Results are out! Find what you need...fast. Get quick advice or join the chat
Hey there Sign in to join this conversationNew here? Join for free

STEP Maths I, II, III 1989 solutions

Announcements Posted on
Complete this short survey for a chance to win an iPad mini! 22-09-2014
Got a question about Student Finance? Ask the experts this week on TSR! 14-09-2014
    • Thread Starter
    • 1 follower
    Offline

    ReputationRep:
    STEP I (Mathematics)
    1:
    2: Solution by kabbers
    3: Solution by Dystopia
    4: Solution by Dystopia
    5: Solution by Dystopia
    6: Solution by Swayam
    7: Solution by squeezebox
    8: Solution by squeezebox
    9: Solution by nota bene
    10:
    11: Solution by Glutamic Acid
    12:
    13:
    14:
    15:
    16:

    STEP II (F.Maths A)
    1: Solution by squeezebox
    2: Solution by kabbers
    3: Solution by squeezebox
    4:
    5: Solution by Dystopia
    6: Solution by *bobo*
    7:
    8:
    9:
    10:
    11:Solution by *bobo*
    12:
    13:
    14:Solution by squeezebox
    15:
    16:

    STEP III (F.Maths B)
    1: Solution by squeezebox
    2: Solution by *bobo*
    3:
    4:
    5:
    6:
    7:
    8: Solution by Dystopia
    9: Solution by squeezebox
    10: Solution by squeezebox
    11:
    12:
    13:
    14:
    15:
    16:


    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
    • Thread Starter
    • 1 follower
    Offline

    ReputationRep:
    I am no expert at STEP, so don't hesitate to correct me, it is more than likely made that I have made a mistake.

    STEP I - Question 8

    Using good old de Moivre's theorem;

    \cos(4\theta) = \Re( \cos(\theta) + i\sin(\theta))^{4} = \Re(\cos^{4}(\theta) + 4i\cos^{3}(\theta)\sin(\theta) - 6\cos^{2}(\theta)\sin^{2}(\theta  ) - 4i\sin^{3}(\theta)\cos(\theta) + \sin^{4}(\theta) ) = cos^{4}(\theta) - 6\cos^{2}(\theta)\sin^{2}(\theta  ) = 8\cos^{4}(\theta) - 8\cos^{2}(\theta) + 1


    (using  \cos^{2}(\theta) + \sin^{2}(\theta) \equiv 1

    and similary;

    

\cos(6\theta) = \Re( \cos(\theta) + i\sin(\theta))^{6} = \Re(\cos^{6}(\theta) + 6i\cos^{5}(\theta)\sin(\theta) - 15\cos^{4}(\theta)\sin^{2}(\thet  a) - 20i\sin^{3}(\theta)\cos^{3}(\the  ta) + 15\sin^{4}(\theta)\cos^{2}(\thet  a) + 6i\cos(\theta)\sin^{5}(\theta) - \sin^{6}(\theta) ) = 32\cos^{6}(\theta) - 48\cos^{4}(\theta) +18\cos^{2}(\theta) - 1

    Now, consider:  \frac{1}{2}\cos(6\theta) - \frac{1}{2}cos(4\theta)

     \frac{1}{2}\cos(6\theta) - \frac{1}{2}cos(4\theta) = 16c^{6} - 28c^{4} + 13c^{2} -1

    Where c is  \cos(\theta) .

    Now let x = cos(\theta),

    so:

     16x^{6} - 28x^{4} + 13x^{2} -1 = 16c^{6} - 28c^{4} + 13c^{2} -1 =  \frac{1}{2}\cos(6\theta) - \frac{1}{2}cos(4\theta) = 0

    \Rightarrow \cos(6\theta) = \cos(4\theta)

     \Rightarrow 6\theta = 2n\pi \pm 4\theta

    Which gives;

     \theta = \frac{n\pi}{5} ( n= 1,2,3,4 ).
    These values of \theta give distinct values of \cos(\theta)

    or

     \theta = n\pi ( n= 0,1 )


    Hence the roots of the equation are:

     x = \cos(0), \cos(\pi), \cos(\frac{\pi}{5}), \cos(\frac{2\pi}{5}), \cos(\frac{3\pi}{5}) and  \cos(\frac{4\pi}{5}) .
    • Thread Starter
    • 1 follower
    Offline

    ReputationRep:
    STEP III - Question 10

    Lets assume that the result is true for n = k;

    \displaystyle\sum_{r=1}^k r(r+1)(r+2)(r+3)(r+4) = \frac{1}{6}k(k+1)(k+2)(k+3)(k+4)  (k+5) (*)

    

\Rightarrow \displaystyle\sum_{r=1}^{k+1} r(r+1)(r+2)(r+3)(r+4) = \frac{1}{6}k(k+1)(k+2)(k+3)(k+4)  (k+5) + (k+1)(k+2)(k+3)(k+4)(k+5)

    \Rightarrow \displaystyle\sum_{r=1}^{k+1} r(r+1)(r+2)(r+3)(r+4) = \frac{1}{6}(k+1)(k+2)(k+3)(k+4)(  k+5)(k+6)

    This is the same as (*) except k+1 replaces k. Hence if the result is true for n=k, its true for n= k+1.

    When n=1,

    LHS of (*) = 1x2x3x4x5 = 120
    RHS of (*) = (1/6)x1x2x3x4x5x6 = 120.

    So (*) is true for n=1.
    Hence, by induction;

    \displaystyle\sum_{r=1}^n r(r+1)(r+2)(r+3)(r+4) = \frac{1}{6}n(n+1)(n+2)(n+3)(n+4)  (n+5)  \forall n  \geq 1 .


    Since;

     r^{5} < r(r+1)(r+2)(r+3)(r+4)

     \Rightarrow \displaystyle\sum_{r=1}^n r^{5} < \displaystyle\sum_{r=1}^n r(r+1)(r+2)(r+3)(r+4)

     \Rightarrow \displaystyle\sum_{r=1}^n r^{5} < \frac{1}{6}n(n+1)(n+2)(n+3)(n+4)  (n+5).

    Using (*);

    \displaystyle\sum_{r=0}^{n-1} r(r-1)(r-2)(r-3)(r-4) = \frac{1}{6}n(n-1)(n-2)(n-3)(n-4)(n-5)


     r(r-1)(r-2)(r-3)(r-4) < r^{5}

     \Rightarrow \displaystyle\sum_{r=0}^{n-1} r^{5} > \displaystyle\sum_{r=0}^{n-1} r(r-1)(r-2)(r-3)(r-4)

     \Rightarrow \displaystyle\sum_{r=0}^{n-1} r^{5} > \frac{1}{6}(n-5)(n-4)(n-3)(n-2)(n-1)n


    In this case, f(x) = x^{5}

    From the previous parts, we know that:

    \displaystyle\sum_{r=0}^{n-1} \frac{a^{6}}{n^{6}}r^{5} > \frac{a^{6}}{6n^{6}}(n-5)(n-4)(n-3)(n-2)(n-1)n

    and clearly, also:

     \displaystyle\sum_{r=0}^{n-1} \frac{a^{6}}{n^{6}}r^{5} < \frac{a^{6}}{6n^{6}}n(n+1)(n+2)(  n+3)(n+4)(n+5)

    as n\longrightarrow \infty ,

     \frac{a^{6}}{6n^{6}}(n-5)(n-4)(n-3)(n-2)(n-1)n and \frac{a^{6}}{6n^{6}}n(n+1)(n+2)(  n+3)(n+4)(n+5) \longrightarrow \frac{a^{6}}{6}

    So,

    \lim_{n\to \infty} \displaystyle\sum_{r=0}^{n-1} \frac{a^{6}}{n^{6}}r^{5} = \frac{a^{6}}{6}

    Using a similar arguement, we can show that:

    \lim_{n\to \infty} \displaystyle\sum_{r=1}^{n} \frac{a^{6}}{n^{6}}r^{5} = \frac{a^{6}}{6}

    We have shown that the limits exist and are equal to \frac{a^{6}}{6}.

    Hence;


    \displaystyle\int^a_0 x^{5} \, \mathrm{d}x = \frac{a^{6}}{6}
    • 0 followers
    Offline

    ReputationRep:
    STEP I, Q2

    The For x > 0, find \displaystyle\int x\ln x dx

    Using integration by parts (\displaystyle\int u'v dx = uv - \int uv' dx ):

    \displaystyle\int x\ln x dx = \frac{1}{2}x^2 \cdot \ln x  -  \int \frac{1}{2}x^2 \cdot \frac{1}{x} dx

    \displaystyle = \frac{1}{2}(x^2 \cdot \ln x  -  \int x dx)

    \displaystyle = \frac{1}{2}(x^2 \cdot \ln x  -  \frac{1}{2} x^2) + C

    \displaystyle = \frac{1}{4}x^2(2 \ln x - 1) + C



    By approximating the area corresponding to \int^1_0 x\ln(1/x) dx by n rectangles of equal width and with their top right hand vertices on the curve y = x\ln(1/x), show that, as n \to \infty,

    \displaystyle\frac{1}{2}(1 + \frac{1}{n})\ln n - \frac{1}{n^2}[ln(\frac{n!}{0!}) + ln(\frac{n!}{1!}) + ln(\frac{n!}{2!}) + ... ln(\frac{n!}{(n-1)!})] \to \frac{1}{4}



    (*) Firstly, note that

    \displaystyle \int^1_0 x\ln(1/x) dx

    \displaystyle= \int^1_0 x(-\ln(x)) dx

    \displaystyle= -\int^1_0 x\ln(x) dx

    \displaystyle= -[\frac{1}{4}x^2(2 \ln x - 1)]^1_0 (by the above indefinite integral)

    \displaystyle= \lim_{t \to 0} -[\frac{1}{4}x^2(2 \ln x - 1)]^1_t

    \displaystyle= \lim_{t \to 0} -\frac{1}{4}(1^2(\ln 1 - 1) - t^2(2 \ln t - 1))

    \displaystyle= \lim_{t \to 0} \frac{1}{4}(1 + 2t^2\ln t - t^2))

    \displaystyle= \lim_{t \to 0} \frac{1}{4}(1 + 2t(t\ln t) - t^2))

    \displaystyle= \frac{1}{4}



    The sum of the areas of our rectangles is going to be

    \displaystyle (**) \sum^n_{r=1} \frac{1}{n} (\frac{r}{n}\ln(\frac{1}{r/n}))

    \displaystyle = \sum^n_{r=1} \frac{1}{n} (\frac{r}{n}\ln(\frac{n}{r}))

    \displaystyle = \frac{1}{n^2} \sum^n_{r=1} r\ln(\frac{n}{r})

    \displaystyle = \frac{1}{n^2} (1\ln(\frac{n}{1}) + 2\ln(\frac{n}{2}) + 3\ln(\frac{n}{3}) + ... + n\ln(\frac{n}{n}))

    \displaystyle = \frac{1}{n^2} \ln((\frac{n}{1})(\frac{n}{2})^2  (\frac{n}{3})^3 \cdot \cdot \cdot (\frac{n}{n})^n)

    \displaystyle = \frac{1}{n^2} \ln(\frac{n\cdot n^2\cdot n^3\cdot\cdot\cdot n^n}{1\cdot 2^2\cdot 3^3\cdot\cdot\cdot n^n})

    \displaystyle = \frac{1}{n^2} \ln(\frac{n^{1+2+3+4+...+n}}{1\c  dot 2^2\cdot 3^3\cdot\cdot\cdot n^n})

