Mathematics

What does \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} represent?

The sine of an angle in a right triangle equals the ratio of the opposite side to the hypotenuse. It defines the vertical projection relative to the hypotenuse.

When is \tan(\theta) undefined?

When the adjacent side (or x-component) is zero — this occurs at \theta = 90^\circ + k180^\circ where cosine is zero.

Given 3x^2 - 5x + 2 = 0, find its roots.

Use the quadratic formula x = \frac{5 \pm \sqrt{25 - 24}}{6} → x = 1 or x = \frac{2}{3}.

What is Euler’s formula relating exponential and trigonometric forms of complex numbers?

e^{j\theta} = \cos(\theta) + j\sin(\theta) [HB] — used to convert between polar and rectangular forms.

Find the conjugate of z = 3 - 4i.

The conjugate is z^* = 3 + 4i. Flip the sign of the imaginary part.

If z = 4(\cos30^\circ + i\sin30^\circ), compute z^3.

z^3 = 4^3[\cos(90^\circ) + i\sin(90^\circ)] = 64i.

What is |3 - 4i|?

The modulus is \sqrt{3^2 + (-4)^2} = 5. It represents the distance from the origin in the complex plane.

Simplify (1 - i)(1 + i).

Multiply: 1 - i^2 = 1 - (-1) = 2.

What is i^{2025}?

Divide 2025 by 4 → remainder 1 → i^{2025} = i since powers of i repeat every 4.

Find

z1 \times z2 for z1 = 3 + 2i, z2 = 1 - 4i

Multiply and combine: (3 + 2i)(1 - 4i) = 3 - 12i + 2i - 8i^2 = 11 - 10i.

What is a strongly connected component in a directed graph?

A set of vertices where every vertex is reachable from every other vertex within the set.

What is the Cartesian product A \times B for A = {1,2}, B = {x,y}?

{(1,x),(1,y),(2,x),(2,y)} — all possible ordered pairs combining elements of A and B.

What is the degree sum of a simple undirected graph with 10 vertices and 21 edges?

The sum of degrees = 2E = 42.

What does an adjacency matrix with zeros on its diagonal represent?

A simple graph with no self-loops, since diagonal elements indicate self-connections.

What is a function from set A to set B?

A relation where each element of A is assigned to exactly one element of B.

What is the slope of a line passing through (1, 2) and (3, 8)?

m = \frac{8 - 2}{3 - 1} = 3.

Find the distance between (2, 3, –1) and (–1, –3, 4).

d = \sqrt{(3)^2 + (6)^2 + (-5)^2} = \sqrt{70}.

Find the equation of a circle centered at (1, –2) with radius 3.

(x - 1)^2 + (y + 2)^2 = 9.

Determine if the lines y = 2x + 3 and y = -\frac{1}{2}x + 1 are perpendicular.

Yes — their slopes multiply to –1, so they are perpendicular.

Find the slope of a line perpendicular to 3x + 4y = 12.

Rewrite as y = -\frac{3}{4}x + 3 → perpendicular slope = \frac{4}{3}.

What is the general equation of a plane with normal vector \langle a,b,c\rangle passing through point

(x0, y0, z_0)

a(x - x0) + b(y - y0) + c(z - z_0) = 0

[HB].

Evaluate \lim_{x\to0}\frac{\sin x}{x}.

Limit = 1 — a standard trigonometric limit often used in derivatives. [HB]

Evaluate \lim_{x\to0}\frac{1 - \cos x}{x^2}.

Use 1 - \cos x = 2\sin^2(x/2) → limit = \frac{1}{2}.

Find f'(x) for f(x) = 3x^2 - 5x + 1.

Differentiate: f'(x) = 6x - 5.

If f'(x) = 3x^2 + 1 and f(1) = 5, find f(2).

Integrate to get f(x) = x^3 + x + C → plug x=1 → 5 = 1 + 1 + C \Rightarrow C = 3 → f(2)=8+2+3=13.

Find the area under y = 4 - x^2 between x = -2 and x = 2.

