4u practice papers 2005 - 2000 PDF

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2000-2005 past papers...

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12/09/2014

Ext 2 2005-2000 CSA Trials Yearly\Yr12-4U\cat-4u_sol.05 Qn1) ¦1)! 4U05-1a The diagram below shows the graph y  f ( x) where f ( x)  3 

1 . x2

y y =3

x 1 0 1 3 3 On separate diagrams, sketch the following graphs, in each case showing any intercepts on the coordinate axes and the equations of any asymptotes: i. y = {f(x)}2 ii. y2 = f(x)†



¦2)!

Yearly\Yr12-4U\cat-4u_sol.05 Qn2)

4U05-1b i. The polynomial equation P(x) = 0 has a double root . Show that  is also a root of the equation P (x) = 0. 1 ii. The line y = mx is a tangent to the curve y  3  2 . Show that the equation x mx3 – 3x2 + 1 = 0 has a double root and hence find any values of m.† ¦3)!

Yearly\Yr12-4U\cat-4u_sol.05 Qn3)

4U05-1c The diagram below shows the graph y = f(x) where f(x) = x3 – 3x, x ≥ 1. y y = f(x)

0

3

x

(1, –2)

i. ii. ¦4)!

Copy the diagram. On your diagram sketch the graph of the inverse function y = f –1(x) showing any intercepts on the coordinate axes on the coordinates of any endpoints. Draw the line y = x. Find the coordinates of any points of intersection of the curves y = f(x) and y = f –1(x). Hence find the area of the region in the first quadrant bounded by the curves y = f(x) and y = f –1(x) and the coordinate axes.†

Yearly\Yr12-4U\cat-4u_sol.05 Qn4)

4U05-2a 1  sin x Find  dx † cos 2 x ¦5)!

Yearly\Yr12-4U\cat-4u_sol.05 Qn5)

4U05-2b

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Find ( ex  e

¦6)!



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1 x 2 2

) dx †

Yearly\Yr12-4U\cat-4u_sol.05 Qn6)

4U05-2c 25

Use the substitution u  x , to evaluate 

1

¦7)!

1 x x

dx , expressing the answer in simplest exact form.†

Yearly\Yr12-4U\cat-4u_sol.05 Qn7)

4U05-2d Use the substitution t  tan

 x 1 , to evaluate  3 dx , expressing the answer in simplest exact 0 5  4 cos x 2

form.† ¦8)!

Yearly\Yr12-4U\cat-4u_sol.05 Qn8)

4U05-2e

¦9)!

1

i.

If I n   x(1  x) n dx , n  0 , 1, 2, ... , show that I n 

ii.

Hence show that I n 

0

n I n1 , n  1, 2, 3, ... n2

1 , n  1, 2, 3, ... † 2  n 2 C2

Yearly\Yr12-4U\cat-4u_sol.05 Qn9)

4U05-3a Show that the complex number z  ¦10)!

6  2i 6  is real.† 3  4i 5i

Yearly\Yr12-4U\cat-4u_sol.05 Qn10)

4U05-3b z 1  4(cos

12

 i sin

 12

) and z 2  2( cos

5 5  i sin ). 12 12

On an Argand diagram draw the vectors OA,OB,OC representing z1, z2, z1 + z2 respectively. Hence find |z1 + z2| in simplest exact form.†

i. ii. ¦11)!



Yearly\Yr12-4U\cat-4u_sol.05 Qn11)

4U05-3c The quadratic equation z2 + kz + 4 = 0, k real and –4 < k < 4, has two non-real roots , . i. Explain why  ,  are complex conjugates. Hence show that || = || = 2. If ,  have arguments

ii. ¦12)!

 4

,

 4

, find the value of k.†

Yearly\Yr12-4U\cat-4u_sol.05 Qn12)

4U05-3d

¦13)!

i.

On an Argand diagram shade the region where both | z  (1  i) | 2 and 0  arg z 

ii.

Find the exact perimeter and the exact area of the shaded region.†

 2

hold.

Yearly\Yr12-4U\cat-4u_sol.05 Qn13)

4U05-4a

x2 y2   1 showing the intercepts on the axes, the coordinates of the 4 3 foci and the equations of the directrices.† Sketch the graph of the ellipse

¦14)!

Yearly\Yr12-4U\cat-4u_sol.05 Qn14)

4U05-4b

x2 y 2 The hyperbola 2  2 1, a b 0 has eccentrici tye. a b i.

Show that the line through the focus F(ae, 0) that is perpendicular to the asymptote y  equation ax + by – a2e = 0.

©EDUDATA SOFTWARE PTY LTD:1995-2014¤©BOARD OF STUDIES NSW 1984-2011 †©CSSA NSW 1984-2011

bx has a

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ii. ¦15)!

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Show that this line meets the asymptote at a point on the corresponding directrix.†

Yearly\Yr12-4U\cat-4u_sol.05 Qn15)

4U05-4c 1 1 P ( p , ) and Q(q , ) are two variable points on the rectangular hyperbola xy = 1 such that the chord PQ p q passes through the point A(0, 2). M is the midpoint of PQ. i. Show that PQ has equation x + pqy – (p + q) = 0. Hence deduce that p + q = 2pq. ii. Deduce that the tangent drawn from the point A to the rectangular hyperbola touches the curve at the point (1, 1). iii. Sketch the rectangular hyperbola showing the points P, Q, A and M. Find the equation of the locus of M and state restrictions on the domain of this locus.† ¦16)!

Yearly\Yr12-4U\cat-4u_sol.05 Qn16)

4U05-5a y y=x

y = x – e–x 0 –1

X

1

x

The diagram shows the graph of the curve y = x – e–x, x ≥ 0. This curve makes an intercept X on the xaxis, where 0 < X < 1. The region bounded by the curve and the line y = x between x = 0 and x = X is rotated through one complete revolution about the y-axis. i. Use the method of cylindrical shells to show that the volume V of the solid of revolution is X

given by V  2  xe x dx . 0

ii. ¦17)!

Hence show that V = 2(1 – X – X 2 )†

Yearly\Yr12-4U\cat-4u_sol.05 Qn17)

4U05-5b z = cos + i sin . i. Express 1 + z in modulus argument form. Hence show that

ii.



(cos 2  i sin 2 ). 2 Use the Binomial Theorem expansion of (1 + z)4 to show that (1  z) 4  16 cos 4

1  4 cos   6 cos 2  4 cos 3  cos 4  16 cos 4

iii.

¦18)!



2

cos 2 , and find the corresponding

expression for 4 sin + 6 sin2  + 4 sin3 + sin4 4 sin   6 sin 2  4 sin 3  sin 4 Hence show that  tan 2 and 1  4 cos   6 cos 2  4 cos 3  cos 4 4 sin   4 sin 3  sin 4  tan 2 . † 1  4 cos   4 cos 3  cos 4

Yearly\Yr12-4U\cat-4u_sol.05 Qn18)

4U05-6a A particle of mass m kg is dropped from rest in a medium in which the resistance to motion has 1 mv 2 when the velocity of the particle is v ms–1. After t seconds the particle has fallen magnitude 10 x metres and has velocity v ms–1 and acceleration a ms–2. Take the acceleration due to gravity as 10 ms–2. 100  v 2 . i. Draw a diagram showing the forces acting on the particle. Hence show that a  10 ©EDUDATA SOFTWARE PTY LTD:1995-2014¤©BOARD OF STUDIES NSW 1984-2011 †©CSSA NSW 1984-2011

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ii. iii. iv. ¦19)!

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1 10  v ). ln ( 2 10  v Find expressions in terms of t for v and x. Show that the terminal velocity is 10 ms–1. Hence find the exact time taken and the exact distance fallen by the particle in reaching a speed equal to 80% of its terminal velocity.†

Show that t 

Yearly\Yr12-4U\cat-4u_sol.05 Qn19)

4U05-6b The equation x3 + px + q = 0 (where p and q are real) has roots, , . i. Show that the monic cubic equation with roots 2 , 2, 2 is x3 + 2px2 + p2x – q2 = 0. 1 1 1 1 1 1 1 1 2 ii. Show that   . Hence find a cubic equation with roots  ,  and  . † q         ¦20)!

Yearly\Yr12-4U\cat-4u_sol.05 Qn20)

4U05-7a

L A P M

B N

C

ABC is an acute-angled triangle inscribed in a circle. P is a point on the minor arc AB of the circle. PL and PN are the perpendiculars from P to CA (produced) and CB respectively. LN cuts AB at M. i. Copy the diagram. ii. Explain why PNCL is a cyclic quadrilateral. iii. Show that PBM = PNM. iv. Hence show that PM is perpendicular to AB.† ¦21)!

Yearly\Yr12-4U\cat-4u_sol.05 Qn21)

4U05-7b The equation x2 + x + 1 = 0 has roots  , . Tn = n + n, n = 1, 2, 3,… i. Show that T1 = T2 = –1. ii. Show that Tn = –Tn – 1 – Tn – 2, n = 3, 4, 5,… 2n ,n  1, 2, 3, ... † iii. Hence use Mathematical Induction to show that Tn  2 cos 3 ¦22)!

Yearly\Yr12-4U\cat-4u_sol.05 Qn22)

4U05-8a A die is biased so that on any single roll the probability of getting an even score is p where p  05. In 12 rolls of this die the probability of getting exactly 4 even scores is three times the probability of getting exactly 3 even scores. Find the value of p.† ¦23)!

Yearly\Yr12-4U\cat-4u_sol.05 Qn23)

4U05-8b

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y y = ex P

M Q

x 0 P(a, ea) and Q(b, eb), where a > b, are two points on the curve y = ex. M is the midpoint of PQ.

¦24)!

1 ( a b)

i.

Use the diagram to show that ea  eb  2e 2

ii.

Hence show that if a > b > c > d then e  e  e  e  4 e a

b

. c

d

1 ( a bc  d ) 4

†

Yearly\Yr12-4U\cat-4u_sol.05 Qn24)

4U05-8c A closed hollow right cone with radius r and height h has volume V and surface area A. i. Show that 9V2 = r2A2 – 2 r4A. ii. ¦1)!

Hence show that if A is fixed then the maximum value of V is

A3 .† 72

Yearly\Yr12-4U\cat-4u_sol.04 Qn1)

4U04-1a Consider the function f(x) = x(x – 3)2. i. Sketch the graph of the curve y = f(x) showing clearly the coordinates and the nature of any turning points and the intercepts on the x and y axes. Find the set of possible values of the real number k such that the equation f(x) = k has three real, distinct solutions. ii. On separate axes, sketch the graphs of the following curves, showing clearly the coordinates and nature of any turning points and the equations of any asymptotes: 1 y  [ f (x )]2 , y  , y2  f ( x) . † f (x ) ¦2)!

Yearly\Yr12-4U\cat-4u_sol.04 Qn2)

4U04-1b A curve is given parametrically in terms of the real number t by the equations 2 3t 3t . x y and  3 1 t 3 1 t i. Express t in terms of x and y. Hence show that the curve has Cartesian equation x3 + y3 = 3xy. Deduce that the curve is symmetrical about the line y = x. 2 dy y  x ii. Show that . Hence show that the curve has a horizontal tangent when x  3 2 .  2 dx y  x Write down the coordinates of a point on the curve where the tangent is vertical.† ¦3)!

Yearly\Yr12-4U\cat-4u_sol.04 Qn3)

4U04-2a i. Find ii. ¦4)!

Find

 (cos x  sin x ) 1  1  x 2 dx . †

2

dx .

Yearly\Yr12-4U\cat-4u_sol.04 Qn4)

4U04-2b Use the substitution u = ex – 1 to find ¦5)!

e2 x

 (e x  1)2 dx †

Yearly\Yr12-4U\cat-4u_sol.04 Qn5)

4U04-2c ©EDUDATA SOFTWARE PTY LTD:1995-2014¤©BOARD OF STUDIES NSW 1984-2011 †©CSSA NSW 1984-2011

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x to evaluate  2  sin x Hence find the value of  2 dx † 0 1  sin x Use the substitution t  tan

i. ii. ¦6)!

12/09/2014

π 2 0 1

1 dx .  sin x

Yearly\Yr12-4U\cat-4u_sol.04 Qn6)

4U04-2d

¦7)!



e

i.

If In 

ii.

Hence find the value of  x 3 (ln x )2 dx . †

1

x 3 (ln x )n dx for n  0 , 1, 2 , ..., show that In 

e4 n  I  for n  1, 2, 3, ... 4 4 n 1

e

1

Yearly\Yr12-4U\cat-4u_sol.04 Qn7)

4U04-3a Solve the equation | z |2 2iz  4i  7 , expressing any answers in the form z = a + ib where a and b are real.† ¦8)!

Yearly\Yr12-4U\cat-4u_sol.04 Qn8)

4U04-3b A, B and C are the angles of a triangle. Show that (cos A + isin A)(cos B + isin B) + (cos C – isin C) = 0.† ¦9)!

Yearly\Yr12-4U\cat-4u_sol.04 Qn9)

4U04-3c

i.

2π 2π π π  i sin ) and z 2  2(cos  i sin ) are two complex numbers. 3 6 6 3 On an Argand diagram draw the vectors

ii.

OA, OB and OS to represent z1 , z2 and z1  z 2 respectively. Hence express z1 + z2 in modulus / argument form.†

z1  2(cos

¦10)!

Yearly\Yr12-4U\cat-4u_sol.04 Qn10)

4U04-3d i. On an Argand diagram shade the region where both |z|  4 and |z – 4|  4. ii. Find the exact area of the shaded region. ¦11)!

Yearly\Yr12-4U\cat-4u_sol.04 Qn11)

4U04-4a

y

x

2

2

x y  2  1, where a  b  0. The tangent to the ellipse at P 2 a b 2 2 x y passes through a focus of the hyperbola 2  2  1 with eccentricity e. a b i. Show that the tangent to the ellipse at P has equation bx cos  + ay sin  = ab. ii. Show that P lies on a directrix of the hyperbola. iii. Show that the tangent to the ellipse at P has gradient  1.† P(a cos  , b sin ) lies on the ellipse

¦12)!

Yearly\Yr12-4U\cat-4u_sol.04 Qn12)

4U04-4b ©EDUDATA SOFTWARE PTY LTD:1995-2014¤©BOARD OF STUDIES NSW 1984-2011 †©CSSA NSW 1984-2011

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i. ii.

iii. ¦13)!

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Sketch the graph of the rectangular hyperbola xy = 1, showing clearly the coordinates of the foci and the equations of the directrices. 1 1 P ( p , ) and Q(q , ) are two points on the rectangular hyperbola xy = 1. Show that the chord p q PQ has equation x + pqy – (p + q) = 0† | p2  q2 | square units. If O is the origin, show that OPQ has area 2 | pq |

Yearly\Yr12-4U\cat-4u_sol.04 Qn13)

4U04-5a 10

i.

Use the substitution x  10 2 sin  to show that 

ii.

geometrical argument to verify this result. A mould for a model railway tunnel is made by rotating the region bounded by the curve

-10

2 200  x dx  100  50 , then use a

y  200  x 2 and the x-axis between the lines x = –10 and x = 10 through 180 about the line x = 100 (where all measurements are in cm). Use the method of cylindrical shells to show that 10

the volume V cm3 of the tunnel is given by V   

10

(100  x ) 200  x 2 dx . Hence find the

volume of the tunnel in m3 correct to 2 significant figures.† ¦14)!

Yearly\Yr12-4U\cat-4u_sol.04 Qn14)

4U04-5b

¦15)!

i.

Show that the roots of the equation z10 = 1 are given by z  cos

ii.

Explain why the equation (

r r , r = 0, 1, 2, …, 9.  i sin 5 5

z  1 10 )  1 has only nine roots. Show that the roots of z z 1 10 1 r ( )  1 are given by z  (1  i cot ), r 1, 2, 3, ..., 9. † z 2 10

Yearly\Yr12-4U\cat-4u_sol.04 Qn15)

4U04-6a A particle of mass m is moving vertically in a resisting medium in which the resistance to motion has 1 mv 2 where the particle has speed v ms–1. The acceleration due to gravity is g ms–2. magnitude 10 i. If the particle falls vertically downwards from rest, show that its acceleration a ms–2 is given by 1 a  g  v 2 . Hence show that its terminal speed V ms–1 is given by V  10 g . 10 ii. If the particle is projected vertically upwards with speed V tan  ms–1 π 1 2 v ) . Hence show that it (0  α  ), show that its acceleration a ms 2 is given by a  ( g  2 10 reaches a maximum height H metres given by H = 5ln sec2 , and that it returns to its point of projection with speed V sin  ms–1.† ¦16)!

Yearly\Yr12-4U\cat-4u_sol.04 Qn16)

4U04-6b The equation x3 + 2x + 1 = 0 has roots ,  and . 1 1 1 i. Find the monic cubic equation with roots , and .

 

ii. ¦17)!

Find the monic cubic equation with roots 

1



,



1



and 

1



.†

Yearly\Yr12-4U\cat-4u_sol.04 Qn17)

4U04-7a

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i.

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Newton’s method is being used to find an approximation to the positive root x  c of the equation x2 – c = 0. The initial approximation is x = a (for some a > 0). The error in this initial approximation is 0  | a  c | . Show that the error 1 in the next approximation (obtained by one application of Newton’s method) is given by 1 

ii. ¦18)!

Find the values of a (in terms of c) such that 1 = 0†

0 2 2a

.

Yearly\Yr12-4U\cat-4u_sol.04 Qn18)

4U04-7b i. If 4 – tan  = 5sin  cos , show that x = tan  is a root of the equation x3 – 4x2 + 6x – 4 = 0. ii. Solve the equation 4 – tan  = 5sin  cos  for 0    360 giving answers correct to the nearest degree.† ¦19)!

Yearly\Yr12-4U\cat-4u_sol.04 Qn19)

4U04-7c i. By writing n! as a product, show that n! < (n + 1)n for all positive integers n. ii. Hence show that n n!  n 1 ( n  1)! for all positive integers n.† ¦20)!

Yearly\Yr12-4U\cat-4u_sol.04 Qn20)

4U04-8a PQ and RS are two chords of a circle which intersect at M inside the circle. MN is the perpendicular from M to SQ. L is the point on NM produced such that LP is perpendicular to PQ. L

R

M P

Q

N S

i. ii. iii. ¦21)!

Copy the diagram. Show that PML ||| NMQ. Hence show that LR  RS.†

Yearly\Yr12-4U\cat-4u_sol.04 Qn21)

4U04-8b The number x and the real number  are such that x  n that x 

¦1)!

1 xn

1  2 cos  . Use Mathematical Induction to show x

 2 cos n for all positive integers n  2.†

Yearly\Yr12-4U\cat-4u_sol.03 Qn1)

4U03-1a 2 The diagram shows the graph of y = f(x) where f ( x)  1  2e  x .

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y

y=1 y = f(x) x

O –1 i. ii. ¦2)!

Find the values of the x-intercepts. On separate diagrams sketch the graphs of y = {f(x)}2, y2 = f(x), y = cos–1 f(x), in each case showing the intercepts on the axes and the equations of any asymptotes.†

Yearly\Yr12-4U\cat-4u_sol.03 Qn2)

4U03-1b x . 1  x2 Show that the function is increasing for all values of x in its domain. Sketch the graph of y = f(x) showing the intercepts on the axes and the equations of any asymptotes. x Find the values of k such that the equation  kx has three distinct real roots.† 1 x2

Consider the function f ( x)  i. ii. iii. ¦...