Find roots and the area under a parabola

Questions about complex arithmetic; argument and modulus of complex numbers; complex roots of polynomials; de Moivre's theorem.

Find roots and the area under a parabola

Find roots and the area under a parabola

Factorise a quadratic equation where the coefficient of the $x^2$ term is greater than 1 and then write down the roots of the equation

Straightforward question: student must find the general solution to a second order constant coefficient ODE. Uses custom marking algorithm to check that both roots appear and that the solution is in the correct form (e.g. two arbitrary constants are present). Arbitrary constants can be any non space-separated string of characters. The algorithm also allows for the use of $e^x$ rather than $\exp(x)$.

Unit tests are also included, to check whether the algorithm accurately marks when the solution is correct; when it's correct but deviates from the 'answer'; when one or more roots is incorrect; or when the roots are correct but constants of integration have been forgotten.

Find roots and the area under a parabola

Solve a quadratic equation by completing the square. The roots are not pretty!

This is the question for week 5 of the MA100 course at the LSE. It looks at material from chapters 9 and 10.

The following describes how we define our revenue and cost functions for part b of the question.

We have variables c, f, m, h.

The revenue function is R(q) = -c q^2 + 2mf q .
The cost function is C(q) = f q^2 - 2mc q + h .

The "revenue - cost" function is -(c+f) q^2 +2m(c+f) q - h

Differentiating, we see that there is a maximum point at m.

We pick each one of f, m, h randomly from the set {2, .. 6}, and we pick c randomly from {h+1 , ... , h+5}. This ensures that the discriminant of the "revenue - cost" function is positive, meaning there are two real roots, meaning the maximum point lies above the x-axis. I.e. we can actually make a profit.

A graph (of a cubic) is given. The question is to determine the number of roots.

Questions about complex arithmetic; argument and modulus of complex numbers; complex roots of polynomials; de Moivre's theorem.

Questions about complex arithmetic; argument and modulus of complex numbers; complex roots of polynomials; de Moivre's theorem.

Straightforward question: student must find the general solution to a second order constant coefficient ODE. Uses custom marking algorithm to check that both roots appear and that the solution is in the correct form (e.g. two arbitrary constants are present). Arbitrary constants can be any non space-separated string of characters. The algorithm also allows for the use of $e^x$ rather than $\exp(x)$.

Unit tests are also included, to check whether the algorithm accurately marks when the solution is correct; when it's correct but deviates from the 'answer'; when one or more roots is incorrect; or when the roots are correct but constants of integration have been forgotten.

Apply the quadratic formula to find the roots of a given equation. The quadratic formula is given in the steps if the student requires it.

Solve a quadratic equation by completing the square. The roots are not pretty!

Quiz covering basic arithmetic with complex numbers and solving roots for a quadratic with complex solutions

Practice of basic transpositions. Doesn't include roots

A graph (of a cubic) is given. The question is to determine the number of roots and number of stationary points the graph has. Non-calculator. Advice is given.

A graph (of a cubic) is given. The question is to determine the number of roots and number of stationary points the graph has. Non-calculator. Advice is given.

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Factorise three quadratic equations of the form $x^2+bx+c$.

The first has two negative roots, the second has one negative and one positive, and the third is the difference of two squares.

Solve: $\displaystyle \frac{d^2y}{dx^2}+2a\frac{dy}{dx}+a^2y=0,\;y(0)=c$ and $y(1)=d$.  (Equal roots example).

Solve for $x(t)$, $\displaystyle\frac{dx}{dt}=\frac{a}{(x+b)^n},\;x(0)=0$

Quiz covering basic arithmetic with complex numbers and solving roots for a quadratic with complex solutions

Factorise a quadratic equation where the coefficient of the $x^2$ term is greater than 1 and then write down the roots of the equation

Find the roots of this bad boy

Find modulus and argument of the complex number $z_1$ and find the $n$th roots of $z_1$ where $n=5,\;6$ or $7$.

Apply the quadratic formula to find the roots of a given equation. The quadratic formula is given in the steps if the student requires it.

Factorise three quadratic equations of the form $x^2+bx+c$.

The first has two negative roots, the second has one negative and one positive, and the third is the difference of two squares.

Apply the quadratic formula to find the roots of a given equation. The quadratic formula is given in the steps if the student requires it.