Hints offered by N Hopley, with video solutions by 'DLBmaths'
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Paper 1
Question 1
1a) Hint 1: recognise that you need to use the chain rule and a standard differential from the Formula List
1b) Hint 2: recognise that you need to use the quotient rule with the chain rule
1c) Hint 3: consider writing y as y(x) to emphasise that y is a function of x, and then complete the implicit differentiation
Hint 4: and here is a video of the solution:
Question 2
Hint 1: factorise the denominator into two linear factors
Hint 2: use the standard method of partial fractions on the integrand
Hint 3: integrate each fraction on its own, bringing in natural logarithms
Hint 4: and here is a video of the solution:
Question 3
3a) Hint 1: consider writing out the general term to the expansion of (a+b)^n first
3a) Hint 2: then substitute the a for 2x and the b for 5/x² and the n for 9
3a) Hint 3: simplify each term to obtain a term with factorials, numerical terms to powers of r, and x to the power of a linear expression in r
3b) Hint 4: know that the term independent of x is the term whose power of x is zero
3b) Hint 5: set the linear expression in r to be equal to zero, and solve for r
3b) Hint 6: evaluate your answer from part (a) with r taking on the value that you just obtained
Hint 7: and here is a video of the solution:
Question 4
4a) Hint 1: know that the bar over z means the complex conjugate of z
4a) Hint 2: multiply the two complex numbers together and gather the real terms and the imaginary terms
4a) Hint 3: factorise i out of the imaginary terms
4b) Hint 4: know that a real number has an imaginary term of 0
Hint 5: and here is a video of the solution:
Question 5
Hint 1: use the standard method of the euclidean algorithm
Hint 2: the first line is 306 = 119 × 2 + 68
Hint 3: and here is a video of the solution:
Question 6
Hint 1: recognise that we want dy/dx when t = -1/3
Hint 2: work out dy/dt and dx/dt
Hint 3: know that dt/dx is the reciprocal of dx/dt
Hint 4: know that dy/dx is (dy/dt)×(dt/dx)
Hint 5: evaluate dy/dx when t = -1/3, to obtain the gradient of the curve at that point
Hint 6: evaluate x and y when t = -1/3 to obtain the x and y coordinates
Hint 7: use these coordinates and the gradient to obtain the equation of the line
Hint 8: and here is a video of the solution:
Question 7
7a) Hint 1: obtain the resulting 3×3 matrix that has an expression in terms of k in row 2, column 1
7b) Hint 2: use a standard method to obtain the determinant of D, in terms of k
7b) Hint 3: know that the inverse of D does not exist if the determinant of D is equal to zero
Hint 4: and here is a video of the solution:
Question 8
Hint 1: work out du/dθ
Hint 2: work out the values of u for both of the values of θ from the limits of the integral
Hint 3: use a standard method of integration by substitution to simplify the integral to a simple polynomial in u, with u limits
Hint 4: and here is a video of the solution:
Question 9
9a) Hint 1: know that consecutive integers can be written a n, n+1 and n+2, where n is an integer
9a) Hint 2: write words to explain the logic behind what your algebraic terms mean in the context of divisibility
9b) Hint 3: know that an odd integer can be written as 2n+1 where n is an integer
9b) Hint 4: write words to explain the meaning of your algebraic terms
Hint 5: and here is a video of the solution:
Question 10
Hint 1: know that z = x + iy
Hint 2: replace z with x + iy in the given modulus equation
Hint 3: square both sides of the modulus equation to prevent square roots appearing
Hint 4: know that |a+ib|² = a² + b²
Hint 5: simplify the expression for y in terms of x to obtain the equation of a straight line, which is the locus required
Hint 6: sketch the locus, noting points of intercept with the axis, and plotting the numbers 0 [at (0,0)] and 2-2i [at (2,-2)]
Hint 7: and here is a video of the solution:
Question 11
11a) Hint 1: refer to the Formula List for the 2x2 matrix that represents a rotation of θ anticlockwise around the origin
11a) Hint 2: use exact value triangles to evaluate each term when θ= π/3
11b) Hint 3: know that a reflection in the x-axis transforms the point (x,y) to the point (x,-y)
11c) Hint 4: know that P = B A, and not P = A B