A = \int{-2}^{2}(4 - x^2)dx = [4x - \frac{x^3}{3}]{-2}^{2} = \frac{32}{3}

Find the derivative of g(x) = \ln(2x + 1).

g'(x) = \frac{2}{2x + 1} [HB].

Find \int (3x^2 + 2x)dx.

x^3 + x^2 + C — integrate each term by power rule.

Compute \int_{0}^{\pi/2}(\sin x + 2\cos x)dx.

[-\cos x + 2\sin x]_{0}^{\pi/2} = 3.

Find the tangent line to y = x^2 - 4x + 3 at x = 1.

y' = 2x - 4 → slope = –2 → point (1, 0) → y = -2x + 2.

Evaluate \int e^{2x}dx.

\frac{1}{2}e^{2x} + C [HB].

What is the integrating factor for y' + 3y = 6e^{-x}?

\mu = e^{\int3dx} = e^{3x}.

Solve y' + 5y = 0.

Separate variables: \frac{dy}{y} = -5dx → integrate → y = Ce^{-5x}.

Solve y' - 2y = 0, y(0)=3.

y = Ce^{2x} → plug initial → 3 = C → y = 3e^{2x}.

Solve y'' - 5y' + 6y = 0.

Characteristic: r^2 - 5r + 6=0 → r=2,3 →

y=c_1e^{2x}+c_2e^{3x}

Solve y'' + 9y = 0.

Characteristic: r = \pm 3i → y =

c_1\cos(3x) + c_2\sin(3x)

For y'' + 2y' + y = 0, find the general solution.

r^2 + 2r + 1 = 0 → r=-1 (double root) →

y = (c_1 + c_2x)e^{-x}

Find the determinant of A = \left[\begin{array}{cc}2 & 1 \\ 4 & 3\end{array}\right]

|A| = (2)(3) - (1)(4) = 2.

Compute A^{-1} for A = \left[\begin{array}{cc}1 & 2 \\ 3 & 4\end{array}\right]

A^{-1} = \frac{1}{(1)(4)-(2)(3)}\begin{bmatrix}4 & -2 \ -3 & 1\end{bmatrix} = \begin{bmatrix}-2 & 1 \ 1.5 & -0.5\end{bmatrix}.

Find eigenvalues of \left[\begin{array}{cc}4 & 1 \\ 2 & 3\end{array}\right].

|A - \lambda I| = (4-\lambda)(3-\lambda) - 2 = \lambda^2 - 7\lambda + 10 = 0 → \lambda = 5, 2.

When are vectors \mathbf{a} and \mathbf{b} orthogonal?

When their dot product \mathbf{a}\cdot\mathbf{b} = 0.

Find \mathbf{a}\cdot\mathbf{b} if \mathbf{a} = \langle 2, -1, 3\rangle and \mathbf{b} = \langle 1, 4, 0\rangle.

2(1) + (-1)(4) + 3(0) = -2.

Find \mathbf{a}\times\mathbf{b} for \mathbf{a} = \langle 1,2,3\rangle, \mathbf{b} = \langle 4,5,6\rangle.

\langle -3, 6, -3\rangle — computed via determinant expansion.

Compute the magnitude of \mathbf{v} = \langle 3, -4, 12\rangle.

|\mathbf{v}| = \sqrt{3^2 + (-4)^2 + 12^2} = 13.

Find the unit vector of \mathbf{v} = \langle 6, 8\rangle.

\hat{v} = \frac{\mathbf{v}}{|\mathbf{v}|} = \langle \frac{3}{5}, \frac{4}{5}\rangle.

Evaluate \nabla \cdot \mathbf{F} for \mathbf{F} = \langle x, y, z\rangle.

\frac{\partial x}{\partial x} + \frac{\partial y}{\partial y} + \frac{\partial z}{\partial z} = 3.

Evaluate \nabla \times \mathbf{F} for \mathbf{F} = \langle y, -x, 0\rangle.

Curl = \langle 0, 0, -2\rangle.

What is the gradient of f(x,y,z) = x^2 + y^2 + z^2?

\nabla f = \langle 2x, 2y, 2z\rangle.

Find the directional derivative of f(x,y)=x^2y at (1,2) in direction \mathbf{v}=\langle3,-4\rangle.

\nabla f = \langle 2xy, x^2\rangle = \langle4,1\rangle → D_{\hat{v}}f = \frac{(4,1)\cdot(3,-4)}{5} = \frac{8}{5}.

What is the equation of a plane tangent to z = x^2 + y^2 at (1,1,2)?

z - 2 = 2(x - 1) + 2(y - 1) → z = 2x + 2y - 2.

State the divergence theorem conceptually.

It relates flux through a closed surface to the volume integral of divergence:

\iiint_V (\nabla\cdot \mathbf{F})dV = \iint_S \mathbf{F}\cdot d\mathbf{A} [HB]

What is the curl physically interpreted as?

Curl represents rotation — how much and about which axis a field “swirls” around a point.

Define a conservative vector field.

A field where \nabla \times \mathbf{F} = 0; path-independent work; exists a potential function \phi such that \mathbf{F} = \nabla\phi.

Find the determinant of \left[\begin{array}{ccc}1 & 0 & 2 \\ -1 & 3 & 1 \\ 2 & -2 & 1\end{array}\right].

Expand → 1(3\cdot1 - 1\cdot(-2)) - 0(...) + 2((-1)(-2) - 3\cdot2) = 1(5) + 2(-4) = -3.

If \mathbf{A}\mathbf{x} = \mathbf{b} has more equations than unknowns, what type of system is it?

Overdetermined — often solved using least squares approximation.

What is the sum of an arithmetic series

Sn = \frac{n}{2}(a1 + a_n)

used for?

Finding the sum of equally spaced terms; useful in discrete sequences. [HB]

What is \sum_{k=1}^{n} k = ?

\frac{n(n+1)}{2} — standard summation identity.

What is \sum_{k=1}^{n} k^2 = ?

\frac{n(n+1)(2n+1)}{6} [HB].

If

an = 2a{n-1} + 3 with a1=1, find a2, a_3

a2=5, a3=13

— plug iteratively using recurrence relation.

Find the inverse of y = 3x + 2.

Swap x and y → x = 3y + 2 \Rightarrow y = \frac{x - 2}{3}.

If f(x) = |x|, what is f'(0)?

Undefined — left derivative = –1, right derivative = +1, so discontinuous slope.

For f(x,y) = x^2y + 3y^3, find f_{xy}.

fx = 2xy → f{xy} = 2x \text{ at any point.}

Find \frac{d}{dx}(\sin^{-1}x).

\frac{1}{\sqrt{1 - x^2}} [HB].

Find \frac{d}{dx}(e^{3x}).

3e^{3x} — chain rule. [HB]

Find \int \frac{1}{x}dx.

\ln|x| + C [HB]

What is \int \sin x\,dx?

-\cos x + C [HB]

Find \int \frac{dx}{x^2 + 1}.

\tan^{-1}x + C [HB]

State the condition for a local extremum using the second derivative.

If f''(x)>0 → local minimum; if f''(x)<0 → local maximum.

If f(x,y)=x^2+y^2, find the critical point.

\nabla f = 0 \Rightarrow (x,y)=(0,0); minimum at the origin.

When is a differential equation considered linear?

When dependent variable and derivatives appear only to the first power and not multiplied together.

What is a homogeneous ODE?

An equation where every term involves the dependent variable or its derivatives — can be written as y' = f(y/x).

What type of ODE is \frac{dy}{dx}=x^2+y^2?

Nonlinear — y^2 term makes it nonlinear.

What is the Laplace transform of e^{at}?

\frac{1}{s - a} [HB].

What is the Laplace transform of \sin(at)?

\frac{a}{s^2 + a^2} [HB].

What is the Laplace transform of \cos(at)?

\frac{s}{s^2 + a^2} [HB].
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