Sample HSC Paper

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Maths In Focus Mathematics Extension 1 HSC Course

Sample Examination Papers This chapter contains two sample Mathematics HSC papers and two sample Extension 1 papers. They are designed to give you some practice in working through an examination. These papers contain work from the Year 11 Preliminary Course, as up to 20% of questions are allowed to contain this work. You may need to revise this work before you try these papers. If you can, set yourself a time limit and work under examination conditions. Give yourself 3 hours to do each Mathematics paper and 2 hours for each Extension 1 paper. Try not to look up any notes while working through these papers.

MATHEMATICS—PAPER 1

(e) Two cities are 3295 km apart. Write this number correct to 2 significant figures. (f) Solve | x + 3| < 7. QUESTION 2 (a)

(i) Copy the diagram into your examination booklet. (ii) Show that AC = CE. (iii) Find the length of DE. (b)

Time allowed—Three hours (Plus 5 minutes’ reading time)

DIRECTIONS TO CANDIDATES • • •

All questions may be attempted. All questions are of equal value. All necessary working should be shown in every question, as marks are awarded for this. Badly arranged or careless work may not receive marks.

QUESTION 1 (a) Find, correct to 2 decimal places, the 28.3 . 15.7 ´ 2.4 (b) Factorise 3x2 - 11x + 6. x x−1 (c) Solve the equation − = 5. 2 3 (d) The volume of a cone is given by 1 V = π r 2. If a cone has volume 12 m3, 3 find its radius correct to 2 decimal places. value of

(i) Copy the diagram into your examination booklet. (ii) Use the sine rule to calculate MO correct to 1 decimal place. (iii) Find MP to the nearest metre. QUESTION 3 (a) Differentiate (i) x + 5x 3 + 1 1

(ii) 3 ln x + x (iii) (2x + 3)5 (b) Find (i) (ii)

# (x − e − x ) dx π #0 (sin θ + 1) dθ

SAMPLE EXAMINATION PAPERS

(c) (i)

Rationalise the denominator of

5 . 2−1 (ii) Find integers a and b such that 5 = a + b. 2−1 QUESTION 4 (a) (i) Plot points A(3, 2) and B(-1, 5) on a number plane. (ii) Show that line AB has equation 3x + 4y - 17 = 0. (iii) Find the perpendicular distance from the origin to line AB. (iv) Find the area of triangle OAB where O is the origin. (b)

QUESTION 6 2 (a) A plant has a probability of of 7 producing white flowers. If 3 plants are chosen at random, find the probability that (i) 2 plants will produce white flowers (ii) at least 1 plant will produce white flowers. 3 (b) Solve sin x = for 0° ≤ x ≤ 360°. 2 (c) A particle P moves so that it is initially at rest at the origin, and its acceleration is given by a = 6t + 4 cms-2. (i) Find the velocity of P when t = 5. (ii) Find the displacement of P when t = 2. QUESTION 7 (a)

Find the length of side BC using the cosine rule and give your answer correct to the nearest centimetre. QUESTION 5 (a) Kate earns $24 600 p.a. In the following year she receives a pay increase of $800. Each year after that, she receives a pay increase of $800. (i) What percentage increase did Kate receive in the first year? (ii) What will be Kate’s salary after 12 years? (iii) What will Kate’s total earnings be over the 12 years? (b) A function is given by y = x3 - 3x2 - 9x + 2. (i) Find the coordinates of the stationary points and determine their nature. (ii) Find the point of inflexion. (iii) Draw a sketch of the function.

ABCD is a parallelogram with AD = 1, AB = 2 and +ADX = 60°. AX is drawn so that it bisects DC. (i) Copy the diagram into your examination booklet. (ii) Show that ADX is an equilateral triangle. (iii) Show that AXB is a right-angled triangle. (iv) Find the exact length of side BX. (b) On a number plane, shade in the region given by the two conditions y ³ x2 and x + y £ 3. (c) The number of people P with chickenpox is increasing but the rate at which the disease is spreading is slowing down over t weeks. (i) Sketch a graph showing this information. dP d2 P and 2 for this (ii) Describe dt dt information.

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QUESTION 8 (a) Consider the function given by y = log10 x. (i) Copy and complete the table, to 3 decimal places, in your examination booklet. x

1

y

0

2

3

4

(b) Kim invests $500 at the beginning of each year in a superannuation fund. The money earns 12% interest per annum. If she starts the fund at the beginning of 1996, what will the fund be worth at the end of 2025?

5

(ii) Apply the trapezoidal rule with 4 subintervals to find an approximation, to 2 decimal 5 places, of # log10 x dx.

QUESTION 10 (a)

1

(b) A population of mice at time t in weeks is given by P = P0ekt , where k is a constant and P0 is the population when t = 0. (i) Given that 20 mice increase to 100 after 6 weeks, calculate the value of k, to 3 decimal places. (ii) How many mice will there be after 10 weeks? (iii) After how many weeks will there be 500 mice? QUESTION 9 (a)

The surface area of a cylinder is given by the formula S = 2π r (r + h) . A cylinder is to have a surface area of 160 cm2. (i) Show that the volume is given by V = 80r − π r3. (ii) Find the value of r, to 2 decimal places, that gives the maximum volume. (iii) Find the maximum volume, to 1 decimal place.

(i) Find the exact point of intersection of the curve y = ex and the line y = 4. (ii) Find the exact shaded area enclosed between the curve and the line. (b) (i) The quadratic equation x2 + (k - 1)x + k = 0 has real and equal roots. Find the exact values of k in simplest surd form. (ii) If k = 5, show that x2 + (k - 1)x + k > 0 for all x. (c) A parabola has equation y = x2 + 2px + q. (i) Show that the coordinates of its vertex are (-p, q - p2). (ii) Find the coordinates of its focus. (iii) Find the distance between the point (m, 3m2 + q) and point P vertically below it on the parabola x2 = 8y when q > 0.

SAMPLE EXAMINATION PAPERS

y (m, 3m2 + q)

x2 = 8y

P x y = x2 + 2px + q

(iv) Find the minimum distance between these two points when m + q = 5.

MATHEMATICS—PAPER 2 Time allowed—Three hours (Plus 5 minutes’ reading time) QUESTION 1 (a) Solve | x − 3 | = 5. (b) If f (x) = 5 - x2, find x when f (x) = -4. 5π (c) Find the exact value of sin . 6 3 2 (d) Factorise fully a - 2a - 4a + 8. (e) Simplify 2 24 − 150 . (f) Evaluate loga 50 if loga 5 = 1.3 and loga 2 = 0.43. (g) Find the midpoint of (-3, 4) and (0, -2). QUESTION 2 1 1 Plot points A e 5, − 1 o , B e 1, 1 o and 2 2 C(-4, -1) on a number plane. (a) Show that the equation of line AB is given by 3x + 4y - 9 = 0. (b) Find the equation of the straight line l through C that is perpendicular to AB. (c) Find the point of intersection P of the two lines, AB and l. (d) Find the area of triangle ABC. (e) Find the coordinates of D such that ADCB is a rectangle.

QUESTION 3 (a) Differentiate (i) x cos x (ii) e5x (iii) loge (2x2 − 1) (b) Find the indefinite integral (primitive function) of (i) (3x - 2)4 (ii) 3 sin 2x (c) Evaluate

3

#0

(ex − e− x ) dx. d2y

= 18x − 6. If dx 2 there is a stationary point at (2, −1), find the equation of the curve. (d) For a certain curve,

QUESTION 4 (a) Mark plays a game of chance that has 2 a probability of of winning the game. 5 Hong plays a different game in which 3 there is a probability of of winning. 8 Find the probability that (i) both Mark and Hong win their games (ii) one of them wins the game (iii) at least one of them wins the game. (b) (i) On a number plane, shade the region where y ³ 0, x £ 3 and y £ x + 2. (ii) Find the area of this shaded region. (iii) This area is rotated about the x-axis. Find the volume of the solid formed. (c) (i) Sketch the graph of y = 2x − 1 on a number plane. (ii) Hence solve 2x − 1 < 3. QUESTION 5 (a) For the curve y = 2x3 - 9x2 + 12x - 7 (i) find any stationary points on the curve and determine their nature (ii) find any points of inflexion (iii) sketch the curve in the domain -3 £ x £ 3.

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Maths In Focus Mathematics Extension 1 HSC Course

(b) A plane leaves Bankstown airport and flies for 850 km on a bearing of 100°. It then turns and flies for 1200 km on a bearing of 040°. (i) Draw a diagram showing this information. (ii) How far from the airport is the plane, to the nearest km? (c) Simplify (cosec θ + cot θ )(cosec θ − cot θ ). QUESTION 6 (a) (i) Find the equation of the normal to the curve y = x2 at point P(-2, 4). (ii) This normal cuts the parabola again at point Q. Find the coordinates of Q. (iii) Find the shaded area enclosed between the parabola and the normal, to 3 significant figures. (b) The graph below shows the displacement of a particle over time t.

(ii) Differentiate ln (cos x).

#

π 4

tan x dx to (iii) Hence evaluate 0 2 decimal places. (b) A bridge is 40 metres long, held by wires at A and C with angles of elevation of 47° and 23° as shown.

(i) Find the length of AD to 3 significant figures. (ii) Find the height of the bridge BD to 1 decimal place. (c) Triangle BEC is isosceles with BC = CE. Also +BEC = 50°, +ABE = 130°, and +ADC = 80°.

Prove that ABCD is a parallelogram.

(i) When is the particle at the origin? (ii) When is the particle at rest? (c) The temperature T of a metal is cooling exponentially over t minutes. It cools down from 97°C to 84°C after 5 minutes. Find (i) the temperature after 15 minutes (ii) when it cools down to 20°C. QUESTION 7 (a) (i) Find the area bounded by the curve y = tan x, the x-axis and the lines π x = 0 and x = , by using Simpson’s rule 4 with 5 function values (to 2 decimal places).

QUESTION 8 (a) (i) Sketch the curve y = 3 sin 2x for 0 ≤ x ≤ 2π . (ii) On the same set of axes, sketch x y= . 2 (iii) How many roots does the equation x 3 sin 2x = have in this domain? 2 (b) Solve 2 sin x - 1 = 0 for 0° £ x £ 360°. (c) If logx 3 = p and logx 2 = q, write in terms of p and q (i) logx 12 (ii) logx 2x (d) A stack of oranges has 1 orange at the top, 3 in the next row down, then each row has 2 more oranges than the previous one.

SAMPLE EXAMINATION PAPERS

(i) How many oranges are in the 12th row? (ii) If there are 289 oranges stacked altogether, how many rows are there?

(iii) Find the value of k if the tangent has an x-intercept of 2. (e) Find the area of the sector to 1 decimal place.

QUESTION 9 (a) The diagram below shows the graph of a function y = f (x).

QUESTION 10 (a) The graph of y = (x + 2)2 is drawn below. Copy the graph into your writing booklet.

(i) Copy this diagram into your writing booklet. (ii) On the same set of axes, draw a sketch of the derivative f l(x) of the function. (b) ‘A bag contains yellow, blue and white marbles. Therefore if I choose one marble at random from the bag, the 1 probability that it is blue is .’ 3 Is this statement true or false? Explain why in no more than one sentence. (c) Solve the equation 2 ln x = ln (2x + 3). (d)

(i) Shade the region bounded by the curve, the x-axis and the line x = 1. (ii) This area is rotated about the x-axis. Find the volume of the solid of revolution formed. (b) The accountant at Acme Business Solutions calculated that the hourly cost of running a business car is s2 + 7500 cents where s is the average speed of the car. The car travels on a 3000 km journey. (i) Show that the cost of the journey is given by C = 3000 b s +

The diagram shows the graph of y = ex and the tangent to the curve at x = k. (i) Find the gradient of the tangent at x = k. (ii) Find the equation of the tangent at x = k.

7500 s l.

(ii) Find the speed that minimises the cost of the journey. (iii) Find the cost of the trip to the nearest dollar.

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Maths In Focus Mathematics Extension 1 HSC Course

EXTENSION 1—PAPER 1 Time allowed—Two hours (Plus 5 minutes’ reading time) QUESTION 1 (a) Solve for x:

3 ≤ 1. x+5 (b) Find the coordinates of the point that divides the interval AB with A(-2, 8) and B(3, 3) in the ratio 3:2. 2 dx . (c) Find the exact value of # 0 4 − x2 1 . (d) Differentiate 1 − x2

(e) Find # x2 (x3 − 5) 5 dx, using the substitution u = x3 - 5. QUESTION 2 (a)

In the diagram A, B, C and E are points on the circle with centre O. AE is produced to D such that BE = DE. (i) Show that +BEO = 2+CDE. π (ii) Show that +BAO = − +BEO. 2 (iii) Tangent DF is drawn to meet the circle at F. If BE = 5 cm and the circle has radius 3.5 cm, find the length of DF in exact form. (b) A root of ex - x2 = 0 lies near x = -0.5. Use Newton’s method to find a second approximation to the root, correct to 2 decimal places (use x = -0.5 as your first approximation). (c) Consider the function f (x) = 2 sin− 1 x. 1 (i) Find the exact value of f e o. 2 (ii) Sketch y = f(x).

(iii) Find the equation of the tangent to the curve at the point where 1 x= . 2 QUESTION 3 (a)

ABCD is a triangular pyramid with AB = AC = AD = 10 cm, BD = 16 cm, CD = 5 cm and +BCA = 55°. (i) Calculate the length of BC, to 1 decimal place. (ii) Find +BCD, to the nearest minute. (b) Find all solutions of the equation cos2 x = cos 2x for 0 ≤ x ≤ 2π . (c) A group of 6 girls and 5 boys are to be arranged in a straight line. Find how many ways they can be arranged: (i) with no restrictions on the order (ii) if boys and girls are to alternate (iii) if 2 particular girls are to stand together. QUESTION 4 (a) A particle is projected from the origin O with velocity 15 ms-1, at an angle of θ.

SAMPLE EXAMINATION PAPERS

(i) Neglecting air resistance and assuming acceleration due to gravity is 10 ms-2, show that the equations for the horizontal (x) and vertical ( y) components of the particle’s displacement from O after t seconds are given by x = 15t cos θ and y = − 5t2 + 15t sin θ. (ii) Show that the Cartesian equation for displacement is given by − x2 y= sec2 θ + x tan θ . 45 (iii) The particle just clears an object 2 m high standing out 5 m from the origin. Find 2 possible values of θ for this to happen. (b) The probability that a piece of space junk will crash in Australia is estimated at 0.01. If 18 pieces of space junk are due to crash, find the probability that 10 of them will crash in Australia. Leave your answer in index form. QUESTION 5 (a) Two points, P (2ap, ap2) and Q (2aq, aq2), lie on the parabola x2 = 4ay. (i) Find the equation of the tangent (l) to the parabola at Q. (ii) Derive the equation of chord PQ and show that pq = -1 if PQ is a focal chord. (iii) Find the acute angle between tangent l and chord PQ if p = 3 and q = -0.2. (b) Use mathematical induction to prove that for all integers n with n ³ 1, 12 + 22 + 32 + . . . + n2 =

1 n(n + 1) (2n + 1) . 6

QUESTION 6 (a) The acceleration of a particle moving .. 900 in a straight line is given by x = − 3 , x where x metres is the displacement from the origin after t seconds. Initially the particle is 10 m to the right of the origin, with velocity 3 ms-1.

(i) Find an equation for the velocity of the particle. (ii) Find the time when the particle is 100 m from the origin. (b) The rate at which a body cools in air is proportional to the difference between the constant air temperature, C, and its own temperature, T. This can be expressed by the differential equation. dT = − k(T − C), where t is time in dt hours and k is a constant. (i) Show that T = C + Ae − kt is a solution of the differential equation, where A is a constant. (ii) A heated piece of metal cools from 90°C to 70°C in 1 hour. The air temperature C is 25°C. Find the temperature (to the nearest degree) of the body after another 2 hours. QUESTION 7 (a) Assume that for all real numbers x and all positive integers n, (1 + x) n =

/ _ nk i xk n

k =0

Show that:

/ _ nk i = 2n n

(i)

k= 0

/ k _ nk i = n2n −1 n

(ii)

k= 1

(b) A particle moving in simple harmonic motion has maximum speed 4 ms-1 and maximum acceleration 8 ms-2. (i) Find the amplitude and period of the motion. (ii) The particle is at the origin after π seconds. Find an equation for the 6 displacement of the particle. .. (iii) Show that x = − 4x . (iv) Show that the velocity is given by v2 = 4(4 – x2).

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Maths In Focus Mathematics Extension 1 HSC Course

EXTENSION 1—PAPER 2 Time allowed—Two hours (Plus 5 minutes’ reading time) QUESTION 1 (a) Factorise 3x3 + 24. (b) Find the exact value of 3 4 (i) dx 1 + x2 0

#

(ii)

#0

2

x

dx, using the 1 + x2 substitution u = 1 + x2. (c) Find the number of different arrangements possible for the letters in the word CERTIFICATE. (d) Find, to the nearest degree, the size of the acute angle between the lines x - 3y + 5 = 0 and 2x + y - 4 = 0. QUESTION 2 (a)

AB is a tangent to circle ACD, and AB = 7.2 cm, BC = 9.3 cm. Find the length of chord CD, correct to 1 decimal place. (b) Find the coefficient of x4 in the 2 8 expansion of b 3x2 − x l . (c) By using the principle of mathematical induction, prove n n / 5k = 5 c 5 4− 1 m . k =1 QUESTION 3 (a) (i) Find the equation of the normal to the curve x2 = 4ay at the point P(2ap, ap2).

(ii) This normal meets the directrix at the point M. Find the coordinates of M. (b) (i) Differentiate y = sin− 1 x + x 1 − x2 . (ii) Hence, or otherwise, evaluate

# 0

1 2

1 − x2 dx 1 − x2

QUESTION 4 (a) If α, β and γ are the roots of x3 - 3x2 - 2x + 1 = 0, find (i) α + β + γ (ii) αβ + βγ + αγ (iii) αβγ 1 1 1 (iv) α + + γ β (v) α 2 + β 2 + γ 2 (b) The volume of an expanding balloon is increasing by a constant rate of 10 cm3 s-1. Find the rate of increase in its surface area when the balloon’s radius is 8 cm. (c) Divide the interval AB, with A(-2, 5) and B(7, 1), in the external ratio 1:4. QUESTION 5 (a) A particle undergoes simple harmonic motion about the origin O. Its displacement x metres from O at time t π seconds is given by x = 2 cos b 3t + l . 3 (i) Write the acceleration as a function of x. (ii) Write down the amplitude and period of the motion. (iii) Determine when the particle is at the origin. (b) (i) Divide the polynomial P(x) = 2x5 − x3 + 5x2 + x − 4 by A(x) = x2 + x − 1. (ii) Hence write P(x) = A(x) Q (x) + R(x), where Q(x) and R(x) are polynomials and R(x) has degree less than 2.

SAMPLE EXAMINATION PAPERS

(iii) Find P(1) and hence, or otherwise, find the remainder when P(x) is divided by x − 1. (iv) Apply Newton’s method once to find an approximate value for a root of P(x), beginning with an initial approximation of x = 0.5. (c) Show that π 1 sin b 2 θ − l = 3 sin θ cos θ − cos2 θ + . 6 2 QUESTION 6 (a) Assume (2 + 5x)12 =

tk + 1 tk

(iii) Find the exact area bounded by the curve y = f (x), the x-axis and the 1 lines x = 0 and x = . 2 (iv) Find the volume of the solid formed if the curve f (x) = cos− 1 x is rotated about the x-axis between 1 x = 0 and x = , using Simpson’s rule 2 with 3 function values. Give your answer correct to 2 decimal places.

12

/ tk xk .

k =0

(i) Use the binomial theorem to write an expression for tk, where 0 ≤ k ≤ 12. (ii) Show that

QUESTION 7 (a) (i) Sketch the function f (x) = cos− 1 x. 1 (ii) Find the exact value of f c m . 2

=

(b)

5 (12 − k) . 2 (k + 1 )

(iii) Hence, or otherwise, find the largest coefficient t k (you may leave your answer in the form 12Ck 2a5b) . (b) (i) Find in how many different ways 8 people can be seated around a round table. (ii) Two people wish to sit opposite one another. In how many different ways can this be arranged? (iii) If people are allocated seats randomly, find the probability that these 2 people will not sit opposite one another.

The diagram shows a cylinder of height H and radius R. Point X is at one end of the cylinder, on the bottom. Point Y is on the other side, halfway up the cylinder. Length XY is D. (i) Show that the volume of the cylinder is given by πΗ V= (4D2 − H2) . 16 (ii) Find the maximum volume of the cylinder in terms of D if D is fixed.

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