Integrated response (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)).

Integrated response (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 (1 mark)

Apply the current TASC one-mark-item outcomes. A correct answer with or without working receives 1 mark. An incorrect answer with some incorrect working receives 0.5 mark.

Mark-by-mark checkpoints:

1. Obtains u_3=3 / 4 .

Authority-specific partial-mark outcomes:

• 0.5 marks: Incorrect answer with some incorrect working.

Acceptable alternatives: Credit an equivalent mathematically correct method only when it establishes all 1 checkpoints and reaches u_3=3 / 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: Obtains u_3=3 / 4 . Giving the final result without establishing: Obtains u_3=3 / 4 .

Part b (2 marks)

Award one mark for each numbered checkpoint below, to a maximum of 2. A correct answer without relevant working is capped at 1.5 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. Selects and substitutes into the finite geometric-series formula.
2. Simplifies the exact sum to 93 / 16 .

Acceptable alternatives: Credit an equivalent mathematically correct method only when it establishes all 2 checkpoints and reaches 93 / 16. 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: Selects and substitutes into the finite geometric-series formula. Giving the final result without establishing: Simplifies the exact sum to 93 / 16 .

Part c (2 marks)

Award one mark for each numbered checkpoint below, to a maximum of 2. A correct answer without relevant working is capped at 1.5 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. Establishes |r|<1 and therefore convergence.
2. Obtains S_infty=6 .

Acceptable alternatives: Credit an equivalent mathematically correct method only when it establishes all 2 checkpoints and reaches It converges because |r|=1 / 2<1, and S_infty=6. 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: Establishes |r|<1 and therefore convergence. Giving the final result without establishing: Obtains S_infty=6 .

Part d (3 marks)

Award one mark for each numbered checkpoint below, to a maximum of 3. 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. States the quantified varepsilon-N definition.
2. Chooses a valid integer threshold for the arbitrary positive epsilon.
3. Verifies |v_n-0|<varepsilon and concludes convergence.

Acceptable alternatives: Credit an equivalent mathematically correct method only when it establishes all 3 checkpoints and reaches For varepsilon>0, choose an integer N>varepsilon⁻¹-1. Then ngeq N gives |v_n|=1 / (n+1)<varepsilon, so v_nto0. 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: States the quantified varepsilon-N definition. Giving the final result without establishing: Verifies |v_n-0|<varepsilon and concludes convergence.

Integrated response (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.

Integrated response (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 (1 mark)

Apply the current TASC one-mark-item outcomes. A correct answer with or without working receives 1 mark. An incorrect answer with some incorrect working receives 0.5 mark.

Mark-by-mark checkpoints:

1. Obtains the exact sum 15 .

Authority-specific partial-mark outcomes:

• 0.5 marks: Incorrect answer with some incorrect working.

Acceptable alternatives: Credit an equivalent mathematically correct method only when it establishes all 1 checkpoints and reaches 15. 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: Obtains the exact sum 15 . Giving the final result without establishing: Obtains the exact sum 15 .

Part b (2 marks)

Award one mark for each numbered checkpoint below, to a maximum of 2. A correct answer without relevant working is capped at 1.5 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. Verifies the base case 1=1^2.
2. Uses the induction hypothesis and obtains (k+1)² before stating the conclusion.

Acceptable alternatives: Credit an equivalent mathematically correct method only when it establishes all 2 checkpoints and reaches The statement is true for n=1. Assuming it for n=k, adding 2k+1 gives k²+2k+1=(k+1)², so the result follows. 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: Verifies the base case 1=1^2. Giving the final result without establishing: Uses the induction hypothesis and obtains (k+1)² before stating the conclusion.

Part c (2 marks)

Award one mark for each numbered checkpoint below, to a maximum of 2. A correct answer without relevant working is capped at 1.5 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. Decomposes the summand as 1 / r-1 / (r+1).
2. Cancels the intermediate terms and obtains 4 / 5 .

Acceptable alternatives: Credit an equivalent mathematically correct method only when it establishes all 2 checkpoints and reaches 4 / 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: Decomposes the summand as 1 / r-1 / (r+1). Giving the final result without establishing: Cancels the intermediate terms and obtains 4 / 5 .

Part d (3 marks)

Award one mark for each numbered checkpoint below, to a maximum of 3. 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. Obtains the constant and linear terms 1+x.
2. Obtains the quadratic term x² / 2.
3. Obtains the cubic term x³ / 6 with an appropriate remainder.

