https://pop.com.sgExplosive pop stories with broad entertainment appeal.en-usFri, 14 Aug 2026 16:47:10 -0400https://pop.com.sg/which-statements-about-a-linear-transformation-would-be-correct.htmlhttps://pop.com.sg/which-statements-about-a-linear-transformation-would-be-correct.htmlLet T be a linear transformation. If we have this then what statements would hold true? T contains $\vec{0}$ (i.e $\vec{0} \in T$) T is a 3 x 4 matrix The columns of T are linearly independent T(......Fri, 14 Aug 2026 20:38:17 -0400https://pop.com.sg/calculate-this-triple-integral-over-d.htmlhttps://pop.com.sg/calculate-this-triple-integral-over-d.html$$\iiint \sqrt{10+y^2-x^2} \,dx \,dy \, dz$$ over the region bounded by $D$: $D=\{(x,y,z)\mid x^2+z^2\le y^2+10,x^2+y^2\le 9\}$ So It's a cylinder intersecting a hyperboloid. Is It correct to sa......Fri, 14 Aug 2026 20:31:34 -0400https://pop.com.sg/why-do-we-only-solve-for-0-on-the-numerator-of-this-fraction.htmlhttps://pop.com.sg/why-do-we-only-solve-for-0-on-the-numerator-of-this-fraction.htmlI have this function where I need to find the $x$-intercepts I see they are $x = 0, 4$. And know that this is found by doing: $x=0, (x-4) = 0$ [Solve]. But why is this only done for the numerator?...Fri, 14 Aug 2026 20:19:09 -0400https://pop.com.sg/computing-a-revenue-for-vcg-auction.htmlhttps://pop.com.sg/computing-a-revenue-for-vcg-auction.htmlI would like your help with the following question regarding computing a revenue for a seller of an VCG (vickrey clarke groves) auction, I'm really new to this auctions\game theory so I'd really ...Fri, 14 Aug 2026 20:12:11 -0400https://pop.com.sg/probability-of-getting-a-7-in-minesweeper.htmlhttps://pop.com.sg/probability-of-getting-a-7-in-minesweeper.htmlLet's say you have a $30$ by $16$ grid and $99$ mines. What is the probability of having at least one empty block surrounded by exactly $7$ mines? For the sake of this question, assume that the mi......Fri, 14 Aug 2026 20:08:59 -0400https://pop.com.sg/how-to-solve-the-equation-x-2-2x-sqrt-2x-2-4x-3.htmlhttps://pop.com.sg/how-to-solve-the-equation-x-2-2x-sqrt-2x-2-4x-3.htmlHow to solve the equation $x^2-2x=\sqrt{ 2x^2-4x+3} $ Will squaring both sides introduce a $x^4$ term that may complicate the equation?...Fri, 14 Aug 2026 20:06:12 -0400https://pop.com.sg/is-sin-2-x-the-same-as-sin-x-2.htmlhttps://pop.com.sg/is-sin-2-x-the-same-as-sin-x-2.htmlI'm working with derivatives and need to know if $\sin^2(x)$ the same as $\sin(x^2)$? I almost don't want to ask because my last question was closed. It was a valid question and so is this one. I've ...Fri, 14 Aug 2026 20:04:45 -0400https://pop.com.sg/sum-of-series-1-11-111.htmlhttps://pop.com.sg/sum-of-series-1-11-111.htmlSum of series $1+11+111+\cdots+11\cdots11$ ($n$ digits) We have: $1=\frac {10-1}9,$ $11=\frac {10^2-1}9$ . . . $11...11= \frac {10^n-1}9$ (number with $n$ digits) and summing them we find ......Fri, 14 Aug 2026 19:55:06 -0400https://pop.com.sg/prove-10n-8-9n-6-n-2-is-divisible-by-45.htmlhttps://pop.com.sg/prove-10n-8-9n-6-n-2-is-divisible-by-45.htmlBasically, I have to use Euler theorem to prove that $10n^8 -9n^6 -n^2$ is divisible by $45$ So my approach so far is to say $10n^8 - 9n^6 - n^2 = 0 \bmod 45$ Now $45$ can be factored into $5$ ......Fri, 14 Aug 2026 19:45:33 -0400https://pop.com.sg/how-many-squares-are-in-this-image-is-there-a-method-to-check.htmlhttps://pop.com.sg/how-many-squares-are-in-this-image-is-there-a-method-to-check.htmlIn this image I have counted 14 but others say 18. Is there a method to check exactly?...Fri, 14 Aug 2026 19:33:19 -0400https://pop.com.sg/how-to-find-the-greatest-value-of-this-expression.htmlhttps://pop.com.sg/how-to-find-the-greatest-value-of-this-expression.html"This expression" is $$\frac{5\sin^2\alpha+4\cos^2\alpha}{4\cos^2\beta+5\sin^2\beta}.