MAT Domain 5: Mathematics

If \(a\) and \(b\) are roots of \(x^2 - x - 1 = 0\), what is \(a^2 + b^2\)?

Consider a sequence where \(T_n = n^2 + n + 1\). What is \(T_{10} - T_9\)?

In group theory, the order of an element \(g\) in a group \(G\) is the smallest positive integer \(n\) such that \(g^n = e\), where \(e\) is the identity element. If the order of \(g\) is 15, which of the following could be the order of the group \(G\)?

What is the remainder when \(2^{2017}\) is divided by 5?

A set of all x for which \(x^2 - 6x + 9 < 0\) is:

What is the derivative of \(f(x) = x^x\) at \(x = 2\)?

The sum of the series \(1 + \frac{1}{3} + \frac{1}{5} + \frac{1}{7} + \cdots\) up to the \(n\)-th odd denominator is:

A matrix \(A\) has eigenvalues 2, 3, and 4. What is the determinant of \(A\)?

For a real number \(x\), the expression \(\frac{d}{dx}(e^{x^2})\) equals:

The integral of \(f(x) = \frac{1}{x\ln(x)}\) from \(e\) to \(e^2\) is:

In topology, the Euler characteristic \(\chi\) of a convex polyhedron with \(V\) vertices, \(E\) edges, and \(F\) faces is given by \(\chi = V - E + F\). For a tetrahedron, \(\chi\) is:

If the radius of a circle is tripled, by what factor does the area of the circle increase?

The binomial coefficient \(\binom{n}{k}\) represents the number of ways to choose \(k\) items from \(n\) items without regard to order. What is \(\binom{5}{2}\)?

The number of solutions to the equation \(sin(x) = 0.5\) in the interval \(0 \leq x \leq 2\pi\) is:

A function \(f(x) = x^3 - 3x^2 + 4\) has a critical point at \(x =\):

If the prime factorization of \(n\) is \(2^3 \cdot 5^2 \cdot 7\), how many positive divisors does \(n\) have?

What is the sum of the infinite geometric series \(1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \cdots\)?

If \(i\) is the imaginary unit where \(i^2 = -1\), what is \(i^{2020}\)?

What is the limit of \(f(x) = \frac{x^2 - 4}{x - 2}\) as \(x\) approaches 2?

Given the matrix \(A = \begin{pmatrix}2 & 3\\ 4 & 5\end{pmatrix}\), what is the trace of \(A^2\)?