6: combine all of the above information together to give the expression for vector AG

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

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