Question Paper 1 Question Paper 2

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Paper 1

Question 1

Hint 1: write out (a + b)⁴ in its binomial expansion

Hint 2: use the substitution of a = (1/x) and b = (-3x), remembering to use the brackets in each term

Hint 3: slowly and carefully expand each term's powers, taking care with negative signs

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

Question 2

Hint 1: recognise that this is a situation for the quotient rule

Hint 2: carefully apply the quotient rule, keeping brackets around each component's terms

Hint 3: expand out the numerator's terms and simplify

Hint 4: do not expand the denominator, leaving it as (3 + 2x)²

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

Question 3

Hint 1: write out the fraction, with z and w replaced by their values

Hint 2: multiply the fraction by the number 1 in disguise...

Hint 3: ... the disguise being based on the complex conjugate of the denominator

Hint 4: so multiply the z/w fraction by (3 + 2i)/(3 + 2i)

Hint 5: taking care with signs, multiply out both numerator and denominator

Hint 6: factorise the numerator, so that the final answer can be simplified

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

Question 4

4a) Hint 1: know that multiplication of a matrix by a scalar means that each element is multiplied by the scalar

4b)i) Hint 2: know that the transpose of a matrix is remembered by this rhyme: 'rows to columns, columns to rows, A to A transpose'

4b)i) Hint 3: use the standard process for multiplying two matrices together, taking great care with signs

4b)ii) Hint 4: know that the determinant of a 2×2 matrix with first row (a b) and second row (c d) is = ad - bc

4b)iii) Hint 5: know that if a matrix is singular, then it does not have an inverse

4b)iii) Hint 6: know that if a matrix that does not have an inverse, then it has a determinant of zero

4b)iii) Hint 7: form an equation, using your response to (b)(ii), and then solve for λ

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

Question 5

Hint 1: carefully examine the form of the integral, and compare to the various 'templates' on the formula sheet given on the inside cover of the question paper

Hint 2: re-write 4x² as (2x)²

Hint 3: recognise that this integral takes on the form that matches the derivative of tan^-1x, but with 'x' replaced by '2x'

Hint 4: know therefore that the answer is likely to contain tan^-1(2x)

Hint 5: however, if this were to be differentiated, then a multiplier of 2 would come into play, due to the chain rule

Hint 6: hence, include a correcting fraction of (1/2)

Hint 7: and don't forget the constant of integration!

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

Question 6

6a) Hint 1: perform polynomial long division with x² + x + 5 being divided by x - 2

6a) Hint 2: interpret the non-zero remainder as the constant 'C' in their suggested format

6b) Hint 3: know that a vertical asymptote occurs when a function is undefined for a given value of x

6b) Hint 4: examine the function f(x) and decide what value of x would cause the denominator to be zero, and thus would cause the function to be undefined.

6b) Hint 5: know that non-vertical asymptotes are what the function tends towards when x tends to ±∞

6b) Hint 6: using the response to part (a), know that as x tends to either +∞ or -∞, then C / (x - 2) will tend to zero, as the denominator will be very much larger than the numerator

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

Question 7

Hint 1: examine the format of the differential equation and consider if it matches any 'standard templates' that you know

Hint 2: know that the equation matches that for which the method of separation of variables would be used

Hint 3: separate the variables, so that the 'dy' and 'y' terms are on one side, and the 'dx' and 'x' terms are on the other side

Hint 4: integrate each side, taking care to check that your answers - when differentiated - do indeed return you to the original integrand expressions

Hint 5: did you remember to check for the chain rule, which should introduce a fraction of (1/2) into one of your expressions?

Hint 6: also, don't forget the constant of integration!

Hint 7: use the values of x = 5 and y = 12 to fix the value of the constant

Hint 8: replace the constant with the value just calculated and then carefully use the laws of logarithms to merge the expressions

Hint 9: aim to present a final answer which starts: y = ...

Hint 10: and here is a video of the solution:

Question 8

8a) Hint 1: Use the standard process for Gaussian elimination, as the question instructs

8a) Hint 2: clearly state the coordinates of the intersection point, T(2, 4, -1)

8b) Hint 3: re-write the equation of line L₁ from cartesian form into parametric form

8b) Hint 4: extract expressions for x = ... , y = ... and z = ... each in terms of your chosen parameter letter

8b) Hint 5: substitute these three expressions into the equation of plane π₃

8b) Hint 6: expand out the brackets and solve for your parameter letter

8b) Hint 7: using your parameter value, substitute it back into your expressions for x, y and z to obtain the coordinates of P

8c) Hint 8: know that L₂ goes through T and P, means that we need the direction vector of TP

8c) Hint 9: using the coordinates of T and P from parts (a) and (b), calculate the vector TP, and factorise out a scalar multiple (we are only interested in the direction of this vector, not its magnitude)

8c) Hint 10: consider a general point on line L₂ that starts from either point T or point P, and has a parameter multiple of the direction vector TP

8c) Hint 11: write out each of the x = ..., y = ... and z = ... equations in terms of whatever parameter letter you just chose

Hint 12: and here is a video of the solution:

Paper 2

Question 1

Hint 1: look at the formula sheet for the template required when differentiating the inverse cosine function

Hint 2: recognise that you shall also need the chain rule, due to the (4x)

Hint 3: keep the brackets around the (4x), so that when you simplify part of the denominator, you obtain (4x)² = 16x²

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

Question 2

Hint 1: recognise that this situation will require implicit differentiation

Hint 2: to help see all of the embedded functions, consider re-writing the equation as: 2[y(x)]² + 4.x.exp[2y(x)] = 3x

Hint 3: notice that both the chain rule and the product rule will be needed here

Hint 4: differentiate implicity to obtain: 2.2[y(x)]¹.y'(x) + 4.exp[2y(x)] + 4.x.exp[2y(x)].2y'(x) = 3

Hint 5: now remove all of the extra brackets and notational choices to obtain: 2.2.y.y' + 4e^2y + 4x.e^2y.2y' = 3

Hint 6: rearrange terms to support factorising out y' from two expressions, with the remaining expressions on the other side of the equation

Hint 7: finish off by making y' the subject of the formala

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

Question 3

Hint 1: look at the denominator and see if there are any repeated factors

Hint 2: there aren't any repeated factors, so this is a straight-forward partial fractions process, aiming for A/(x - 1) + B/(x - 3) + C/(x + 5)

Hint 3: proceed with your usual partial fraction process

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

Question 4

4a) Hint 1: start off with 1118 = p × 416 + q, for some values of p and q

4a) Hint 2: continue with the Euclidean algorithm, line by line, until your final row has a remainder of zero

4a) Hint 3: the greatest common divisor will be the remainder of the penultimate row

4b) Hint 4: rearrange your penultimate row from part (a), to give 26 = 286 - 2 × 130

4b) Hint 5: rearrange the next row up from part (a) to give 130 = 416 - 1 × 286, and then use this to substitute in for 130

4b) Hint 6: repeat the rearrange and substitute process until you obtain a final line of 26 = a × 1118 + b × 416

4b) Hint 7: clearly state the final values of 'a' and 'b' (and note that 'b' will be negative)

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

Question 5

Hint 1: know that the equation y = x^cot(x) cannot be differentiated as it stands

Hint 2: take the natural logarithm of each side of the equation and use laws of logarithms to obtain ln(y) = cot(x).ln(x)

Hint 3: differentiate the entire equation implicity, with respect to x

Hint 4: know that the product rule will need to be used

Hint 5: rearrange to make y'(x) the subject

Hint 6: substitute y = x^cot(x) into your expression, so that y'(x) is only in terms of x

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

Question 6

6a) Hint 1: refer to page 2 of the question paper formula sheet for the Maclaurin Series expansion

6a) Hint 2: define f(x) = cos(3x) and calculate f'(x), f''(x), f'''(x) and f^iv(x)

6a) Hint 3: evaluate f(0), f'(0), f''(0), f'''(0) and f^iv(0)

6a) Hint 4: substitute all calculated values into the Maclaurin Series expansion

6b) Hint 5: from part (a), you have cos(3x) = (.....)

