(Updated as far as #40) SimonM - 28.03.2009

STEP I:

1: Solution by Glutamic Acid

2: Solution by Aurel-Aqua

3: Solution by Aurel-Aqua

4: Solution by SimonM

5: Solution by SimonM

6: Solution by Unbounded

7: Solution by brianeverit

8: Solution by SimonM

9: Solution by sonofdot

10: Solution by brianeverit

11: Solution by brianeverit

12: Solution by brianeverit

13: Solution by Mark13

14: Solution by SimonM

STEP II:

1: Solution by Glutamic Acid

2: Solution by ForGreatJustice

3: Solution by SimonM

4: Solution by Dadeyemi

5: Solution by Glutamic Acid

6: Solution by sonofdot

7: Solution by Glutamic Acid

8: Solution by Glutamic Acid

9: Solution by Glutamic Acid

10: Solution to tommm

11: Solution by Glutamic Acid

12: Solution by brianeverit

13: Solution by Aurel-Aqua

14: Solution by Aurel-Aqua

STEP III:

1: Solution by Aurel-Aqua

2: Solution by SimonM

3: Solution by tommm

4: Solution by tommm

5: Solution by SimonM

6: Solution by Dadeyemi

7: Solution by tommm

8: Solution by SimonM

9: Solution by tommm

10: Solution by brianeverit

11: Solution by tommm

12: Solution by brianeverit

13: Solution by Aurel-Aqua

14: Solution by SimonM

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

STEP I:

1: Solution by Glutamic Acid

2: Solution by Aurel-Aqua

3: Solution by Aurel-Aqua

4: Solution by SimonM

5: Solution by SimonM

6: Solution by Unbounded

7: Solution by brianeverit

8: Solution by SimonM

9: Solution by sonofdot

10: Solution by brianeverit

11: Solution by brianeverit

12: Solution by brianeverit

13: Solution by Mark13

14: Solution by SimonM

STEP II:

1: Solution by Glutamic Acid

2: Solution by ForGreatJustice

3: Solution by SimonM

4: Solution by Dadeyemi

5: Solution by Glutamic Acid

6: Solution by sonofdot

7: Solution by Glutamic Acid

8: Solution by Glutamic Acid

9: Solution by Glutamic Acid

10: Solution to tommm

11: Solution by Glutamic Acid

12: Solution by brianeverit

13: Solution by Aurel-Aqua

14: Solution by Aurel-Aqua

STEP III:

1: Solution by Aurel-Aqua

2: Solution by SimonM

3: Solution by tommm

4: Solution by tommm

5: Solution by SimonM

6: Solution by Dadeyemi

7: Solution by tommm

8: Solution by SimonM

9: Solution by tommm

10: Solution by brianeverit

11: Solution by tommm

12: Solution by brianeverit

13: Solution by Aurel-Aqua

14: Solution by SimonM

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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I Question 2

Part (i) $1+2x-x^2>\frac{2}{x}$, $x\neq 0$.

First it might help to find the points of equality: multiplying by x and rearranging, we get $x^3-2x^2-x+2 = (x-2)(x-1)(x+1)=0$, $x\neq 0$. The roots are -1, 1 and 2. Now we must sketch the graph and notice that the inequality is satisfied for $-1 < x < 0$ (the zero due to the asymptote) and $1<x<2$.

Part (ii) $\sqrt{3x+10} > 2 + \sqrt{x+4}$, $x \geq -\frac{10}{3}$.

Because of the condition, squaring both sides still leaves an iff, giving: $3x + 10 > 4 + x + 4 + 4\sqrt{x+4}\implies 2x + 2 > 4\sqrt{x+4} \implies x + 1 > 2\sqrt{x+4}$. Now by the condition, we can deduce that x > -1 because of the square root being always positive. Squaring again, $x^2+2x+1 > 4(x + 4) \implies x^2 - 2x - 15 > 0$, giving $(x+3)(x-5)>0$. From this, we can deduce that $x > 5$ and $x<-3$, but our new condition, $x>-1$ only gives us $x>5$ as the final solution.

Part (i) $1+2x-x^2>\frac{2}{x}$, $x\neq 0$.

First it might help to find the points of equality: multiplying by x and rearranging, we get $x^3-2x^2-x+2 = (x-2)(x-1)(x+1)=0$, $x\neq 0$. The roots are -1, 1 and 2. Now we must sketch the graph and notice that the inequality is satisfied for $-1 < x < 0$ (the zero due to the asymptote) and $1<x<2$.

Part (ii) $\sqrt{3x+10} > 2 + \sqrt{x+4}$, $x \geq -\frac{10}{3}$.

Because of the condition, squaring both sides still leaves an iff, giving: $3x + 10 > 4 + x + 4 + 4\sqrt{x+4}\implies 2x + 2 > 4\sqrt{x+4} \implies x + 1 > 2\sqrt{x+4}$. Now by the condition, we can deduce that x > -1 because of the square root being always positive. Squaring again, $x^2+2x+1 > 4(x + 4) \implies x^2 - 2x - 15 > 0$, giving $(x+3)(x-5)>0$. From this, we can deduce that $x > 5$ and $x<-3$, but our new condition, $x>-1$ only gives us $x>5$ as the final solution.

STEP II Question 2

Sketch not included, but pretty simple (y=0 for 0<x<N, and y=1 for N<x<2N)

i) $[k/N]$ for $1<k<2N$ can be split into 3 regions:

$[k/N]=0$ for $1<k<N-1$

$[k/N]=1$ for $N<k<2N-1$ and

$[k/N]=2$ for $k=2N$

So by splitting the summation into these three ranges we get

$\displaystyle\sum_{k=1}^{2N} (-1)^{[k/2N]} k = \displaystyle\sum_{k=1}^{N-1} k - \displaystyle\sum_{k=N}^{2N-1} k + 2N$

$= \displaystyle\frac{1}{2}(N-1)(N) -(\displaystyle\frac{1}{2}(2N-1)(2N) -\displaystyle\frac{1}{2}(N-1)(N)) +2N$

$=(N-1)(N)-(2N-1)(N)+2N$

$=N^2-N-2N^2+N+2N$

$\displaystyle\sum_{k=1}^{2N} (-1)^{[k/2N]} k =2N-N^2$ as required

ii) $S_N= \displaystyle\sum_{k=1}^{2N} (-1)^{[k/2N]} 2^{-k}$

Splitting into 3 as before:

$S_N=\displaystyle\sum_{k=1}^{N-1} 2^{-k} - \displaystyle\sum_{k=N}^{2N-1} 2^{-k} + 2^{-2N}$

Using the formula for the sum of a geometric progression

$S_n=\displaystyle\frac{a(1-r^n)}{1-r}$ we get

$S_N=\displaystyle\frac{2^{-1}(1-(1/2)^{N-1})}{1-2^{-1}} +\displaystyle\frac{2^{-N}(1-(1/2)^{N})}{1-2^{-1}} + 2^{-2N}$

$S_N=1-(1/2)^{N-1}+2^{1-N}-2^{1-2N}+2^{-2N}$

Which could probably be made simpler, so now we can see as$N\to\infty$ then $S_N\to1$

Sketch not included, but pretty simple (y=0 for 0<x<N, and y=1 for N<x<2N)

i) $[k/N]$ for $1<k<2N$ can be split into 3 regions:

$[k/N]=0$ for $1<k<N-1$

$[k/N]=1$ for $N<k<2N-1$ and

$[k/N]=2$ for $k=2N$

So by splitting the summation into these three ranges we get

$\displaystyle\sum_{k=1}^{2N} (-1)^{[k/2N]} k = \displaystyle\sum_{k=1}^{N-1} k - \displaystyle\sum_{k=N}^{2N-1} k + 2N$

$= \displaystyle\frac{1}{2}(N-1)(N) -(\displaystyle\frac{1}{2}(2N-1)(2N) -\displaystyle\frac{1}{2}(N-1)(N)) +2N$

$=(N-1)(N)-(2N-1)(N)+2N$

$=N^2-N-2N^2+N+2N$

$\displaystyle\sum_{k=1}^{2N} (-1)^{[k/2N]} k =2N-N^2$ as required

ii) $S_N= \displaystyle\sum_{k=1}^{2N} (-1)^{[k/2N]} 2^{-k}$

Splitting into 3 as before:

