Hints offered by N Hopley, with video solutions by 'DLBmaths'
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Paper 1
Question 1
Hint 1: consider starting with (a+b)³ and write the general term of the expansion
Hint 2: replace a with (2/y²) and b with (-5y), being careful to include brackets
Hint 3: simplify each of the final 4 terms
Hint 4: and here is a video of the solution:
Question 2
Hint 1: notice that one term in the denominator is repeated
Hint 2: know that the general form will be A/(x+1) + B/(x-2) + C/(x-2)²
Hint 3: procede with a standard method for obtaining A, B and C
Hint 4: and here is a video of the solution:
Question 3
Hint 1: recognise that this requires quotient rule
Hint 2: recognise that this also requires chain rule
Hint 3: and here is a video of the solution:
Question 4
4a) Hint 1: write down the values of u5 and u12
4a) Hint 2: write down general nth term in arithmetic series in terms of a and d
4a) Hint 3: replace n with 5 and u5 with its value & similarly for n being 12 and u12
4a) Hint 4: recognise that you have two simultaneous equations in a and d
4b) Hint 5: write down general formula for sum of first n terms of an arithmetic sequence
4b) Hint 6: using values of a and d from part (a), and -144, substitue in all of the relevant values
4b) Hint 7: obtain a quadratic equation in n, and solve it
4b) Hint 8: determine which of the two solutions to discard, and state the reason for discarding it
Hint 9: and here is a video of the solution:
Question 5
5a)i) Hint 1: no hint - standard gaussian elimination method to be used
5a)ii) Hint 2: know that 'inconsistent' means that there is no solution
5a)ii) Hint 3: consider what value of λ would result in z not being defined
5b) Hint 4: evaluate z for the given value of λ
5b) Hint 5: then evaluate y, and then x
Hint 6: and here is a video of the solution:
Question 6
Hint 1: using the given substitution, work out du/dx
Hint 2: rearrange this to obtain an expression for dx
Hint 3: using the given substitution, work out the values of u for the two values of the limits of the integral that's in terms of x
Hint 4: re-write the integral in terms of u's limits and have the integrand in terms of u only
Hint 5: recognise a standard inverse trig function integrand
Hint 6: use trigonometric exact value knowledge to calcuate the exact value of the integral
Hint 7: and here is a video of the solution:
Question 7
7a)i) Hint 1: equate the corresponding components of matrices P and Q
7a)ii) Hint 2: use standard knowledge of the formula for an inverse of a 2×2 matrix
7a)iii) Hint 3: know that 'transpose' means rows to columns, and columns to rows
7a)iii) Hint 4: procede with matrix multiplication, obtaining a single 2×2 matrix that's in terms of y
7b)) Hint 5: recall that a singular matrix has no inverse
7b)) Hint 6: recall that a matrix has no inverse if its determinant is equal to zero
Hint 7: and here is a video of the solution:
Question 8
Hint 1: the first line of the Euclidean Algorithm should read 1595 = 1218 + 377
Hint 2: The second line is 1218 = 3 x 377 + 87
Hint 3: Continue with standard method to obtain the values of a and b
Hint 4: and here is a video of the solution:
Question 9
Hint 1: recognise that each side of the equation has a single expression
Hint 2: recognise that this means that the method of separation of variables will be used
Hint 3: after obtaining the y terms on one side and the x terms on the other, integrate
Hint 4: recognise the standard inverse trig function integrand for the expression in y
Hint 5: use the given initial conditions to fix the value of the constant
Hint 6: be sure to present the final answer in terms of y = …
Hint 7: and here is a video of the solution:
Question 10
10a) Hint 1: know that the Σ summation operator is a linear operator
10a) Hint 2: know that Σ (A+B) = Σ A + Σ B
10a) Hint 3: know that Σ kB = k Σ B
10a) Hint 4: know standard formula for Σ r² and Σ r
10a) Hint 5: resist the urge to multiply out all the brackets - look for common factors first
10b) Hint 6: know that (sum from 10 to 2p) = (sum from 1 to 2p) - (sum from 1 to 9)
10b) Hint 7: evaluate both expressions using the answer from part (a)
10b) Hint 8: you should end up with a cubic in p
Hint 9: and here is a video of the solution:
Question 11
Hint 1: know that logarithmic differentiation means to first take natural logs of both sides
Hint 2: consider re-writing y as y(x) to emphasise that it is a function of x
Hint 3: implicitly differentiate both sides of the equation with respect to x
Hint 4: recognise that the product rule is needed for the expression in x
Hint 5: rearrange to make dy/dx the subject and replace y with its original expression
Hint 6: and here is a video of the solution:
Question 12
12a) Hint 1: know that an odd function has half turn rotational symmetry about the origin
12a) Hint 2: rotate the dotted line, the function curve and the point (-1, -2) a half turn about the origin
12b) Hint 3: know that the absolute, or magnitude, function (|…|) only returns positive values
12b) Hint 4: the graph of |f(x)| will only exist above the x-axis
12b) Hint 5: reflect all the parts of f(x) above the x-axis. Do this NEATLY!
