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STEP Maths I, II, III 1996 Solutions

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    (Updated as far as #86.) SimonM - 23.03.2009

    For explanation, scroll down:

    STEP Mathematics I
    1: Solution by ukgea
    2: Solution by Speleo
    3: Solution by Speleo
    4: Solution by Datr
    5: Solution by DFranklin
    6: Solution by Speleo
    7: Solution by Speleo
    8: Solution by Speleo
    9: Solution by dr_98_98
    10: Solution by brianeverit
    11: Solution by ad absurdum
    12: Solution by brianeverit
    13: Solution by Datr
    14: Solution by brianeverit


    STEP Mathematics II:
    1: Solved in Siklos Booklet on p20
    2: Solved in Siklos Booklet on p24//Solution by Speleo
    3: Solution by DFranklin
    4: Solution by Datr
    5: Solution by Speleo
    6: Solution by ukgea
    7: Solution by DFranklin
    8: Solution by Speleo (1), (2)
    9: Solution by brianeverit
    10: Solution by brianeverit
    11: Solution by brianeverit
    12: Solution by Datr
    13: Solution by brianeverit
    14: Solution by brianeverit


    STEP Mathematics III
    1: Solution by DFranklin
    2: Solution by Insparato (Incomplete)
    3: Solution by Rabite (1), (2)
    4: Solution by ukgea
    5: Solution by dvs
    6: Solution by dvs
    7: Solution by DFranklin
    8: Solution by Dvs
    9: Solution by DFranklin
    10: Solution by DFranklin
    11: Solution by DFranklin
    12: Solution by brianeverit
    13: Solution by DFranklin
    14: Solution by DFranklin



    Explanation:
    In this thread, we will try to collectively provide solutions for the questions from the STEP Maths I, II and III papers from 1996, seeing as no solutions are available on the net.

    If you want to contribute, simply find a question which is marked with "unsolved" above (you might want to check for new solutions after the time the list was last updated as well), solve it, and post your solution in the thread. I will try to update the list as often as possible. If you have a solution to a question that uses a different idea from one already posted, submit it as well, as it's often useful to see different approaches to the same problem.

    If you find an error in one of the solutions, simply say so, and hope that whoever wrote the solution in the first place corrects it.

    Solutions written by TSR members:
    1987 - 1988 - 1989 - 1990 - 1991 - 1992 - 1993 - 1994 - 1995 - 1996 - 1997 - 1998 - 1999 - 2000 - 2001 - 2002 - 2003 - 2004 - 2005 - 2006 - 2007
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    Im going to search for any answers in Siklos booklet.
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    Nevermind, apparently Siklos did that one too.
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    STEP II Q4

    \displaystyle cos4u = 2cos^22u - 1 = 2(2cos^2u - 1)^2 = 2(4cos^4u - 4cos^2u + 1) - 1
    \displaystyle cos4u = 8cos^4u - 8cos^2u + 1

    \displaystyle I = \int_{-1}^{1} \frac{1}{\sqrt{1+x} + \sqrt{1 - x} + 2} dx

    x = cost
    \frac{dx}{dt} = -sint

    \displaystyle I = \int_{\pi}^{0} \frac{-sint}{\sqrt{1+cost} + \sqrt{1 - cost} + 2} dt

    \displaystyle I = \int_{0}^{\pi} \frac{sint}{\sqrt{2cos^2\frac{t}  {2}} + \sqrt{2sin^2\frac{t}{2}} + 2} dt = \int_{0}^{\pi} \frac{sint}{\sqrt{2}cos\frac{t}{  2} + \sqrt{2}sin\frac{t}{2} + 2} dt

    \displaystyle Rcos\left(\frac{t}{2} + \alpha\right) = Rcos\left(\frac{t}{2}\right)cos\  alpha - Rsin\left(\frac{t}{2}\right)sin\  alpha

