text:
# Difference between revisions of "1981 AHSME Problems"

## Problem 1

If $\sqrt{x+2}=2$, then $(x+2)^{2}$ equals

$\textbf{(A)}\ \sqrt{2}\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 4\qquad\textbf{(D)}\ 8\qquad\textbf{(E)}\ 16$

## Problem 2

Point $E$ is on side $AB$ of square $ABCD$. If $EB$ has length one and $EC$ has length two, then the area of the square is

$\textbf{(A)}\ \sqrt{3}\qquad\textbf{(B)}\ \sqrt{5}\qquad\textbf{(C)}\ 3\qquad\textbf{(D)}\ 2\sqrt{3}\qquad\textbf{(E)}\ 5$

## Problem 3

For $x\neq0$, $\dfrac{1}{x}+\dfrac{1}{2x}+\dfrac{1}{3x}$ equals

$\textbf{(A)}\ \dfrac{1}{2x}\qquad\textbf{(B)}\ \dfrac{1}{6}\qquad\textbf{(C)}\ \dfrac{5}{6x}\qquad\textbf{(D)}\ \dfrac{11}{6x}\qquad\textbf{(E)}\ \dfrac{1}{6x^3}$

## Problem 4

If three times the larger of two numbers is four times the smaller and the difference between the numbers is 8, the the larger of two numbers is

$\textbf{(A)}\ 16\qquad\textbf{(B)}\ 24\qquad\textbf{(C)}\ 32\qquad\textbf{(D)}\ 44\qquad\textbf{(E)}\ 52$

## Problem 5

In trapezoid $ABCD$, sides $AB$ and $CD$ are parallel, and diagonal $BD$ and side $AD$ have equal length. If $m\angle DCB=110^\circ$ and $m\angle CBD=30^\circ$, then $m\angle ADB=$

$\textbf{(A)}\ 80^\circ\qquad\textbf{(B)}\ 90^\circ\qquad\textbf{(C)}\ 100^\circ\qquad\textbf{(D)}\ 110^\circ\qquad\textbf{(E)}\ 120^\circ$

## Problem 6

If $\dfrac{x}{x-1}=\dfrac{y^2+2y-1}{y^2+2y-2}$, then $x$ equals

$\textbf{(A)}\ y^2+2y-1\qquad\textbf{(B)}\ y^2+2y-2\qquad\textbf{(C)}\ y^2+2y+2 \qquad \\ \textbf{(D)}\ y^2+2y+1\qquad\textbf{(E)}\ -y^2-2y+1$

## Problem 7

How many of the first one hundred positive integers are divisible by all of the numbers $2$, $3$, $4$, and $5$?

$\textbf{(A)}\ 0\qquad\textbf{(B)}\ 1\qquad\textbf{(C)}\ 2\qquad\textbf{(D)}\ 3\qquad\textbf{(E)}\ 4$

## Problem 8

For all positive numbers $x$, $y$, $z$, the product $(x+y+z)^{-1}(x^{-1}+y^{-1}+z^{-1})(xy+yz+xz)^{-1}[(xy)^{-1}+(yz)^{-1}+(xz)^{-1}]$ equals $\textbf{(A)}\ x^{-2}y^{-2}z^{-2}\qquad\textbf{(B)}\ x^{-2}+y^{-2}+z^{-2}\qquad\textbf{(C)}\ (x+y+z)^{-1}\qquad \textbf{(D)}\ \dfrac{1}{xyz}\qquad \\ \textbf{(E)}\ \dfrac{1}{xy+yz+xz}$

## Problem 9

In the adjoining figure, $PQ$ is a diagonal of the cube. If $PQ$ has length $a$, then the surface area of the cube is

$[asy] import three; unitsize(1cm); size(200); currentprojection=orthographic(1/3,-1,1/2); draw((0,0,0)--(1,0,0)--(1,1,0)--(0,1,0)--cycle,black); draw((0,0,0)--(0,0,1),black); draw((0,1,0)--(0,1,1),black); draw((1,1,0)--(1,1,1),black); draw((1,0,0)--(1,0,1),black); draw((0,0,1)--(1,0,1)--(1,1,1)--(0,1,1)--cycle,black); draw((0,0,0)--(1,1,1),black); label("P",(0, 0, 0),NW); label("Q",(1, 1, 1),NE); [/asy]$

$\textbf{(A)}\ 2a^2\qquad\textbf{(B)}\ 2\sqrt{2}a^2\qquad\textbf{(C)}\ 2\sqrt{3}a^2\qquad\textbf{(D)}\ 3\sqrt{3}a^2\qquad\textbf{(E)}\ 6a^2$

## Problem 10

The lines $L$ and $K$ are symmetric to each other with respect to the line $y=x$. If the equation of the line $L$ is $y=ax+b$ with $a\neq 0$ and $b\neq 0$, then the equation of $K$ is $y=$

$\textbf{(A)}\ \dfrac{1}{a}x+b\qquad\textbf{(B)}\ -\dfrac{1}{a}x+b\qquad\textbf{(C)}\ \dfrac{1}{a}x-\dfrac{b}{a}\qquad\textbf{(D)}\ \dfrac{1}{a}x+\dfrac{b}{a}\qquad\textbf{(E)}\ \dfrac{1}{a}x-\dfrac{b}{a}$

## Problem 11

The three sides of a right triangle have integral lengths which form an arithmetic progression. One of the sides could have length

$\textbf{(A)}\ 22\qquad\textbf{(B)}\ 58\qquad\textbf{(C)}\ 81\qquad\textbf{(D)}\ 91\qquad\textbf{(E)}\ 361$

## Problem 12

If $p$, $q$, and $M$ are positive numbers and $q<100$, then the number obtained by increasing $M$ by $p\%$ and decreasing the result by $q\%$ exceeds $M$ if and only if

$\textbf{(A)}\ p>q \qquad\textbf{(B)}\ p>\dfrac{q}{100-q}\qquad\textbf{(C)}\ p>\dfrac{q}{1-q}\qquad \textbf{(D)}\ p>\dfrac{100q}{100+q}\qquad \\ \textbf{(E)}\ p>\dfrac{100q}{100-q}$

## Problem 13

Suppose that at the end of any year, a unit of money has lost $10\%$ of the value it had at the beginning of that year. Find the smallest integer $n$ such that after $n$ years, the money will have lost at least $90\%$ of its value (To the nearest thousandth $\log_{10} 3 = 0.477$).

$\textbf{(A)}\ 14\qquad\textbf{(B)}\ 16\qquad\textbf{(C)}\ 18\qquad\textbf{(D)}\ 20\qquad\textbf{(E)}\ 22$

## Problem 14

In a geometric sequence of real numbers, the sum of the first $2$ terms is $7$, and the sum of the first $6$ terms is $91$. The sum of the first $4$ terms is

$\textbf{(A)}\ 28\qquad\textbf{(B)}\ 32\qquad\textbf{(C)}\ 35\qquad\textbf{(D)}\ 49\qquad\textbf{(E)}\ 84$

## Problem 15

If $b>1$, $x>0$, and $(2x)^{\log_b 2}-(3x)^{\log_b 3}=0$, then $x$ is

$\textbf{(A)}\ \dfrac{1}{216}\qquad\textbf{(B)}\ \dfrac{1}{6}\qquad\textbf{(C)}\ 1\qquad\textbf{(D)}\ 6\qquad\textbf{(E)}\ \text{not uniquely determined}$

## Problem 16

The base three representation of $x$ is $$12112211122211112222$$ The first digit (on the left) of the base nine representation of $x$ is

$\textbf{(A)}\ 1\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 3\qquad\textbf{(D)}\ 4\qquad\textbf{(E)}\ 5$

## Problem 17

The function $f$ is not defined for $x=0$, but, for all non-zero real numbers $x$, $f(x)+f\left(\dfrac{1}x\right)=x$. The equation $f(x)=f(-x)$ is satisfied by

$\textbf{(A)}\ \text{exactly one real number} \qquad \textbf{(B)}\ \text{exactly two real numbers} \qquad\textbf{(C)}\ \text{no real numbers}\qquad \\ \textbf{(D)}\ \text{infinitely many, but not all, non-zero real numbers} \qquad\textbf{(E)}\ \text{all non-zero real numbers}$

## Problem 18

The number of real solutions to the equation $$\dfrac{x}{100}=\sin x$$ is

$\textbf{(A)}\ 61\qquad\textbf{(B)}\ 62\qquad\textbf{(C)}\ 63\qquad\textbf{(D)}\ 64\qquad\textbf{(E)}\ 65$

## Problem 19

In $\triangle ABC$, $M$ is the midpoint of side $BC$, $AN$ bisects $\angle BAC$, and $BN\perp AN$. If sides $AB$ and $AC$ have lengths $14$ and $19$, respectively, then find $MN$.

$[asy] size(150); defaultpen(linewidth(0.7)+fontsize(10)); pair B=origin, A=14*dir(42), C=intersectionpoint(B--(30,0), Circle(A,19)), M=midpoint(B--C), b=A+14*dir(A--C), N=foot(A, B, b); draw(N--B--A--N--M--C--A^^B--M); markscalefactor=0.1; draw(rightanglemark(B,N,A)); pair point=N; label("A", A, dir(point--A)); label("B", B, dir(point--B)); label("C", C, dir(point--C)); label("M", M, dir(point--M)); label("N", N, dir(30)); label(rotate(angle(dir(A--C)))*"19", A--C, dir(A--C)*dir(90)); label(rotate(angle(dir(A--B)))*"14", A--B, dir(A--B)*dir(90)); [/asy]$

$\textbf{(A)}\ 2\qquad\textbf{(B)}\ \dfrac{5}{2}\qquad\textbf{(C)}\ \dfrac{5}{2}-\sin\theta\qquad\textbf{(D)}\ \dfrac{5}{2}-\dfrac{1}{2}\sin\theta\qquad\textbf{(E)}\ \dfrac{5}{2}-\dfrac{1}{2}\sin\left(\dfrac{1}{2}\theta\right)$

## Problem 20

A ray of light originates from point $A$ and travels in a plane, being reflected $n$ times between lines $AD$ and $CD$ before striking a point $B$ (which may be on $AD$ or $CD$) perpendicularly and retracing its path back to $A$ (At each point of reflection the light makes two equal angles as indicated in the adjoining figure. The figure shows the light path for $n=3$). If $\measuredangle CDA=8^\circ$, what is the largest value $n$ can have?

$[asy] unitsize(1.5cm); pair D=origin, A=(-6,0), C=6*dir(160), E=3.2*dir(160), F=(-2.1,0), G=1.5*dir(160), B=(-1.4095,0); draw((-6.5,0)--D--C,black); draw(A--E--F--G--B,black); dotfactor=4; dot("A",A,S); dot("C",C,N); dot("R_1",E,N); dot("R_2",F,S); dot("R_3",G,N); dot("B",B,S); markscalefactor=0.015; draw(rightanglemark(G,B,D)); draw(anglemark(C,E,A,12)); draw(anglemark(F,E,G,12)); draw(anglemark(E,F,A)); draw(anglemark(E,F,A,12)); draw(anglemark(B,F,G)); draw(anglemark(B,F,G,12)); draw(anglemark(E,G,F)); draw(anglemark(E,G,F,12)); draw(anglemark(E,G,F,16)); draw(anglemark(B,G,D)); draw(anglemark(B,G,D,12)); draw(anglemark(B,G,D,16)); [/asy]$

$\textbf{(A)}\ 6\qquad\textbf{(B)}\ 10\qquad\textbf{(C)}\ 38\qquad\textbf{(D)}\ 98\qquad\textbf{(E)}\ \text{There is no largest value.}$

## Problem 21

In a triangle with sides of lengths $a$, $b$, and $c$, $(a+b+c)(a+b-c) = 3ab$. The measure of the angle opposite the side length $c$ is

$\textbf{(A)}\ 15^\circ\qquad\textbf{(B)}\ 30^\circ\qquad\textbf{(C)}\ 45^\circ\qquad\textbf{(D)}\ 60^\circ\qquad\textbf{(E)}\ 150^\circ$

## Problem 22

How many lines in a three dimensional rectangular coordiante system pass through four distinct points of the form $(i,j,k)$, where $i$, $j$, and $k$ are positive integers not exceeding four?

$\textbf{(A)}\ 60\qquad\textbf{(B)}\ 64\qquad\textbf{(C)}\ 72\qquad\textbf{(D)}\ 76\qquad\textbf{(E)}\ 100$

## Problem 23

Equilateral $\triangle ABC$ is inscribed in a circle. A second circle is tangent internally to the circumcircle at $T$ and tangent to sides $AB$ and $AC$ at points $P$ and $Q$. If side $BC$ has length $12$, then segment $PQ$ has length

$[asy] defaultpen(linewidth(.8pt)); pair B = origin; pair A = dir(60); pair C = dir(0); pair circ = circumcenter(A,B,C); pair P = intersectionpoint(circ--(circ + (-1,0)),A--B); pair Q = intersectionpoint(circ--(circ + (1,0)),A--C); label("A",A,N); label("B",B,SW); label("C",C,SE); label("P",P,NW); label("Q",Q,NE); label("T",(0.5,-0.3),S); draw(A--B--C--cycle); draw(circumcircle(A,B,C)); draw(P--Q); draw(Circle((0.5,0.09),0.385)); [/asy]$

$\textbf{(A)}\ 6\qquad\textbf{(B)}\ 6\sqrt{3}\qquad\textbf{(C)}\ 8\qquad\textbf{(D)}\ 8\sqrt{3}\qquad\textbf{(E)}\ 9$

## Problem 24

If $\theta$ is a constant such that $0 < \theta < \pi$ and $x + \dfrac{1}{x} = 2\cos{\theta}$, then for each positive integer $n$, $x^n + \dfrac{1}{x^n}$ equals

$\textbf{(A)}\ 2\cos\theta\qquad \textbf{(B)}\ 2^n\cos\theta\qquad \textbf{(C)}\ 2\cos^n\theta\qquad \textbf{(D)}\ 2\cos n\theta\qquad \textbf{(E)}\ 2^n\cos^n\theta$

## Problem 25

In $\triangle ABC$ in the adjoining figure, $AD$ and $AE$ trisect $\angle BAC$. The lengths of $BD$, $DE$ and $EC$ are $2$, $3$, and $6$, respectively. The length of the shortest side of $\triangle ABC$ is

$[asy] defaultpen(linewidth(.8pt)); pair A = (0,11); pair B = (2,0); pair D = (4,0); pair E = (7,0); pair C = (13,0); label("A",A,N); label("B",B,SW); label("C",C,SE); label("D",D,S); label("E",E,S); label("2",midpoint(B--D),N); label("3",midpoint(D--E),NW); label("6",midpoint(E--C),NW); draw(A--B--C--cycle); draw(A--D); draw(A--E); [/asy]$

$\textbf{(A)}\ 2\sqrt{10}\qquad \textbf{(B)}\ 11\qquad \textbf{(C)}\ 6\sqrt{6}\qquad \textbf{(D)}\ 6\qquad \textbf{(E)}\ \text{not uniquely determined by the given information}$

## Problem 26

Alice, Bob, and Carol repeatedly take turns tossing a die. Alice begins; Bob always follows Alice; Carol always follows Bob; and Alice always follows Carol. Find the probability that Carol will be the first one to toss a six. (The probability of obtaining a six on any toss is $\dfrac{1}{6}$, independent of the outcome of any other toss.)

$\textbf{(A)}\ \dfrac{1}{3}\qquad \textbf{(B)}\ \dfrac{2}{9}\qquad \textbf{(C)}\ \dfrac{5}{18}\qquad \textbf{(D)}\ \dfrac{25}{91}\qquad \textbf{(E)}\ \dfrac{36}{91}$

## Problem 27

In the adjoining figure triangle $ABC$ is inscribed in a circle. Point $D$ lies on $\stackrel{\frown}{AC}$ with $\stackrel{\frown}{DC} = 30^\circ$, and point $G$ lies on $\stackrel{\frown}{BA}$ with $\stackrel{\frown}{BG}\, > \, \stackrel{\frown}{GA}$. Side $AB$ and side $AC$ each have length equal to the length of chord $DG$, and $\angle CAB = 30^\circ$. Chord $DG$ intersects sides $AC$ and $AB$ at $E$ and $F$, respectively. The ratio of the area of $\triangle AFE$ to the area of $\triangle ABC$ is

$[asy] defaultpen(linewidth(.8pt)); pair C = origin; pair A = 2.5*dir(75); pair B = A + 2.5*dir(-75); path circ =circumcircle(A,B,C); pair D = waypoint(circ,(7/12)); pair G = waypoint(circ,(1/6)); pair E = intersectionpoint(D--G,A--C); pair F = intersectionpoint(A--B,D--G); label("A",A,N); label("B",B,SE); label("C",C,SW); label("D",D,SW); label("G",G,NE); label("E",E,NW); label("F",F,W); label("30^\circ",A,12S+E,fontsize(6pt)); draw(A--B--C--cycle); draw(circ); draw(Arc(A,0.25,-75,-105)); draw(D--G); [/asy]$

$\textbf{(A)}\ \dfrac {2 - \sqrt {3}}{3}\qquad \textbf{(B)}\ \dfrac {2\sqrt {3} - 3}{3}\qquad \textbf{(C)}\ 7\sqrt {3}-12\qquad \textbf{(D)}\ 3\sqrt {3}-5\qquad\\ \textbf{(E)}\ \dfrac {9-5\sqrt {3}}{3}$

## Problem 28

Consider the set of all equations $x^3 + a_2x^2 + a_1x + a_0 = 0$, where $a_2$, $a_1$, $a_0$ are real constants and $|a_i| < 2$ for $i = 0,1,2$. Let $r$ be the largest positive real number which satisfies at least one of these equations. Then

$\textbf{(A)}\ 1 < r < \dfrac{3}{2}\qquad \textbf{(B)}\ \dfrac{3}{2} < r < 2\qquad \textbf{(C)}\ 2 < r < \dfrac{5}{2}\qquad \textbf{(D)}\ \dfrac{5}{2} < r < 3\qquad \\ \textbf{(E)}\ 3 < r < \dfrac{7}{2}$

## Problem 29

If $a > 1$, then the sum of the real solutions of

$\sqrt{a - \sqrt{a + x}} = x$

is equal to

$\textbf{(A)}\ \sqrt{a} - 1\qquad \textbf{(B)}\ \dfrac{\sqrt{a}- 1}{2}\qquad \textbf{(C)}\ \sqrt{a - 1}\qquad \textbf{(D)}\ \dfrac{\sqrt{a - 1}}{2}\qquad \textbf{(E)}\ \dfrac{\sqrt{4a- 3} - 1}{2}$

## Problem 30

If $a$, $b$, $c$, and $d$ are the solutions of the equation $x^4 - bx - 3 = 0$, then an equation whose solutions are $$\dfrac {a + b + c}{d^2}, \dfrac {a + b + d}{c^2}, \dfrac {a + c + d}{b^2}, \dfrac {b + c + d}{a^2}$$is

$\textbf{(A)}\ 3x^4 + bx + 1 = 0\qquad \textbf{(B)}\ 3x^4 - bx + 1 = 0\qquad \textbf{(C)}\ 3x^4 + bx^3 - 1 = 0\qquad \\\textbf{(D)}\ 3x^4 - bx^3 - 1 = 0\qquad \textbf{(E)}\ \text{none of these}$


text:
# Is there a standardized way to assign a label to the symbol of a chemical element (in a published work)?

When referring to a particular atom, both in text and when using a graphical representation, several conventions are being applied.

In the field of crystallography an arabic numeral either follows the element symbol without any space, e.g. $$\mathrm{C}42$$ for carbon atom number $$42$$ (usually assigned during solve/refine routine by SHELX), or is being placed in braces, e.g. $$\mathrm{C}(42).$$ Sometimes hydrogens attached to the same carbon atom inherit its number and receive an additional letter, e.g. $$\mathrm{H42c}$$ for the third hydrogen in $$\mathrm{C42}$$ methyl group.

In NMR spectroscopy an element is normally labeled with a script-sized symbol (arabic number or lowercase letter) in the bottom right corner, e.g. $$\ce{H_3}$$ or $$\ce{H_c}.$$ Mestrelab Mnova places numerical labels underneath the element symbol, e.g. $$\underset{3}{\ce{H}}$$, or directly in place of the element if the symbol for the latter is hidden (carbon, in most cases): $${\scriptsize 3}.$$

These labeling schemes used in both instrumental methods (which I cherry-picked solely for the illustrative purposes) seem to work OK in a sense that they are compatible with expanded AZE-notation required to depict any element and its attributes, such as chemical symbol $$\ce{El},$$ mass number $$A,$$ atomic number $$Z,$$ oxidation number O.N., charge $$q$$ and number of atoms of given kind $$x$$:

$$\Large\ce{\overset{\text{O.N.}}{^{A}_{Z}El^{q}_x}}$$

and won't cause any confusion (except for $$\ce{H_3},$$ maybe). However, it would be nice to know whether there is a standardized way to attribute a label to an element that can be uniformly displayed and recognized (main text, titles, structures, illustrations etc.). Does such robust labeling scheme exist?

P.S. If you are a software developer and read this, also feel free to comment or add an answer as to which labeling scheme you've came up with for calling out a particular atom.

• IUPAC's green book "Quantities Units and Symbols in Physical Chemistry" might be worth a look. It is a 165-paged book. – M. Farooq Dec 23 '19 at 16:41
• @M.Farooq Already did. According to the Green Book, labels (in general) must be typed in upright typeface, that's all I found. – andselisk Dec 23 '19 at 16:44
• I can imagine that this can become a headache for a reader if the numbering scheme is not obvious, but that does not appear to be a consideration in this question. To draw a comparison, protein sequences are numbered N->C ie Ala42 is the C-terminal residue in the A$\beta$(1-42) peptide. This rarely is a problem although numbering conventions may refer to the original gene rather than a specific construct such as the A$\beta$ in question. But since the sequences are linear the numbering convention is otherwise clear. Now consider branching and rings in compounds and the story changes. – Buck Thorn Dec 23 '19 at 19:54
• In any case your question is about convention, not about the best way to do this for the sake of clarity. I would say stick to the crystallography convention but if the molecule is complex provide a structure file or figure with numbering. – Buck Thorn Dec 23 '19 at 19:56
• @BuckThorn Yeah, I have the same feeling that crystallographic notation would be easier to maintain. And I actually do care about reader's ergonomics, but I'd put standards/recommendations/guidelines first, if there are any, to avoid confusion and develop a coherent labeling scheme. Ideally, the one that is interdisciplinary understood:) – andselisk Dec 23 '19 at 20:00

format: plain


text:
Ji Sunn spent $24 at the local farmer’s market. She bought apples that cost$4 per pound and raspberries that cost $6 per pound. Which equat Question Ji Sunn spent$24 at the local farmer’s market. She bought apples that cost $4 per pound and raspberries that cost$6 per pound. Which equation can be used to represent the amount of raspberries, r, in terms of apples, a?
r = 4 minus two-thirds a
r = 4 + two-thirds a
r = 6 minus three-halves a
r = 6 + three-halves aThe formula for electrical power is P = I V. Which is the equivalent equation solved for i?

in progress 0
6 months 2021-07-19T00:24:17+00:00 1 Answers 9 views 0


text:
# Sorting based on pairwise comparison

## Input

1. An ordered sequence $S$ of length $n$.
2. A definition of comparison on the component type of $S$.

## Output

A permutation of $S$ such that $S[i] \le S[i+1]$ for all $i\in \{1,...,n-1\}$.

N/A

## Complexity

Polynomial if the comparison is polynomial.

## Remark

Some algorithms could be formulated in a much simpler way in case each key occurs at most once in $S$. Here all algorithms are formulated for the general case of arbitrary multiplicities. Alternatively, there are three options to make each key unique:

1. An additional generated key such as, for example, the student ID or the tax file number.
2. If the keys are associated with values, they are additionally compared if the keys of two items are identical (e.g. date of birth and home address, if last and first names of two persons are identical).
3. On the other hand, if the keys are not associated with values, we may remove the duplicates and store their multiplicities separately.


text:
22 views

Consider communication over a memoryless binary symmetric channel using a $(7, 4)$ Hamming code. Each transmitted bit is received correctly with probability $(1 – \in),$ and flipped with probability $\in.$ For each codeword transmission, the receiver performs minimum Hamming distance decoding, and correctly decodes the message bits if and only if the channel introduces at most one bit error.

For $\in = 0.1,$ the probability that a transmitted codeword is decoded correctly is ___________(rounded off to two decimal places).


text:
A Second Look at Slew Rate Limiters

January 14, 2022

I recently had to pick a slew rate for a current waveform, and I got this feeling of déjà vu… hadn’t I gone through this effort already? So I looked, and lo and behold, way back in 2014 I wrote an article titled Slew Rate Limiters: Nonlinear and Proud of It! where I explored the effects of two types of slew rate limiters, one feedforward and one feedback, given a particular slew rate $R$.

Here was one figure I published at the time:

This...

Supply Chain Games: What Have We Learned From the Great Semiconductor Shortage of 2021? (Part 1)

December 31, 20211 comment

So by now I’m sure you’ve heard about the semiconductor shortage of 2021. For a few complicated reasons, demand is greater than supply, and not everybody who wants to buy integrated circuits can do so. Today we’re going to try to answer some hard questions:

• Why are we in the middle of a semiconductor shortage?
• Why is it taking so long to get my [insert part number here]?
• Did this shortage suddenly sneak up on everybody? If not, what were the signs, and why...

Definite Article: Notes on Traceability

September 6, 2021

Electronic component distibutor Digi-Key recently announced part tracing for surface-mount components purchased in cut-tape form. This is a big deal, and it’s a feature that is a good example of traceability. Some thing or process that has traceability basically just means that it’s possible to determine an object’s history or provenance: where it came from and what has happened to it since its creation. There are a...

Painting with Light to Measure Time

December 26, 2020

Recently I was faced with a dilemma while working from home. I needed to verify an implementation of first-order sigma-delta modulation used to adjust LED brightness. (I have described this in more detail in Modulation Alternatives for the Software Engineer.) I did not, however, have an oscilloscope.

And then I remembered something, about a technique called “light painting”: basically a long-exposure photograph where a...

Scorchers, Part 3: Bare-Metal Concurrency With Double-Buffering and the Revolving Fireplace

July 25, 2020

This is a short article about one technique for communicating between asynchronous processes on bare-metal embedded systems.

A: to To other side. get the

— Jason Whittington

There are many reasons why concurrency is

Tolerance Analysis

May 31, 2020

Today we’re going to talk about tolerance analysis. This is a topic that I have danced around in several previous articles, but never really touched upon in its own right. The closest I’ve come is Margin Call, where I discussed several different techniques of determining design margin, and ran through some calculations to justify that it was safe to allow a certain amount of current through an IRFP260N MOSFET.

Tolerance analysis...

Scorchers, Part 2: Unknown Bugs and Popcorn

This is a short article about diminishing returns in the context of software releases.

Those of you who have been working professionally on software or firmware have probably faced this dilemma before. The scrum masters of the world will probably harp on terms like the Definition of Done and the Minimum Viable Product. Blah blah blah. In simple terms, how do you know when your product is ready to release? This is both an easy and a difficult question to answer.

What makes...

Racing to Sleep

December 30, 2019

Today we’re going to talk about low-power design.

Suppose I’m an electrical engineer working with wildlife biologists who are gathering field data on the Saskatchewan ringed-neck mountain goat. My team has designed a device called the BigBrotherBear 2000 (BBB2000) with a trip cable and a motor and a camera and a temperature sensor and a hot-wire anemometer and a real-time clock and an SD card and a battery and a LoRa transceiver. The idea is something like...

Jaywalking Around the Compiler

Our team had another code review recently. I looked at one of the files, and bolted upright in horror when I saw a function that looked sort of like this:

void some_function(SOMEDATA_T *psomedata) { asm volatile("push CORCON"); CORCON = 0x00E2; do_some_other_stuff(psomedata); asm volatile("pop CORCON"); }

There is a serious bug here — do you see what it is?

Shibboleths: The Perils of Voiceless Sibilant Fricatives, Idiot Lights, and Other Binary-Outcome Tests

September 29, 2019

AS-SALT, JORDAN — Dr. Reza Al-Faisal once had a job offer from Google to work on cutting-edge voice recognition projects. He turned it down. The 37-year-old Stanford-trained professor of engineering at Al-Balqa’ Applied University now leads a small cadre of graduate students in a government-sponsored program to keep Jordanian society secure from what has now become an overwhelming influx of refugees from the Palestinian-controlled West Bank. “Sometimes they visit relatives...

Adventures in Signal Processing with Python

Author’s note: This article was originally called Adventures in Signal Processing with Python (MATLAB? We don’t need no stinkin' MATLAB!) — the allusion to The Treasure of the Sierra Madre has been removed, in deference to being a good neighbor to The MathWorks. While I don’t make it a secret of my dislike of many aspects of MATLAB — which I mention later in this article — I do hope they can improve their software and reduce the price. Please note this...

My Love-Hate Relationship with Stack Overflow: Arthur S., Arthur T., and the Soup Nazi

Warning: In the interest of maintaining a coherent stream of consciousness, I’m lowering the setting on my profanity filter for this post. Just wanted to let you know ahead of time.

I’ve been a user of Stack Overflow since December of 2008. And I say “user” both in the software sense, and in the drug-addict sense. I’m Jason S, user #44330, and I’m a programming addict. (Hi, Jason S.) The Gravatar, in case you were wondering, is a screen...

Understanding and Preventing Overflow (I Had Too Much to Add Last Night)

December 4, 2013

Happy Thanksgiving! Maybe the memory of eating too much turkey is fresh in your mind. If so, this would be a good time to talk about overflow.

In the world of floating-point arithmetic, overflow is possible but not particularly common. You can get it when numbers become too large; IEEE double-precision floating-point numbers support a range of just under 21024, and if you go beyond that you have problems:

for k in [10, 100, 1000, 1020, 1023, 1023.9, 1023.9999, 1024]: try: ...

How to Estimate Encoder Velocity Without Making Stupid Mistakes: Part I

Here's a common problem: you have a quadrature encoder to measure the angular position of a motor, and you want to know both the position and the velocity. How do you do it? Some people do it poorly -- this article is how not to be one of them.

Well, first we need to get position. Quadrature encoders are incremental encoders, meaning they can only measure relative changes in position. They produce a pair of pulse trains, commonly called A and B, that look like...

How to Read a Power MOSFET Datasheet

One of my pet peeves is when my fellow engineers misinterpret component datasheets. This happened a few times recently in separate instances, all involving power MOSFETs. So it’s time for me to get on my soapbox. Listen up!

I was going to post an article on how to read component datasheets in general. But MOSFETs are a good place to start, and are a little more specific. I’m not the first person to write something about how to read datasheets; here are some other good...

Ten Little Algorithms, Part 2: The Single-Pole Low-Pass Filter

Other articles in this series:

I’m writing this article in a room with a bunch of other people talking, and while sometimes I wish they would just SHUT UP, it would be...

Thermistor signal conditioning: Dos and Don'ts, Tips and Tricks

In an earlier blog entry,  I mentioned this circuit for thermistor signal conditioning:

It is worth a little more explanation on thermistor signal conditioning; it's something that's often done poorly, whereas it's among the easiest applications for signal conditioning.

The basic premise here is that there are two resistors in a voltage divider: Rth is the thermistor, and Rref is a reference resistor. Here Rref is either R3 alone, or R3 || R4, depending on the gain...

Chebyshev Approximation and How It Can Help You Save Money, Win Friends, and Influence People

Well... maybe that's a stretch. I don't think I can recommend anything to help you win friends. Not my forte.

But I am going to try to convince you why you should know about Chebyshev approximation, which is a technique for figuring out how you can come as close as possible to computing the result of a mathematical function, with a minimal amount of design effort and CPU power. Let's explore two use cases:

• Amy has a low-power 8-bit microcontroller and needs to compute $\sqrt{x}$...


text:
##### The number of candidate keys possible ?

If there are 'n' attribute in table ,then number of candidate keys possible ?

meet @meet
8 Jan 2016 02:10 pm

according to me candidate key is always depends on FDS not on attributes.

meet @meet
8 Jan 2016 02:22 pm

let's consider the example : A → B, B → C, C → D, D → A
then candidate key is A,B,C,D.

so total 4 which is our total attributes.

jayendra
11 Jan 2016 12:13 am

they have asked number of candidate keys "possible"... so we can assume that they are interested in maximum number of candidate keys..

Now definition of candidate key is: CK is a single field or least combination of fields that uniquely defines a row.

Super keys:

So possible number of combinations are same as possible number of binary strings.

n + nC2 + nC3 + .. + nCn

candidate keys:

For this we have to consider maximum number of combinations with same number of attributes.

for example.. if we have five attributes(A,B,C,D,E) then if we take individual attributes we can not take any other. But if we focus on AB, BC.. then we can consider all combinations with 2 attributes but not with 3 or 4. In this way we are getting maximum candidate keys.

If there are 6 attributes then we will get maximum CK at 6C3..

In general we can say, nCfloor(n/2) will give maximum combinations that we can consider as CK..

Arul
11 Jan 2016 12:19 am

just for the completeness, n + nC2 + nC3 + .. + nCn will be (2n - 1)

Parimal Andhalkar
10 Jan 2016 11:27 pm

but if we say A is candidate key then AB will not be a candidate key . AB will be super key .

then how we count AB also ?

meet @meet
10 Jan 2016 11:52 pm

yaa u r right @parimal

Arul
11 Jan 2016 12:07 am

This question looks ambiguous to me. I can understand this question in two ways:

1. What are all the combinations of those 'n' attributes that can be candidate key to this table? (Jayendra's answer)
2. What is the maximum possible number of candidate keys at a particular instant for this table? (meet's answer)

meet @meet
11 Jan 2016 12:11 am

ok bro......