- Write ` 1 + i` in the form `r (cos theta + i sin theta).` (2 marks)
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- Hence, or otherwise, find `(1 + i)^17` in the form `a + ib`, where `a` and `b` are integers. (3 marks)
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Calculus, EXT2 C1 2007 HSC 1e
It can be shown that
`2/(x^3 + x^2 + x + 1) = 1/(x + 1) - x/(x^2 + 1) + 1/(x^2 + 1).` (Do NOT prove this.)
Use this result to evaluate `int_(1/2)^2 2/(x^3 + x^2 + x + 1)\ dx.` (4 marks)
Calculus, EXT2 C1 2007 HSC 1c
Evaluate `int_0^pi x cos x\ dx.` (3 marks)
Calculus, EXT2 C1 2007 HSC 1b
Find `int tan^2 x sec^2 x\ dx.` (2 marks)
Functions, EXT1′ F1 2015 HSC 8 MC
Complex Numbers, EXT2 N1 2015 HSC 5 MC
Given that `z = 1 − i`, which expression is equal to `z^3 ?`
(A) `sqrt 2 (cos((-3 pi)/4) + i sin((-3 pi)/4))`
(B) `2 sqrt 2 (cos((-3 pi)/4) + i sin((-3 pi)/4))`
(C) `sqrt 2 (cos((3 pi)/4) + i sin((3 pi)/4))`
(D) `2 sqrt 2 (cos((3 pi)/4) + i sin((3 pi)/4))`
Conics, EXT2 2015 HSC 1 MC
Which conic has eccentricity `sqrt 13/3?`
(A) `x^2/3 + y^2/2 = 1`
(B) `x^2/3^2 + y^2/2^2 = 1`
(C) `x^2/3 - y^2/2 = 1`
(D) `x^2/3^2 - y^2/2^2 = 1`
Functions, EXT1′ F1 2014 HSC 5 MC
Complex Numbers, EXT2 N1 2014 HSC 4 MC
Given `z = 2(cos\ pi/3 + i sin\ pi/3)`, which expression is equal to `(bar {:z:})^(−1)`?
(A) `1/2(cos\ pi/3 − i sin\ pi/3)`
(B) `2(cos\ pi/3 − i sin\ pi/3)`
(C) `1/2(cos\ pi/3 + i sin\ pi/3)`
(D) `2(cos\ pi/3 + i sin\ pi/3)`
Conics, EXT2 2014 HSC 3 MC
What is the eccentricity of the ellipse `9x^2 + 16y^2 = 25`?
(A) `7/16`
(B) `sqrt7/4`
(C) `sqrt15/4`
(D) `5/4`
Polynomials, EXT2 2014 HSC 2 MC
The polynomial `P(z)` has real coefficients, and `z = 2 − i` is a root of `P(z)`.
Which quadratic polynomial must be a factor of `P(z)`?
(A) `z^2 −4z +5`
(B) `z^2 +4z +5`
(C) `z^2 −4z +3`
(D) `z^2 +4z +3`
Volumes, EXT2 2013 HSC 8 MC
The base of a solid is the region bounded by the circle `x^2 + y^2 = 16`. Vertical cross-sections are squares perpendicular to the `x`-axis as shown in the diagram.
Which integral represents the volume of the solid?
(A) `int_-4^4 4x^2\ dx`
(B) `int_-4^4 4 pi x^2\ dx`
(C) `int_-4^4 4 (16 - x^2)\ dx`
(D) `int_-4^4 4 pi (16 - x^2)\ dx`
Integration, EXT2 2013 HSC 6 MC
Which expression is equal to `int 1/sqrt (x^2 - 6x + 5)\ dx?`
(A) `sin^-1 ((x - 3)/2) + C`
(B) `cos^-1 ((x - 3)/2) + C`
(C) `ln (x - 3 + sqrt ((x - 3)^2 + 4)) + C`
(D) `ln (x - 3 + sqrt ((x - 3)^2 - 4)) + C`
Functions, EXT1′ F2 2013 HSC 4 MC
The polynomial equation `4x^3 + x^2 − 3x + 5 = 0` has roots `alpha, beta and gamma.`
Which polynomial equation has roots `alpha + 1, beta + 1 and gamma + 1?`
(A) `4x^3 - 11x^2 + 7x + 5 = 0`
(B) `4x^3 + x^2 - 3x + 6 = 0`
(C) `4x^3 + 13x^2 + 11x + 7 = 0`
(D) `4x^3 - 2x^2 - 2x + 8 = 0`
Conics, EXT2 2013 HSC 2 MC
Which pair of equations gives the directrices of `4x^2 - 25y^2 = 100?`
(A) `x = +- 25/sqrt 29`
(B) `x = +- 1/sqrt 29`
(C) `x = +- sqrt 29`
(D) `x = +- (sqrt 29)/25`
Complex Numbers, EXT2 N1 2006 HSC 2b
- Express `sqrt 3 - i` in modulus-argument form. (2 marks)
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- Express `(sqrt 3 - i)^7` in modulus-argument form. (2 marks)
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- Hence express `(sqrt 3 - i)^7` in the form `x + iy.` (1 mark)
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Complex Numbers, EXT2 N1 2006 HSC 2a
Let `z = 3 + i` and `w = 2 - 5i`. Find, in the form `x + iy`,
- `z^2.` (1 mark)
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- `bar z w.` (1 mark)
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- `w/z.` (1 mark)
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Calculus, EXT2 C1 2006 HSC 1e
Use the substitution `t = tan\ theta/2` to show that
`int_(pi/2)^((2 pi)/3) (d theta)/(sin theta) = 1/2 log 3.` (3 marks)
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Calculus, EXT2 C1 2006 HSC 1d
Evaluate `int_0^2 te^-t\ dt.` (3 marks)
Calculus, EXT2 C1 2006 HSC 1c
- Given that `(16x - 43)/((x - 3)^2 (x + 2))` can be written as
`qquad (16x - 43)/((x - 3)^2 (x + 2)) = a/(x - 3)^2 + b/(x - 3) + c/(x + 2)`,
where `a, b` and `c` are real numbers, find `a, b and c.` (3 marks)
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- Hence find `int (16x - 43)/((x - 3)^2 (x + 2))\ dx.` (2 marks)
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Harder Ext1 Topics, EXT2 2009 HSC 5a
In the diagram `AB` is the diameter of the circle. The chords `AC` and `BD` intersect at `X`. The point `Y` lies on `AB` such that `XY` is perpendicular to `AB`. The point `K` is the intersection of `AD` produced and `YX` produced.
Copy or trace the diagram into your writing booklet.
- Show that `/_ AKY = /_ ABD.` (2 marks)
- Show that `CKDX` is a cyclic quadrilateral. (2 marks)
- Show that `B, C and K` are collinear. (2 marks)
Functions, EXT1′ F1 2009 HSC 3a
The diagram shows the graph `y = f(x).`
Draw separate one-third page sketches of the graphs of the following:
- `y = 1/(f(x)) .` (2 marks)
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- `y = f(x)\ f(x)` (2 marks)
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- `y = f(x^2).` (2 marks)
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Graphs, EXT2 2009 HSC 3b
Find the coordinates of the points where the tangent to the curve `x^2 + 2xy + 3y^2 = 18` is horizontal. (3 marks)
Complex Numbers, EXT2 N2 2009 HSC 2f
- Find the square roots of `3 +4i.` (3 marks)
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- Hence, or otherwise, solve the equation
`z^2 + iz - 1 - i = 0.` (2 marks)
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Complex Numbers, EXT2 N2 2009 HSC 2d
Sketch the region in the complex plane where the inequalities `| z - 1 | <= 2` and `-pi/4 <= text(arg) (z - 1) <= pi/4` hold simultaneously. (2 marks)
Complex Numbers, EXT2 N1 2009 HSC 2c
The points `P` and `Q` on the Argand diagram represent the complex numbers `z` and `w` respectively.
Copy the diagram into your writing booklet, and mark on it the following points:
- the point `R` representing `iz.` (1 mark)
- the point `S` representing `bar w.` (1 mark)
- the point `T` representing `z + w.` (1 mark)
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Calculus, EXT2 C1 2009 HSC 1c
Find `int x^2/(1 + 4x^2)\ dx.` (3 marks)
Calculus, EXT2 C1 2009 HSC 1b
Find `int x e^(2x)\ dx.` (2 marks)
Calculus, EXT2 C1 2009 HSC 1a
Find `int (ln x)/x\ dx.` (2 marks)
Polynomials, EXT2 2010 HSC 6c
- Expand `(cos theta + i sin theta)^5` using the binomial theorem. (1 mark)
- Expand `(cos theta + i sin theta)^5` using de Moivre’s theorem, and hence show that
- `sin 5theta = 16 sin^5 theta − 20sin^3 theta + 5 sin theta`. (3 marks)
- Deduce that
- `x = sin (pi/10)` is one of the solutions to
- `16x^5 − 20x^3 + 5x − 1 = 0`. (1 mark)
- Find the polynomial `p(x)` such that `(x − 1) p(x) = 16x^5 − 20x^3 + 5x − 1`. (1 mark)
- Find the value of `a` such that `p(x) = (4x^2 + ax − 1)^2`. (1 mark)
- Hence find an exact value for
- `sin (pi/10)`. (1 mark)
Conics, EXT2 2010 HSC 5a
The diagram shows two circles, `C_1` and `C_2`, centred at the origin with radii `a` and `b`, where `a > b`.
The point `A` lies on `C_1` and has coordinates `(a cos theta, a sin theta)`.
The point `B` is the intersection of `OA` and `C_2`.
The point `P` is the intersection of the horizontal line through `B` and the vertical line through `A`.
- Write down the coordinates of `B`. (1 mark)
- Show that `P` lies on the ellipse
`(x^2)/(a^2) + (y^2)/(b^2) = 1`. (1 mark) - Find the equation of the tangent to the ellipse
`(x^2)/(a^2) + (y^2)/(b^2) = 1` at `P`. (2 marks) - Assume that `A` is not on the `y`-axis.
- Show that the tangent to the circle `C_1` at `A`, and the tangent to the ellipse
`(x^2)/(a^2) + (y^2)/(b^2) = 1` at `P`, intersect at a point on the `x`-axis. (2 marks)
Mechanics, EXT2 2010 HSC 4b
A bend in a highway is part of a circle of radius `r`, centre `O`. Around the bend the highway is banked at an angle `α` to the horizontal.
A car is travelling around the bend at a constant speed `v`. Assume that the car is represented by a point `P` of mass `m`. The forces acting on the car are a lateral force `F`, the gravitational force `mg` and a normal reaction `N` to the road, as shown in the diagram.
- By resolving forces, show that
`F = mg sin α − (mv^2)/r cos α`. (3 marks)
- Find an expression for `v` such that the lateral force `F` is zero. (1 mark)
Graphs, EXT2 2010 HSC 4a
- A curve is defined implicitly by `sqrtx + sqrty = 1`.
- Use implicit differentiation to find `(dy)/(dx)`. (2 marks)
- Sketch the curve `sqrtx + sqrty = 1`. (2 marks)
- Sketch the curve `sqrt(|\ x\ |) + sqrt(|\ y\ |) = 1` (1 mark)
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)
Complex Numbers, EXT2 N2 2010 HSC 2c
Sketch the region in the complex plane where the inequalities `1 ≤ |\ z\ | ≤ 2` and `0 ≤ z + bar z ≤ 3` hold simultaneously. (2 marks)
Complex Numbers, EXT2 N1 2010 HSC 2b
- Express `-sqrt3 − i` in modulus–argument form. (2 marks)
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- Show that `(-sqrt3 − i)^6` is a real number. (2 marks)
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Calculus, EXT2 C1 2010 HSC 1d
Using the substitution `t = tan\ x/2`, or otherwise, evaluate `int_0^(pi/2) (dx)/(1 + sin\ x)`. (4 marks)
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Calculus, EXT2 C1 2010 HSC 1b
Evaluate `int_0^(pi/4) tan\ x\ dx`. (3 marks)
Calculus, EXT2 C1 2010 HSC 1a
Find `int x/(sqrt(1 + 3x^2))\ dx`. (2 marks)
Calculus, EXT2 C1 2011 HSC 7b
Let `I = int_1^3 (cos^2(pi/8 x))/(x(4 - x))\ dx.`
- Use the substitution `u = 4 - x` to show that
`I = int_1^3 (sin^2(pi/8 u))/(u(4 - u))\ du.` (2 marks)
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- Hence, find the value of `I`. (3 marks)
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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)
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- 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)
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- 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)
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Mechanics, EXT2 2011 HSC 5a
A small bead of mass `m` is attached to one end of a light string of length `R`. The other end of the string is fixed at height `2h` above the centre of a sphere of radius `R`, as shown in the diagram. The bead moves in a circle of radius `r` on the surface of the sphere and has constant angular velocity `omega > 0`. The string makes an angle of `theta` with the vertical.
Three forces act on the bead: the tension force `F` of the string, the normal reaction force `N` to the surface of the sphere, and the gravitational force `mg`.
- By resolving the forces horizontally and vertically on a diagram, show that
- `F sin theta - N sin theta = m omega^2 r`
- and
- `F cos theta + N cos theta = mg.` (2 marks)
- Show that
- `N = 1/2 mg sec theta - 1/2 m omega^2 r\ text(cosec)\ theta.` (2 marks)
- Show that the bead remains in contact with the sphere if
- `omega <= sqrt (g/h).` (2 marks)
Harder Ext1 Topics, EXT2 2011 HSC 4b
In the diagram, `ABCD` is a cyclic quadrilateral. The point `E` lies on the circle through the points `A, B, C` and `D` such that `AE\ text(||)\ BC`. The line `ED` meets the line `BA` at the point `F`. The point `G` lies on the line `CD` such that `FG\ text(||)\ BC.`
Copy or trace the diagram into your writing booklet.
- Prove that `FADG` is a cyclic quadrilateral. (2 marks)
- Explain why `/_ GFD =/_ AED.` (1 mark)
- Prove that `GA` is a tangent to the circle through the points `A, B, C` and `D.` (2 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)
Functions, EXT1′ F1 2011 HSC 3a
- Draw a sketch of the graph
`quad y = sin\ pi/2 x` for `0 < x < 4.` (1 mark)
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- Find `lim_(x -> 0) x/(sin\ pi/2 x).` (1 mark)
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- Draw a sketch of the graph
`quad y = x/(sin\ pi/2 x)` for `0 < x < 4.` (2 marks) -
(Do NOT calculate the coordinates of any turning points.)
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Complex Numbers, EXT2 N2 2011 HSC 2d
- Use the binomial theorem to expand `(cos theta + i sin theta)^3.` (1 mark)
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- Use de Moivre’s theorem and your result from part (i) to prove that
`cos^3 theta = 1/4 cos 3 theta + 3/4 cos theta.` (3 marks)
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- Hence, or otherwise, find the smallest positive solution of
`4 cos^3 theta - 3 cos theta = 1.` (2 marks)
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Complex Numbers, EXT2 N2 2011 HSC 2c
Find, in modulus-argument form, all solutions of `z^3 = 8.` (2 marks)
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Complex Numbers, EXT2 N2 2011 HSC 2b
On the Argand diagram, the complex numbers `0, 1 + i sqrt 3 , sqrt 3 + i` and `z` form a rhombus.
- Find `z` in the form `a + ib`, where `a` and `b` are real numbers. (1 mark)
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- An interior angle, `theta`, of the rhombus is marked on the diagram.
Find the value of `theta.` (2 marks)
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Calculus, EXT2 C1 2011 HSC 1e
Evaluate `int_-1^1 1/(5 - 2t + t^2) \ dt.` (3 marks)
Calculus, EXT2 C1 2011 HSC 1d
Find `int cos^3 theta\ d theta` (3 marks)
Calculus, EXT2 C1 2011 HSC 1c
- Find real numbers `a, b` and `c` such that
`1/(x^2 (x - 1)) = a/x + b/x^2 + c/(x - 1).` (2 marks)
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- Hence, find `int 1/(x^2 (x - 1))\ dx` (2 marks)
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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)
Integration, EXT2 2012 HSC 14a
Find `int(3x^2 + 8)/(x(x^2 +4))\ dx`. (3 marks)
Harder Ext1 Topics, EXT2 2012 HSC 13b
Mechanics, EXT2 M1 2012 HSC 13a
An object on the surface of a liquid is released at time `t = 0` and immediately sinks. Let `x` be its displacement in metres in a downward direction from the surface at time `t` seconds.
The equation of motion is given by
`(dv)/(dt) = 10 − (v^2)/40`,
where `v` is the velocity of the object.
- Show that `v = (20(e^t − 1))/(e^t + 1)`. (4 marks)
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- Use `(dv)/(dt) = v (dv)/(dx)` to show that
`x = 20\ log_e(400/(400 − v^2))` (2 marks)
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- How far does the object sink in the first 4 seconds? (2 marks)
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Calculus, EXT2 C1 2012 HSC 12c
For every integer `n ≥ 0` let `I_n = int_1^(e^2)(log_e x)^n\ dx`.
Show that for `n ≥ 1,`
`I_n = e^2 2^n − nI_(n − 1)`. (3 marks)
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Conics, EXT2 2012 HSC 12b
The diagram shows the ellipse `(x^2)/(a^2) + (y^2)/(b^2) = 1` with `a > b`. The ellipse has focus `S` and eccentricity `e`. The tangent to the ellipse at `P(x_0, y_0)` meets the `x`-axis at `T`. The normal at `P` meets the `x`-axis at `N`.
- Show that the tangent to the ellipse at `P` is given by the equation
- `y − y_0 = -(b^2x_0)/(a^2y_0)(x − x_0)`. (2 marks)
- Show that the `x`-coordinate of `N` is `x_0e^2`. (2 marks)
- Show that `ON xx OT = OS^2` (2 marks)
Calculus, EXT2 C1 2012 HSC 12a
Using the substitution `t = tan\ theta/2`, or otherwise, find `int(d theta)/(1 − cos\ theta)`. (3 marks)
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Calculus, EXT2 C1 2012 HSC 11e
Evaluate `int_0^1 (e^(2x))/(e^(2x) + 1)\ dx`. (3 marks)
Complex Numbers, EXT2 N2 2012 HSC 11b
Shade the region on the Argand diagram where the two inequalities
`|\ z + 2\ | ≥ 2` and `|\ z − i\ | ≤ 1`
both hold. (2 marks)
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