    Rcos\alpha = \sqrt{2}
    Rsin\alpha = -\sqrt{2}

    \alpha = -\frac{\pi}{4}
    R = 2

    \displaystyle I = \int_{0}^{\pi} \frac{sint}{2cos(\frac{t}{2} - \frac{\pi}{4}) + 2} dt = \int_{0}^{\pi} \frac{sint}{4cos^2(\frac{t}{4} - \frac{\pi}{8})} dt

    \displaystyle u = \frac{t}{4} - \frac{\pi}{8}
    \displaystyle \frac{du}{dt} = \frac{1}{4}

    \displaystyle I = \int_0^\pi \frac{sint}{4cos^2(\frac{t}{4} - \frac{\pi}{8})} dt = \int_{-\frac{\pi}{8}}^{\frac{\pi}{8}} \frac{sin(4u + \frac{\pi}{2})}{cos^2u} du

    sin(4u + \frac{\pi}{2}) = sin4ucos\frac{\pi}{2} + cos4usin\frac{\pi}{2} = cos4u

    \displaystyle I = \int_{-\frac{\pi}{8}}^{\frac{\pi}{8}} \frac{8cos^4u - 8cos^2u + 1}{cos^2u} du = \int_{-\frac{\pi}{8}}^{\frac{\pi}{8}} 8cos^2u - 8 + sec^2u \,du

    \displaystyle I = \int_{-\frac{\pi}{8}}^{\frac{\pi}{8}} 4cos2u - 4 + sec^2u \,du = \left[2sin2u - 4u + tanu\right]_{-\frac{\pi}{8}}^{\frac{\pi}{8}}

    I = 4\sqrt{2} - \pi - 2
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    (Original post by datr)
    I'm just typing up STEP II Q4
    I managed to get that one out too. But it was a while back now.
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    Yeah it's a nice question, most of the integration/differentiation ones are - I normally just go straight for them first.
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    Trying to post it all in one go isn't working.

    STEP I Question 3

    i) f(n) = n^5 - n^3 = n^3(n^2 - 1)
    = n^3(n+1)(n-1)
    One of n-1, n, n+1 must be divisible by three, therefore f(n) is.
    If n is even, n^3 is divisible by 8, so f(n) is divisible by 24.
    If n is odd, n+1, n-1 are divisible by 2 and one of them is divisible by 4.
    Therefore f(n) is divisible by 2*4*3 = 24.

    ii) g(n) = (2^n + 1)(2^n - 1)
    2^n is not divisible by 3, but one of (2^n - 1), 2^n and 2^n + 1 must be.
    Therefore g(n) must be divisible by 3.

    iii) h(n) = n^3 - 1 = (n-1)(n^2+n+1). n-1 = 0 mod 3. n = 1 mod 3.
    n^2 + n + 1 = 1 + 1 + 1 = 0 mod 3.
    h(n) has two factors each divisible by 3 and is therefore divisible by 9.
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    This one looks like it may be wrong :/

    STEP I Question 7

    i) dy/dt = -ky; k>=1
    lny = C - kt
    y = e^(C-kt) = Ae^(-kt)
    When t = 0, y = 1
    1 = A
    y = e^(-kt)
    y = [e^(-k)]^t
    Let b = e^(-k)
    y = b^t
    Since k>=1, b<=1/e, b<1

    ii) dy/dt = a - ky
    dy/dt = a + ylnb
    dy/dt - ylnb = a
    Integrating factor = e^(-tlnb)
    e^(-tlnb)dy/dt + e^(-tlnb)y(-lnb) = ae^(-tlnb)
    d/dt[ye^(-tlnb)] = ae^(-tlnb)
    ye^(-tlnb) = -(a/lnb)e^(-tlnb) + C
    When t = 0. y = 1
    1 = C - (a/lnb)
    C = 1 + (a/lnb)
    ye^(-tlnb) = -(a/lnb)e^(-tlnb) + (a/lnb) + 1
    y = -(a/lnb) + (a/lnb)e^(tlnb) + e^(tlnb)
    y = [1 + (a/lnb)]b^t - (a/lnb)
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    STEP I Question 8

    i) S = 0.38383838...
    = 38/100 + 38/(100)^2 + 38/(100)^3 + ...
    = [38/100]/[1-(1/100)]
    = (38/100)/(99/100)
    = 38/99.