11d) Hint 5: refer to the general matrix for a rotation, noting which elements have to be the same sign
Hint 6: and here is a video of the solution:
Question 12
Hint 1: evaluate a base case when n = 1, to verify the statement is true
Hint 2: use a standard method for proof by induction, for the inductive step
Hint 3: write words to draw together how the base case and the inductive step together mean the statement is true for all positive integers
Hint 4: and here is a video of the solution:
Question 13
13a) Hint 1: use pythagoras' theorem on one of the right angled triangles
13a) Hint 2: solve for h, giving a reason for rejecting the possible negative solution
13b) Hint 3: recognise that a decreasing rate means that it will be negative
13b) Hint 4: deduce that dx/dt = -0.3
13b) Hint 5: recognise that we want dh/dt when x = 30
13b) Hint 6: implicitly differentiate with respect to t, the relationship h² + x² = 2500, from part (a)'s workings
13b) Hint 7: calculate the value for h, when x = 30, using part(a)
13b) Hint 8: replace in your implicitly differentiated equation the values for x, h and dx/dt
13b) Hint 9: rearrange to make dh/dt the subject
Hint 10: and here is a video of the solution:
Question 14
14a) Hint 1: use standard formulae to calculate each of u7 and S∞
14b)i) Hint 2: use the standard formula for Sn and replace the values for Sn, n and a, to then solve for d
14b)ii) Hint 3: use a standard formula, now that a and d are known
14c) Hint 4: use a similar approach to part (b), but this time substituting in values for Sn, a and d to then obtain an equation in n
14c) Hint 5: rearrange to make the quadratic in n equal to zero
14c) Hint 6: factorise the quadratic by first taking out a common factor of 16 from all terms
Hint 7: and here is a video of the solution:
Question 15
15a) Hint 1: use a standard method for integration by parts
15a) Hint 2: remember to include the constant of integration
15b) Hint 3: recognise that this equation requires an integrating factor, or...
15b) Hint 4: ...alternatively, in order to obtain part (a)'s integrand in part (b), just divide all terms through by x
15b) Hint 5: integrate by a standard method and use the initial conditions to fix the value of the constant of integration
15b) Hint 6: present your final answer in the required form
Hint 7: and here is a video of the solution:
Question 16
16a) Hint 1: use the standard method of gaussian elimination
16a) Hint 2: carefully interpret the final row of your augmented matrix that should be (a-8)z = 0
16a) Hint 3: consider which values of a would give an infinite number of solutions (that would give the intersection line)
16b) Hint 4: work out expressions for y and x, each in terms of z
16b) Hint 5: write the x, y and z components in vector form
16b) Hint 6: extract the constant vector and a parametric multiple of a direction vector, introducing a parameter, instead of z
16c) Hint 7: know that the angle between two planes is the angle between their two normal vectors
16c) Hint 8: know that we need the acute angle, so careful consideration with the help of an angle diagram should help
16d) Hint 9: compare the two planes' direction vectors
16d) Hint 10: know that one vector being the multiple of another means that they are parallel vectors
16d) Hint 11: know what parallel normal vectors mean in terms of the planes themselves
16d) Hint 12: know how to check that the two planes are not coincident with each other (i.e. they are the same plane in the same space)
Hint 13: and here is a video of the solution:
Question 17
17a) Hint 1: use the standard method for calculating each term of a Maclaurin Series
17b)i) Hint 2: recognise that this will require repeated use of both the chain rule and the product rule
17b)ii) Hint 3: evaluate the answers from (b)(i) when x = 0
17c) Hint 4: write down the two series up to and including their cubic powers of x
17c) Hint 5: multiply the two polynomials together to obtain all 8 terms
17c) Hint 6: discard all the terms that have powers higher than 3
17d) Hint 7: recognise that the expression given is the derivative of that from part (c)
17d) Hint 8: differentiate the series answer from part (c), term by term.
Hint 9: and here is a video of the solution: