Math HL test - February 8, 2010.

Polynomials, long division, factor and remainder theorem, logarithms, inequalities.

1. Determine which of (x - 3), (x - 1), (x + 2) are factors of P(x) = 2x^3+7x^2+7x+2,
and hence factorise P(x) completely.

We can factorise right away and that will also answer the first question:

(%i383) factor(2*x^3+7*x^2+7*x+2);

2. Given that (x - 1) and (x - 2) are factors of 6x^4+ax^3-17x^2+bx-4;
find a and b and any remaining factors.

(%i384) f(x):=6*x^4+a*x^3-17*x^2+b*x-4;

(%i385) solve([f(1)=0,f(2)=0],[a,b]);

(%i386) factor( subst(%, f(x)));

Maxima can't factor 6x^2+9x-2 so we find its roots:

(%i387) solve(6*x^2+9*x-2=0);

So f(x) = (x-2)(x-1)(x+(sqrt(129)+9)/12)(x-(sqrt(129)-9)/12)

3. The number of bacteria, N, t hours after infection is given by the formula N(t)=4.7^(t+1).

(a) What is the number of bacteria after 5 hours?

(%i388) N(t):=4.7^(t+1);

(%i389) round(N(5));

(b) After how many hours will the number of bacteria exceed 5000?

(%i390) eq: 4.7^(t+1)=5000;

We have to help Maxima by taking log (log stands for ln in Maxima) on both sides:

(%i391) map(log,eq),logexpand=super;

(%i392) float(solve(%,t));

4. Solve for x: 6^(2*x+1) - 17*6^x + 12 = 0. Find approximate solutions to 3 decimal places.

(%i393) f(x):=6^(2*x+1) - 17*6^x + 12 ;

If we graph the function, after a while we find that the zeros are close to .2:

(%i394) plot2d(f(x),[x,.1,.3]);

So, then we can find a numerical solution by choosing x values where f(x) has different signs:

(%i395) find_root(f(x),.2,.3);

(%i396) find_root(f(x),-2,.2);

I can not get solve() to work:

(%i397) solve(f(x)=0);

So I manually do the substitution z = 6^x.

(%i398) solve(6*z^2-17*z+12);

So the first solution is:

(%i399) solve(6^x=4/3);

(%i400) float(%);

and the second solution is:

(%i401) solve(6^x=3/2);

(%i402) float(%);

5. Simplify these expressions: (to one log function)
(a) log(xy^2) - log(y) (Note that log() below stands for ln().)

(%i403) logcontract(log(x*y^2) - log(y));

(b) ln(ab) + 2lna - ln{b^2}

(%i404) logcontract(log(a*b) + 2*log(a) - log(b^2));

6. Solve the inequality:(x + 1) / 4 < 2 / (x - 1).

(%i405) load(solve_rat_ineq);

(%i406) solve_rat_ineq((x+1)/4 < 2/(x-1));