2U HSC Questions - Geometric Applications of Calculus (Selected questions 2012-2016 ) 2018 PDF

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very helpful- for practise and examination preparation material. Preferably done timed and as quizzes. Helpful for those studying any maths topics....

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WORKSHEET - Mathematics (Advanced)

Questions 1. Geometry and Calculus, 2UA 2013 HSC 8 MC

10. Geometrical Applications of Differentiation Curve Sketching and The Primitive Function Maxima and Minima

The diagram shows points

,

,

At which point is

and

and

on the graph

.

Teacher: Derek Stokes Exam Equivalent Time: 66 minutes (based on HSC allocation of 1.5 minutes approx. per mark)

?

(A) (B) HISTORICAL CONTRIBUTION T10 Geometry and Calculus is the single largest topic within the Mathematics course, contributing 15.0% to each paper, on average, in the last 10 years. This topic has been split into three sub-categories: 1-Maxima and Minima (5.3%), 2-Curve Sketching and The Primitive Function (7.6%), and 3-Tangents and Normals (2.1%). 2017 HSC ANALYSIS - What to expect and common pitfalls Maxima and Minima (5.3%) problems appear on every examination paper and have produced sub-50% mean marks every year except for 2016. Area/Volume themes have dominated in recent times, being asked in 6 of the last 7 years. Other themes have been distance (2013) and a cost equation (2009). Curve Sketching and The Primitive Function (7.6%). The most popular curve is easily the cubic, asked 7 times since 2003, including 2015. Sketches of polynomials of degree 4 are the next most popular, asked in 2007, 2008, 2012 and 2016. Tangents and Normals (2.1%) questions make up the smallest sub-category within Topic 10 and are generally a great topic area for high scoring. Examined in 7 of the last 8 years.

(C) (D)

2. Geometry and Calculus, 2UA 2013 HSC 12a The cubic Show that

has a point of inflexion at .

.

(2 marks)

3. Geometry and Calculus, 2UA 2015 HSC 13c Consider the curve

.

(i) Find the stationary points and determine their nature. (ii)

Given that the point

(4 marks)

lies on the curve, prove that there is a point of

inflexion at . (2 marks) (iii) Sketch the curve, labelling the stationary points, point of inflexion and -intercept. marks)

(2

4. Geometry and Calculus, 2UA 2016 HSC 14c A farmer wishes to make a rectangular enclosure of area m². She uses an existing straight boundary as one side of the enclosure. She uses wire fencing for the remaining three sides and also to divide the enclosure into four equal rectangular areas of width m as shown.

5. Geometry and Calculus, 2UA 2014 HSC 16c The diagram shows a window consisting of two sections. The top section is a semicircle of diameter m. The bottom section is a rectangle of width m and height m. The entire frame of the window, including the piece that separates the two sections, is made using m of thin metal.

i. Show that the total length, m, of the wire fencing is given by (1 mark)

ii. Find the minimum length of wire fencing required, showing why this is the minimum length. (3 marks)

The semicircular section is made of coloured glass and the rectangular section is made of clear glass. Under test conditions the amount of light coming through one square metre of the coloured glass is unit and the amount of light coming through one square metre of the clear glass is units. The total amount of light coming through the window under test conditions is (i) Show that (ii)

Show that

.

units.

(2 marks)

.

(2 marks)

(iii) Find the values of and that maximise the amount of light coming through the window under test conditions. (3 marks)

6. Geometry and Calculus, 2UA 2014 HSC 14e The diagram shows the graph of a function

.

The graph has a horizontal point of inflexion at turning point at .

Sketch the graph of the derivative

.

8. Geometry and Calculus, 2UA 2013 HSC 14b

, a point of inflexion at

and a maximum

Two straight roads meet at at an angle of . At time car leaves road, and car is 100km from on the other road. Car travels away from speed of km/h, and car travels towards at a speed of km/h.

The distance between the cars at time

(3 marks)

hours is

(i) Show that (ii) Find the minimum distance between the cars.

7. Geometry and Calculus, 2UA 2012 HSC 14a A function is given by

.

(i) Find the coordinates of the stationary points of

and determine their nature.

(3

marks)

(ii) Hence, sketch the graph

showing the stationary points.

(iii) For what values of

is the function increasing?

(iv) For what values of

will

(1 mark)

(2 marks)

(1 mark)

have no solution?

km. .

(2 marks)

(3 marks)

on one at a

9. Geometry and Calculus, 2UA 2015 HSC 16c The diagram shows a cylinder of radius and height height , where and are constants.

Worked Solutions

inscribed in a cone of radius

and

1. Geometry and Calculus, 2UA 2013 HSC 8 MC ♦ Mean mark 48%

2. Geometry and Calculus, 2UA 2013 HSC 12a

The volume of a cone of radius

and height

The volume of a cylinder of radius (i) Show that the volume,

is

and height

is

, of the cylinder can be written as (3 marks)

(ii) By considering the inscribed cylinder of maximum volume, show that the volume of any inscribed cylinder does not exceed

of the volume of the cone.

(4 marks)

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3. Geometry and Calculus, 2UA 2015 HSC 13c (i)

(iii)

(ii)

4. Geometry and Calculus, 2UA 2016 HSC 14c

5. Geometry and Calculus, 2UA 2014 HSC 16c

i.

(i)

ii.

(ii)

♦ Mean mark 35%

6. Geometry and Calculus, 2UA 2014 HSC 14e

(iii)

♦ Mean mark 38% COMMENT: A sanity check for your answer could be to compare your answers to the perimeter restriction of 10m.

7. Geometry and Calculus, 2UA 2012 HSC 14a (i)

8. Geometry and Calculus, 2UA 2013 HSC 14b (i) ♦♦ Mean mark 26%

(ii)

(ii) ♦♦ Mean mark 27% ALGEBRA TIP: Finding the derivative of (rather than making the subject), makes calculations much easier. ENSURE you apply the test to confirm a minimum.

(iii)

(iv)

♦ Mean mark 42% MARKER'S COMMENT: Be careful to use the correct inequality signs, and not carelessly include or by mistake.

♦♦♦ Mean mark 12%. This part was the second most poorly answered question in the 2012 exam.

9. Geometry and Calculus, 2UA 2015 HSC 16c (i)

♦♦ Mean mark 16%.

(ii)

♦♦ Mean mark 21%.

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