Acceptable alternatives: Credit an equivalent mathematically correct method only when it establishes all 3 checkpoints and reaches 1+x+((x²) / (2))+((x³) / (6))+O(x⁴). 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: Obtains the constant and linear terms 1+x. Giving the final result without establishing: Obtains the cubic term x³ / 6 with an appropriate remainder.

Integrated response (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.

Integrated response (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 (1 mark)

Apply the current TASC one-mark-item outcomes. A correct answer with or without working receives 1 mark. An incorrect answer with some incorrect working receives 0.5 mark.

Mark-by-mark checkpoints:

1. Obtains det A_lambda=lambda-1 .

Authority-specific partial-mark outcomes:

• 0.5 marks: Incorrect answer with some incorrect working.

Acceptable alternatives: Credit an equivalent mathematically correct method only when it establishes all 1 checkpoints and reaches lambda-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: Obtains det A_lambda=lambda-1 . Giving the final result without establishing: Obtains det A_lambda=lambda-1 .

Part b (2 marks)

Award one mark for each numbered checkpoint below, to a maximum of 2. A correct answer without relevant working is capped at 1.5 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 the reciprocal factor 1 / (lambda-1).
2. Forms the correct adjugate matrix.

Acceptable alternatives: Credit an equivalent mathematically correct method only when it establishes all 2 checkpoints and reaches dfrac(1)(lambda-1)begin(bmatrix)lambda&-1-1&1end(bmatrix). 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 the reciprocal factor 1 / (lambda-1). Giving the final result without establishing: Forms the correct adjugate matrix.

Part c (2 marks)

Award one mark for each numbered checkpoint below, to a maximum of 2. A correct answer without relevant working is capped at 1.5 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. Applies R_2arrow R_2-R_1 to the full augmented row.
2. Obtains (lambda-1)y=mu .

Acceptable alternatives: Credit an equivalent mathematically correct method only when it establishes all 2 checkpoints and reaches Use R_2arrow R_2-R_1, giving (lambda-1)y=mu. 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: Applies R_2arrow R_2-R_1 to the full augmented row. Giving the final result without establishing: Obtains (lambda-1)y=mu .

Part d (3 marks)

Award one mark for each numbered checkpoint below, to a maximum of 3. 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 the unique intersection when the determinant is non-zero.
2. Identifies the coincident-line case and infinitely many solutions.
3. Identifies the distinct parallel-line case and no solution.

Acceptable alternatives: Credit an equivalent mathematically correct method only when it establishes all 3 checkpoints and reaches For lambdane1 the lines meet once. For lambda=1,mu=0 they coincide. For lambda=1,mune0 they are distinct and parallel. 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 unique intersection when the determinant is non-zero. Giving the final result without establishing: Identifies the distinct parallel-line case and no solution.

Integrated response (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).

Integrated response (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 (1 mark)

Apply the current TASC one-mark-item outcomes. A correct answer with or without working receives 1 mark. An incorrect answer with some incorrect working receives 0.5 mark.

Mark-by-mark checkpoints:

1. States a non-zero scalar multiple of (2,-1,2) .

Authority-specific partial-mark outcomes:

• 0.5 marks: Incorrect answer with some incorrect working.

Acceptable alternatives: Credit an equivalent mathematically correct method only when it establishes all 1 checkpoints and reaches For example, (2,-1,2). 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: States a non-zero scalar multiple of (2,-1,2) . Giving the final result without establishing: States a non-zero scalar multiple of (2,-1,2) .

Part b (2 marks)

Award one mark for each numbered checkpoint below, to a maximum of 2. A correct answer without relevant working is capped at 1.5 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. Obtains both diagonal entries correctly.
2. Obtains the signed off-diagonal entries correctly.

Acceptable alternatives: Credit an equivalent mathematically correct method only when it establishes all 2 checkpoints and reaches begin(bmatrix)cosalphacosbeta-sinalphasinbeta&-(cosalphasinbeta+sinalphacosbeta)sinalphacosbeta+cosalphasinbeta&cosalphacosbeta-sinalphasinbetaend(bmatrix). 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: Obtains both diagonal entries correctly. Giving the final result without establishing: Obtains the signed off-diagonal entries correctly.

Part c (2 marks)

Award one mark for each numbered checkpoint below, to a maximum of 2. A correct answer without relevant working is capped at 1.5 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. States the cosine addition identity.
2. States the sine addition identity.

Acceptable alternatives: Credit an equivalent mathematically correct method only when it establishes all 2 checkpoints and reaches cos(alpha+beta)=cosalphacosbeta-sinalphasinbeta and sin(alpha+beta)=sinalphacosbeta+cosalphasinbeta. 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: States the cosine addition identity. Giving the final result without establishing: States the sine addition identity.