12c) Hint 6: notice that the gradient of f(x) is at its steepest when passing through the origin
12c) Hint 7: notice that the gradient of f(x) is at its shallowest when approaching its asymptotes
12c) Hint 8: notice that the gradient of the asymptotes is 1/2
12c) Hint 9: consider carefully whether the gradient of f(x) will ever take on the values of either 1/2 or 2
Hint 10: and here is a video of the solution:
Question 13
Hint 1: know that if P ⇒ Q then the contrapositive is ¬Q ⇒ ¬P
Hint 2: know that ¬(even) is (odd) and ¬(odd) is (even)
Hint 3: know that the general form for an odd number is 2k+1, where k is an integer
Hint 4: replace n with 2k+1 in your contrapositive statement to establish that the quadratic expression is odd
Hint 5: write words to describe the logic behind why this means that the statement being proved has been proven
Hint 6: and here is a video of the solution:
Question 14
Hint 1: obtain the auxiliary equation
Hint 2: solve the auxiliary quadratic equation to give two equal roots
Hint 3: know that the particular integral will be a linear sum of trigonometric functions
Hint 4: proceed with standard method for solving the differential equation, using the provided information to determine the four constants
Hint 5: and here is a video of the solution:
Question 15
15a) Hint 1: obtain the vector BT for the direction vector of the line
15a) Hint 2: consider dividing this vector by -5 to obtain the simplest values for the direction vector of the line
15a) Hint 3: use either point B or point T to act as the 'starting point' of the line, and introduce a parameter letter
15b) Hint 4: obtain vector PQ and vector PR
15b) Hint 5: calculate the vector product of vector PQ and vector PR
15b) Hint 6: this vector product is the normal vector to the plane containing points P, Q and R
15b) Hint 7: use the coordinates of any of the points P, Q or R to determine the constant term in the equation of the plane
15c) Hint 8: realise that we are seeking the intersection of the line from part (a) with the plane from part (b)
15c) Hint 9: substitute the values for x, y and z that are in terms of a parameter, into the equation of the plane
15c) Hint 10: solve the resulting linear equation that's in terms of the parameter
15c) Hint 11: subsitute the parameter value back into the expressions for x, y and z from part (a) to obtain the coordinates of intersection
Hint 12: and here is a video of the solution:
Question 16
Hint 1: obtain the point of intersection of the curve with the y-axis
Hint 2: rearrange the equation of the curve to make x² the subject
Hint 3: know the standard technique for a volume of revolution, to allow you to substitute in the expression for x²
Hint 4: leave your final answer in terms of π to keep it exact.
Hint 5: and here is a video of the solution:
Question 17
17a) Hint 1: know that a polynomial with real coefficients has roots that are complex conjugates
17a) Hint 2: know that the complex conjugate of (a + ib) is (a - ib)
17b) Hint 3: know that if z1 and z2 are values of roots, then (z - z1)(z - z2) will give a quadratic in z that will divide into the original polynomial
17b) Hint 4: perform polynomial long division of the original expression using your quadratic in z
17b) Hint 5: know that the remainder should be zero, thus giving the value for q
17b) Hint 6: factorise the quadratic expression that came from the quotient of the division, to obtain two further complex roots of z
17c) Hint 7: plot the four complex numbers on an argand diagram, expecting there to be two pairs of complex conjugates
Hint 8: and here is a video of the solution:
Question 18
18a) Hint 1: know that speed is the magnitude of a velocity vector
18a) Hint 2: know that velocity vector is the rate of change of the position vector, with respect to time
18a) Hint 3: calculate the rates of change of each of x and y, with respect to t (i.e. work out dx/dt and dy/dt)
18a) Hint 4: know that the magnitude of a vector is the √[ (x component)² + (y component)² ]
18a) Hint 5: use a trigonometric identity to simplify the sin²(t) and cos²(t) terms
18b) Hint 6: know that point A has a y coordinate of zero
18b) Hint 7: determine the values of t which might give a y coordinate of zero
18b) Hint 8: observe that the motion started at the origin, when t = 0
18b) Hint 9: determine which value of t corresponds to when the path passes through point A
18b) Hint 10: evaluate the expression in part (a) for this value of t, to obtain the instantaneous speed at point A
Hint 11: and here is a video of the solution: