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Troubleshooting: Evaluating an Trigonometric Integral Algebraically

Hehe…we are always amused by calculus students who manage to mis-integrate a function to the point of yielding nonsensical answers. To illustrate what we mean by this, here is Steve, who — at the point of this writing at least — still haven’t got a clue where his computations went awry: How do you evaluate […]

Modular Arithmetic: Some Advanced Tricks in Finding the Least Positive Residue

We got a least-residue question from David a few weeks ago, but never really took the time to address it properly. Here’s the bombshell David dropped us a bit earlier: Find the least positive residue of $\displaystyle 12^{345}$ in mod $35$. Wow. That seems like a huge number. How on earth do you reduce it […]

How to Solve a System of Linear Equations with an Additional Parameter?

Today, Connor asked the following question involving a system of linear equation with an additional parameter: Given that $x_1 + hx_2 = 4$  and $3x_1 + 6x_2 =8$, find all pairs of the form $(x_1,x_2)$ satisfying both equations. OK…What do we do with that additional $h$? Well, read on!

How to Solve the Triangle with $A=30^{\circ}$, $b=40$, and $a=10$?

We got asked this trigonometry question, which we immediately converted into drawing for better readability. Before we begin, what does it mean to solve a triangle? Well, it’s just a short way of saying finding all the missing side lengths and angles of a triangle. So how on earth do we actually solve this triangle, with […]

Given a bunch of boxes and balls, what is the chance that randomly dropping the balls will fill all the boxes?

Carl asked the following discrete-math-style probability question: $N$ balls are randomly dropped into $k$ boxes $(k \le N)$. What is the probability that all boxes are filled? Want to know the answer? Here it is!

How to Solve the Logarithmic Inequality $\log_{0.3} (x^2+8) > \log_{0.3} (9x)$?

Subham asked the following question involving a logarithmic inequality: Solve $\log_{0.3} (x^2+8) > \log_{0.3} (9x)$. And here is our answer: