Click HERE for the 2015 'Revised' Higher Maths Examination
Hints offered by N Hopley, with video solutions by 'DLBmaths'
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Paper 1
Question 1
Hint 1: perpendicular vectors are at right angles to each other
Hint 2: if the scalar product of two vectors is equal to zero, then they are perpendicular
Hint 3: set up u.v = 0, and create an equation in terms of t
Hint 4: and here is a video of the solution:
Question 2
Hint 1: the equation of a tangent needs a gradient and a point to go through
Hint 2: the gradient is obtained from differentiating the function, and evaluating it at x = -2
Hint 3: the y coordinate of the point is obtained from evaluating the original function when x = -2
Hint 4: and here is a video of the solution:
Question 3
Hint 1: know that if (x + 3) is a factor, then x = -3 is a root
Hint 2: know that if x = -3 is a root, then evaluating the polynomial when x = -3 should give the answer of zero
Hint 3: make sure to clearly state that the value of zero means that (x + 3) is a factor
Hint 4: use either polynomial long division, or synthetic division, to divide the given polynomial by (x + 3)
Hint 5: remember to factorise the resulting quadratic, to give a final factorised form of three linear factors
Hint 6: and here is a video of the solution:
Question 4
Hint 1: look at the highest and lowest values of the function on the graph. The difference between these is double the amplitude
Hint 2: notice that the 'centre line' of the function will be half way between the highest and lowest values of the function
Hint 3: look at the π/2 and notice that it is the period of the function. So, how many cycles would there be in 2π?
Hint 4: know that p is the amplitude, q is the number of cycles in 2π and r the vertical translation
Hint 5: and here is a video of the solution:
Question 5
5a) Hint 1: use a standard method to work out an inverse function
5b) Hint 2: know that g(g-1(x)) is the function of its inverse and by definition this should return the original value
Hint 3: and here is a video of the solution:
Question 6
Hint 1: aim to combine all the terms into a single log expression
Hint 2: use the following log rule on the fraction one third: n logap = logapn
Hint 3: know that the power of one third is the same as ∛
Hint 4: use the log rule: log(x) + log(y) = log (xy), so long as the logs have the same base (which they do, here)
Hint 5: keep simplifying the expression to give a single integer as a final answer.
Hint 6: and here is a video of the solution:
Question 7
Hint 1: recognise that the function is not ready to differentiate in its current form
Hint 2: multiply out the brackets, simplify each term and re-write any roots or denominators as terms in x to a power
Hint 3: after differentiating, re-write the x terms with fractional powers back in terms of square roots, and any negative powers back as fractions
Hint 4: when the expression is as simple as possible, substitute the value of x = 4 into the expression for f'(x)
Hint 5: and here is a video of the solution:
Question 8
Hint 1: know that the formula for the area of a rectangle is length × breadth
Hint 2: set up an inequation for this formula to be < 15
Hint 3: expand out the brackets and rearrange terms so that the inequation is of the form ... < 0
Hint 4: factorise the quadratic expression
Hint 5: sketch a graph of the quadratic expression, as if it were a function, noting the x-axis intercepts and whether it has a maximum or minimum turning point
Hint 6: decide from your graph which region(s) represent the values of x when the quadratic expression is < 0
Hint 7: and here is a video of the solution:
Question 9
Hint 1: if AB is parallel to the given equation, then rearrange that equation to then read off its gradient
Hint 2: if BC makes an angle of 150°, then we will use m = tan(θ) to obtain the gradient of BC
Hint 3: know that the value for tan(150°) is related to the value for tan(30°). Draw an accurate sketch of the function y = tan(x) to see why.
Hint 4: use exact value triangle knowledge to obtain the exact value of tan(30°) and thus the exact value of the gradient BC
Hint 5: compare the gradients of AB and BC and clearly state whether this shows if the points A, B and C might be collinear, or not.
Hint 6: and here is a video of the solution:
Question 10
10a) Hint 1: sketch a right-angled triangle with angle '2x' and opposite length 3 and adjacent length 4.
10a) Hint 2: use Pythagoras' Theorem to work out the length of the hypotenuse (or just notice a familiar Pythagorean triplet?)
10a) Hint 3: your diagram can now be used to write down the value of cos(2x)
10b) Hint 4: know that cos(2x) can be written in terms of cos²(x)
10b) Hint 5: use your answer from part (a) to then obtain an expression for cos²(x)
10b) Hint 6: know that this will give two values for cos(x)
10b) Hint 7: from the graph of y = cos(x), realise that the restriction given in the question that 0 < x < π/4 means that we can reject one of the values of cos(x)
10b) Hint 8: simplify the remaining expression for cos(x) as far as possible.