    \displaystyle = \frac{1}{n^2} (ln(n^{1+2+3+4+...+n})-\ln(1\cdot 2^2\cdot 3^3\cdot\cdot\cdot n^n))

    \displaystyle = \frac{1}{n^2} (ln(n^{\frac{1}{2}n(n+1)})-\ln(1\cdot 2^2\cdot 3^3\cdot\cdot\cdot n^n))

    \displaystyle = \frac{1}{n^2} ({\frac{1}{2}n(n+1)}ln(n)-\ln(1\cdot 2^2\cdot 3^3\cdot\cdot\cdot n^n))

    \displaystyle = \frac{1}{n^2}{\frac{1}{2}n(n+1)}  ln(n) - \frac{1}{n^2} \ln(1\cdot 2^2\cdot 3^3\cdot\cdot\cdot n^n)

    \displaystyle = \frac{1}{2}(1+\frac{1}{n})ln(n) - \frac{1}{n^2} \ln(1\cdot 2^2\cdot 3^3\cdot\cdot\cdot n^n)



    Claim: 1\cdot 2^2\cdot 3^3\cdot\cdot\cdot k^k = \frac{k!}{0!}\cdot\frac{k!}{1!}\  cdot\frac{k!}{2!} \cdot\cdot\cdot\frac{k!}{k!}

    So for k + 1, we must prove that \frac{k!}{0!}\cdot\frac{k!}{1!}\  cdot\frac{k!}{2!} \cdot \cdot \cdot \frac{(k+1)!}{(k+1)!} is the equation we arrive at.

    Basis case (let k=1): 1^1 = \frac{1!}{0!}\cdot\frac{1!}{1!} = 1

    Inductive step: 1\cdot 2^2 \cdot 3^3 \cdot \cdot \cdot k^k\cdot (k+1)^{k+1} = (k+1)^{k+1}(\frac{k!}{0!} \cdot \frac {k!}{1!} \cdot \frac{k!}{2!} \cdot \cdot \cdot \frac{k!}{k!})

     = ((k+1)\frac{k!}{0!}(k+1) \cdot \frac{k!}{1!}(k+1) \cdot \frac{k!}{2!} \cdot\cdot\cdot (k+1)\frac{k!}{k!})

     = (\frac{(k+1)!}{0!}\cdot\frac{(k+  1)!}{1!}\cdot\frac{(k+1)!}{2!}\c  dot\cdot\cdot\frac{(k+1)!}{k!})

     = (\frac{(k+1)!}{0!}\cdot\frac{(k+  1)!}{1!}\cdot\frac{(k+1)!}{2!}\c  dot\cdot\cdot\frac{(k+1)!}{k!}\c  dot \frac{(k+1)!}{(k+1)!})



    So,

    \displaystyle (**) = \frac{1}{2}(1+\frac{1}{n})ln(n) - \frac{1}{n^2} \ln(\frac{n!}{0!}\cdot\frac{n!}{  1!}\cdot\frac{n!}{2!}\cdot\cdot\  cdot\frac{n!}{n!})

    \displaystyle = \frac{1}{2}(1+\frac{1}{n})ln(n) - \frac{1}{n^2} (\ln(\frac{n!}{0!}) + \ln(\frac{n!}{1!}) + \ln(\frac{n!}{2!}) + ... + \ln(\frac{n!}{(n-1)!}) + \ln(\frac{n!}{n!}))

    \displaystyle = \frac{1}{2}(1+\frac{1}{n})ln(n) - \frac{1}{n^2} (\ln(\frac{n!}{0!}) + \ln(\frac{n!}{1!}) + \ln(\frac{n!}{2!}) + ... + \ln(\frac{n!}{(n-1)!}))


    As n \to \infty, our sum tends to the integral (*). So,

    \displaystyle \frac{1}{2}(1+\frac{1}{n})ln(n) - \frac{1}{n^2} (\ln(\frac{n!}{0!}) + \ln(\frac{n!}{1!}) + \ln(\frac{n!}{2!}) + ... + \ln(\frac{n!}{(n-1)!})) \to \frac{1}{4}


    is this ok? have i missed anything out or not explained each step in enough detail?
    • 1 follower
    Offline

    ReputationRep:
    Question 4, STEP I

    With six points, each point is attached to five other points, so at least three lines radiating from A must be of the same colour (gold, say). The points which are joined to A must then be joined to each other by silver lines, or an entirely gold triangle would be made. However, then a triangle with entirely silver edges can be made from these three points.

    Diagram showing five points is attached.
    Attached Thumbnails
    Click image for larger version. 

Name:	fivepoints.jpg 
Views:	134 
Size:	4.8 KB 
ID:	48337  
    • 1 follower
    Offline

    ReputationRep:
    Question 5, STEP I

    \displaystyle (1+x)^{n} = \binom{n}{0} + \binom{n}{1}x + \binom{n}{2}x^{2} + \cdots + \binom{n}{n}x^{n}

    a)
    Let x = 1.

    \displaystyle 2^{n} = \binom{n}{0} + \binom{n}{1} + \binom{n}{2} + \cdots + \binom{n}{n}

    Let x = -1

    \displaystyle 0 = \binom{n}{0} - \binom{n}{1} + \binom{n}{2} - \cdots + \binom{n}{n}

    As n is even. So

    \displaystyle \binom{n}{0} + \binom{n}{2} + \cdots + \binom{n}{n} = \binom{n}{1} + \binom{n}{3} + \cdots + \binom{n}{n-1} = 2^{n-1}

    As their sum is \displaystyle 2^{n}

    b) Suppose that

    \displaystyle \binom{k}{1} + 2\binom{k}{2} + \cdots + k\binom{k}{k} = k2^{k-1}

    Then \displaystyle \binom{k}{0} + 2\binom{k}{1} + 3\binom{k}{2} + \cdots + (k+1)\binom{k}{k} = k2^{k-1} + 2^{k}

    Note that \displaystyle \binom{k}{r} + \binom{k}{r-1} = \binom{k+1}{r} and that \displaystyle \binom{k}{k} = \binom{k+1}{k+1}

    So, upon adding these two expressions, we get

    \displaystyle \binom{k+1}{1} + 2\binom{k+1}{2} + \cdots + (k+1)\binom{k+1}{k+1} = k2^{k-1} + k2^{k-1} + 2^{k} = (k+1)2^{k}

    Also, \displaystyle \binom{1}{1} = 1 = 1 \times 2^{0}

    So true by induction.

    \displaystyle \displaystyle \sum_{r=0}^{n} \left(r + (-1)^{r}\right) \binom{n}{r} = \left(\binom{n}{1} + 2\binom{n}{2} + \cdots + n \binom{n}{n}\right) + \left(\binom{n}{0} - \binom{n}{1} + \cdots + (-1)^{n}\binom{n}{n}\right)

    \displaystyle  = n2^{n-1}
    • Thread Starter
    • 1 follower
    Offline

    ReputationRep:
    STEP II Q1

    

\cos(3\theta) = \cos(2\theta + \theta) = \cos(2\theta)\cos(\theta) - \sin(2\theta)\sin(\theta) = (2cos^{2}(\theta) - 1)\cos(\theta) - 2\sin(\theta)\cos(\theta)sin(\th  eta) = 2cos^{3}(\theta) - \cos(\theta) - 2(1-\cos^{2}(\theta))\cos(\theta) = 4cos^{3}(\theta) - 3\cos(\theta).

    Substituting x= y-a into the equation:

     24x^{3} - 72x^{2} + 66x -19 = 24(y-a)^{3} - 72(y-a)^{2} + 66(y-a) -19 = 0

    Notice that the reduced form shown in the question has no z^{2} term, so we need to find the value of a which will get rid of the y^{2} term.

    Expanding out and collectiing terms we get:

     24y^{3} - 72(a+1)y^{2} + (72a^{2} + 144a +66)y - 24a^{3} - 72a^{2} -66a -19 = 0

    So the value of a to get rid of the y^{2} is -1.

    So we now have:

     24y^{3} - 72(a+1)y^{2} + (72a^{2} + 144a +66)y - 24a^{3} - 72a^{2} -66a -19 = 24t^{3} - 6y -1 = 0

     \Rightarrow 24y^{3} - 6y = 1 (*)

    Substituting y=z/b into (*);

    24(\frac{z}{b})^{3} - 6(\frac{z}{b}) = 1

    dividing both sides by 2 and multiplying by b;

    12\frac{z^{3}}{b^{2}} - 3z = \frac{b}{2} .

    And so to make this equal to;

    4z^{3}- 3z = \frac{b}{2}

    we make b = \pm \sqrt3.

    Lets take b = \sqrt3 and let z = \cos(\theta)

    so;

     4z^{3}- 3z = 4\cos^{3}(\theta) - 3\cos(\theta) = \cos(3\theta) = \frac{\sqrt3}{2}

    \Rightarrow 3\theta = \frac{pi}{6}, \frac{11\pi}{6} and  \frac{13\pi}{6} (these give distinct values of cos\theta, and hence, three distinct solutions.)


    we know that; x = y - a = (z/b) - a.

    Hence the solutions of the equation are:

     x = \frac{\cos(\frac{\pi}{18})}{\sqr  t3} + 1 , \frac{\cos(\frac{11\pi}{18})}{\s  qrt3} +1 and  \frac{\cos(\frac{13\pi}{18})}{\s  qrt3} + 1
    • 10 followers
    Offline

    ReputationRep:
    Question 11, STEP I.

    Vertical component = v \sin \theta
    Using s = ut + 1/2at^2.
    -h = v \sin \theta - 1/2gt^2
    1/2gt^2 - v \sin \theta - h = 0
    Quadratic in t. Ignoring the negative root.
    t = \dfrac{v \sin \theta + \sqrt{v^2 \sin^2 \theta + 2hg}}{g}
    R = horizontal component x time.
    Horizontal component = v \cos \theta
    R = v \cos \theta (\dfrac{v \sin \theta + \sqrt{v^2 \sin^2 \theta + 2hg}}{g}
    R = \dfrac{v^2 \sin \theta \cos \theta + v \cos \theta \sqrt{v^2 \sin^2 \theta + 2hg}}{g}
    Multiply both numerator and denominator by 2.
    R = \dfrac{2v^2 \sin \theta \cos \theta + 2v \cos \theta \sqrt{v^2 \sin^2 \theta + 2hg}}{2g}
    Using \sin (2\theta) = 2 \sin \theta \cos \theta
    We can simplify it.
    R = \dfrac{v^2 \sin 2\theta + 2v \cos \theta \sqrt{v^2 \sin^2 \theta + 2hg}}{2g}
    Splitting up Mr. Big Fraction.
    R = \dfrac{v^2 \sin 2\theta}{2g} + \dfrac{2v \cos \theta \sqrt{v^2 \sin^2 \theta + 2hg}}{2g}

    Ignore the first fraction for the moment, we'll work on the second one.
    \dfrac{2v \cos \theta \sqrt{v^2 \sin^2 \theta (1 + \dfrac{2hg}{v^2 \sin^2 \theta})}}{2g}

    Unparseable or potentially dangerous latex formula. Error 4: no dvi output from LaTeX. It is likely that your formula contains syntax errors or worse.
    \dfrac{2v^2 \cos \theta \sin \theta}{2g} \times \sqrt{1 + \dfrac{2hg}{v^2 \sin^2 \theta}