    ii) x = 0.a1a2...aNb1b2b3...bkb1b2...bkb 1..
    = 0.a1a2...aN + 0.000...000b1b2...bkb1...
    = a1a2...aN/(10^N) + 0.b1b2...bkb1.../(10^N)
    But 0.b1b2...bkb1...
    = b1b2...bk/(10^k) + b1b2...bk(10^k)^2 + ...
    = [b1b2...bk/10^k]/[1 - 1/(10^k)]
    = b1b2...bk/(10^k - 1)
    So x = a1a2...aN/(10^N) + b1b2...bk/(10^N)(10^k - 1)
    Since a1a2...aN and b1b2...bk integers, x is rational.
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    I haven't checked to see if the last part is consistent with the Siklos book.

    STEP II Question 2

    Let a = yz, b = zx, c = xy.

    I: 2a + b - 5c = 2
    II: a - b + 2c = 1
    III: a - 2b + 6c = 3

    I-2II = 3b - 9c = 0, b = 3c
    I-2III = 5b - 17c = -4
    -2c = -4
    c = 2
    b = 6
    a = 1 - 4 + 6 = 3
    abc = 36
    (xyz)^2 = 36
    xyz = +- 6

    Factors of +- 6 must be +- 1, 2, 3 or 1, 1, 6.
    Since yz = 3 and xy = 2; 1, 1, 6 is not possible.
    Therefore a, b, c are +- 1, 2, 3.
    |y| = 1 clearly, so |z| = 3 and |x| = 2.
    Since xy, yz, zx are all > 0,
    (x,y,z) = (2,1,3) or (-2,-1,-3).
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    This is ridiculous, TSR won't post this in one go:

    STEP II Question 8

    g(0) = f(0) + f(-0) = 2

    g'(x) = f'(x) - f'(-x)
    g'(0) = -1 + 1 = 0
    g''(x) = f''(x) + f''(-x)
    f''(x) = x + 3cos2x - f(-x)
    f''(-x) = -x + 3cos2x - f(x)
    g''(x) = 6cos2x - f(x) - f(-x)
    g''(x) = 6cos2x - g(x)
    g''(x) + g(x) = 6cos2x

    Aux quadratic -> m^2 + 1 = 0
    m = +-i
    g(x) = Acosx + Bsinx + ...
    g(x) = pcos2x
    -4p + p = 6
    p = -2
    g(x) = Acosx + Bsinx - 2cos2x
    g(0) = 2.
    A - 2 = 2
    A = 4
    g'(x) = -4sinx + Bcosx + 4sin2x
    g'(0) = 0 = B
    g(x) = 4cosx - 2cos2x
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    h(x) = f(x) - f(-x)
    h(0) = 0
    h'(x) = f'(x) + f'(-x)
    h'(0) = -2
    h''(x) = f''(x) - f''(-x)
    f''(x) = x + 3cos2x - f(-x)
    f''(-x) = -x + 3cos2x - f(x)
    h''(x) = 2x - f(-x) + f(x)
    h''(x) = 2x + h(x)
    h''(x) - h(x) = 2x.
    Aux quadratic is m²-1 = 0 ie m=±1.
    If h(x) = kx, then h(x) = -2x.
    So h(x) = Ae^x + Be^-x -2x.
    h(0) = 0=A+B -0
    h`(0) = -2 =A-B - 2
    So A+B = 0 and A-B = 0, so A and B are both 0.

    h(x) = -2x

    2f(x) = g(x) + h(x)

    f(x) = 2cosx - cos2x - x

    Correction in italics thanks to Rabite
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    After several attempts, I have a solution to Paper I, Q5. I think I must have missed something obvious though - this was very hard work. (Though largely because it is so easy to make a mistake; the written up answer probably won't look too bad).
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    Step QI, P5.