6b) Hint 6: so, cos²(3x) = [cos(3x)]² = (.....)² = (.....)(.....)

6b) Hint 7: multiply out the terms in the two brackets, discarding any that have a power that is greater than 4

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

Question 7

7a) Hint 1: know that to work out dy/dx, you will first need to calculate dx/dt and dy/dt

7a) Hint 2: using dy/dx = dy/dt × dt/dx, know that dt/dx = the reciprocal of dx/dt

7a) Hint 3: substitute in all expressions and simplify the answer as far as possible, ready for further differentiation in part (b)

7b) Hint 4: know that d²y/dx² = d/dx (dy/dx)

7b) Hint 5: know that d²y/dx² = d/dx (dy/dx) = d/dt (dy/dx) × dt/dx

7b) Hint 6: calculate the derivative, with respect to t, of your response to part (a), recognising that you'll need the quotient rule and the chain rule

7b) Hint 7: for the numerator, consider re-writing sec²(t) = [sec(t)]² to help you see both the 'inside' and 'outside' functions for the chain rule

7b) Hint 8: tidy up the answer slightly, but accept that it's not going to simplify very much

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

Question 8

8a) Hint 1: know that use of either pre- or post- multiplication by A is going to be the technqiue to use

8a) Hint 2: use either type of multiplication by A, to obtain A³ = 6A² - A

8a) Hint 3: substitute in A² = 6A - I

8a) Hint 4: expand brackets and simplify terms in A and I, clearly stating the values of p and q

8b) Hint 5: know that if a matrix is non-singular, it means that the matrix's inverse exists

8b) Hint 6: know that use of either pre- or post- multiplication by A^-1 is going to be the technique to use

8b) Hint 7: use either type of multiplication by A^-1, to obtain A = 6I - A^-1

8b) Hint 8: rearrange to make A^-1 the subject

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

Question 9

9a) Hint 1: know that displacement, s(t), will be the integral of the velocity, v(t), with respect to time

9a) Hint 2: after integrating, do not forget the constant of integration

9a) Hint 3: fix the value of the constant using the provided information that when t = 0, s(0) = 0

9a) Hint 4: present a final expression for s(t) in terms of t, and the evaluated constant

9b) Hint 5: know that the acceleration, a(t), is the derivative of the velocity, v(t), with respect to time

9b) Hint 6: after differentating, you ought to have a(t) = 2 + 5.e^5t

9b) Hint 7: know that the task is to show that this expression is always positive

9b) Hint 8: looking at the 'core part' of this expression, we can start by stating that e^t > 0, for all values of t, using our knowledge of the graph of the exponential function

9b) Hint 9: it therefore follows that e^5t > 0

9b) Hint 10: it therefore follows that 5.e^5t > 0

9b) Hint 11: it therefore follows that 2 + 5.e^5t > 2 > 0

9b) Hint 12: therefore we can conclude that a(t) > 0 for all values of t, and therefore it is always positive.

Hint 13: and here is a video of the solution:

Question 10

Hint 1: know that laws of sigma notation mean that Σ (r³ - 3r) = Σ r³ - Σ (3r)

Hint 2: know that Σ (3r) = 3 Σ r

Hint 3: use the formula provided on page 2 of the question paper to substitute in expressions for Σ r³ and Σ r

Hint 4: factorise out at least (n) and (n + 1) from the resulting expression

Hint 5: factorise out (1/4) as well to leave all other coefficients as integers

Hint 6: simplify and factorise the remaining quadratic

Hint 7: your final answer should look like: (1/4).n.(n + 1).(n - ...).(n + ...)

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

Question 11

11a) Hint 1: know that to use the suggested substitution, you will need to calculate du/dx

11a) Hint 2: rearrange your expression for du/dx to make dx the subject

11a) Hint 3: proceed with substituting the 2x² and dx terms only, as the remaining x in the integrand will simplify with the denominator of dx's substitution

11a) Hint 4: perform the integration, with respect to u

11a) Hint 5: remember to re-substitute the terms in u back into terms in x, for your final answer

11b) Hint 6: know that the volume of revolution is the definite integral, from 0 to 1, of πy², with respect to x.