$S_N=\displaystyle\sum_{k=1}^{N-1} 2^{-k} - \displaystyle\sum_{k=N}^{2N-1} 2^{-k} + 2^{-2N}$

Using the formula for the sum of a geometric progression

$S_n=\displaystyle\frac{a(1-r^n)}{1-r}$ we get

$S_N=\displaystyle\frac{2^{-1}(1-(1/2)^{N-1})}{1-2^{-1}} +\displaystyle\frac{2^{-N}(1-(1/2)^{N})}{1-2^{-1}} + 2^{-2N}$

$S_N=1-(1/2)^{N-1}+2^{1-N}-2^{1-2N}+2^{-2N}$

Which could probably be made simpler, so now we can see as$N\to\infty$ then $S_N\to1$

(edited 13 years ago)

I Question 2

$f(x) = (x-p)(x-q)(x-r)$, $p<q<r$. Well, we sketch the graph to show the two stationary points and crossing the x-axis at $x = p$, $x=q$, $x=r$. We can conclude the two stationary points exist, so there must be a real x-value for them to be found.

$f'(x) = 3x^2+2x(p+q+r)+(pq+qr+rp) = 0$.

Since the two stationary points are distinct and real, by $b^2-4ac>0$, we get $(2(p+q+r))^2-4\cdot 3 \cdot (pq+qr+rp) > 0 \implies (p+q+r)^2 > 3(pq+qr+rp)$.

Considering $(x^2+gx+h)(x-k)$, to have three roots, there must be a real solution to the quadratic, giving $g^2>4h$. This implies that the two other roots are $\alpha = \frac{-g-\sqrt{g^2-2h}}{2}$ and $\beta = \frac{-g+\sqrt{g^2-2h}}{2}$, and $\gamma = k$. Letting $p = \alpha$, $q = \beta$, $r = \gamma$ in any order (since they can be changed around due to triple-symmetry ). Knowing that $p+q+r= (k-g)$ and $pq+qr+rp = \alpha\beta + \alpha\gamma + \gamma\beta= h +\gamma(\alpha+\beta) = h + k(-g) = h - gk$. Substituting into our previous result, we obtain: $(k-g)^2 > 3(h-gk) \implies (g-k)^2>3(h-gk)$, as required. We can show that it is not necessary by letting $g=1$, $k = 1$, $(1-1)^2>3(h-1)\implies h > 1 \implies 1^2 > 4h \text{ is not true}$.

$f(x) = (x-p)(x-q)(x-r)$, $p<q<r$. Well, we sketch the graph to show the two stationary points and crossing the x-axis at $x = p$, $x=q$, $x=r$. We can conclude the two stationary points exist, so there must be a real x-value for them to be found.

$f'(x) = 3x^2+2x(p+q+r)+(pq+qr+rp) = 0$.

Since the two stationary points are distinct and real, by $b^2-4ac>0$, we get $(2(p+q+r))^2-4\cdot 3 \cdot (pq+qr+rp) > 0 \implies (p+q+r)^2 > 3(pq+qr+rp)$.

Considering $(x^2+gx+h)(x-k)$, to have three roots, there must be a real solution to the quadratic, giving $g^2>4h$. This implies that the two other roots are $\alpha = \frac{-g-\sqrt{g^2-2h}}{2}$ and $\beta = \frac{-g+\sqrt{g^2-2h}}{2}$, and $\gamma = k$. Letting $p = \alpha$, $q = \beta$, $r = \gamma$ in any order (since they can be changed around due to triple-symmetry ). Knowing that $p+q+r= (k-g)$ and $pq+qr+rp = \alpha\beta + \alpha\gamma + \gamma\beta= h +\gamma(\alpha+\beta) = h + k(-g) = h - gk$. Substituting into our previous result, we obtain: $(k-g)^2 > 3(h-gk) \implies (g-k)^2>3(h-gk)$, as required. We can show that it is not necessary by letting $g=1$, $k = 1$, $(1-1)^2>3(h-1)\implies h > 1 \implies 1^2 > 4h \text{ is not true}$.

STEP I - Q13

Given that plate is broken, the probability that it is broken by the mathematician is 0.25.

Therefore, the number of plates, $X$, broken by the mathematician can be represented by the following binomial distribution:

$X$~$B(n,0.25)$

We need to find $P(X \geq 3)$ when n=5.

$P(X \geq 3) = P(X=3) + P(X=4) + P(X=5)$

$=\displaystyle \binom{5}{3}(0.25)^3(0.75)^2 + \displaystyle \binom{5}{4}(0.25)^4(0.75) + \displaystyle \binom{5}{5}(0.25)^5$

$=0.0879+0.0146+0.0010$

$=0.1035$

Given that plate is broken, the probability that it is broken by the mathematician is 0.25.

Therefore, the number of plates, $X$, broken by the mathematician can be represented by the following binomial distribution:

$X$~$B(n,0.25)$

We need to find $P(X \geq 3)$ when n=5.

$P(X \geq 3) = P(X=3) + P(X=4) + P(X=5)$

$=\displaystyle \binom{5}{3}(0.25)^3(0.75)^2 + \displaystyle \binom{5}{4}(0.25)^4(0.75) + \displaystyle \binom{5}{5}(0.25)^5$

$=0.0879+0.0146+0.0010$

$=0.1035$

II/11:

Highest point of trajectory when v = 0, so 0 = v - gt and t = v/g. The horizontal distance travelled will be uv/g at this time, and the vertical will be $\dfrac{v^2}{2g}$.

P will hit the ground when its vertical displacement is $-\dfrac{v^2}{2g}$, so one must solve the quadratic equation $v_et - \dfrac{1}{2}gt^2 = -\dfrac{v^2}{2g}$ giving $t = \dfrac{v_e + \sqrt{v_e^2 + v^2}}{g}$ (using the quadratic formula). The horizontal distance travelled in this time will be $\dfrac{u(v_e + \sqrt{v_e^2 + v^2})}{g}$.

Adding the distances: $R = \dfrac{uv}{g} + \dfrac{u(v_e + \sqrt{v_e^2 + v^2}}{g} \Rightarrow \dfrac{gR}{u} = v + v_e + \sqrt{v_e^2 + v^2}$, as required.

If R > D then $v + v_e + \sqrt{v_e^2 + v^2} > \dfrac{Dg}{u} \Rightarrow \sqrt{v_e^2 + v^2} > \dfrac{Dg}{u} - (v + v_e) \Rightarrow v_e^2 + v^2 > \dfrac{D^2g^2}{u^2} - \dfrac{2Dg}{u}(v + v_e) + v^2 + 2vv_e + v_e^2$

$\Rightarrow \dfrac{2Dgv_e}{u} - 2vv_e > \dfrac{D^2g^2}{u^2} - \dfrac{2Dgv}{u} \Rightarrow v_e \left( \dfrac{2Dg}{u} - 2v \right) > \dfrac{Dg}{u} \left( \dfrac{Dg}{u} - 2v \right) \Rightarrow v_e > \dfrac{Dg}{2u} \left( \dfrac{\frac{Dg}{u} - 2v}{\frac{Dg}{u} - v} \right)$

$\Rightarrow v_e > \dfrac{Dg}{2u} \left( \dfrac{Dg - 2vu}{Dg - vu} \right)$, as required.

Highest point of trajectory when v = 0, so 0 = v - gt and t = v/g. The horizontal distance travelled will be uv/g at this time, and the vertical will be $\dfrac{v^2}{2g}$.

P will hit the ground when its vertical displacement is $-\dfrac{v^2}{2g}$, so one must solve the quadratic equation $v_et - \dfrac{1}{2}gt^2 = -\dfrac{v^2}{2g}$ giving $t = \dfrac{v_e + \sqrt{v_e^2 + v^2}}{g}$ (using the quadratic formula). The horizontal distance travelled in this time will be $\dfrac{u(v_e + \sqrt{v_e^2 + v^2})}{g}$.

Adding the distances: $R = \dfrac{uv}{g} + \dfrac{u(v_e + \sqrt{v_e^2 + v^2}}{g} \Rightarrow \dfrac{gR}{u} = v + v_e + \sqrt{v_e^2 + v^2}$, as required.

If R > D then $v + v_e + \sqrt{v_e^2 + v^2} > \dfrac{Dg}{u} \Rightarrow \sqrt{v_e^2 + v^2} > \dfrac{Dg}{u} - (v + v_e) \Rightarrow v_e^2 + v^2 > \dfrac{D^2g^2}{u^2} - \dfrac{2Dg}{u}(v + v_e) + v^2 + 2vv_e + v_e^2$

$\Rightarrow \dfrac{2Dgv_e}{u} - 2vv_e > \dfrac{D^2g^2}{u^2} - \dfrac{2Dgv}{u} \Rightarrow v_e \left( \dfrac{2Dg}{u} - 2v \right) > \dfrac{Dg}{u} \left( \dfrac{Dg}{u} - 2v \right) \Rightarrow v_e > \dfrac{Dg}{2u} \left( \dfrac{\frac{Dg}{u} - 2v}{\frac{Dg}{u} - v} \right)$

$\Rightarrow v_e > \dfrac{Dg}{2u} \left( \dfrac{Dg - 2vu}{Dg - vu} \right)$, as required.

III Question 1

$\displaystyle y = \ln\left(x+\sqrt{x^2+1}\right) \implies \frac{\mathrm{d}y}{\mathrm{d}x} = \frac{1+\frac{x}{\sqrt{x^2+1}}}{x+\sqrt{x^2+1}} =$

$\displaystyle = \frac{1}{\sqrt{x^2+1}}\frac{\sqrt{x^2+1}+x}{\sqrt{x^2+1}+x} = \frac{1}{\sqrt{x^2+1}} \text{, given } \sqrt{x^2+1}+x \neq 0 \text{, which is clearly true}$

Case when $n = 0$:

$(x^2+1)(-x(x^2+1)^{-3/2}) + x(x^2+1)^{-1/2} = x(x^2+1)^{-1/2}(-1+1) = 0$.

Differentiating the given relation, we get:

$(x^2+1)y^{(n+3)}+(2+(2n+1))xy^{(n+2)} + (2n+1+n^2)y^{(n+1)} = 0 =$

$= (x^2+1)y^{((n+1)+2)} + (2(n+1)+1)xy^{(n+2)} + (n+1)^2y^{(n+1)} = 0$.

Since it is true for n = 0, and it is also true for n + 1, by mathematical induction we conclude that it must also be true for n = 1, n = 2, and so forth for all positive values of n.

Substituting $x = 0$ into the relation, we get $y^{(n+2)}(0) = -n^2y^{(n)}(0)$. Thus $y^{(0)}(0) = 0\implies y^{(2k)}(0) = 0$, all even powers are zero. $y^{(1)}(0) = 1\implies y^{(3)}(0) = -1 \implies y^{(5)}(0) = 9 \implies y^{(7)}(0) = -225$.

Maclaurin series with terms of even powers equal to zero is $y(x) \approx xy^{(1)}(0) + \frac{1}{3!}x^3y^{(3)}(0) + \frac{1}{5!}x^5y^{(5)}(0) + \frac{1}{7!}x^7y^{(7)}(0)$, so $y(x)\approx x + \frac{-1}{3!}x^2 + \frac{9}{5!}x^5+\frac{-225}{7!}x^7 = x-\frac{1}{6}x^3 + \frac{3}{40}x^5 - \frac{5}{112}x^7$.

$\displaystyle y = \ln\left(x+\sqrt{x^2+1}\right) \implies \frac{\mathrm{d}y}{\mathrm{d}x} = \frac{1+\frac{x}{\sqrt{x^2+1}}}{x+\sqrt{x^2+1}} =$

$\displaystyle = \frac{1}{\sqrt{x^2+1}}\frac{\sqrt{x^2+1}+x}{\sqrt{x^2+1}+x} = \frac{1}{\sqrt{x^2+1}} \text{, given } \sqrt{x^2+1}+x \neq 0 \text{, which is clearly true}$

Case when $n = 0$:

$(x^2+1)(-x(x^2+1)^{-3/2}) + x(x^2+1)^{-1/2} = x(x^2+1)^{-1/2}(-1+1) = 0$.

Differentiating the given relation, we get:

$(x^2+1)y^{(n+3)}+(2+(2n+1))xy^{(n+2)} + (2n+1+n^2)y^{(n+1)} = 0 =$

$= (x^2+1)y^{((n+1)+2)} + (2(n+1)+1)xy^{(n+2)} + (n+1)^2y^{(n+1)} = 0$.

Since it is true for n = 0, and it is also true for n + 1, by mathematical induction we conclude that it must also be true for n = 1, n = 2, and so forth for all positive values of n.

Substituting $x = 0$ into the relation, we get $y^{(n+2)}(0) = -n^2y^{(n)}(0)$. Thus $y^{(0)}(0) = 0\implies y^{(2k)}(0) = 0$, all even powers are zero. $y^{(1)}(0) = 1\implies y^{(3)}(0) = -1 \implies y^{(5)}(0) = 9 \implies y^{(7)}(0) = -225$.

Maclaurin series with terms of even powers equal to zero is $y(x) \approx xy^{(1)}(0) + \frac{1}{3!}x^3y^{(3)}(0) + \frac{1}{5!}x^5y^{(5)}(0) + \frac{1}{7!}x^7y^{(7)}(0)$, so $y(x)\approx x + \frac{-1}{3!}x^2 + \frac{9}{5!}x^5+\frac{-225}{7!}x^7 = x-\frac{1}{6}x^3 + \frac{3}{40}x^5 - \frac{5}{112}x^7$.

Mark13

Given that plate is broken, the probability that it is broken by the mathematician is 0.25.

Therefore, the number of plates, $X$, broken by the mathematician can be represented by the following binomial distribution:It seems to me you are assuming independance between the plates - that is, knowing that a particular plate was broken by the mathematician doesn't affect the probability that another plate was broken by the mathematician. It's not at all obvious to me such an assumption is justified.

(It isn't justified for general distributions; imagine the case where each student either breaks 5 plates with probability k, or otherwise breaks no plates at all. Then if you know the mathematican broke a plate, you know he broke 5 of them, and you'd end up with an answer for 'broke > 3 plates' of 1/4).

On the other hand, it wouldn't surprise me if you were right, either. I confess I have a tendancy to overcomplicate these problems. You'd still need some kind of justification though (based on the distribution being Poisson).

I did try doing this the 'brute force' way by looking at all possibilities (e.g. 1st student breaks 0 plates, 2nd student 1 plate, 3rd student 1 plate, mathmo breaks 3 plates), and it feels a little tough for STEP I. Assuming I made no mistakes, I do get a different answer, however. I'll have another go when I'm not literally writing on the back of an envelope in the kitchen.

II/5:

dy/dx = 0 when x = 0, so there's a stationary point. $\dfrac{\text{d}^2y}{\text{d}x^2} = e^{-x^2}(2x^4 - 5x^2 + 1)$, which equals 1 at x = 0, so it's a minimum point. Consider $\displaystyle \int x(1 - x^2)e^{-x^2} \, \text{d}x$, letting u = x^2 gives du = 2x dx so it becomes

$\displaystyle \int \frac{1}{2}e^{-u} - \frac{1}{2}ue^{-u} \, \text{d}u$. From inspection, $y = \frac{1}{2}ue^{-u} \Rightarrow y = \frac{1}{2}x^2e^{-x^2}$. No constant as the curve passes through the origin. At x = 1, y = $\frac{1}{2}e^{-1}$ and $\dfrac{\text{d}^2y}{\text{d}x^2} = -2e^{-1}$; negative, so a maximum.

The other stationary point is $(-1, \dfrac{1}{2}e^{-1}$ as it's an even function (which is to be expected as the derivative is odd).

For C_2: $\dfrac{\text{d}^2y}{\text{d}x^2} = e^{-x^3}(3x^5 - 3x^3 - 3x^2 + 1)$. At x = 0, this equals 1; so minimum. At x = 1, this equals $-2e^{-1}$, negative, so maximum.

The derivatives of C_1 and C_2 are equal at x = 0 and x = 1. If the derivative of C_2 is greater than C_1 for 0 < x < 1 then it follows that $k > \frac{1}{2}e^{-1}$. This relies on showing that $e^{-x^3} > e^{-x^2} \Leftrightarrow e^{x^2} > e^{x^3} \Leftrightarrow x^2 < x^3 \Leftrightarrow x^2 (x - 1) < 0$ for 0 < x < 1, which can be shown with a very simple sketch. As e^(x^2) and e^(x^3) < 1 for 0 < x < 1, when taking logarithms the inequality is reversed.

dy/dx = 0 when x = 0, so there's a stationary point. $\dfrac{\text{d}^2y}{\text{d}x^2} = e^{-x^2}(2x^4 - 5x^2 + 1)$, which equals 1 at x = 0, so it's a minimum point. Consider $\displaystyle \int x(1 - x^2)e^{-x^2} \, \text{d}x$, letting u = x^2 gives du = 2x dx so it becomes

$\displaystyle \int \frac{1}{2}e^{-u} - \frac{1}{2}ue^{-u} \, \text{d}u$. From inspection, $y = \frac{1}{2}ue^{-u} \Rightarrow y = \frac{1}{2}x^2e^{-x^2}$. No constant as the curve passes through the origin. At x = 1, y = $\frac{1}{2}e^{-1}$ and $\dfrac{\text{d}^2y}{\text{d}x^2} = -2e^{-1}$; negative, so a maximum.

The other stationary point is $(-1, \dfrac{1}{2}e^{-1}$ as it's an even function (which is to be expected as the derivative is odd).

For C_2: $\dfrac{\text{d}^2y}{\text{d}x^2} = e^{-x^3}(3x^5 - 3x^3 - 3x^2 + 1)$. At x = 0, this equals 1; so minimum. At x = 1, this equals $-2e^{-1}$, negative, so maximum.

The derivatives of C_1 and C_2 are equal at x = 0 and x = 1. If the derivative of C_2 is greater than C_1 for 0 < x < 1 then it follows that $k > \frac{1}{2}e^{-1}$. This relies on showing that $e^{-x^3} > e^{-x^2} \Leftrightarrow e^{x^2} > e^{x^3} \Leftrightarrow x^2 < x^3 \Leftrightarrow x^2 (x - 1) < 0$ for 0 < x < 1, which can be shown with a very simple sketch. As e^(x^2) and e^(x^3) < 1 for 0 < x < 1, when taking logarithms the inequality is reversed.

STEP III 2001, QUESTION 3

Consider the equation

$x^2 - bx + c = 0$

where b and c are real numbers.

(i) Show that the roots of the equation are real and positive if and only if $b > 0$ and $b^2 \geq 4c > 0$, and sketch the region of the b-c plane in which these conditions hold.

(ii) Sketch the region of the b-c plane in which the roots are real and less than 1 in magnitude.

(Apologies for any dodgy copypasting, notepad is a bitch to work with sometimes.)

Consider the equation

$x^2 - bx + c = 0$

where b and c are real numbers.

(i) Show that the roots of the equation are real and positive if and only if $b > 0$ and $b^2 \geq 4c > 0$, and sketch the region of the b-c plane in which these conditions hold.

Spoiler

(ii) Sketch the region of the b-c plane in which the roots are real and less than 1 in magnitude.

Spoiler

(Apologies for any dodgy copypasting, notepad is a bitch to work with sometimes.)

DFranklin

It seems to me you are assuming independance between the plates - that is, knowing that a particular plate was broken by the mathematician doesn't affect the probability that another plate was broken by the mathematician. It's not at all obvious to me such an assumption is justified.

(It isn't justified for general distributions; imagine the case where each student either breaks 5 plates with probability k, or otherwise breaks no plates at all. Then if you know the mathematican broke a plate, you know he broke 5 of them, and you'd end up with an answer for 'broke > 3 plates' of 1/4).

On the other hand, it wouldn't surprise me if you were right, either. I confess I have a tendancy to overcomplicate these problems. You'd still need some kind of justification though (based on the distribution being Poisson).

I did try doing this the 'brute force' way by looking at all possibilities (e.g. 1st student breaks 0 plates, 2nd student 1 plate, 3rd student 1 plate, mathmo breaks 3 plates), and it feels a little tough for STEP I. Assuming I made no mistakes, I do get a different answer, however. I'll have another go when I'm not literally writing on the back of an envelope in the kitchen.

(It isn't justified for general distributions; imagine the case where each student either breaks 5 plates with probability k, or otherwise breaks no plates at all. Then if you know the mathematican broke a plate, you know he broke 5 of them, and you'd end up with an answer for 'broke > 3 plates' of 1/4).

On the other hand, it wouldn't surprise me if you were right, either. I confess I have a tendancy to overcomplicate these problems. You'd still need some kind of justification though (based on the distribution being Poisson).

I did try doing this the 'brute force' way by looking at all possibilities (e.g. 1st student breaks 0 plates, 2nd student 1 plate, 3rd student 1 plate, mathmo breaks 3 plates), and it feels a little tough for STEP I. Assuming I made no mistakes, I do get a different answer, however. I'll have another go when I'm not literally writing on the back of an envelope in the kitchen.

Hmm, I got 53/512, which is rounded to 0.1035., using the brute force method.

STEP I 2001, Question 5

Which happens to be true. (For more info, look into differentiation under the integral sign)

Spoiler

Which happens to be true. (For more info, look into differentiation under the integral sign)

Aurel-Aqua

I Question 2

Part (i) $1+2x-x^2>\frac{2}{x}$, $x\neq 0$.

First it might help to find the points of equality: multiplying by x and rearranging, we get $x^3-2x^2-x+2 = (x-2)(x-1)(x+1)=0$, $x\neq 0$. The roots are -1, 1 and 2. Now we must sketch the graph and notice that the inequality is satisfied for $-1 < x < 0$ (the zero due to the asymptote) and $1<x<2$.

Part (ii) $\sqrt{3x+10} > 2 + \sqrt{x+4}$, $x \geq -\frac{10}{3}$.

Because of the condition, squaring both sides still leaves an iff, giving: $3x + 10 > 4 + x + 4 + 4\sqrt{x+4}\implies 2x + 2 > 4\sqrt{x+4} \implies x + 1 > 2\sqrt{x+4}$. Now by the condition, we can deduce that x > -1 because of the square root being always positive. Squaring again, $x^2+2x+1 > 4(x + 4) \implies x^2 - 2x - 15 > 0$, giving $(x+3)(x-5)>0$. From this, we can deduce that $x > 5$ and $x<-3$, but our new condition, $x>-1$ only gives us $x>5$ as the final solution.

Part (i) $1+2x-x^2>\frac{2}{x}$, $x\neq 0$.

First it might help to find the points of equality: multiplying by x and rearranging, we get $x^3-2x^2-x+2 = (x-2)(x-1)(x+1)=0$, $x\neq 0$. The roots are -1, 1 and 2. Now we must sketch the graph and notice that the inequality is satisfied for $-1 < x < 0$ (the zero due to the asymptote) and $1<x<2$.

Part (ii) $\sqrt{3x+10} > 2 + \sqrt{x+4}$, $x \geq -\frac{10}{3}$.

Because of the condition, squaring both sides still leaves an iff, giving: $3x + 10 > 4 + x + 4 + 4\sqrt{x+4}\implies 2x + 2 > 4\sqrt{x+4} \implies x + 1 > 2\sqrt{x+4}$. Now by the condition, we can deduce that x > -1 because of the square root being always positive. Squaring again, $x^2+2x+1 > 4(x + 4) \implies x^2 - 2x - 15 > 0$, giving $(x+3)(x-5)>0$. From this, we can deduce that $x > 5$ and $x<-3$, but our new condition, $x>-1$ only gives us $x>5$ as the final solution.

You're genius you know, I've got 98% in my C2 - but I was clueless in this question - you're really a maths genius

Glutamic Acid

Hmm, I got 53/512, which is rounded to 0.1035., using the brute force method.

Can someone do 2001 III Q8?

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