Part d (3 marks)

Award one mark for each numbered checkpoint below, to a maximum of 3. 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. Finds the original triangle area.
2. Calculates |det S|=1.
3. Obtains the unchanged image area.

Acceptable alternatives: Credit an equivalent mathematically correct method only when it establishes all 3 checkpoints and reaches 1 square unit. 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: Finds the original triangle area. Giving the final result without establishing: Obtains the unchanged image area.

Integrated response (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.

Integrated response (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 (1 mark)

Apply the current TASC one-mark-item outcomes. A correct answer with or without working receives 1 mark. An incorrect answer with some incorrect working receives 0.5 mark.

Mark-by-mark checkpoints:

1. f'(x)=1e^(1x)

Authority-specific partial-mark outcomes:

• 0.5 marks: Incorrect answer with some incorrect working.

Acceptable alternatives: Credit an equivalent mathematically correct method only when it establishes all 1 checkpoints and reaches f'(x)=1e^(1x). 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: f'(x)=1e^(1x). Giving the final result without establishing: f'(x)=1e^(1x).

Part b (2 marks)

Award one mark for each numbered checkpoint below, to a maximum of 2. A correct answer without relevant working is capped at 1.5 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 implicit differentiation to obtain tangent slope -1.
2. States both the tangent and perpendicular normal equations.

Acceptable alternatives: Credit an equivalent mathematically correct method only when it establishes all 2 checkpoints and reaches The tangent is y-1=-(x-1), and the normal is y-1=x-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: Uses implicit differentiation to obtain tangent slope -1. Giving the final result without establishing: States both the tangent and perpendicular normal equations.

Part c (2 marks)

Award one mark for each numbered checkpoint below, to a maximum of 2. A correct answer without relevant working is capped at 1.5 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. Obtains g'(x)=3(x²-1).
2. Solves to obtain x=pm1.

Acceptable alternatives: Credit an equivalent mathematically correct method only when it establishes all 2 checkpoints and reaches x=pm1. 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: Obtains g'(x)=3(x²-1). Giving the final result without establishing: Solves to obtain x=pm1.

Part d (3 marks)

Award one mark for each numbered checkpoint below, to a maximum of 3. 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 the second derivative to classify every stationary point.
2. Solves the second-derivative condition for all possible inflection points.
3. Verifies the required concavity change and states every inflection.

Acceptable alternatives: Credit an equivalent mathematically correct method only when it establishes all 3 checkpoints and reaches There is a local maximum at x=-1, a local minimum at x=1, and an inflection at x=0, where g''(x)=6x changes sign. 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 the second derivative to classify every stationary point. Giving the final result without establishing: Verifies the required concavity change and states every inflection.

Integrated response (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.

Integrated response (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 (1 mark)

Apply the current TASC one-mark-item outcomes. A correct answer with or without working receives 1 mark. An incorrect answer with some incorrect working receives 0.5 mark.

Mark-by-mark checkpoints:

1. States int_0^2x(2-x),dx.

Authority-specific partial-mark outcomes:

• 0.5 marks: Incorrect answer with some incorrect working.

Acceptable alternatives: Credit an equivalent mathematically correct method only when it establishes all 1 checkpoints and reaches int_0^2x(2-x),dx. 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: States int_0^2x(2-x),dx. Giving the final result without establishing: States int_0^2x(2-x),dx.

Part b (2 marks)

Award one mark for each numbered checkpoint below, to a maximum of 2. A correct answer without relevant working is capped at 1.5 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 the area integrand correctly.
2. Applies the limits and obtains ((4) / (3)).

Acceptable alternatives: Credit an equivalent mathematically correct method only when it establishes all 2 checkpoints and reaches ((4) / (3)) square 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 the area integrand correctly. Giving the final result without establishing: Applies the limits and obtains ((4) / (3)).

Part c (2 marks)

Award one mark for each numbered checkpoint below, to a maximum of 2. A correct answer without relevant working is capped at 1.5 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 the disk method with the correct limits.
2. Squares and simplifies the radius function correctly.

Acceptable alternatives: Credit an equivalent mathematically correct method only when it establishes all 2 checkpoints and reaches piint_0²[x(2-x)]²,dx. 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 the disk method with the correct limits. Giving the final result without establishing: Squares and simplifies the radius function correctly.

Part d (3 marks)

Award one mark for each numbered checkpoint below, to a maximum of 3. 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. Finds a correct antiderivative including the factor of π.
2. Applies both limits without a sign or power error.
3. Simplifies to the exact volume ((16π) / (15)).

Acceptable alternatives: Credit an equivalent mathematically correct method only when it establishes all 3 checkpoints and reaches ((16π) / (15)) 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: Finds a correct antiderivative including the factor of π. Giving the final result without establishing: Simplifies to the exact volume ((16π) / (15)).

Integrated response (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.

Integrated response (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 (1 mark)

Apply the current TASC one-mark-item outcomes. A correct answer with or without working receives 1 mark. An incorrect answer with some incorrect working receives 0.5 mark.

Mark-by-mark checkpoints:

1. Obtains dfrac1x-dfrac1(x+1) .

Authority-specific partial-mark outcomes:

• 0.5 marks: Incorrect answer with some incorrect working.

Acceptable alternatives: Credit an equivalent mathematically correct method only when it establishes all 1 checkpoints and reaches dfrac1x-dfrac1(x+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: Obtains dfrac1x-dfrac1(x+1) . Giving the final result without establishing: Obtains dfrac1x-dfrac1(x+1) .

Part b (2 marks)

Award one mark for each numbered checkpoint below, to a maximum of 2. A correct answer without relevant working is capped at 1.5 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 each simple fraction with the correct coefficient and absolute value.
2. States ln|x|-ln|x+1|+C .

Acceptable alternatives: Credit an equivalent mathematically correct method only when it establishes all 2 checkpoints and reaches ln|x|-ln|x+1|+C. 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 each simple fraction with the correct coefficient and absolute value. Giving the final result without establishing: States ln|x|-ln|x+1|+C .

Part c (2 marks)

Award one mark for each numbered checkpoint below, to a maximum of 2. A correct answer without relevant working is capped at 1.5 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. Matches the integrand to the supplied inverse-sine form.
2. Includes the constant and obtains the exact antiderivative.

Acceptable alternatives: Credit an equivalent mathematically correct method only when it establishes all 2 checkpoints and reaches sin⁻¹(x / 2)+C. 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: Matches the integrand to the supplied inverse-sine form. Giving the final result without establishing: Includes the constant and obtains the exact antiderivative.

Part d (3 marks)

Award one mark for each numbered checkpoint below, to a maximum of 3. 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. Chooses u and v consistently.
2. Applies the integration-by-parts formula with the correct sign.
3. Simplifies to the stated antiderivative.

Acceptable alternatives: Credit an equivalent mathematically correct method only when it establishes all 3 checkpoints and reaches e^(2x)(x / 2-1 / 4)+C. 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: Chooses u and v consistently. Giving the final result without establishing: Simplifies to the stated antiderivative.

Integrated response (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.

Integrated response (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 (1 mark)

Apply the current TASC one-mark-item outcomes. A correct answer with or without working receives 1 mark. An incorrect answer with some incorrect working receives 0.5 mark.

Mark-by-mark checkpoints:

1. States a consistent substitution and differential.

Authority-specific partial-mark outcomes:

• 0.5 marks: Incorrect answer with some incorrect working.

Acceptable alternatives: Credit an equivalent mathematically correct method only when it establishes all 1 checkpoints and reaches u=2x+1, with dx=du / 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: States a consistent substitution and differential. Giving the final result without establishing: States a consistent substitution and differential.

Part b (2 marks)

Award one mark for each numbered checkpoint below, to a maximum of 2. A correct answer without relevant working is capped at 1.5 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. Finds the general antiderivative.
2. Uses the initial condition correctly.

Acceptable alternatives: Credit an equivalent mathematically correct method only when it establishes all 2 checkpoints and reaches y=x²+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: Finds the general antiderivative. Giving the final result without establishing: Uses the initial condition correctly.

Part c (2 marks)

Award one mark for each numbered checkpoint below, to a maximum of 2. A correct answer without relevant working is capped at 1.5 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. Makes the homogeneous substitution.
2. Integrates and back-substitutes.

Acceptable alternatives: Credit an equivalent mathematically correct method only when it establishes all 2 checkpoints and reaches y=x(ln x+C). 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: Makes the homogeneous substitution. Giving the final result without establishing: Integrates and back-substitutes.

Part d (3 marks)

Award one mark for each numbered checkpoint below, to a maximum of 3. 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. Obtains the exponential solution family.
2. Applies the initial value.
3. Solves the doubling equation.

Acceptable alternatives: Credit an equivalent mathematically correct method only when it establishes all 3 checkpoints and reaches P(t)=100e^t and t=ln2. 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: Obtains the exponential solution family. Giving the final result without establishing: Solves the doubling equation.

Integrated response (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.

Integrated response (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 (1 mark)

Apply the current TASC one-mark-item outcomes. A correct answer with or without working receives 1 mark. An incorrect answer with some incorrect working receives 0.5 mark.

Mark-by-mark checkpoints:

1. States overline z=sqrt3-i .

Authority-specific partial-mark outcomes:

• 0.5 marks: Incorrect answer with some incorrect working.

Acceptable alternatives: Credit an equivalent mathematically correct method only when it establishes all 1 checkpoints and reaches overline z=sqrt3-i. 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: States overline z=sqrt3-i . Giving the final result without establishing: States overline z=sqrt3-i .

Part b (2 marks)

Award one mark for each numbered checkpoint below, to a maximum of 2. A correct answer without relevant working is capped at 1.5 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. Finds the modulus.
2. Finds the principal argument and states the Euler form.

Acceptable alternatives: Credit an equivalent mathematically correct method only when it establishes all 2 checkpoints and reaches z=2e^(ipi / 6). 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: Finds the modulus. Giving the final result without establishing: Finds the principal argument and states the Euler form.

Part c (2 marks)

Award one mark for each numbered checkpoint below, to a maximum of 2. A correct answer without relevant working is capped at 1.5 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. Squares the modulus and doubles the argument.
2. Converts the result back to Cartesian form.

Acceptable alternatives: Credit an equivalent mathematically correct method only when it establishes all 2 checkpoints and reaches z²=2+2sqrt3i. 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: Squares the modulus and doubles the argument. Giving the final result without establishing: Converts the result back to Cartesian form.

Part d (3 marks)

Award one mark for each numbered checkpoint below, to a maximum of 3. 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 the conjugate root.
2. Uses the sum and product of the roots.
3. Obtains the real monic factor.

Acceptable alternatives: Credit an equivalent mathematically correct method only when it establishes all 3 checkpoints and reaches The paired root is sqrt3-i and the factor is x²-2sqrt3x+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: Identifies the conjugate root. Giving the final result without establishing: Obtains the real monic factor.

Integrated response (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.

Integrated response (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.

Part a (1 mark)

Apply the current TASC one-mark-item outcomes. A correct answer with or without working receives 1 mark. An incorrect answer with some incorrect working receives 0.5 mark.

Mark-by-mark checkpoints:

1. Obtains z³=1 while excluding z=1 .

Authority-specific partial-mark outcomes:

• 0.5 marks: Incorrect answer with some incorrect working.

Acceptable alternatives: Credit an equivalent mathematically correct method only when it establishes all 1 checkpoints and reaches z³=1, with zne1. 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: Obtains z³=1 while excluding z=1 . Giving the final result without establishing: Obtains z³=1 while excluding z=1 .

Part b (2 marks)

Award one mark for each numbered checkpoint below, to a maximum of 2. A correct answer without relevant working is capped at 1.5 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. Lists all three cube roots of unity before applying the exclusion.
2. States exactly the two non-real solutions without duplication.

Acceptable alternatives: Credit an equivalent mathematically correct method only when it establishes all 2 checkpoints and reaches z=operatorname(cis)(2π / 3) or z=operatorname(cis)(4π / 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: Lists all three cube roots of unity before applying the exclusion. Giving the final result without establishing: States exactly the two non-real solutions without duplication.

Part c (2 marks)

Award one mark for each numbered checkpoint below, to a maximum of 2. A correct answer without relevant working is capped at 1.5 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. Converts the modulus condition to the correct disk inequality.
2. Converts the imaginary-part condition to the correct half-plane.

Acceptable alternatives: Credit an equivalent mathematically correct method only when it establishes all 2 checkpoints and reaches (x-1)²+y^2leq4 and ygeq0. 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: Converts the modulus condition to the correct disk inequality. Giving the final result without establishing: Converts the imaginary-part condition to the correct half-plane.

Part d (3 marks)

Award one mark for each numbered checkpoint below, to a maximum of 3. 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 the correct upper half-disk.
2. States that the arc and diameter are included.
3. Obtains both diameter endpoints.

Acceptable alternatives: Credit an equivalent mathematically correct method only when it establishes all 3 checkpoints and reaches It is the closed upper half of the disk centred at (1,0) with radius 2. Both boundary pieces are included, and they meet at (-1,0) and (3,0). 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: Identifies the correct upper half-disk. Giving the final result without establishing: Obtains both diameter endpoints.