11b) Hint 7: after substituting for y, and factorising out the 16π you should be left with the same integrand from part (a)

11b) Hint 8: use your answer from part (a) to move swiftly to the evaluation of the integral between the limits 0 and 1

11b) Hint 9: take care with the evalutation of each term, remembering that e⁰ is not equal to zero!

11b) Hint 10: given that the volume of revolution will be a positive quantity, it would be a good habit to present your answer with a positive coefficient, and with the terms inside the bracket ordered so that they clearly give a positive value.

Hint 11: and here is a video of the solution:

Question 12

Hint 1: proceed with your standard process for solving a non-homogeneous second order differential equation...

Hint 2: ... obtain the auxiliary equation that should give solutions of both 3 and 5

Hint 3: ... write down the complementary function y = A.e^3x + B.e^5x, for some constants A and B

Hint 4: ... for the particular integral, consider y = C.x² + D.x + E, for some constants C, D and E, as it is a generic quadratic function that matches the form of the right hand side of the original differential equation

Hint 5: calculate the first and second derivatives of the particular integral

Hint 6: substitute y, y' and y'' into the original differential equation, and then sequentially isolate terms in x², x and constants in order to compare coefficients and determine the values of C, D and E

Hint 7: write down the particular integral, using the newly found values of C, D and E

Hint 8: write down the general solution, y = complementary function + particular integral, that still has unknown constants A and B in it

Hint 9: use the given information of y(0) = 4 and y'(0) = 13 to determine the values of A and B by doing the following ....

Hint 10: substitute in x = 0 and y(0) = 4 into the general solution, y(x)

Hint 11: substitute in x = 0 and y'(0) = 13 into the derivative of the general solution, y'(x)

Hint 12: use the resulting simultaneous equations in A and B to fix their values

Hint 13: finally (!), state the specific solution for this differential equation, which has all coefficients now known

Hint 14: and here is a video of the solution:

Question 13

13a)i) Hint 1: know that from the question stem, we have u₂ = 100 and u₄ = 16

13a)i) Hint 2: know that u₄ = u₂ × r²

13a)i) Hint 3: substitute in values for u₄ and u₂

13a)i) Hint 4: solve for r, to obtain two possible solutions

13a)i) Hint 5: note from the question stem that we are told that the sequence only contains positive numbers, so this means that one of the two possible solutions for r can be discarded. So, discard it and give the reason for discarding it.

13a)ii) Hint 6: know that u_n = a.r^n-1

13a)ii) Hint 7: replace n with 2, u₂ with its value, and r with the value from part (a)(i) to obtain an equation in the first term, 'a'

13b) Hint 8: know that a sum to infinity exists if -1 < r < 1

13b) Hint 9: clearly communicate that the value of r from part (a)(i) satisfies this inequality

13c) Hint 10: know that S_∞ = a / (1 - r)

13c) Hint 11: substitute in the values of a and r, and carefully evaluate the expression

13d)i) Hint 12: consider the original sequence to be u₁ u₂ u₃ u₄

13d)i) Hint 13: consider the new sequence to be k.u₁ k.u₂ k.u₃ k.u₄

13d)i) Hint 14: so the new sequence's common ratio will be k.u₂ / (k.u₁)

13d)i) Hint 15: simplify the k's and see that you are left with u₂ / u₁, so decide what this means for the common ratio

13d)ii) Hint 16: so the new sequence's sum to infinity, S_∞ = k.u₂ / (1 - r)

13d)ii) Hint 17: which can be re-written as, S_∞ = k [u₂ / (1 - r)], so decide what this means for the sum to infinity

Hint 18: and here is a video of the solution:

Question 14

Hint 1: examine the format of the differential equation and consider if it matches any 'standard templates' that you know

Hint 2: recognise that this is a case for using an integrating factor

Hint 3: identify that the coefficient of y, P(x) = -2/x

Hint 4: calculate the integral of P(x) with respect to x, and rearrange it to have the form ln(....)