    We can use our old friend Mr. Double Angle Formula to simplify it:
    \dfrac{v^2 \sin 2\theta}{2g} \sqrt{1 + \dfrac{2hg}{v^2 \sin^2 \theta}}
    The whole fraction looks like:
    Unparseable or potentially dangerous latex formula. Error 4: no dvi output from LaTeX. It is likely that your formula contains syntax errors or worse.
    R = (\dfrac{v^2\sin 2\theta}{2g}) + (\dfrac{v^2 \sin 2\theta}{2g}) \sqrt{1 + \dfrac{2hg}{v^2 \sin^2 \theta}

    And lo and behold, there's a common factor to both fractions. Let's take it out:
    R = \dfrac{v^2}{2g}\sin 2\theta(1 + \sqrt{1 + \dfrac{2hg}{v^2 \sin^2 \theta}})
    And that's what we're looking for.
    *I'll finish off the last small bit in a few minutes*
    • 16 followers
    Offline

    ReputationRep:
    STEP I Question 6

    y = f(x)
    dy/dx = f'(x)

    The normal to the curve will therefore have gradient -1/f'(x). The equation of the normal would be

    y - f(x) = \frac{-1}{f'(x)}(x' - x) (where x' is the general x coordinate of the normal).

    At the point Q, x' = 0 as it cuts the y axis.

    y - f(x) = \frac{x}{f'(x)}

    y = \frac{x}{f'(x)} + f(x)

    The distance PQ can be worked out using Pythagoras' theorem.

    PQ^2 = (f(x) - (\frac{x}{f'(x)} + f(x)))^2 + x^2

    = \frac{x^2}{f'(x)^2} + x^2

    It's given that PQ^2 = e^{x^2} + x^2

    So

    \frac{x^2}{f'(x)^2} + x^2 =  e^{x^2} + x^2

    \frac{x^2}{f'(x)^2} = e^{x^2}

    \frac{x^2}{e^{x^2}} = f'(x)^2

    \frac{x}{e^{0.5x^2}} = f'(x)

    \int 1 \mathrm{d}y = \int \frac{x}{e^{0.5x^2}} \mathrm{d}x

    y = \int xe^{-0.5x^2} \mathrm{d}x

    y = -e^{-0.5x^2} + c (use a substitution of u = x^2 if you can't see why)

    -2 = -1 + c

    c = -1

    y = -e^{-0.5x^2} - 1
    • 1 follower
    Offline

    ReputationRep:
    STEP III, Q8

    \frac{\mathrm{d}^{2}x}{\mathrm{d  }t^{2}} = 4\frac{\mathrm{d}x}{\mathrm{d}t} - 4\frac{\mathrm{d}y}{\mathrm{d}t}

    \frac{\mathrm{d}^{2}x}{\mathrm{d  }t^{2}} - 4\frac{\mathrm{d}x}{\mathrm{d}t} + 4x = 48e^{2t} + 48e^{-2t}

    Let u = \frac{\mathrm{d}x}{\mathrm{d}t} - 2x

    Then \frac{\mathrm{d}u}{\mathrm{d}t} - 2u = 48e^{2t} + 48e^{-2t}

    Using an integrating factor, e^{-2t}, we get

    ue^{-2t} = \int 48 + 48e^{-4t} \; \mathrm{d}t = 48t - 12e^{-4t} + c

    u = e^{2t}(48t + c) - 12e^{-2t}

    \frac{\mathrm{d}x}{\mathrm{d}t} - 2x = e^{2t}(48t + c) - 12e^{-2t}

    xe^{-2t} = \int 48t + c - 12e^{-4t} \; \mathrm{d}t = 24t^{2} + ct + 3e^{-4t} + d

    x = e^{2t}(24t^{2} + ct + d) + 3e^{-2t}

    Applying boundary conditions:

    0 = d + 3 \Rightarrow d = -3

    \frac{\mathrm{d}x}{\mathrm{d}t} = 2e^{2t}(24t^{2} + ct + d) + e^{2t}(48t + c) - 6e^{-2t} = 4(x-y)

    2d + c - 6 = 0 \Rightarrow c = 6 - 2d = 6 + 6 = 12

    x = e^{2t}(24t^{2} + 12t - 3) + 3e^{-2t}

    4y = 4x - \frac{\mathrm{d}x}{\mathrm{d}t} = e^{2t}(48t^{2} - 4t - 18) + 18e^{-2t} = 4te^{2t}(12t - 1) - 36\sinh 2t

    y = te^{2t}(12t - 1) - 9\sinh 2t

    ---

    I think that I shall restrict myself to a maximum of three solutions per day (at least at the start), so that everyone has a chance to answer some if they want.
    • 0 followers
    Offline

    ReputationRep:
    Hang on a sec, mine was mislabelled, should be Step I not step II.. sorry about that!

    edit: thanks.. now typing up II/2 :p:
    • 0 followers
    Offline

    ReputationRep:
    STEP II/2

    \displaystyle \tan x = \sum^{\infty}_0 a_n x^n


    \displaystyle x \cot x = 1 + \sum^{\infty}_1 b_n x^n

    \displaystyle (*) \cot x = \frac{1}{x} + \sum^{\infty}_1 b_n x^{n-1}

    \displaystyle = \frac{1}{x} + \sum^{\infty}_0 b_{n+1} x^n



    \displaystyle 2 \cot 2x = \cot x - \tan x

    Therefore, by (*),

    \displaystyle 2 \cot 2x = \frac{1}{x} + \sum^{\infty}_0 b_{n+1} x^n - \sum^{\infty}_0 a_n x^n

    \displaystyle = \frac{1}{x} + \sum^{\infty}_0 (b_{n+1} - a_n) x^n


    we can also use (*) to express 2cot2x as \displaystyle 2(\frac{1}{2x} + \sum^{\infty}_0 b_{n+1} (2x)^n) = \frac{1}{x} + 2\sum^{\infty}_0 b_{n+1} 2^n x^n


    Therefore,

    \displaystyle \frac{1}{x} + 2\sum^{\infty}_0 b_{n+1} 2^n x^n = \frac{1}{x} + \sum^{\infty}_0 (b_{n+1} - a_n) x^n

    \displaystyle \sum^{\infty}_0 b_{n+1} 2^{n+1} x^n = \sum^{\infty}_0 (b_{n+1} - a_n) x^n

    we can compare the coefficients of the respective summations and say

    \displaystyle b_{n+1} 2^{n+1} = b_{n+1} - a_n

    \displaystyle a_n = b_{n+1} - b_{n+1} 2^{n+1}

    \displaystyle a_{n-1} = b_n(1 - 2^n)


    for the second part, note that

    \displaystyle x \csc x = 1 + \sum^{\infty}_1 c_n x^n

    \displaystyle \csc x = \frac{1}{x} + \sum^{\infty}_1 c_n x^{n-1}

    \displaystyle = \frac{1}{x} + \sum^{\infty}_0 c_{n+1} x^n


    we have to find an identity for 2csc2x

    \displaystyle 2 \csc 2x = \frac{2}{\sin 2x} = \frac{2}{2 \sin x \cos x} = \frac{1}{\sin x \cos x}

    (using partial fractions)
    \displaystyle = \frac{\sin^2 x + cos^2 x}{\sin x \cos x} = \frac{\cos x}{\sin x} + \frac{\sin x}{\cos x} = \cot x + \tan x


    Also,
    \displaystyle 2 \csc 2x = 2(\frac{1}{2x} + \sum^{\infty}_0 c_{n+1} (2x)^n)

    \displaystyle = 2(\frac{1}{2x} + \sum^{\infty}_0 c_{n+1} 2^n x^n)

    \displaystyle = \frac{1}{x} + \sum^{\infty}_0 c_{n+1} 2^{n+1} x^n



    \displaystyle \cot x + \tan x = \frac{1}{x} + \sum^{\infty}_0 b_{n+1} x^n + \sum^{\infty}_0 a_n x^n

    \displaystyle 2 \csc 2x = \frac{1}{x} + \sum^{\infty}_0 b_{n+1} x^n + \sum^{\infty}_0 a_n x^n


    2 \csc 2x = \cot x + \tan x, so:


    \displaystyle \frac{1}{x} + \sum^{\infty}_0 c_{n+1} 2^{n+1} x^n = \frac{1}{x} + \sum^{\infty}_0 b_{n+1} x^n + \sum^{\infty}_0 a_n x^n

    \displaystyle \sum^{\infty}_0 c_{n+1} 2^{n+1} x^n = \sum^{\infty}_0 b_{n+1} x^n + \sum^{\infty}_0 a_n x^n

    \displaystyle \sum^{\infty}_0 c_{n+1} 2^{n+1} x^n = \sum^{\infty}_0 (b_{n+1} + a_n) x^n


    Comparing coefficients of the polynomials (in x) of the expansion,

    \displaystyle c_{n+1} 2^{n+1} = b_{n+1} + a_n

    Rearranging and substituting in the earlier definition of a_{n-1} = b_n(1-2^n) \Longrightarrow \frac{a_{n-1}}{1-2^n} = b_n,


    \displaystyle c_{n+1} 2^{n+1} = \frac{a_{n}}{1-2^{n+1}} + a_n

    \displaystyle c_{n+1} = (\frac{1}{1 - 2^{n+1}}+1)(\frac{1}{2^{n+1}})a_  n

    \displaystyle c_{n+1} = (\frac{2 - 2^{n+1}}{1 - 2^{n+1}})(\frac{1}{2^{n+1}})a_n

    \displaystyle c_{n+1} = (\frac{1 - 2^n}{1 - 2^{n+1}})(\frac{1}{2^n})a_n

    \displaystyle c_{n+1} = (\frac{2^n - 1}{2^{n+1} - 1})(\frac{1}{2^n})a_n

    \displaystyle c_{n+1} = (\frac{2^{n-1} - 1}{2^{n} - 1})(\frac{1}{2^{n-1}})a_{n-1}

    any corrections welcome !
    • 0 followers
    Offline

    ReputationRep:
    STEP II, (11) attached

    attachments 2&3 are in the wrong order
    Attached Files
  1. File Type: doc step1989q11.doc (540.0 KB, 89 views)
  2. File Type: doc step1989q11p3.doc (205.5 KB, 90 views)
  3. File Type: doc step1989q11p2.doc (158.5 KB, 82 views)
    • 0 followers
    Offline

    ReputationRep:
    Step III (2)
    Attached Files
  4. File Type: doc step1989q2p1.doc (113.5 KB, 116 views)
  5. File Type: doc step1989q2p2.doc (79.0 KB, 101 views)
  6. File Type: doc step1989q2p3.doc (71.5 KB, 93 views)
    • 0 followers
    Offline

    ReputationRep:
    STEP II (6) attached
    Attached Files
  7. File Type: doc step1989q6p1.doc (79.5 KB, 114 views)
  8. File Type: doc step1989q62.doc (95.0 KB, 119 views)
    • Thread Starter
    • 1 follower
    Offline

    ReputationRep:
    STEP II Question 3

     2(u + iv) = e^{x+iy} - e^{-(x+iy)} = 2\sinh(x + iy) = 2\sinh(x)\cosh(iy) + 2\cosh(x)\sinh(yi) = 2\sinh(x)\cos(y) + 2i\sin(y)\cosh(x)

    Equating real and imaginary parts:

    u = \sinh(x)\cos(y) (*)

    v = \sin(y)\cosh(x) (**)

    if x=a, then:

    u = \sinh(a)\cos(y) (1)

    v = \sin(y)\cosh(a) (2)

    Now, subbing (1) and (2) into:

     \cos^{2}y + \sin^{2}y = 1

    We get:

     (\frac{u}{\sinh(a)})^{2} + (\frac{v}{\cosh(a)})^{2} = 1 (\triangle)

    As required.

    Substituting y=b into (*) and (**):

    u = \sinh(x)\cos(b) (3)

    v = \cosh(x)\sin(b) (4).

    This time, using  \cosh^{2}x - \sinh^{2}x = 1, we end up with:

     (\frac{v}{\sin(b)})^{2} - (\frac{u}{\cos(b)})^{2} = 1  (\heartsuit) .

    Since 0 < \sin(b) < 1 and  \cosh(x) \geq 1, \Rightarrow v>0. Hence, (\heartsuit) corresponds to a single hyeprbola branch, which is above the line v=0.

    One point of intersection is:  u = \sinh(a)\cos(b) and  v = \sin(b)\cosh(a) .


    Differentiating \triangle implicitly we get:

     \frac{dv}{du} = (\frac{u}{v})\coth^{2}(a) .

    Which is equal to:  -\cot(b)\coth(a) at u = \sinh(a)\cos(b) and  v = \sin(b)\cosh(a).

    Differentiating \heartsuit implicitly we get:

    \frac{dv}{du} = (\frac{u}{v})\tan^{2}(b) .

    Which is, \tanh(a)\tan(b) when u = \sinh(a)\cos(b) and  v = \sin(b)\cosh(a).

    \tanh(a)\tan(b)X-\cot(b)\coth(a) = -1

    Hence the curves intersect and right angles at this point.

    Regarding the sketch, I think the elipse becomes a vertical straight line from v = -1 to v = 1, and the hyperbola becomes the line v = 1.
    • 1 follower
    Offline

    ReputationRep:
    STEP 1, Q3

    \textbf{x} = x\textbf{a}, \; \textbf{y} = (1-x)\textbf{b}

    \textbf{c} = \frac{1}{3}(2\textbf{a} + \textbf{b}), \textbf{d} = \frac{1}{3}(\textbf{a} + 2\textbf{b})

    (By the ratio theorem.)

    \overrightarrow{CY} = (\frac{2}{3} - x)\textbf{b} - \frac{2}{3}\textbf{a}
    \overrightarrow{DX} = (x - \frac{1}{3})\textbf{a} - \frac{2}{3}\textbf{b}

    These vectors are perpendicular, so the scalar product is zero.

    (\frac{2}{9} + x - x^{2})\textbf{a}.\textbf{b} - \frac{2}{9} = 0

    (Note that the scalar product is commutative and distributive.)

    \textbf{a}.\textbf{b} = \frac{-2}{9x^{2} - 9x - 2}

    Let f(x) = \frac{-2}{9x^{2} - 9x - 2}, \; 0 \leq x \leq 1

    f'(x) = \frac{2(18x - 9)}{(9x^{2}-9x-2)^{2}}

    Minimum and maximum values occur at the endpoint of a range or at a turning point. There is a turning point when x = 1/2.

    f(0) = 1, \; f(\frac{1}{2}) = \frac{8}{17}, \; f(1) = 1

    So \frac{8}{17} \leq \textbf{a}.\textbf{b} \leq 1

    Note that \textbf{a}.\textbf{b} = |a||b|\cos\theta = \cos\theta

    Where \theta = \angle AOB, \; 0 < \theta < \pi

    The maximum value is \theta occurs when \cos\theta = \frac{8}{17}

    Suppose that \lambda \overrightarrow{CY} = \overrightarrow{CE},\; \mu \overrightarrow{DX} = \overrightarrow{DE}

    Then \lambda (\frac{1}{6} \textbf{b} - \frac{2}{3} \textbf{a}) = \frac{1}{3} ( \textbf{b} - \textbf{a} ) + \mu ( \frac{1}{6} \textbf{a} - \frac{2}{3} \textbf{b} )

    So \frac{1}{6} \lambda = \frac{1}{3} - \frac{2}{3}\mu

    -\frac{2}{3}\lambda = -\frac{1}{3} + \frac{1}{6}\mu

    \lambda = \mu = \frac{2}{5}

    \textbf{e} = \textbf{c} + \frac{2}{5}\overrightarrow{CY} = \frac{2}{5}\textbf{a} + \frac{2}{5}\textbf{b}

    By the cosine rule, AB^{2} = \frac{18}{17}
    Let F be the midpoint of AB. By Pythagoras, OF^{2} = 1 - \frac{9}{34} = \frac{25}{34}, \; OF = \frac{5}{\sqrt{34}}

    OE = \frac{3}{5} \times \frac{5}{\sqrt{34}} = \frac{3}{\sqrt{34}}
    • Thread Starter
    • 1 follower
    Offline

    ReputationRep:
    *bump* (just to remind people that this thread is still here ) - I have solutions that I can type up if no one else wants to, but I really don't want to take over the thread..
    • Thread Starter
    • 1 follower
    Offline

    ReputationRep:
    STEP I - Question 7

    *Graph Attached*

    If we work out the area of a quarter of the rectangle, which is in the top right hand bit of the axes, and maximise this, it make calculation soooo much eaiser. (Since we can forget about modulus signs and y being negative) :

    Let the area of the rectangle be A.

    Base of smaller rectangle = 1-x
    height = \sqrt(1-(1-x)) = \sqrt x

    so \frac{1}{4}A = (1-x)\sqrt x

    which we want to maximise.

    \frac{d\frac{1}{4}A}{dx} = \frac{1-3x}{2\sqrt x}

    Which is zero when  x = \frac{1}{3}

    Finding the second derivative and subsituting x = \frac{1}{3} in shows \frac{1}{4}A is a maximum at this point.

    So A_{max} = 4 \times [(1-\frac{1}{3})\frac{1}{\sqrt 3}] = \frac{8}{\sqrt 27} .

    ________

    We can do the second bit in the same way, since its a closed curve that is symmetrical in both axes.

    Let the area of the second rectangle be B.
    Working in the top right quadrant again;
    Base = 1-x
    height = (1-(1-x))^{\frac{n}{2m}} = x^{\frac{n}{2m}}.

    So \frac{1}{4}B = x^{\frac{n}{2m}}(1-x) .

     \frac{d\frac{1}{4}B}{dx} = x^{\frac{n}{2m} -1}(\frac{n}{2m} - x(1 + \frac{n}{2m}))

    Which is zero when x = 0 or x = \frac{n}{2m + n} .

    Clearly the area is a minimum when x= 0, and so is a maximum when x =\frac{n}{2m + n} .

    \Rightarrow B_{max} = 4 \times [(\frac{n}{2m+n})^{\frac{n}{2m}}(  1 - \frac{n}{2m +n})] = 4(\frac{n}{2m + n})^{\frac{n}{2m}}(\frac{2m}{2m+  n}) .
    Attached Files
  9. File Type: doc STEP I Q7 graph.doc (196.0 KB, 101 views)
    • 3 followers
    Offline

    ReputationRep:
    I've got a solution to I question 9 , but don't have a scanner available so will add graphs when I get the opportunity.

    Question 9 STEP I

    y=\frac{x^3}{8}-\frac{3}{2}x and y'=\frac{3}{8}x^2-\frac{3}{2} Setting y'=0 \Rightarrow x^2=4 \Rightarrow x=\pm2
    f(2)=-2, f(-2)=2, also for graphing purposes it may be worth to note that f(0)=0 and f(4)=2 and f(-4)=-2. y=0 has solutions x=0 or x=\pm\sqrt{12}=\pm2\sqrt{3} f''(-2)=- i.e. local max f''(2)=+ i.e. local min
    Graph, see attachment

    (a) X=\frac{1}{2}x\,\Y=y i.e. 2X=x therefore the equation of y in this X-Y plane is Y=\frac{(2X)^3}{8}-\frac{3}{2}2X=X^3-3X=X(X^2-3)
    Y'=3X^2-3=3(x^2-1) setting it equal to 0 gives x=\pm1 f(1)=-2 ans f(-1)=2
    For graphing it can also be good to see that f(2)=2, f(-2)=-2 and y=0 \Rightarrow x=0, x=\pm\sqrt{3}
    Graph, see attachment

    (b) X=x, Y=\frac{1}{2}y i.e. 2Y=y therefore the equation of y in this X-Y plane is Y=\frac{X^3}{16}-\frac{3}{4}X=\frac{X}{4}(\frac{X  ^2}{4}-3)
    Y'=\frac{3}{16}X^2-\frac{3}{4}=\frac{3}{4}(\frac{X^  2}{4}-1) Setting this equal to 0 gives X=\pm2. f(2)=-1 and f(-2)=1
    For graphing, also see that Y=0 \Rightarrow x=0, x= \pm2\sqrt{3}, f(4)=1 and f(-4)=-1
    Graph, see attachment.

    (c) X=\frac{1}{2}x+1, Y=y i.e. x=2(X-1) therefore the equation in the X-Y plane is Y=\frac{(2(X-1))^3}{8} - \frac{3}{2}2(X-1)=(X-1)^3 - 3(X-1) = (X-1)(X^2-2X-2)
    Y'=3(X-1)^2-3=3((X-1)^2-1)=3(X^2-2X)=3X(X-2) Setting this equal to 0 gives  x=0, x=2 f(0)=2, f(2)=-2.
    For graphing, also see that Y=0 \Rightarrow X=1, X=1\pm\sqrt{3} and f(-1)=-2, f(3)=2

    (d) X=x, Y=\frac{1}{2}y+1 i.e. y=2(Y-1) therefore the equation in the X-Y plane is Y=\frac{X^3}{16}-\frac{3}{4}X+1
    Y'=\frac{3}{16}X^2-\frac{3}{4}=\frac{3}{4}(\frac{X^  2}{4}-1) setting this equal to 0 gives  X=\pm2 f(2)=0 f(-2)=2
    For graphing I'll check f(4) and f(-4) too, which give 2 and 0 respectively.

    For the last part, we're looking for a graph in the X-Y plane with local min (0,0) and local max (1,1). To obtain this on the form X=ax+b and Y=cy+d, first observe that for a graph of a cubic, changing the sign means a reflection in the x-axis i.e. local max and min swap places, this is what we need to do. That means either a or c is negative (but not both!). a squeezes the graph along the X-axis and c squeezes it along the Y-axis, c moves the graph to the left/right and d moves the graph up/down. Knowing these we can see that X=-\frac{1}{4}x+\frac{1}{2} and Y=\frac{1}{4}y+\frac{1}{2} will produce the desired graph.

    Okay, so I hope I've interpreted everything correct in the question. Seems pretty nice as a question.

Reply

Submit reply

Register

Thanks for posting! You just need to create an account in order to submit the post
  1. this can't be left blank
    that username has been taken, please choose another Forgotten your password?
  2. this can't be left blank
    this email is already registered. Forgotten your password?
  3. this can't be left blank

    6 characters or longer with both numbers and letters is safer

  4. this can't be left empty
    your full birthday is required
  1. By joining you agree to our Ts and Cs, privacy policy and site rules

  2. Slide to join now Processing…

Updated: September 21, 2013
New on TSR

TSR Freshers' blogs 2014

Read what TSR's freshers have to say as they head off to uni

Article updates
Reputation gems:
You get these gems as you gain rep from other members for making good contributions and giving helpful advice.