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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 product rule and the chain rule

1b) Hint 2: recognise that the quotient rule is needed

1c) Hint 3: recognise that the chain rule is needed along with a standard differential from the Formula List

1c) Hint 4: substitute the value given into f'(x) and carefully simplify the expression

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

Question 2

2a) Hint 1: use a standard method to work out the determinant of the 3x3 matrix in terms of p

2a) Hint 2: equate the expression to the number 3 and solve for p

2b) Hint 3: carefully work out the product of a 3x3 matrix with a 3x2 matrix to obtain a 3x2 matrix, in terms of p and q

2c) Hint 4: know that inverse matrices only exist for square matrices i.e. n×n matrices

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

Question 3

3a) Hint 1: knowing f(x), work out f(-x) and determine if it is the same as either f(x) or -f(x) or neither

3b) Hint 2: know that the absolute value function, or modulus function, only returns positive values

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

Question 4

4a) Hint 1: use polynomial long division to extract the value of p from the rational function

4b) Hint 2: factorise the denominator into two distinct linear factors

4b) Hint 3: apply a standard method to decompose the algebraic fraction into partial fractions

4b) Hint 4: be sure to re-assemble the entire expression together that will be the sum of 3 terms

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

Question 5

5a) Hint 1: work out dy/dt and dx/dt

5a) Hint 2: know that dt/dx is the reciprocal of dx/dt

5a) Hint 3: use (dy/dx) = (dy/dt) × (dt/dx)

5b) Hint 4: use d²y/dx² = (d/dx)(dy/dx)

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

Question 6

Hint 1: know that deflating means reducing, which implies that dV/dt will be negative

Hint 2: recognise that we want dr/dt when r = 3

Hint 3: start from the formulae for the volume of a sphere (given) and write it as V(r) = …. to highlight that V is a function of r

Hint 4: using implicit differentiation, differentiate the volume formulae with respect to t

Hint 5: replace dV/dt with its value and rearrange to make dr/dt the subject, giving a formula that's in terms of r

Hint 6: replace r with 3 and evaluate dr/dt

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

Question 7

7a) Hint 1: know that sigma is a linear operator so that Σ (6r+13) = Σ 6r + Σ 13

7a) Hint 2: know that Σ 6r equals 6 Σ r

7a) Hint 3: know that Σ 13 equals 13 Σ 1 (and think carefully what Σ 1 is equal to …. it's not 1)

7b) Hint 4: recognise that to sum from (p+1) up to 20 involves (sum from 1 up to 20) and then subtracting the (sum from 1 up to p)

7b) Hint 5: be careful with your negatives and brackets when subtracting the Σ (6r+13) term

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

Question 8

Hint 1: obtain the auxiliary equation

Hint 2: solve the auxiliary quadratic equation to give two distinct roots

Hint 3: proceed with standard method for solving the differential equation, using the provided information to determine the two constants

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

Question 9

9a) Hint 1: use the binomial theorem to write an expression for the r th term of the expansion

9a) Hint 2: simplify each part of the expression so that you have a single d term and a single x term, both of which have a power in terms of r

9b) Hint 3: equate the power of x to be equal to -1, and solve for r

9b) Hint 4: substitute the value for r into the expression, as well as the number -70000, to then solve for d

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

Question 10

10a) Hint 1: consider writing the y terms as y(x) terms, to emphasise that y is a function of x

10a) Hint 2: use implicit differentiation to differentiate the equation, with respect to x

10a) Hint 3: rearrange the equation to make y'(x) the subject

10b) Hint 4: know that if a tangent has equation x = k, then it is a vertical line

10b) Hint 5: know that a vertical line has an undefined gradient

10b) Hint 6: look at your answer to part(a) and determine what would give rise to an undefined gradient

10b) Hint 7: rearrange the conditional equation to make y the subject

10b) Hint 8: substitute this expression for y back into the original equation of the curve, to obtain a quadratic in x

10b) Hint 9: solve for the two values of x and relate them back to the question in part (b)

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

Question 11

11a) Hint 1: pick a value for n that provides a counterexample. Try n = 1, then 2, then 3, then 4, etc