Hint 9: and here is a video of the solution:
Question 11
11a) Hint 1: draw a sketch of the circle and plot point T on its circumference
11a) Hint 2: know that a tangent requires a point (point T) and a gradient
11a) Hint 3: calculate the gradient of the line from the centre of the circle to point T
11a) Hint 4: calculate the perpendicular gradient to this gradient
11a) Hint 5: use a standard method to work out the equation of the tangent using point T and the gradient just calculated
11b) Hint 6: know that a tangent to a parabola will intersect it in the same place, twice
11b) Hint 7: realise that this means that the equation formed from equating the quadratic with the equation of the tangent through T will have equal roots
11b) Hint 8: know that equal roots of a quadratic mean that the discriminant will equal zero
11b) Hint 9: the discriminant equation gives a quadratic in p which can be solved to give two values for p
11b) Hint 10: read the question to see the restriction on values of p, and thus reject one of the calculated solutions, clearly stating the reason why
Hint 11: and here is a video of the solution:
Question 12
Hint 1: know that the graph of a cos(bx) is symmetrical about the x-axis
Hint 2: therefore the shaded region above the x-axis is connected to the non-shaded region below the x-axis
Hint 3: know that the non-shaded region would give an integral value that is negative
Hint 4: realise that the non-shaded region is -2 lots of the shaded region
Hint 5: and here is a video of the solution:
Question 13
13a) Hint 1: point P(1, b) has x = 1 and y = b
13a) Hint 2: substitute these values of x and y into the equation y = 2x + 3, and then solve for b
13b)i) Hint 3: know that the graph of the f-1(x) is geometrically related to the graph of f(x)
13b)i) Hint 4: the graph of the f-1(x) is the reflection of the graph of f(x) in the line y = x
13b)ii) Hint 5: the image of point (x, y) is the point (y, x)
13c) Hint 6: know that y = 4 - f(x + 1) has three transformations involved: from the 4, the -1 multiplier and the +1 in the brackets
13c) Hint 7: working from the 'x' term outwards, it is first affected by +1, then by -1, then by 4
13c) Hint 8: know that the +1 moves the graph left one unit
13c) Hint 9: know that the -1 multiplier reflects the function in the x-axis
13c) Hint 10: know that the +4 translates the graph upwards by 4
13c) Hint 11: track the point R(3, 11) through each of these transformations
Hint 12: and here is a video of the solution:
Question 14
Hint 1: either complete the square in the x terms, and the y terms, or use the circle equation formula to determine the centre and the radius
Hint 2: from the previous step, you should know the coordinates of the centre of the circle
Hint 3: sketch a set of axes and draw in a circle that would meet the coordinate axes at exactly 3 points
Hint 4: your sketch should be a circle going through the origin, plus one point on the y-axis and one point on the x-axis
Hint 5: realise that the radius of the circle will be the distance from the origin to the centre of the circle
Hint 6: match this value of the radius with your expression for the radius from the first step, and work out the value of k
Hint 7: and here is a video of the solution:
Question 15
Hint 1: realise that you will need to integrate the expression and then use the given information to determine the constant of integration
Hint 2: integrate T'(t) with respect to t, to give T(t), with a constant of integration
Hint 3: use the information that when t = 0, T(0) = 100 in order to fix the value of the constant
Hint 4: use the information that when t = 10, T(10) = 82 in order to fix the value of k
Hint 5: clearly state the expression for T(t) in terms of t.
Hint 6: and here is a video of the solution:
Paper 2
Question 1
1a) Hint 1: the altitude through C has a gradient that is perpendicular to the gradient of AB
1a) Hint 2: work out mAB and then take the negative reciprocal to give the gradient of the altitude
1a) Hint 3: use a standard method to work out the equation of the line through T with the required gradient
1b) Hint 4: know that the median from B will go through the midpoint of AC
1b) Hint 5: calculate the coordinates of M, the mid-point of AC
1b) Hint 6: calculate the gradient of line BM
1b) Hint 7: use a standard method to work out the equation of the line through B with the gradient mBM
1c) Hint 8: realise that you have simultaneous equations, using your answers from parts (a) and (b)
1c) Hint 9: be sure to state the coordinates in brackets, with a comma between them.
Hint 10: and here is a video of the solution:
Question 2
2a) Hint 1: know that f(g(x)) means to do g(x) first, then f(…) second
2a) Hint 2: replace g(x) in f(g(x)) with what g(x) is equal to
2a) Hint 3: apply function f(…) to what its input is, then expand and simplify the quadratic expression
2b) Hint 4: use a standard method to complete the square for your answer from part (a)
2c) Hint 5: know that h(x) will not be defined if its denominator has the value of zero.
2c) Hint 6: use your answer from part (b) to determine the values of x that would make f(g(x)) equal to zero.