    (i) (r+s\sqrt{3})=4-2\sqrt{3} \implies r^2+3s^2+2rs\sqrt{3} = 4-2\sqrt{3}. Suppose rs \neq -1. Then
    r^2+3s^2-4 = -2(1+rs)\sqrt{3} would give us a rational expression for \sqrt{3}, contradiction.

    So rs=1, \text{ so } s = -1/r \text { and so } r^2+3/r^2 = 4 \implies r^4-4r^2+3 = 0\\

\implies r^2 = 1, 3.

    As r is rational, r^2 = 1, \text{ so } r = \pm 1, s = -r. (i.e. our solutions are \pm(1-\sqrt{3}).

    (ii) (p+qi)^2 = p^2-q^2+2pqi = 3-2\sqrt{3} + 2(1-\sqrt{3})i. So
    p^2-q^2 = 3-2\sqrt{3}, pq = 1-\sqrt{3}. Again, we set up the quadratic and get:
    p^4-(3-2\sqrt{3})p^2-(1-\sqrt{3})^2 = 0. Use the quadratic formula to get:
    2p^2 = (3-2\sqrt{3}) \pm \sqrt{(3-2\sqrt{3})^2+4(1-\sqrt{3}))^2}
    Now  (3-2\sqrt{3})^2 = 3^2+4\cdot 3 - 12\sqrt{3} = 21 - 12\sqrt{3}, \quad (1-\sqrt{3})^2 = 1 + 3 - 2 \sqrt{3} = 4-2\sqrt{3}.
    So (3-2\sqrt{3})^2+4(1-\sqrt{3}))^2 = 37 - 20 \sqrt{3}.

    As in (i), we need to find the square root of this, so look for rational \alpha, \beta with (\alpha+\beta\sqrt{3})^2 = 37 - 20\sqrt{3}.

    We find \beta = -10/\alpha and end up with a quadratic \alpha^4 - 37\alpha^2+300=0. Solve to find \alpha^2 = 12, 25 and since we want \alpha rational deduce \alpha = \pm 5 and so the square root is \pm(5-2\sqrt{3}).

    Plugging this back in, we have

    2p^2 = (3-2\sqrt{3}) \pm (5-2\sqrt{3}) = -2, 8-4\sqrt{3}.

    Since p is real, we take the latter root, dividing by 2 and using (i) we find p = \pm (1-\sqrt{3}). Use pq = (1-\sqrt{3}) to get final solutions \pm (1-\sqrt{3}+i).

    (iii) Again, use the quadratic formula:
    Unparseable or potentially dangerous latex formula. Error 4: no dvi output from LaTeX. It is likely that your formula contains syntax errors or worse.
    2z=2\pm\sqrt{4 - 4(1+i)(2\sqrt{3}-2)} \\

    \implies z = 1 \pm \sqrt{1 - (1+i)(2\sqrt{3}-2)} = 1 \pm \sqrt{3-2\sqrt{3}+2(1-\sqrt{3})i)
    .

    Using (ii),
     z = 1 \pm  (1-\sqrt{3}+i) = \sqrt{3} - i, 2-\sqrt{3}+i.
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    On the first page of the tests:

    "To be brought by the candidate: electronic calculators, [...]"

    Wtf? Were they allowed to use calculators back then? On all the later papers it says "Candidates may not use electronic calculators" instead. But anyone know why were they allowed previously and not now?
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    Step II, Q3:

    F0 = 0, F1 =1, F2 = F0+F1 = 1, F3 = F1 + F2 = 2, F4 = F3 + F2 = 3, F5 = 5, F6 = 8, F7 = 13.

    F0.F2-F1^2 = -1, F1.F3-F2^2 = 1, F2.F4-F3^2 = -1.

    Claim F_{n+1}F_{n-1}-F_n^2 = (-1)^n. Proof by induction on n.
    By above, true for n<=3. Assume true for n = k. Then
    F_{k+2}F_k - F_{k+1}^2 \\

= (F_{k+1}+F_k)F_k - F_{k+1}^2 \\ 

= F_k^2 +F_{k+1}F_k - F_{k+1}^2 = F_k^2 - F_{k+1}(F_{k+1} - F_k) \\

= [F_{k+1}F_{k-1}+(-1)^{k+1}]  - F_{k+1}(F_{k+1} - F_k) \\

= (-1)^{k+1} + F_{k+1}F_{k-1} - F_{k+1}(F_{k-1})\\

= (-1)^{k+1}F_{k+1}^2

    So true for n=k+1 and so true for all n by induction.

    Want to show F_{n+k} = F_kF_{n+1}+F_{k-1}F_n. Assume true for 1 < k <=m. Then
    F_{n+k+1} = F_{n+k}+F_{n+k-1}\\

= F_kF_{n+1}+F_{k-1}F_n + F_k-1F_{n+1}+F_{k-2}F_n\\

=(F_k+F_{k-1})F_{n+1}+(F_{k-1}+F_{k-2}F_n \\

=F_{k+1}F_{n+1}+F_k F_n.

    So true for k=m+1.

    Explictly when k=1 we have F_{n+1} = F_1F_{n+1}+F_0F_n = F_{n+1}. So true for k=1, so true for all k.
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    (Original post by ukgea)
    On the first page of the tests:

    "To be brought by the candidate: electronic calculators, [...]"

    Wtf? Were they allowed to use calculators back then? On all the later papers it says "Candidates may not use electronic calculators" instead. But anyone know why were they allowed previously and not now?
    I would guess it's a response to the increasing capabilities of calculators: graphing, algebra, programmability. Rather than try to rule out the "super calculators", simpler to just ban them all, and make sure the questions don't need them.

    I think Siklos also says there was a tendancy for the weaker candidates to just blindly throw everything into the calculator without thinking too much.
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    Paper II question 6

    We can easily see that the proper factors of 12 are 2, 3, 4, 6 and those of 16 are 2, 4, 8.

    Let M be a positive factor of N (not necessarily proper), Then f can be written as

    M = p_1^{n_1}p_2^{n_2}p_3^{n_3}\cdot P_r^{n_r}

    where 0 \leq n_i \leq m_i for all  1 \leq i \leq r.

    For each of the i, there are m_i + 1 ways of choosing the integers n_i, for a total number of F possible factors of N:

    F = (m_1+1)(m_2+1)(m_3+1)\cdot(m_r+1  )

    Each of these must be unique since prime factorisation is unique. But now we've counted the two "improper" factors 1 and N as well, so we need to remove these, and the total number of proper factors is

    (m_1+1)(m_2+1)(m_3+1)\cdot(m_r+1  ) - 2.

    (i) For the total number of proper factors to be 12, we must have

    (m_1+1)(m_2+1)(m_3+1)\cdot(m_r+1  ) = 14

    which means that, r must be 2 and m_1 + 1 = 2, m_2 + 1 = 7, or vice versa, or m_1 + 1 = 14 In the latter case, the smallest integer that has only one prime factor repeated 13 times is N = 2^13. In the former case, clearly, to minimise N, we need to take as small prime factors as possible, and make sure that the smaller of the two factors is raised to power 6, i.e. the minimum is N = 2^6\cdot3. This is less that 2^13, and so N = 2^6\cdot3 is our answer.

    (ii) Okay. The smallest integer is 120 = 2^3\cdot3\cdot5. But it seems that every attempt to prove this either boils down to a huge awful grind, or a complete orgy of hand-waving. The only slightly rigorous proof I've found is the one I'm going to make below, but it uses the inequality between the arithmetic and geometric means, which says (in the case of 3 numbers, which is the case I'm going to use):

    For any positive reals a, b, c, we have

    \displaystyle \frac{a + b + c}{3} \geq \sqrt[3]{abc}

    but then, I'm not sure I'm allowed to use this.

    Anyway, basically, we first note that the number of factors depend only on the number of times the prime factors are occuring in N, but not on what the prime factors actually are. To minimise the number N, we thus need to make sure that we use the smallest prime factors available, and make sure that we have the highest exponents in our smallest prime factors, i.e. we want

    p_1 = 2, p_2 = 3, p_3 = 5, p_4 = 7, \ldots and

    m_1 \geq m_2 \geq m_3 \geq m_4 \geq \cdots

    Now, we note that 2^3\cdot3\cdot5 satisfies the requirements (it has 14 proper factors). This shows that our minimum can't have prime factors higher than or equal to 7, because we have already established that the minimum satisfies m_1 \geq m_2 \geq m_3 \geq m_4 \geq \cdots, which means that the smallest possible N with a prime factor of 7 is 2\cdot3\cdot5\cdot7. But this is larger than 2^3\cdot3\cdot5 and hence is not the minimum. Thus we can assume that

    N = 2^{m_1}\cdot3^{m_2}\cdot5^{m_3}.

    Now, because we want 12 or more proper factors, we have to have

    (m_1 + 1)(m_2 + 1)(m_3 + 1) \geq 14

    Together with the arithmetic-geometric means inequality, this means that

    m_1 + 1 + m_2 + 1 + m_3 + 1 \geq 3\sqrt[3]{14}

    which means that

    m_1 + m_2 + m_3 \geq 5

    (calculators were allowed, remember? :p:)

    Now we basically list the numbers with m_1 + m_2 + m_3 \geq 5 in rising order, and note that none of the ones below 2^3\cdot3\cdot5 can have 12 proper factors or more:

    32 = 2^5 has 4 proper factors
    48 = 2^4*3 has 8 proper factors
    64 = 2^6 has 5 proper factors
    96 = 2^5*3 has 10 proper factors
    120 = 2^3 * 3 * 5 has 14 proper factors

    and thus 120 is our desired minimum.

    So, there ought to be a better way of showing this last part. Ideas, anyone?
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    Paper II,1996, Q7:

    Wlog, let the corners of the square be A(-1,-1), B(1,-1), C(1,1), D(-1,1).

    Let the position of P be (x,y). Then p = |y+1|, q=|x-1|, r=|y-1|, s=|x+1|.

    pr=qs \iff|y+1||y-1| = |x-1||x+1|\\

\iff (y+1)^2(y-1)^2 = (x-1)^2(x+1)^2 \\

\iff [(y+1)(y-1)]^2 - [(x+1)(x-1)]^2 = 0 \\

\iff (y^2-1)^2 - (x^2 - 1)^2 = 0 \\

\iff y^4-2y^2 - x^4 + 2x^2 = 0\\

\iff y^4-x^4 -2(y^2-x^2) = 0\\

\iff (y^2-x^2)[(y^2+x^2)-2] = 0\\

\iff (y+x)(y-x)[(y^2+x^2)-2] = 0

    So either x=y, x=-y or x^2+y^2 = 2, and every line above is reversible so this proves points on these lines or circle satisfy pq=rs as required.
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    (Original post by DFranklin)
    Wlog, let the corners of the square be A(-1,-1), B(1,-1), C(1,1), D(-1,1).

    Let the position of P be (x,y). Then p = |y+1|, q=|x-1|, r=|y-1|, s=|x+1|.

    pr=qs \iff|y+1||y-1| = |x-1||x+1|\\

\iff (y+1)^2(y-1)^2 = (x-1)^2(x+1)^2 \\

\iff [(y+1)(y-1)]^2 - [(x+1)(x-1)]^2 = 0 \\

\iff (y^2-1)^2 - (x^2 - 1)^2 = 0 \\

\iff y^4-2y^2 - x^4 + 2x^2 = 0\\

\iff y^4-x^4 -2(y^2-x^2) = 0\\

\iff (y^2-x^2)[(y^2+x^2)-2] = 0\\

\iff (y+x)(y-x)[(y^2+x^2)-2] = 0

    So either x=y, x=-y or x^2+y^2 = 2, and every line above is reversible so this proves points on these lines or circle satisfy pq=rs as required.
    Which Question/Paper?

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