11b)i) Hint 2: know that if P ⇒ Q then the contrapositive is ¬Q ⇒ ¬P

11b)ii) Hint 3: know that an even number, n, can be written as 2k, where k is an integer

11b)ii) Hint 4: replace n with 2k in your contrapositive statement to establish that the quadratic expression is odd

11b)ii) Hint 5: write words to describe the logic behind why this means that the original statement in (b)(i) must be true

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

Question 12

Hint 1: know that base 11 has 'units' column worth 1, 'tens' column worth 11 and 'hundreds' column worth 121

Hint 2: convert 231 (base 11) into a base 10 number

Hint 3: know that base 7 will involve 1's, 7's and 49's

Hint 4: decompose your base 10 number into multiples of 49, multiples of 7 and multiples of 1

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

Question 13

Hint 1: recognise that you want to integrate both sides of the given equation

Hint 2: recognise that you shall have to use the method of separation of variables

Hint 3: use the given initial conditions to determine the constant of integration

Hint 4: remember of rearrange the equation to make V the subject

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

Question 14

Hint 1: verify the statement with a base case, where n = 1

Hint 2: write down what the statement would look like for the (n+1) case, as this helps you know what you are aiming for

Hint 3: proceed with a standard method of proof by induction

Hint 4: write in words at the end the clear logic behind how the base case and the induction step shown that the statement is true for all positive integers

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

Question 15

15a) Hint 1: replace the given expressions for x, y and z into each of the equations of the planes

15a) Hint 2: write words to explain why the results of this show that the two planes intersect on a line

15b) Hint 3: know that the acute angle between the line and the plane is calculated from using the angle between the normal vector of the plane and the direction vector of the line

15b) Hint 4: use the scalar product to obtain the angle between the normal vector of the plane and the direction vector of the line

15b) Hint 5: sketch a small diagram to help you then deduce the acute angle between the plane itself and the line

15c) Hint 6: recognise that L2 has a direction vector that's the same as the normal to plane π2

15c) Hint 7: write the equations for L1 and L2 in vector form

15c) Hint 8: equate the x, y and z components to obtain three equations, each in terms of two parameters

15c) Hint 9: solve the three equations to determine whether the lines meet, or not

15c) Hint 10: write words to commnunicate your conclusion

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

Question 16

16a) Hint 1: use the standard method of integration by parts TWICE

16b) Hint 2: know the standard formula for the volume of revolution of a function around the x-axis

16b) Hint 3: recognise that expanding (x-1)² gives x²-2x+1

16b) Hint 4: look to use your answer from part (a)

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

Question 17

17a) Hint 1: know that a geometric sequence has a common ratio, between any two consecutive terms

17b)i) Hint 2: know that a geometric series has a sum to infinity is -1 < r < 1

17b)ii) Hint 3: know the formulae for the sum to infinity of a geoemtric series

17c)i) Hint 4: create an equation in terms of x from (u2/u1) = (u3/u2)

17c)ii) Hint 5: solve the resulting quadratic equation in x to obtain two distinct values for x

17c)ii) Hint 6: after recognising one of the values for x, use the other value to determine the value for r by first working out u1, u2 and u3

17c)iii) Hint 7: write out the first few terms of this new sequence to see how they will group together if the sequence continues forever

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

Question 18

18a)i) Hint 1: cartesian form is like x + iy

18a)ii) Hint 2: polar form requires a magnitude, r, and an argument, θ

18a)ii) Hint 3: you will need to use exact value triangles to obtain the exact value of θ in radians, measured anti-clockwise from the positive Real axis

18b)i) Hint 4: use the answer from part (a)(ii) to replace w in z³ = w with the polar form

18b)i) Hint 5: take the cube root of both sides of this equation

18b)i) Hint 6: use de Moivre's theorem with the power of (1/3) to change the modulus and argument accordingly

18b)ii) Hint 7: compare with the given form of z₁, to read off the values of k and m

18b)ii) Hint 8: know that the roots will be regularly spaced around a circle of radius 2 in the complex plane

18b)ii) Hint 9: know that three roots will be spaced 2π/3 radians apart around the circle

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

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