Hint 7: and here is a video of the solution:
Question 3
3a) Hint 1: use the formula for the tn sequence, replacing n with 1
3b) Hint 2: work out the limit of each sequence by a standard method
3b) Hint 3: compare each limit to the number 50 to decide if either the frog or toad can escape
Hint 4: and here is a video of the solution:
Question 4
4a) Hint 1: set f(x) = g(x) and solve for x
4b) Hint 2: realise that integration will be used
4b) Hint 3: realise that the shield is symmetrical, so we can work out the area between f(x) and h(x), and then double it to obtain the total area
4b) Hint 4: treat f(x) as the 'top function' and h(x) as the 'bottom' function
4b) Hint 5: when subtracting h(x) from f(x), be sure to include brackets so that the correct terms are all subtracted
4b) Hint 6: the limits of integration are from zero to the value you found in part(a)
4b) Hint 7: remember to double the integral answer at the end to get the total area!
Hint 8: and here is a video of the solution:
Question 5
5a) Hint 1: complete the square in the x and y terms, or use the circle formula, to obtain the centre and radius of circle C1
5a) Hint 2: work out the distance from the centre of C1 to the centre of C2
5a) Hint 3: this distance is the sum of the two circles' radii, and you know the radius of C1, so you can work out the radius of C2
5b) Hint 4: copy the given diagram of two circles and draw C3 such that it has its centre on the line joining the other two circles' centres
5b) Hint 5: know that the radius of C3 will be the sum of the diameters of C1 and C2
5b) Hint 6: draw another sketch of the line joining the centres of C1 and C2, and extend it out to show the diameters of C1 and C2
5b) Hint 7: know that the centre of C3 will be in the middle of this line that the length of the two diameters
5b) Hint 8: work out the ratio of where the centre point for C3 is relative to line segment joining the centres of C1 and C2
5b) Hint 9: use your ratio technique to obtain the coordinates of the centre of C3 and therefore the equation of C3
Hint 10: and here is a video of the solution:
Question 6
6a) Hint 1: know that p.(q + r) = p.q + p.r
6a) Hint 2: know that the angle between vectors p and q is 60°
6a) Hint 3: use exact value triangle knowledge to give cos(60°) and know what cos(0°) is as well.
6a) Hint 4: know that |q| = |p| = 3
6b) Hint 5: look for a route from point E to point A that goes along segments that you know about
6b) Hint 6: vector EC = vector EA + vector AB + vector BC
6b) Hint 7: know that vector EA is the negative of vector AE, and that vector BC is the same as vector ED (because it is a parallelogram)
6c) Hint 8: use your answer from part (b) and a similar technique to part (a) to expand and simplify a series of scalar products involving p, q and r
6c) Hint 9: you will need to know the angle between q and r, when the vectors are placed 'tail to tail'
6c) Hint 10: you should find that you want the exact value for cos(30°)
6c) Hint 11: substituting all values into the given equation should allow you to solve an equation in |r|
Hint 12: and here is a video of the solution:
Question 7
7a) Hint 1: realise that integrating the 3cos(2x) term will involve the 'opposite' of the chain rule
7a) Hint 2: remember to include the constant of integration
7b) Hint 3: work with the left hand side and chose an identity for cos(2x) that includes terms like those on the right hand side
7c) Hint 4: notice that the integrand is a linear multiple of the right hand side of part (b)
7c) Hint 5: rewrite the integrand in terms of the left hand side of part (b)
7c) Hint 6: use your workings from part (a) to complete the integration
Hint 7: and here is a video of the solution:
Question 8
8a)i) Hint 1: realise that if the crocodile goes by water the whole way, then x = 20
8a)i) Hint 2: evaluate T(x) when x = 20
8a)ii) Hint 3: realise that if the crocodile swims the shortest distance possible, then it will go straight across the river, and therefore x = 0
8a)ii) Hint 4: evaluate T(x) when x = 0
8b) Hint 5: realise that this is an optimisation equation, so it will involve differentiating T(x) to find a stationary point
8b) Hint 6: realise that T(x) is not ready to differentiate, so you need to re-write the √ as a fractional power
8b) Hint 7: when differentiating T(x), you will need to use the chain rule
8b) Hint 8: after differentiating, re-write any terms with fractional powers back in terms of square roots, and any negative powers back as fractions
8b) Hint 9: clearly state that you are seeking T'(x) = 0 in order to find a stationary point
8b) Hint 10: take careful steps to rearrange and solve for x. You should obtain two values: one positive and one negative
8b) Hint 11: from reading the context of the question, give a reason for rejecting one of the values of x
8b) Hint 12: to check the nature of the stationary point, you will need to use a nature table, as to work out T''(x) requires knowledge that is beyond the Higher course
8b) Hint 13: after confirming that you have a minimum turning point, evaluate T(x) for the found value of x.
Hint 14: and here is a video of the solution:
Question 9
Hint 1: use a standard method for re-writing the given expression in the stated form
Hint 2: set the value of h in the first equation to be h = 100
Hint 3: subtract 65 from both sides of the equation and use your earlier work to re-write the equation in terms of a single sine function
Hint 4: check you have your calculator in radian mode, and take inverse sine to obtain two values for (1.5t - 0.395)
Hint 5: add 0.395 to each of your values
Hint 6: divide each of your answers by 1.5, to give the values of t required.
Hint 7: and here is a video of the solution: