Part a (4 marks)

Award one mark for each numbered checkpoint below, to a maximum of 4. A correct answer without relevant working is capped at half marks.

Mark-by-mark checkpoints:

1. Introduces constants A and B in a partial-fraction decomposition.
2. Forms 1=A(r+2)+Br.
3. Solves A=frac12 and B=-frac12.
4. States the correct difference frac12(frac1r-frac1(r+2)).

Acceptable alternatives: Credit an equivalent mathematically correct method only when it establishes all 4 checkpoints and reaches ((1) / (r(r+2)))=frac12(frac1r-frac1(r+2)). Accept an algebraically equivalent exact form. Where a decimal is requested, accept the stated rounding or a value within + / -0.01 unless another tolerance is specified.

Do not credit by itself: Omitting or contradicting: Introduces constants A and B in a partial-fraction decomposition. Giving the final result without establishing: States the correct difference frac12(frac1r-frac1(r+2)).

Part b (6 marks)

Award one mark for each numbered checkpoint below, to a maximum of 6. A correct answer without relevant working is capped at half marks. Apply consequential marking when a correct later method consistently uses an earlier incorrect result, unless the error materially simplifies the task.

Mark-by-mark checkpoints:

1. Substitutes the difference from part (a) into the summation.
2. Writes enough expanded terms to show the two-step cancellation pattern.
3. Retains the uncancelled initial terms 1+frac12.
4. Retains the uncancelled final terms -frac1(n+1)-frac1(n+2).
5. Obtains S_n=frac34-((1) / (2(n+1)))-((1) / (2(n+2))).
6. Uses both remainder terms tending to zero to conclude the limit is frac34.

Acceptable alternatives: Credit an equivalent mathematically correct method only when it establishes all 6 checkpoints and reaches S_n=frac34-((1) / (2(n+1)))-((1) / (2(n+2))), so lim_(ntoinfty)S_n=frac34. Accept an algebraically equivalent exact form. Where a decimal is requested, accept the stated rounding or a value within + / -0.01 unless another tolerance is specified.

Do not credit by itself: Omitting or contradicting: Substitutes the difference from part (a) into the summation. Giving the final result without establishing: Uses both remainder terms tending to zero to conclude the limit is frac34.

Part a (4 marks)

Award one mark for each numbered checkpoint below, to a maximum of 4. A correct answer without relevant working is capped at half marks.

Mark-by-mark checkpoints:

1. Starts from the Maclaurin expansion of e^u.
2. Substitutes u=-2x.
3. Simplifies the quadratic coefficient to 2.
4. Simplifies the cubic coefficient to -frac43.

Acceptable alternatives: Credit an equivalent mathematically correct method only when it establishes all 4 checkpoints and reaches e^(-2x)approx1-2x+2x²-frac43x^3. Accept an algebraically equivalent exact form. Where a decimal is requested, accept the stated rounding or a value within + / -0.01 unless another tolerance is specified.

Do not credit by itself: Omitting or contradicting: Starts from the Maclaurin expansion of e^u. Giving the final result without establishing: Simplifies the cubic coefficient to -frac43.

Part b (6 marks)

Award one mark for each numbered checkpoint below, to a maximum of 6. A correct answer without relevant working is capped at half marks. Apply consequential marking when a correct later method consistently uses an earlier incorrect result, unless the error materially simplifies the task.

Mark-by-mark checkpoints:

1. Substitutes x=0.2 into every term of the cubic polynomial.
2. Evaluates the linear contribution as -0.4.
3. Evaluates the quadratic contribution as 0.08.
4. Evaluates the cubic contribution as -0.010666ldots.
5. Obtains 0.669333ldots and compares it with 0.670320ldots.
6. Calculates the absolute error and correctly identifies the approximation as an underestimate with error below 0.001.

Acceptable alternatives: Credit an equivalent mathematically correct method only when it establishes all 6 checkpoints and reaches The polynomial gives 0.669333ldots. Since e^(-0.4)=0.670320ldots, the absolute error is about 0.000987, an underestimate of less than 0.001. Accept an algebraically equivalent exact form. Where a decimal is requested, accept the stated rounding or a value within + / -0.01 unless another tolerance is specified.

Do not credit by itself: Omitting or contradicting: Substitutes x=0.2 into every term of the cubic polynomial. Giving the final result without establishing: Calculates the absolute error and correctly identifies the approximation as an underestimate with error below 0.001.

Part a (4 marks)

Award one mark for each numbered checkpoint below, to a maximum of 4. A correct answer without relevant working is capped at half marks.

Mark-by-mark checkpoints:

1. Writes the correct 3 × 4 augmented matrix.
2. States R_2arrow R_2-2R_1.
3. Obtains row 2 as [0,-3,1mid-7].
4. Obtains row 3 as [0,3,2mid4] using R_3arrow R_3+R_1.

Acceptable alternatives: Credit an equivalent mathematically correct method only when it establishes all 4 checkpoints and reaches [begin(array)(ccc|c)1&1&1&20&-3&1&-70&3&2&4end(array)]. Accept an algebraically equivalent exact form. Where a decimal is requested, accept the stated rounding or a value within + / -0.01 unless another tolerance is specified.

Do not credit by itself: Omitting or contradicting: Writes the correct 3 × 4 augmented matrix. Giving the final result without establishing: Obtains row 3 as [0,3,2mid4] using R_3arrow R_3+R_1.

Part b (6 marks)

Award one mark for each numbered checkpoint below, to a maximum of 6. A correct answer without relevant working is capped at half marks. Apply consequential marking when a correct later method consistently uses an earlier incorrect result, unless the error materially simplifies the task.

Mark-by-mark checkpoints:

1. Eliminates y from row 3 to obtain an equation in z.
2. Obtains z=-1.
3. Uses row 2 to obtain y=2.
4. Uses row 1 to obtain x=1.
5. Presents the reduced row-echelon matrix or equivalent complete row reduction.
6. States the unique ordered triple (1,2,-1).

Acceptable alternatives: Credit an equivalent mathematically correct method only when it establishes all 6 checkpoints and reaches The reduced matrix is [begin(array)(ccc|c)1&0&0&10&1&0&20&0&1&-1end(array)], so (x,y,z)=(1,2,-1). Accept an algebraically equivalent exact form. Where a decimal is requested, accept the stated rounding or a value within + / -0.01 unless another tolerance is specified.

Do not credit by itself: Omitting or contradicting: Eliminates y from row 3 to obtain an equation in z. Giving the final result without establishing: States the unique ordered triple (1,2,-1).

Part a (5 marks)

Award one mark for each numbered checkpoint below, to a maximum of 5. A correct answer without relevant working is capped at half marks.

Mark-by-mark checkpoints:

1. Identifies the composite as RS, not SR.
2. Forms the first row of RS correctly.
3. Forms the second row of RS correctly.
4. Multiplies Mbegin(bmatrix)12end(bmatrix).
5. Obtains the image P'=(-2,5).

Acceptable alternatives: Credit an equivalent mathematically correct method only when it establishes all 5 checkpoints and reaches M=RS=begin(bmatrix)0&-11&2end(bmatrix) and P'=(-2,5). Accept an algebraically equivalent exact form. Where a decimal is requested, accept the stated rounding or a value within + / -0.01 unless another tolerance is specified.

Do not credit by itself: Omitting or contradicting: Identifies the composite as RS, not SR. Giving the final result without establishing: Obtains the image P'=(-2,5).

Part b (5 marks)

Award one mark for each numbered checkpoint below, to a maximum of 5. A correct answer without relevant working is capped at half marks. Apply consequential marking when a correct later method consistently uses an earlier incorrect result, unless the error materially simplifies the task.

Mark-by-mark checkpoints:

1. Parameterises the original line as (t,t).
2. Maps (t,t) to (-t,3t).
3. Eliminates t to obtain y=-3x.
4. Calculates det M=1.
5. Interprets |det M|=1 as the area scale factor.

Acceptable alternatives: Credit an equivalent mathematically correct method only when it establishes all 5 checkpoints and reaches The image line is y=-3x, and the area scale factor is 1. Accept an algebraically equivalent exact form. Where a decimal is requested, accept the stated rounding or a value within + / -0.01 unless another tolerance is specified.

Do not credit by itself: Omitting or contradicting: Parameterises the original line as (t,t). Giving the final result without establishing: Interprets |det M|=1 as the area scale factor.

Part a (5 marks)

Award one mark for each numbered checkpoint below, to a maximum of 5. A correct answer without relevant working is capped at half marks.

Mark-by-mark checkpoints:

1. Differentiates to obtain f'(x)=1-frac4(x²).
2. Solves f'(x)=0 to obtain x=-2 and x=2.
3. Calculates the stationary-point coordinates (-2,-4) and (2,4).
4. Obtains f''(x)=frac8(x³).
5. Uses f''(-2)<0 and f''(2)>0 to classify the local maximum and local minimum.

Acceptable alternatives: Credit an equivalent mathematically correct method only when it establishes all 5 checkpoints and reaches The stationary points are (-2,-4), a local maximum, and (2,4), a local minimum. Accept an algebraically equivalent exact form. Where a decimal is requested, accept the stated rounding or a value within + / -0.01 unless another tolerance is specified.

Do not credit by itself: Omitting or contradicting: Differentiates to obtain f'(x)=1-frac4(x²). Giving the final result without establishing: Uses f''(-2)<0 and f''(2)>0 to classify the local maximum and local minimum.

Part b (5 marks)

Award one mark for each numbered checkpoint below, to a maximum of 5. A correct answer without relevant working is capped at half marks. Apply consequential marking when a correct later method consistently uses an earlier incorrect result, unless the error materially simplifies the task.

Mark-by-mark checkpoints:

1. Identifies x=0 as the vertical asymptote.
2. Uses f(x)-x=frac4xto0 as |x|toinfty to identify the oblique asymptote y=x.
3. Rearranges y=x+frac4x to the quadratic x²-yx+4=0.
4. Requires discriminant y²-16ge0 for real x.
5. States the range (-infty,-4]cup[4,infty), including the endpoint values attained at the stationary points.

Acceptable alternatives: Credit an equivalent mathematically correct method only when it establishes all 5 checkpoints and reaches The asymptotes are x=0 and y=x. The range is (-infty,-4]cup[4,infty). Accept an algebraically equivalent exact form. Where a decimal is requested, accept the stated rounding or a value within + / -0.01 unless another tolerance is specified.

Do not credit by itself: Omitting or contradicting: Identifies x=0 as the vertical asymptote. Giving the final result without establishing: States the range (-infty,-4]cup[4,infty), including the endpoint values attained at the stationary points.

Part a (4 marks)

Award one mark for each numbered checkpoint below, to a maximum of 4. A correct answer without relevant working is capped at half marks.

Mark-by-mark checkpoints:

1. Identifies the outer radius as 4.
2. Identifies the inner radius as y^2.
3. Uses the washer area π(R²-r²).
4. Uses the correct limits 0le yle2.

Acceptable alternatives: Credit an equivalent mathematically correct method only when it establishes all 4 checkpoints and reaches V=piint_0²(16-y⁴),dy. Accept an algebraically equivalent exact form. Where a decimal is requested, accept the stated rounding or a value within + / -0.01 unless another tolerance is specified.

Do not credit by itself: Omitting or contradicting: Identifies the outer radius as 4. Giving the final result without establishing: Uses the correct limits 0le yle2.

Part b (6 marks)

Award one mark for each numbered checkpoint below, to a maximum of 6. A correct answer without relevant working is capped at half marks. Apply consequential marking when a correct later method consistently uses an earlier incorrect result, unless the error materially simplifies the task.

Mark-by-mark checkpoints:

1. Integrates 16 to 16y.
2. Integrates y⁴ to ((y⁵) / (5)).
3. Applies the upper limit y=2.
4. Applies the lower limit y=0.
5. Simplifies 32-((32) / (5)) to ((128) / (5)).
6. States ((128π) / (5)) with cubic units.

Acceptable alternatives: Credit an equivalent mathematically correct method only when it establishes all 6 checkpoints and reaches V=((128π) / (5)) cubic units. Accept an algebraically equivalent exact form. Where a decimal is requested, accept the stated rounding or a value within + / -0.01 unless another tolerance is specified.

Do not credit by itself: Omitting or contradicting: Integrates 16 to 16y. Giving the final result without establishing: States ((128π) / (5)) with cubic units.

Part a (4 marks)

Award one mark for each numbered checkpoint below, to a maximum of 4. A correct answer without relevant working is capped at half marks.

Mark-by-mark checkpoints:

1. Writes ((A) / (x+1))+((B) / (x+2)).
2. Forms 5x+7=A(x+2)+B(x+1).
3. Solves A=2.
4. Solves B=3 and states the decomposition.

Acceptable alternatives: Credit an equivalent mathematically correct method only when it establishes all 4 checkpoints and reaches ((5x+7) / ((x+1)(x+2)))=frac2(x+1)+frac3(x+2). Accept an algebraically equivalent exact form. Where a decimal is requested, accept the stated rounding or a value within + / -0.01 unless another tolerance is specified.

Do not credit by itself: Omitting or contradicting: Writes ((A) / (x+1))+((B) / (x+2)). Giving the final result without establishing: Solves B=3 and states the decomposition.

Part b (6 marks)

Award one mark for each numbered checkpoint below, to a maximum of 6. A correct answer without relevant working is capped at half marks. Apply consequential marking when a correct later method consistently uses an earlier incorrect result, unless the error materially simplifies the task.

Mark-by-mark checkpoints:

1. Integrates frac2(x+1) as 2ln|x+1|.
2. Integrates frac3(x+2) as 3ln|x+2|.
3. Evaluates the antiderivative at x=1.
4. Evaluates the antiderivative at x=0.
5. Simplifies to 3ln3-ln2.
6. Combines logarithms to obtain ln(((27) / (2))).

Acceptable alternatives: Credit an equivalent mathematically correct method only when it establishes all 6 checkpoints and reaches I=3ln3-ln2=ln(((27) / (2))). Accept an algebraically equivalent exact form. Where a decimal is requested, accept the stated rounding or a value within + / -0.01 unless another tolerance is specified.

Do not credit by itself: Omitting or contradicting: Integrates frac2(x+1) as 2ln|x+1|. Giving the final result without establishing: Combines logarithms to obtain ln(((27) / (2))).

Part a (6 marks)

Award one mark for each numbered checkpoint below, to a maximum of 6. A correct answer without relevant working is capped at half marks.

Mark-by-mark checkpoints:

1. Sets y=vx.
2. Differentiates to obtain y'=v+x((dv) / (dx)).
3. Substitutes into the differential equation and simplifies to x((dv) / (dx))=1.
4. Integrates to obtain v=ln x+C.
5. Returns to y=x(ln x+C).
6. Uses y(1)=2 to obtain C=2 and states x>0.

Acceptable alternatives: Credit an equivalent mathematically correct method only when it establishes all 6 checkpoints and reaches y=x(ln x+2), for x>0. Accept an algebraically equivalent exact form. Where a decimal is requested, accept the stated rounding or a value within + / -0.01 unless another tolerance is specified.

Do not credit by itself: Omitting or contradicting: Sets y=vx. Giving the final result without establishing: Uses y(1)=2 to obtain C=2 and states x>0.

Part b (4 marks)

Award one mark for each numbered checkpoint below, to a maximum of 4. A correct answer without relevant working is capped at half marks. Apply consequential marking when a correct later method consistently uses an earlier incorrect result, unless the error materially simplifies the task.

Mark-by-mark checkpoints:

1. Uses ln e=1.
2. Obtains y(e)=e(1+2)=3e.
3. Substitutes x=e and y=3e into 1+((y) / (x)).
4. Obtains the gradient 4.

Acceptable alternatives: Credit an equivalent mathematically correct method only when it establishes all 4 checkpoints and reaches y(e)=3e and .((dy) / (dx))|_(x=e)=4. Accept an algebraically equivalent exact form. Where a decimal is requested, accept the stated rounding or a value within + / -0.01 unless another tolerance is specified.

Do not credit by itself: Omitting or contradicting: Uses ln e=1. Giving the final result without establishing: Obtains the gradient 4.

Part a (4 marks)

Award one mark for each numbered checkpoint below, to a maximum of 4. A correct answer without relevant working is capped at half marks.

Mark-by-mark checkpoints:

1. Writes -16 with modulus 16 and argument π+2kpi.
2. Takes the fourth root of the modulus to obtain 2.
3. Uses angles ((π+2kpi) / (4)).
4. Lists the four distinct roots for k=0,1,2,3.