Hint 5: determine the integrating factor to be 'e' to the power of your recently calculated expression

Hint 6: simplify this expression, using the knowledge that the exponential function and the logarithmic functions are inverses of each other

Hint 7: multiply the original differential equation through by the integrating factor of 1/x²

Hint 8: for the left hand side, re-order the terms, to more easily see them as the derivative of product of (x^-2.y)

Hint 9: proceed with the integration of both sides of the equation to obtain an expression in the form x^-2.y = ....

Hint 10: make sure that the right hand side has a constant of integration introduced to it

Hint 11: multiply through by x², to obtain a solution of the form y = ....

Hint 12: and here is a video of the solution:

Question 15

Hint 1: proceed with the standard process for a proof by induction, that you should be fluent in

Hint 2: make sure that your solution ends with something like: 'by mathematical induction, as proposition with n = 1 is true, and that a proposition with n = k being true implies that a proposition with n = k + 1 will be true, then the proposition will be true for all n, where n ≥ 1 and n is an integer.'

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

Question 16

Hint 1: recognise that e^x will remain unchanged by the integration by parts process

Hint 2: recognise that sin(5x) will alternate between cos(5x) and then back to sin(5x)

Hint 3: therefore, this will be an integration by parts process that has to be performed twice, in order to return to an expression that contains the original integral

Hint 4: take care at each step of the process to employ the same 'u' and 'v' allocation of terms, to ensure the process works

Hint 5: and don't forget the constant of integration!

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

Question 17

Hint 1: recognise that 'piped in' is a positive, 'filling up', concept, so dV/dt = +6

Hint 2: recognise that 'leaking out' is a negative, 'emptying', concept, so dV/dt = - (1/10)√h

Hint 3: putting these together, gives dV/dt = 6 - (1/10)√h

Hint 4: know that we are aiming for calculating dh/dt when h = 400

Hint 5: start with V = (1/5) h³

Hint 6: re-write this to highlight the functions and their variables: V(t) = (1/5) [h(t)]³

Hint 7: implicitly differentiate this formula with respect to t, knowing that it will involve the chain rule.

Hint 8: this should give dV/dt = (1/5).3.[h(t)]².(dh/dt)

Hint 9: substitute in the expression for dV/dt ...

Hint 10: substitute in the value of h = 400 ...

Hint 11: ... and then carefully simplify and solve for dh/dt

Hint 12: and here is a video of the solution:

Question 18

18a)i) Hint 1: know that if z = x + iy, that z̄ = x - iy

18a)i) Hint 2: write out z̄ + iz in terms of x's and y's

18a)i) Hint 3: gather the real coefficients together, and the imaginary coefficients together, so that z̄ + iz = .... + i × ....

18a)ii) Hint 4: sketch a small argand diagram, with a right angled triangle whose horizontal length is (x - y) and whose vertical length is (x - y) ... these are the two components from the answer to (a)(i)

18a)ii) Hint 5: mark the angle of the triangle to be θ as this will be the argument that you are looking to work out

18a)ii) Hint 6: use right angled trigonometry, and the inverse of a function, to obtain θ (in radians)

18b) Hint 7: note that the argument of z̄ + iz when x < y is given as -3π/4

18b) Hint 8: this is not the only argument that it could be ... there could be integer multiples of 2π added to it and it will still generate the same complex number

18b) Hint 9: so, change the argument to -3π/4 + k.2π, where k is an integer

18b) Hint 10: knowing that a square root is the power of (1/2), use De Moivre's theorem on the complex number with its new argument

18b) Hint 11: the modulus of the answer will be r^1/2

18b) Hint 12: the argument of the answer will be (1/2) times the argument of -3π/4 + k.2π

18b) Hint 13: evaluate this argument for two values of k ... the most obvious to choose are k = 0, and k = 1

18b) Hint 14: clearly present the two complex number values

Hint 15: and here is a video of the solution: