Mechanics, EXT2* M1 2008 HSC 7
A projectile is fired from `O` with velocity `V` at an angle of inclination `theta` across level ground. The projectile passes through the points `L` and `M`, which are both `h` metres above the ground, at times `t_1` and `t_2` respectively. The projectile returns to the ground at `N`.
The equations of motion of the projectile are
`x = Vtcos theta` and `y = Vtsin theta − 1/2 g t^2`. (Do NOT prove this.)
- Show that `t_1 + t_2 = (2V)/g sin theta` AND `t_1t_2 = (2h)/g`. (2 marks)
--- 5 WORK AREA LINES (style=lined) ---
Let `∠LON = α` and `∠LNO = β`. It can be shown that
`tan alpha = h/(Vt_1 cos theta)` and `tan beta = h/(Vt_2 cos theta)`. (Do NOT prove this.)
- Show that `tan alpha + tan beta = tan theta`. (2 marks)
--- 5 WORK AREA LINES (style=lined) ---
- Show that `tan alpha tan beta = (gh)/(2V^2cos^2theta)`. (1 mark)
--- 5 WORK AREA LINES (style=lined) ---
Let `ON = r` and `LM = w`.
- Show that `r = h(cot alpha + cot beta)` and `w = h(cot beta - cot alpha)`. (2 marks)
--- 4 WORK AREA LINES (style=lined) ---
Let the gradient of the parabola at `L` be `tan phi`.
- Show that `tan phi = tan alpha - tan beta`. (3 marks)
--- 8 WORK AREA LINES (style=lined) ---
- Show that `w/(tan phi) = r/(tan theta)`. (2 marks)
--- 5 WORK AREA LINES (style=lined) ---
Calculus, MET2 2013 VCAA 16 MC
The graph of `f: [1, 5] -> R,\ f(x) = sqrt (x - 1)` is shown below.
Which one of the following definite integrals could be used to find the area of the shaded region?
A. `int_1^5 (sqrt (x - 1))\ dx`
B. `int_0^2 (sqrt (x - 1))\ dx`
C. `int_0^5 (2 - sqrt (x - 1))\ dx`
D. `int_0^2 (x^2 + 1)\ dx`
E. `int_0^2 (x^2)\ dx`
Graphs, MET2 2013 VCAA 20 MC
A transformation `T: R^2 -> R^2, T ([(x), (y)]) = [(1, 0), (0, -1)] [(x), (y)] + [(5), (0)]` maps the graph of a function `f` to the graph of `y = x^2, x in R.`
The rule of `f` is
- `f(x) = -(x + 5)^2`
- `f(x) = (5 - x)^2`
- `f(x) = -(x - 5)^2`
- `f(x) = -x^2 + 5`
- `f(x) = x^2 - 5`
Binomial, EXT1 2008 HSC 6c
Let `p` and `q` be positive integers with `p ≤ q`.
- Use the binomial theorem to expand `(1 + x) ^(p+ q)`, and hence write down the term of
- `((1 + x)^(p + q))/(x^q)` which is independent of `x`. (2 marks)
- Given that
- `((1 + x)^(p + q))/(x^q) = (1 + x)^p(1 + 1/x)^q`,
apply the binomial theorem and the result of part(i) to find a simpler expression for
- `1 + ((p),(1))((q),(1)) + ((p),(2))((q),(2)) + … + ((p),(p))((q),(p))`. (3 marks)
Calculus, 2ADV C3 2004 HSC 10b
The diagram shows a triangular piece of land `ABC` with dimensions `AB = c` metres, `AC = b` metres and `BC = a` metres, where `a ≤ b ≤ c`.
The owner of the land wants to build a straight fence to divide the land into two pieces of equal area. Let `S` and `T` be points on `AB` and `AC` respectively so that `ST` divides the land into two pieces of equal area.
Let `AS = x` metres, `AT = y` metres and `ST = z` metres.
- Show that `xy = 1/2 bc`. (1 mark)
--- 2 WORK AREA LINES (style=lined) ---
- Use the cosine rule in triangle `AST` to show that
`z^2 = x^2 + (b^2c^2)/(4x^2) − bc cos A.` (2 marks)
--- 5 WORK AREA LINES (style=lined) ---
- Show that the value of `z^2` in the equation in part (ii) is a minimum when
`x = sqrt((bc)/2)`. (4 marks)
--- 8 WORK AREA LINES (style=lined) ---
- Show that the minimum length of the fence is `sqrt(((P − 2b)(P − 2c))/2)` metres, where `P = a + b + c`.
(You may assume that the value of `x` given in part (iii) is feasible.) (2 marks)
--- 5 WORK AREA LINES (style=lined) ---
Calculus, 2ADV C3 2004 HSC 9c
Consider the function `f(x) = (log_e x)/x`, for `x > 0`.
- Show that the graph of `y = f(x)` has a stationary point at `x = e`. (2 marks)
--- 6 WORK AREA LINES (style=lined) ---
- By considering the gradient on either side of `x = e`, or otherwise, show that the stationary point is a maximum. (1 mark)
--- 2 WORK AREA LINES (style=lined) ---
- Use the fact that the maximum value of `f(x)` occurs at `x = e` to deduce that `e^x ≥ x^e` for all `x > 0`. (2 marks)
--- 6 WORK AREA LINES (style=lined) ---
Calculus, EXT1* C1 2004 HSC 9b
A particle moves along the `x`-axis. Initially it is at rest at the origin. The graph shows the acceleration, `a`, of the particle as a function of time `t` for `0 ≤ t ≤ 5`.
- Write down the time at which the velocity of the particle is a maximum. (1 marks)
--- 2 WORK AREA LINES (style=lined) ---
- At what time during the interval `0 ≤ t ≤ 5` is the particle furthest from the origin? Give brief reasons for your answer. (2 marks)
--- 8 WORK AREA LINES (style=lined) ---
Calculus, 2ADV C4 2004 HSC 8b
The diagram shows the graph of the parabola `x^2 = 16y`. The points `A (4, 1)` and `B (−8, 4)` are on the parabola, and `C` is the point where the tangent to the parabola at `A` intersects the directrix.
- Write down the equation of the directrix of the parabola `x^2 = 16y`. (1 mark)
- Find the equation of the tangent to the parabola at the point `A`. (2 marks)
- Show that `C` is the point `(−6, −4)`. (1 mark)
- Given that the equation of the line `AB` is `y = 2 − x/4`, find the area bounded by the line `AB` and the parabola. (2 marks)
- Hence, or otherwise, find the shaded area bounded by the parabola, the tangent at `A` and the line `BC`. (3 marks)
Probability, MET2 2012 VCAA 20 MC
A discrete random variable `X` has the probability function `text(Pr)(X = k) = (1 -p)^k p`, where `k` is a non-negative integer.
`text(Pr)(X > 1)` is equal to
- `1 - p + p^2`
- `1 - p^2`
- `p - p^2`
- `2p - p^2`
- `(1 - p)^2`
Polynomials, EXT2 2015 HSC 16b
Let `n` be a positive integer.
- By considering `(cos alpha + i sin alpha)^(2n)`, show that
- `cos(2n alpha) = cos^(2n) alpha - ((2n), (2)) cos^(2n - 2) alpha sin^2 alpha + ((2n), (4)) cos^(2n - 4) alpha sin^4 alpha - …`
- `+ … + (-1)^(n - 1) ((2n), (2n - 2)) cos^2 alpha sin^(2n - 2) alpha + (-1)^n sin^(2n) alpha.`
- Let `T_(2n) (x) = cos(2n cos^-1 x)`, for `-1 <= x <= 1.` (2 marks)
- Show that
- `T_(2n)(x) = x^(2n) - ((2n), (2)) x^(2n - 2)(1 - x^2) + ((2n), (4)) x^(2n - 4) (1 - x^2)^2 +`
- `… + (-1)^n (1 - x^2)^n.` (2 marks)
- By considering the roots of `T_(2n) (x)`, find the value of
- `cos(pi/(4n)) cos((3 pi)/(4n)) …\ cos (((4n - 1) pi)/(4n)).` (3 marks)
- Prove that
- `1 - ((2n), (2)) + ((2n), (4)) - ((2n), (6)) + … + (-1)^n ((2n), (2n)) = 2^n cos ((n pi)/2).` (2 marks)
Harder Ext 1 Topics, EXT2 2015 HSC 16a
- A table has `3` rows and `5` columns, creating `15` cells as shown.
- Counters are to be placed randomly on the table so that there is one counter in each cell. There are `5` identical black counters and `10` identical white counters.
- Show that the probability that there is exactly one black counter in each column is `81/1001.` (2 marks)
- The table is extended to have `n` rows and `q` columns. There are `nq` counters, where `q` are identical black counters and the remainder are identical white counters. The counters are placed randomly on the table with one counter in each cell.
- Let `P_n` be the probability that each column contains exactly one black counter.
- Show that `P_n = n^q/(((nq), (q))).` (2 marks)
- Find `lim_(n -> oo) P_n.` (2 marks)
Proof, EXT2 P1 2015 HSC 15b
Suppose that `x >= 0` and `n` is a positive integer.
- Show that `1 - x <= 1/(1 + x) <= 1.` (2 marks)
--- 2 WORK AREA LINES (style=lined) ---
- Hence, or otherwise, show that
`1 - 1/(2n) <= n ln (1 + 1/n) <= 1.` (2 marks)
--- 4 WORK AREA LINES (style=lined) ---
- Hence, explain why
`lim_(n -> oo) (1 + 1/n)^n = e.` (1 mark)
--- 4 WORK AREA LINES (style=lined) ---
Harder Ext1 Topics, EXT2 2006 HSC 8b
For `x > 0`, let `f(x) = x^n e^-x`, where `n` is an integer and `n >= 2.`
- The two points of inflection of `f(x)` occur at `x = a` and `x = b`, where `0 < a < b`. Find `a` and `b` in terms of `n.` (4 marks)
- Show that
- `(f(b))/(f(a)) = ((1 + 1/sqrt n)/(1 - 1/sqrt n))^n e^(-2 sqrt n).` (2 marks)
- `(f(b))/(f(a)) = ((1 + 1/sqrt n)/(1 - 1/sqrt n))^n e^(-2 sqrt n).` (2 marks)
- Using the following
- If `0<=x<=1/sqrt2` then `1 <= ((1 + x)/(1 - x)) e^(-2x) <= e^((4x^3)/3),` (DO NOT prove this)
- show that `1 <= (f(b))/(f(a)) <= e^(4/(3 sqrt n)).` (2 marks)
- show that `1 <= (f(b))/(f(a)) <= e^(4/(3 sqrt n)).` (2 marks)
- What can be said about the ratio `(f(b))/(f(a))` as `n -> oo?` (1 mark)
Calculus, EXT2 C1 2006 HSC 7b
- Let `I_n = int_0^x sec^n t\ dt`, where `0 <= x <= pi/2`.
Show that `I_n = (sec^(n - 2) x tan x)/(n - 1) + (n - 2)/(n - 1) I_(n - 2).` (3 marks)
--- 8 WORK AREA LINES (style=lined) ---
- Hence find the exact value of
`int_0^(pi/3) sec^4 t\ dt.` (2 marks)
--- 4 WORK AREA LINES (style=lined) ---
Harder Ext1 Topics, EXT2 2006 HSC 4d
In the acute-angled triangle `ABC, \ K` is the midpoint of `AB, \ L` is the midpoint of `BC` and `M` is the midpoint of `CA`. The circle through `K, L` and `M` also cuts `BC` at `P` as shown in the diagram.
Copy or trace the diagram into your writing booklet.
- Prove that `KMLB` is a parallelogram. (1 mark)
- Prove that `/_ KPB = /_ KML.` (1 mark)
- Prove that `AP _|_ BC.` (2 marks)
Harder Ext1 Topics, EXT2 2007 HSC 8c
The diagram shows a regular `n`-sided polygon with vertices `X_1, X_2, …, X_n`. Each side has unit length. The length `d_k` of the ‘diagonal’ `X_n X_k` where `k = 1, 2, …, n - 1` is given by
`d_k = (sin\ (k pi)/n)/(sin\ pi/n).` (Do NOT prove this.)
- Show, using the identity,
- `sin\ pi/n + sin\ (2 pi)/n + … + sin\ ((n - 1) pi)/n = cot\ pi/(2n),` (Do NOT prove this.)
-
- that `d_1 + … + d_(n - 1) = 1/(2 sin^2\ pi/(2n)).` (2 marks)
- that `d_1 + … + d_(n - 1) = 1/(2 sin^2\ pi/(2n)).` (2 marks)
- Let `p` be the perimeter of the polygon and
- `q = 1/n (d_1 + … + d_(n - 1))`.
- Show that `p/q = 2 (n sin\ pi/(2n))^2.` (2 marks)
- Hence calculate the limiting value of `p/q` as `n -> oo.` (1 mark)
Complex Numbers, EXT2 N2 2007 HSC 8b
- Let `n` be a positive integer. Show that if `z^2 != 1` then
`1 + z^2 + z^4 + … + z^(2n - 2) = ((z^n - z^-n)/(z - z^-1)) z^(n - 1)`. (2 marks)
--- 5 WORK AREA LINES (style=lined) ---
- By substituting `z = cos theta + i sin theta` where `sin theta != 0`, into part (i), show that
`1 + cos 2 theta + … + cos (2n - 2) theta + i[sin 2 theta + … + sin (2n - 2) theta]`
`= (sin n theta)/(sin theta) [cos (n - 1) theta + i sin (n - 1) theta].` (3 marks)
--- 8 WORK AREA LINES (style=lined) ---
- Suppose `theta = pi/(2n)`. Using part (ii), show that
`sin\ pi/n + sin\ (2 pi)/n + … + sin\ ((n - 1) pi)/n = cot\ pi/(2n).` (2 marks)
--- 8 WORK AREA LINES (style=lined) ---
Calculus, EXT2 C1 2007 HSC 8a
- Using a suitable substitution, show that
`int_0^a f(x)\ dx = int_0^a f(a - x)\ dx.` (1 mark)
--- 6 WORK AREA LINES (style=lined) ---
- A function `f(x)` has the property that `f(x) + f(a - x) = f(a).`
Using part (i), or otherwise, show that
`int_0^a f(x)\ dx = a/2\ f(a).` (2 marks)
--- 8 WORK AREA LINES (style=lined) ---
Proof, EXT2 P1 2007 HSC 7a
- Show that `sin x < x` for `x > 0.` (2 marks)
--- 5 WORK AREA LINES (style=lined) ---
- Let `f(x) = sin x - x + x^3/6`.
Show that the graph of `y = f(x)` is concave up for `x > 0.` (2 marks)
--- 4 WORK AREA LINES (style=lined) ---
- By considering the first two derivatives of `f(x)`,show that `sin x > x - x^3/6` for `x > 0.` (2 marks)
--- 6 WORK AREA LINES (style=lined) ---
Calculus, EXT2 C1 2012 HSC 10 MC
Without evaluating the integrals, which one of the following integrals is greater than zero?
(A) `int_(−pi/2)^(pi/2) x/(2 + cos x)\ dx`
(B) `int_(−pi)^pi x^3 sin x\ dx`
(C) `int_(−1)^1 (e^(−x^2) − 1)\ dx`
(D) `int_(−2)^2 tan^(−1)(x^3)\ dx`
Calculus, EXT2 C1 2014 HSC 10 MC
Which integral is necessarily equal to `int_(−a)^a f(x)\ dx`?
(A) `int_0^a f(x) − f(−x)\ dx`
(B) `int_0^a f(x) − f(a − x)\ dx`
(C) `int_0^a f(x − a) + f(−x)\ dx`
(D) `int_0^a f(x − a) + f(a − x)\ dx`
Harder Ext1 Topics, EXT2 2013 HSC 10 MC
A hostel has four vacant rooms. Each room can accommodate a maximum of four people.
In how many different ways can six people be accommodated in the four rooms?
(A) 4020
(B) 4068
(C) 4080
(D) 4096
Harder Ext1 Topics, EXT2 2009 HSC 8c
A game is being played by `n` people, `A_1, A_2, ..., A_n`, sitting around a table. Each person has a card with their own name on it and all the cards are placed in a box in the middle of the table. Each person in turn, starting with `A_1`, draws a card at random from the box. If the person draws their own card, they win the game and the game ends. Otherwise, the card is returned to the box and the next person draws a card at random. The game continues until someone wins.
Let `W` be the probability that `A_1` wins the game.
Let `p = 1/n and q = 1 - 1/n`.
- Show that `W = p + q^n W.` (1 mark)
- Let `m` be a fixed positive integer and let `W_m` be the probability that `A_1` wins in no more than `m` attempts.
- Use `e^(-n/(n - 1)) < (1 - 1/n)^n < e^-1,`
- to show that, if `n` is large, `W_m/W` is approximately equal to `1 - e^-m.` (3 marks)
Proof, EXT2 P2 2009 HSC 8a
- Using the substitution `t = tan\ theta/2`, or otherwise, show that
`qquad cot theta + 1/2 tan\ theta/2 = 1/2 cot\ theta/2.` (2 marks)
--- 5 WORK AREA LINES (style=lined) ---
- Use mathematical induction to prove that, for integers `n >= 1`,
`qquad sum_(r = 1)^n 1/2^(r - 1) tan x/2^r = 1/2^(n - 1) cot\ x/2^n - 2 cot x.` (3 marks)
--- 8 WORK AREA LINES (style=lined) ---
- Show that `lim_(n -> oo) sum_(r = 1)^n 1/2^(r - 1) tan\ x/2^r = 2/x - 2 cot x.` (2 marks)
--- 6 WORK AREA LINES (style=lined) ---
- Hence find the exact value of
-
`qquad tan\ pi/4 + 1/2 tan\ pi/8 + 1/4 tan\ pi/16 + ….` (2 marks)
--- 4 WORK AREA LINES (style=lined) ---
Complex Numbers, EXT2 N2 2009 HSC 7b
Let `z = cos theta + i sin theta.`
- Show that `z^n + z^-n = 2 cos n theta`, where `n` is a positive integer. (2 marks)
--- 6 WORK AREA LINES (style=lined) ---
- Let `m` be a positive integer. Show that
`(2 cos theta)^(2m) = 2 [cos 2 m theta + ((2m), (1)) cos (2m - 2) theta + ((2m), (2)) cos (2m - 4) theta`
`+ … + ((2m), (m - 1)) cos 2 theta] + ((2m), (m)).` (3 marks)
--- 8 WORK AREA LINES (style=lined) ---
- Hence, or otherwise, prove that
`int_0^(pi/2) cos^(2m) theta\ d theta = pi/(2^(2m + 1)) ((2m), (m))`
where `m` is a positive integer. (2 marks)
--- 6 WORK AREA LINES (style=lined) ---
Harder Ext1 Topics, EXT2 2009 HSC 6c
The diagram shows a circle of radius `r`, centred at the origin, `O`. The line `PQ` is tangent to the circle at `Q`, the line `PR` is horizontal, and `R` lies on the line `x = c`.
- Find the length of `PQ` in terms of `x, y and r.` (1 mark)
- The point `P` moves such that `PQ = PR`.
- Show that the equation of the locus of `P` is
- `y^2 = r^2 + c^2 - 2cx.` (2 marks)
- `y^2 = r^2 + c^2 - 2cx.` (2 marks)
- Find the focus, `S`, of the parabola in part (ii). (2 marks)
- Show that the difference between the length `PS` and the length `PQ` is independent of `x.` (2 marks)
Integration, EXT2 2010 HSC 8
Let
`A_n = int_0^(pi/2) cos^(2n) x\ dx` and `B_n = int_0^(pi/2) x^2cos^(2n)x\ dx`,
where `n` is an integer, `n ≥ 0`. (Note that `A_n > 0`, `B_n > 0`.)
- Show that
- `nA_n = (2n − 1)/2 A_(n − 1)` for `n ≥ 1`. (2 marks)
- Using integration by parts on `A_n`, or otherwise, show that
- `A_n = 2n int_0^(pi/2) x sin x cos^(2n − 1) x\ dx` for `n ≥ 1`. (1 mark)
- `A_n = 2n int_0^(pi/2) x sin x cos^(2n − 1) x\ dx` for `n ≥ 1`. (1 mark)
- Use integration by parts on the integral in part (ii) to show that
- `(A_n)/(n^2) = ((2n − 1))/n B_(n − 1) − 2B_n` for `n ≥ 1`. (3 marks)
- `(A_n)/(n^2) = ((2n − 1))/n B_(n − 1) − 2B_n` for `n ≥ 1`. (3 marks)
- Use parts (i) and (iii) to show that
- `1/(n^2) = 2((B_(n − 1))/(A_(n − 1)) − (B_n)/(A_n))` for `n ≥ 1`. (1 mark)
- `1/(n^2) = 2((B_(n − 1))/(A_(n − 1)) − (B_n)/(A_n))` for `n ≥ 1`. (1 mark)
- Show that
- `sum_(k = 1)^n 1/(k^2) = (pi^2)/6 − 2 (B_n)/(A_n)`. (2 marks)
- `sum_(k = 1)^n 1/(k^2) = (pi^2)/6 − 2 (B_n)/(A_n)`. (2 marks)
- Use the fact that
- `sin x ≥ 2/pi x` for `0 ≤ x ≤ pi/2` to show that
- `B_n ≤ int_0^(pi/2) x^2(1 − (4x^2)/(pi^2))^n dx`. (1 mark)
- `B_n ≤ int_0^(pi/2) x^2(1 − (4x^2)/(pi^2))^n dx`. (1 mark)
- Show that
- `int_0^(pi/2) x^2(1 − (4x^2)/(pi^2))^n dx = (pi^2)/(8(n + 1)) int_0^(pi/2) (1 − (4x^2)/(pi^2))^(n + 1) dx`. (1 mark)
- From parts (vi) and (vii) it follows that
- `B_n ≤ (pi^2)/(8(n + 1)) int_0^(pi/2)(1 − (4x^2)/(pi^2))^(n + 1) dx`.
- `B_n ≤ (pi^2)/(8(n + 1)) int_0^(pi/2)(1 − (4x^2)/(pi^2))^(n + 1) dx`.
- Use the substitution `x = pi/2 sin t` in this inequality to show that
-
- `B_n ≤ (pi^3)/(16(n + 1)) int_0^(pi/2) cos^(2n + 3)t\ dt ≤ (pi^3)/(16(n + 1)) A_n`. (2 marks)
- `B_n ≤ (pi^3)/(16(n + 1)) int_0^(pi/2) cos^(2n + 3)t\ dt ≤ (pi^3)/(16(n + 1)) A_n`. (2 marks)
- Use part (v) to deduce that
- `(pi^2)/6 − (pi^3)/(8(n + 1)) ≤ sum_(k = 1)^n 1/(k^2) < (pi^2)/6`. (1 mark)
- `(pi^2)/6 − (pi^3)/(8(n + 1)) ≤ sum_(k = 1)^n 1/(k^2) < (pi^2)/6`. (1 mark)
- What is
- `lim_(n → ∞) sum_(k = 1)^n 1/(k^2)`? (1 mark)
Polynomials, EXT2 2010 HSC 7c
Let `P(x) = (n − 1)x^n − nx^(n − 1) + 1`, where `n` is an odd integer, `n ≥ 3`.
- Show that `P(x)` has exactly two stationary points. (1 mark)
- Show that `P(x)` has a double zero at `x = 1`. (1 mark)
- Use the graph `y = P(x)` to explain why `P(x)` has exactly one real zero other than `1`. (2 marks)
- Let `α` be the real zero of `P(x)` other than `1`.
- Given that `2^x>=3x-1` for `x>=3`, or otherwise, show that `-1 < α ≤ -1/2`. (2 marks)
- Deduce that each of the zeros of `4x^5 − 5x^4 + 1` has modulus less than or equal to `1`. (2 marks)
Harder Ext1 Topics, EXT2 2010 HSC 7a
In the diagram `ABCD` is a cyclic quadrilateral. The point `K` is on `AC` such that `∠ADK = ∠CDB`, and hence `ΔADK` is similar to `ΔBDC`.
Copy or trace the diagram into your writing booklet.
- Show that `ΔADB` is similar to `ΔKDC`. (2 marks)
- Using the fact that `AC = AK + KC`,
- show that `BD xx AC = AD xx BC + AB xx DC`. (2 marks)
- A regular pentagon of side length `1` is inscribed in a circle, as shown in the diagram.
- Let `x` be the length of a chord in the pentagon.
- Use the result in part (ii) to show that `x = (1 + sqrt5)/2`. (2 marks)
Harder Ext1 Topics, EXT2 2010 HSC 5c
A TV channel has estimated that if it spends `$x` on advertising a particular program it will attract a proportion `y(x)` of the potential audience for the program, where
`(dy)/(dx) = ay(1 − y)`
and `a > 0` is a given constant.
- Explain why
`(dy)/(dx)` has its maximum value when `y = 1/2`. (1 mark) - Using
- `int (dy)/(y(1 − y)) = ln (y/(1 − y)) + c`, or otherwise, deduce that
- `y(x) = 1/(ke^(-ax) + 1)` for some constant `k > 0`. (3 marks)
- The TV channel knows that if it spends no money on advertising the program then the audience will be one-tenth of the potential audience.
- Find the value of the constant `k` referred to in part (c)(ii). (1 mark)
- What feature of the graph
`y = 1/(ke^(-ax) + 1)` is determined by the result in part (c)(i)? (1 mark) - Sketch the graph
`y(x) = 1/(ke^(-ax) + 1)` (1 mark)
Harder Ext1 Topics, EXT2 2010 HSC 4c
Let `k` be a real number, `k ≥ 4`.
Show that, for every positive real number `b`, there is a positive real number `a` such that `1/a + 1/b = k/(a + b)`. (3 marks)
Conics, EXT2 2010 HSC 3d
The diagram shows the rectangular hyperbola `xy = c^2`, with `c > 0`.
The points `A(c, c)`, `R(ct, c/t)` and `Q(-ct, -c/t)` are points on the hyperbola,
with `t ≠ ±1`.
- The line `l_1` is the line through `R` perpendicular to `QA`.
- Show that the equation of `l_1` is
-
- `y = -tx + c(t^2 + 1/t)`. (2 marks)
- `y = -tx + c(t^2 + 1/t)`. (2 marks)
- The line `l_2` is the line through `Q` perpendicular to `RA`.
- Write down the equation of `l_2`. (1 mark)
- Let `P` be the point of intersection of the lines `l_1` and `l_2`.
- Show that `P` is the point `(c/(t^2), ct^2)`. (2 marks)
- Give a geometric description of the locus of `P`. (1 mark)
Harder Ext1 Topics, EXT2 2010 HSC 3c
Two identical biased coins are each more likely to land showing heads than showing tails.
The two coins are tossed together, and the outcome is recorded. After a large number of trials it is observed that the probability that the two coins land showing a head and a tail is `0.48`.
What is the probability that both coins land showing heads? (2 marks)
Polynomials, EXT2 2011 HSC 8c
Let `beta` be a root of the complex monic polynomial
`P(z) = z^n + a_(n - 1) z^(n - 1) + … + a_1z + a_0.`
Let `M` be the maximum value of `|\ a_(n - 1)\ |,\ |\ a_(n - 2)\ |,\ …\ ,\ |\ a_0\ |.`
- Show that
- `|\ beta\ |^n <= M(|\ beta\ |^(n - 1) + |\ beta\ |^(n - 2) + … + |\ beta\ | + 1).` (2 marks)
- Hence, show that for any root `beta` of `P(z)`
- `|\ beta\ | < 1 + M.` (3 marks)
- Let `S(x) = sum_(k = 0)^n c_k(x + 1/x)^k`, where the real numbers `c_k` satisfy `|\ c_k\ | <= |\ c_n\ |` for all `k < n`, and `c_n != 0.`
- Using parts (i) and (ii), or otherwise, show that `S(x) = 0` has no real solutions. (3 marks)
Harder Ext1 Topics, EXT2 2011 HSC 8b
A bag contains seven balls numbered from `1` to `7`. A ball is chosen at random and its number is noted. The ball is then returned to the bag. This is done a total of seven times.
- What is the probability that each ball is selected exactly once? (1 mark)
- What is the probability that at least one ball is not selected? (1 mark)
- What is the probability that exactly one of the balls is not selected? (2 marks)
Calculus, EXT2 C1 2011 HSC 8a
For every integer `m >= 0` let
`I_m = int_0^1 x^m (x^2 - 1)^5\ dx.`
Prove that for `m >= 2`
`I_m = (m - 1)/(m + 11) \ I_(m - 2).` (3 marks)
--- 8 WORK AREA LINES (style=lined) ---
Complex Numbers, EXT2 N2 2011 HSC 6c
On an Argand diagram, sketch the region described by the inequality
`|\ 1 + 1/z\ | <= 1.` (2 marks)
Graphs, EXT2 2011 HSC 6b
Let `f (x)` be a function with a continuous derivative.
- Prove that `y = (f(x))^3` has a stationary point at `x = a` if `f(a) = 0` or `f prime(a) = 0.` (2 marks)
- Without finding `f″(x)`, explain why `y = (f(x))^3` has a horizontal point of inflection at `x = a` if `f(a) = 0` and `f prime (a) != 0.` (1 mark)
- The diagram shows the graph `y = f(x).`
- Copy or trace the diagram into your writing booklet.
-
On the diagram in your writing booklet, sketch the graph `y = (f(x))^3`, clearly distinguishing it from the graph `y = f(x).` (3 marks)
Mechanics, EXT2 M1 2011 HSC 6a
Jac jumps out of an aeroplane and falls vertically. His velocity at time `t` after his parachute is opened is given by `v(t)`, where `v(0) = v_0` and `v(t)` is positive in the downwards direction. The magnitude of the resistive force provided by the parachute is `kv^2`, where `k` is a positive constant. Let `m` be Jac’s mass and `g` the acceleration due to gravity. Jac’s terminal velocity with the parachute open is `v_T.`
Jac’s equation of motion with the parachute open is
`m (dv)/(dt) = mg - kv^2.` (Do NOT prove this.)
- Explain why Jac’s terminal velocity `v_T` is given by
`sqrt ((mg)/k).` (1 mark)
--- 3 WORK AREA LINES (style=lined) ---
- By integrating the equation of motion, show that `t` and `v` are related by the equation
`t = (v_T)/(2g) ln[((v_T + v)(v_T - v_0))/((v_T - v)(v_T + v_0))].` (3 marks)
--- 6 WORK AREA LINES (style=lined) ---
- Jac’s friend Gil also jumps out of the aeroplane and falls vertically. Jac and Gil have the same mass and identical parachutes.
Jac opens his parachute when his speed is `1/3 v_T.` Gil opens her parachute when her speed is `3v_T.` Jac’s speed increases and Gil’s speed decreases, both towards `v_T.`
Show that in the time taken for Jac's speed to double, Gil's speed has halved. (3 marks)
--- 6 WORK AREA LINES (style=lined) ---
Conics, EXT2 2011 HSC 5c
The diagram shows the ellipse `x^2/a^2 + y^2/b^2 = 1`, where `a > b`. The line `l` is the tangent to the ellipse at the point `P`. The foci of the ellipse are `S` and `S prime`. The perpendicular to `l` through `S` meets `l` at the point `Q`. The lines `SQ` and `S prime P` meet at the point `R`.
Copy or trace the diagram into your writing booklet.
- Use the reflection property of the ellipse at `P` to prove that `SQ = RQ.` (2 marks)
- Explain why `S prime R = 2a.` (1 mark)
- Hence, or otherwise, prove that `Q` lies on the circle `x^2 + y^2 = a^2.` (3 marks)
Harder Ext1 Topics, EXT2 2011 HSC 4c
A mass is attached to a spring and moves in a resistive medium. The motion of the mass satisfies the differential equation
`(d^2y)/(dt^2) + 3 (dy)/(dt) + 2y = 0,`
where `y` is the displacement of the mass at time `t.`
- Show that, if `y = f(t)` and `y = g(t)` are both solutions to the differential equation and `A` and `B` are constants, then
- `y = A f (t) + Bg (t)`
- is also a solution. (2 marks)
- A solution of the differential equation is given by `y = e^(kt)` for some values of `k`, where `k` is a constant.
- Show that the only possible values of `k` are `k = -1` and `k = -2.` (2 marks)
- A solution of the differential equation is
- `y = Ae^(-2t) + Be^-t.`
- When `t = 0`, it is given that `y = 0` and `(dy)/(dt) = 1`.
- Find the values of `A` and `B.` (3 marks)
Conics, EXT2 2011 HSC 3d
The equation `x^2/16 - y^2/9 = 1` represents a hyperbola.
- Find the eccentricity `e.` (1 mark)
- Find the coordinates of the foci. (1 mark)
- State the equations of the asymptotes. (1 mark)
- Sketch the hyperbola. (1 mark)
- For the general hyperbola
- `x^2/a^2 - y^2/b^2 = 1`,
- describe the effect on the hyperbola as `e -> oo.` (1 mark)
Proof, EXT2 P1 2012 HSC 16c
Let `n` be an integer where `n > 1`. Integers from `1` to `n` inclusive are selected randomly one by one with repetition being possible. Let `P(k)` be the probability that exactly `k` different integers are selected before one of them is selected for the second time, where `1 ≤ k ≤ n`.
- Explain why `P(k) = ((n − 1)!k)/(n^k(n − k)!)`. (2 marks)
--- 6 WORK AREA LINES (style=lined) ---
- Suppose `P(k) ≥ P(k − 1)`. Show that `k^2- k- n ≤ 0`. (2 marks)
--- 5 WORK AREA LINES (style=lined) ---
- Show that if `sqrt(n + 1/4) > k − 1/2` then the integers `n` and `k` satisfy `sqrtn > k − 1/2`. (2 marks)
--- 6 WORK AREA LINES (style=lined) ---
- Hence show that if `4n + 1` is not a perfect square, then `P(k)` is greatest when `k` is the closest integer to `sqrtn`.
You may use part (iii) and also that `k^2 − k − n >0` if `P(k)< P(k − 1)`. (2 marks)
--- 6 WORK AREA LINES (style=lined) ---
Harder Ext1 Topics, EXT2 2012 HSC 16a
- In how many ways can `m` identical yellow discs and `n` identical black discs be arranged in a row? (1 mark)
- In how many ways can 10 identical coins be allocated to 4 different boxes? (1 mark)
Polynomials, EXT2 2012 HSC 15b
Let `P(z) = z^4 − 2kz^3 + 2k^2z^2 − 2kz + 1`, where `k` is real.
Let `α = x + iy`, where `x` and `y` are real.
Suppose that `α` and `iα` are zeros of `P(z)`, where `bar α ≠ iα`.
- Explain why `bar α` and `-i bar α` are zeros of `P(z)`. (1 mark)
- Show that `P(z) = z^2(z − k)^2 + (kz − 1)^2`. (1 mark)
- Hence show that if `P(z)` has a real zero then
- `P(z) = (z^2 + 1)(z+ 1)^2` or `P(z) = (z^2 + 1)(z − 1)^2.` (2 marks)
- `P(z) = (z^2 + 1)(z+ 1)^2` or `P(z) = (z^2 + 1)(z − 1)^2.` (2 marks)
- Show that all zeros of `P(z)` have modulus `1`. (2 marks)
- Show that `k = x − y`. (1 mark)
- Hence show that `-sqrt2 ≤ k ≤ sqrt2`. (2 marks)
Proof, EXT2 P1 2012 HSC 15a
- Prove that `sqrt(ab) ≤ (a + b)/2`, where `a ≥ 0` and `b ≥ 0`. (1 mark)
--- 3 WORK AREA LINES (style=lined) ---
- If `1 ≤ x ≤ y`, show that `x(y − x + 1) ≥ y`. (2 marks)
--- 5 WORK AREA LINES (style=lined) ---
- Let `n` and `j` be positive integers with `1 ≤ j ≤ n`.
Prove that `sqrtn ≤ sqrt(j(n − j + 1)) ≤ (n + 1)/2.` (2 marks)
--- 4 WORK AREA LINES (style=lined) ---
- For integers `n ≥ 1`, prove that
`(sqrtn)^n ≤ n! ≤ ((n + 1)/2)^n`. (1 mark)
--- 3 WORK AREA LINES (style=lined) ---
Volumes, EXT2 2012 HSC 14c
The solid `ABCD` is cut from a quarter cylinder of radius `r` as shown. Its base is an isosceles triangle `ABC` with `AB = AC`. The length of `BC` is `a` and the midpoint of `BC` is `X`.
The cross-sections perpendicular to `AX` are rectangles. A typical cross-section is shown shaded in the diagram.
Find the volume of the solid `ABCD`. (4 marks)
Mechanics, EXT2 2013 HSC 16b
A small bead `P` of mass `m` can freely move along a string. The ends of the string are attached to fixed points `S` and `S′`, where `S′` lies vertically above `S`. The bead undergoes uniform circular motion with radius `r` and constant angular velocity `omega` in a horizontal plane.
The forces acting on the bead are the gravitational force and the tension forces along the string. The tension forces along `PS` and `PS′` have the same magnitude `T`.
The length of the string is `2a` and `SS′= 2ae`, where `0 < e < 1`. The horizontal plane through `P` meets `SS′` at `Q`. The midpoint of `SS′` is `O` and `beta = /_S′PQ`. The parameter `theta` is chosen so that `OQ =a cos theta.`
- What information indicates that `P` lies on an ellipse with foci `S` and `S′`, and with eccentricity `e`? (1 mark)
- Using the focus–directrix definition of an ellipse, or otherwise, show that
- `SP = a(1 − e cos theta ).` (1 mark)
- Show that
- `sin beta = (e + cos theta)/(1 + e cos theta).` (2 marks)
- By considering the forces acting on `P` in the vertical direction, show that
- `mg = (2T(1 - e^2) cos theta)/(1 - e^2 cos^2 theta).` (2 marks)
- Show that the force acting on `P` in the horizontal direction is
- `mr omega^2 = (2T sqrt(1 - e^2) sin theta)/(1 - e^2 cos^2 theta).` (3 marks)
- Show that
- `tan theta = (r omega^2)/g sqrt(1 - e^2).` (1 mark)
Proof, EXT2 P1 2013 HSC 16a
- Find the minimum value of `P(x) = 2x^3 - 15x^2 + 24x + 16`, for `x >= 0.` (2 marks)
--- 6 WORK AREA LINES (style=lined) ---
- Hence, or otherwise, show that for `x >= 0`,
`(x + 1) (x^2 + (x + 4)^2) >= 25x^2.` (1 mark)
--- 5 WORK AREA LINES (style=lined) ---
- Hence, or otherwise, show that for `m >= 0` and `n >= 0`,
`(m + n)^2 + (m + n + 4)^2 >= (100mn)/(m + n + 1).` (2 marks)
--- 6 WORK AREA LINES (style=lined) ---
Mechanics, EXT2 M1 2013 HSC 15d
A ball of mass `m` is projected vertically into the air from the ground with initial velocity `u`. After reaching the maximum height `H` it falls back to the ground. While in the air, the ball experiences a resistive force `kv^2`, where `v` is the velocity of the ball and `k` is a constant.
The equation of motion when the ball falls can be written as
`m dot v = mg - kv^2.` (Do NOT prove this.)
- Show that the terminal velocity `v_T` of the ball when it falls is
`sqrt ((mg)/k).` (1 mark)
--- 2 WORK AREA LINES (style=lined) ---
- Show that when the ball goes up, the maximum height `H` is
`H = (v_T^2)/(2g) ln (1 + u^2/(v_T^2)).` (3 marks)
--- 6 WORK AREA LINES (style=lined) ---
- When the ball falls from height `H` it hits the ground with velocity `w`.
Show that `1/w^2 = 1/u^2 + 1/(v_T^2).` (2 marks)
--- 8 WORK AREA LINES (style=lined) ---
Harder Ext1 Topics, EXT2 2013 HSC 14d
A triangle has vertices `A, B` and `C`. The point `D` lies on the interval `AB` such that `AD = 3` and `DB = 5`. The point `E` lies on the interval `AC` such that `AE = 4`, `DE = 3` and `EC = 2`.
- Prove that `Delta ABC` and `Delta AED` are similar. (1 mark)
- Prove that `BCED` is a cyclic quadrilateral. (1 mark)
- Show that `CD = sqrt 21`. (2 marks)
- Find the exact value of the radius of the circle passing through the points `B, C, E and D`. (2 marks)
Proof, EXT2 P1 2014 HSC 16b
Suppose `n` is a positive integer.
- Show that
`-x^(2n) ≤ 1/(1 + x^2) − (1 − x^2 + x^4 − x^6 + … + (-1)^(n − 1) x^(2n − 2)) ≤ x^(2n)`. (3 marks)
--- 6 WORK AREA LINES (style=lined) ---
- Use integration to deduce that
`-1/(2n + 1) ≤ pi/4 − (1 − 1/3 + 1/5 − … + (-1)^(n − 1) 1/(2n − 1)) ≤ 1/(2n + 1)`. (2 marks)
--- 6 WORK AREA LINES (style=lined) ---
- Explain why `pi/4 = 1 − 1/3 + 1/5 − 1/7 + …`. (1 mark)
--- 5 WORK AREA LINES (style=lined) ---
Mechanics, EXT2 2014 HSC 15c
A toy aeroplane `P` of mass `m` is attached to a fixed point `O` by a string of length `l`. The string makes an angle `ø` with the horizontal. The aeroplane moves in uniform circular motion with velocity `v` in a circle of radius `r` in a horizontal plane.
The forces acting on the aeroplane are the gravitational force `mg`, the tension force `T` in the string and a vertical lifting force `kv^2`, where `k` is a positive constant.
- By resolving the forces on the aeroplane in the horizontal and the vertical directions, show that
`(sin\ ø)/(cos^2\ ø) = (lk)/m − (lg)/(v^2)`. (3 marks) - Part (i) implies that
`(sin\ ø)/(cos^2\ ø) < (lk)/m`. (Do NOT prove this.) - Use this to show that
- `sin\ ø < (sqrt(m^2 + 4l^2k^2) − m)/(2lk).` (2 marks)
- `sin\ ø < (sqrt(m^2 + 4l^2k^2) − m)/(2lk).` (2 marks)
- Show that
`(sin\ ø)/(cos^2\ ø)` is an increasing function of `ø` for `-pi/2 < ø < pi/2`. (2 marks) - Explain why `ø` increases as `v` increases. (1 mark)
Complex Numbers, EXT2 N2 2014 HSC 15b
- Using de Moivre’s theorem, or otherwise, show that for every positive integer `n`,
`(1 + i)^n + (1 − i)^n = 2(sqrt2)^n\ cos\ (npi)/4`. (2 marks)
--- 6 WORK AREA LINES (style=lined) ---
- Hence, or otherwise, show that for every positive integer `n` divisible by 4,
`((n),(0)) − ((n),(2)) + ((n),(4)) − ((n),(6)) + … + ((n),(n)) = (-1)^(n/4)(sqrt2)^n` (3 marks)
--- 6 WORK AREA LINES (style=lined) ---
Proof, EXT2 P1 2014 HSC 15a
Three positive real numbers `a`, `b` and `c` are such that `a + b + c = 1` and `a ≤ b ≤ c`.
By considering the expansion of `(a + b + c)^2`, or otherwise, show that
`qquad 5a^2 + 3b^2 +c^2 ≤ 1`. (2 marks)
--- 5 WORK AREA LINES (style=lined) ---
Mechanics, EXT2* M1 2004 HSC 7a
The rise and fall of the tide is assumed to be simple harmonic, with the time between successive high tides being 12.5 hours. A ship is to sail from a wharf to the harbour entrance and then out to sea. On the morning the ship is to sail, high tide at the wharf occurs at 2 am. The water depths at the wharf at high tide and low tide are 10 metres and 4 metres respectively.
- Show that the water depth, `y` metres, at the wharf is given by
`y = 7 + 3 cos\ ((4pit)/(25))`, where `t` is the number of hours after high tide. (2 marks)
--- 5 WORK AREA LINES (style=lined) ---
- An overhead power cable obstructs the ship’s exit from the wharf. The ship can only leave if the water depth at the wharf is 8.5 metres or less.
Show that the earliest possible time that the ship can leave the wharf is 4:05 am. (2 marks)
--- 8 WORK AREA LINES (style=lined) ---
- At the harbour entrance, the difference between the water level at high tide and low tide is also 6 metres. However, tides at the harbour entrance occur 1 hour earlier than at the wharf. In order for the ship to be able to sail through the shallow harbour entrance, the water level must be at least 2 metres above the low tide level.
The ship takes 20 minutes to sail from the wharf to the harbour entrance and it must be out to sea by 7 am. What is the latest time the ship can leave the wharf? (2 marks)
--- 6 WORK AREA LINES (style=lined) ---
Mechanics, EXT2* M1 2007 HSC 7b
A small paintball is fired from the origin with initial velocity `14` metres per second towards an eight-metre high barrier. The origin is at ground level, `10` metres from the base of the barrier.
The equations of motion are
`x = 14t\ cos\ theta`
`y = 14t\ sin\ theta – 4.9t^2`
where `theta` is the angle to the horizontal at which the paintball is fired and `t` is the time in seconds. (Do NOT prove these equations of motion)
- Show that the equation of trajectory of the paintball is
`y = mx − ((1 + m^2)/40)x^2`, where `m = tan\ theta`. (2 mark)
--- 6 WORK AREA LINES (style=lined) ---
- Show that the paintball hits the barrier at height `h` metres when
`m = 2 ± sqrt(3 − 0.4h)`.
Hence determine the maximum value of `h`. (2 marks)
--- 8 WORK AREA LINES (style=lined) ---
- There is a large hole in the barrier. The bottom of the hole is `3.9` metres above the ground and the top of the hole is `5.9` metres above the ground. The paintball passes through the hole if `m` is in one of two intervals. One interval is `2.8 ≤ m ≤ 3.2`.
Find the other interval. (2 marks)
--- 8 WORK AREA LINES (style=lined) ---
- Show that, if the paintball passes through the hole, the range is
`(40m)/(1 + m^2)\ \ text(metres.)`
Hence find the widths of the two intervals in which the paintball can land at ground level on the other side of the barrier. (3 marks)
--- 10 WORK AREA LINES (style=lined) ---
L&E, EXT1 2007 HSC 7a
Functions, EXT1 F1 2007 HSC 6b
Consider the function `f(x) = e^x − e^(-x)`.
- Show that `f(x)` is increasing for all values of `x`. (1 mark)
--- 3 WORK AREA LINES (style=lined) ---
- Show that the inverse function is given by
`qquad qquad f^(-1)(x) = log_e((x + sqrt(x^2 + 4))/2)` (3 marks)
--- 7 WORK AREA LINES (style=lined) ---
- Hence, or otherwise, solve `e^x - e^(-x) = 5`. Give your answer correct to two decimal places. (1 mark)
--- 4 WORK AREA LINES (style=lined) ---
- « Previous Page
- 1
- …
- 16
- 17
- 18
- 19
- 20
- 21
- Next Page »