Acceptable alternatives: Credit an equivalent mathematically correct method only when it establishes all 4 checkpoints and reaches z=2operatorname(cis)((π) / (4)),,2operatorname(cis)((3π) / (4)),,2operatorname(cis)((5π) / (4)),,2operatorname(cis)((7π) / (4)). Accept an algebraically equivalent exact form. Where a decimal is requested, accept the stated rounding or a value within + / -0.01 unless another tolerance is specified.

Do not credit by itself: Omitting or contradicting: Writes -16 with modulus 16 and argument π+2kpi. Giving the final result without establishing: Lists the four distinct roots for k=0,1,2,3.

Part b (6 marks)

Award one mark for each numbered checkpoint below, to a maximum of 6. A correct answer without relevant working is capped at half marks. Apply consequential marking when a correct later method consistently uses an earlier incorrect result, unless the error materially simplifies the task.

Mark-by-mark checkpoints:

1. Pairs the roots at angles ((π) / (4)) and ((7π) / (4)).
2. Forms the factor z²-2sqrt2,z+4.
3. Pairs the roots at angles ((3π) / (4)) and ((5π) / (4)).
4. Forms the factor z²+2sqrt2,z+4.
5. Multiplies or otherwise verifies cancellation of the z³ and z terms.
6. States the complete real quadratic factorisation.

Acceptable alternatives: Credit an equivalent mathematically correct method only when it establishes all 6 checkpoints and reaches z⁴+16=(z²-2sqrt2,z+4)(z²+2sqrt2,z+4). Accept an algebraically equivalent exact form. Where a decimal is requested, accept the stated rounding or a value within + / -0.01 unless another tolerance is specified.

Do not credit by itself: Omitting or contradicting: Pairs the roots at angles ((π) / (4)) and ((7π) / (4)). Giving the final result without establishing: States the complete real quadratic factorisation.

Part a (4 marks)

Award one mark for each numbered checkpoint below, to a maximum of 4. A correct answer without relevant working is capped at half marks.

Mark-by-mark checkpoints:

1. Identifies the circle centre 1+i.
2. Identifies the circle radius 2.
3. Identifies the two boundary rays at angles 0 and ((π) / (3)).
4. States that the circle and rays are included but the origin is excluded because operatorname(Arg)0 is undefined.

Acceptable alternatives: Credit an equivalent mathematically correct method only when it establishes all 4 checkpoints and reaches The boundaries are the included circle |z-(1+i)|=2 and the included rays operatorname(Arg)z=0 and operatorname(Arg)z=((π) / (3)), restricted to their common intersection; z=0 is excluded because its argument is undefined. Accept an algebraically equivalent exact form. Where a decimal is requested, accept the stated rounding or a value within + / -0.01 unless another tolerance is specified.

Do not credit by itself: Omitting or contradicting: Identifies the circle centre 1+i. Giving the final result without establishing: States that the circle and rays are included but the origin is excluded because operatorname(Arg)0 is undefined.

Part b (6 marks)

Award one mark for each numbered checkpoint below, to a maximum of 6. A correct answer without relevant working is capped at half marks. Apply consequential marking when a correct later method consistently uses an earlier incorrect result, unless the error materially simplifies the task.

Mark-by-mark checkpoints:

1. Sketches the disc with centre 1+i and radius 2.
2. Restricts the sketch to the sector between angles 0 and ((π) / (3)).
3. Checks 2+i against both conditions and includes it.
4. Checks 1+3i and excludes it because its argument exceeds ((π) / (3)).
5. Checks 3+i as lying on the circle and within the sector.
6. Uses solid boundaries and excludes the origin explicitly.

Acceptable alternatives: Credit an equivalent mathematically correct method only when it establishes all 6 checkpoints and reaches The region is the part of the closed disc lying in the closed sector 0lethetale((π) / (3)), excluding the origin. The points 2+i and 3+i belong to R; 1+3i does not. Accept an algebraically equivalent exact form. Where a decimal is requested, accept the stated rounding or a value within + / -0.01 unless another tolerance is specified.

Do not credit by itself: Omitting or contradicting: Sketches the disc with centre 1+i and radius 2. Giving the final result without establishing: Uses solid boundaries and excludes the origin explicitly.