$$ The answer is $1.25$ I used simple steps to simplify this, but couldn't find the greatest value, since......Fri, 14 Aug 2026 19:31:32 -0400https://pop.com.sg/formula-for-calculating-residue-at-a-simple-pole.htmlhttps://pop.com.sg/formula-for-calculating-residue-at-a-simple-pole.htmlSuppose $f=P/Q$ is a rational function and suppose $f$ has a simple pole at $a$. Then a formula for calculating the residue of $f$ at $a$ is $$ \text{Res}(f(z),a)=\lim_{z\to a}(z-a)f(z)=\lim_{z\t......Fri, 14 Aug 2026 19:28:04 -0400https://pop.com.sg/finding-out-if-a-number-is-rational-or-irrational-duplicate.htmlhttps://pop.com.sg/finding-out-if-a-number-is-rational-or-irrational-duplicate.htmlHow does one find out if a given number is a rational number or irrational number?...Fri, 14 Aug 2026 19:24:20 -0400https://pop.com.sg/finding-vertex-cut-using-menger-039-s-theorem.htmlhttps://pop.com.sg/finding-vertex-cut-using-menger-039-s-theorem.htmlMenger's theorem states that A Graph, $G$, is $k$-connected if and only if for every $x,y \in V(G)$, there exists $k$ pairwise internally disjoint paths. If I need to find size of vertex cut f......Fri, 14 Aug 2026 19:23:44 -0400https://pop.com.sg/user-or-eisenberg-mathematics-stack-exchange.htmlhttps://pop.com.sg/user-or-eisenberg-mathematics-stack-exchange.htmlQ&A for people studying math at any level and professionals in related fields...Fri, 14 Aug 2026 19:17:54 -0400https://pop.com.sg/how-to-calculate-cumulative-probabilities-for-a-discrete-random-variab.htmlhttps://pop.com.sg/how-to-calculate-cumulative-probabilities-for-a-discrete-random-variab.htmlI want to understand the difference between the cumulative distribution function and probability density function for a discrete random variable. I thought I understood the difference, but then I ...Fri, 14 Aug 2026 19:07:59 -0400https://pop.com.sg/how-to-find-unit-tangent-normal-and-binormal-vectors.htmlhttps://pop.com.sg/how-to-find-unit-tangent-normal-and-binormal-vectors.htmlI was given that $$p(t)=(1+2\cos t)\mathbf i + 2(1+\sin t)\mathbf j + (9+4\cos t+8 \sin t)\mathbf k$$ and that I needed to find the tangent, normal, and binormal vectors. The curvature and the ...Fri, 14 Aug 2026 18:53:11 -0400https://pop.com.sg/negating-statements-with-quantifiers.htmlhttps://pop.com.sg/negating-statements-with-quantifiers.htmlI understand that when we want to negate a statement with universal quantifier, that quantifier changes to existential quantifier, and vice versa. For example, negation of $(\exists x\in\Bbb N)(x......Fri, 14 Aug 2026 18:36:14 -0400https://pop.com.sg/finding-the-equation-of-vertical-and-horizontal-asymptotes.htmlhttps://pop.com.sg/finding-the-equation-of-vertical-and-horizontal-asymptotes.htmlI am having some trouble understanding these two questions. Any help is appreciated. Scanned questions are included at the end. 6) We are given the function $ f(x) =\frac{1 - 2x} {2x^2 - 3x - 2}......Fri, 14 Aug 2026 18:35:10 -0400https://pop.com.sg/given-sin-x-3-4-and-pi-2-x-pi-find-sin-x-pi-6.htmlhttps://pop.com.sg/given-sin-x-3-4-and-pi-2-x-pi-find-sin-x-pi-6.htmlGiven $\sin x=3/4$ and $\pi/2<x<\pi$, find $\sin(x+\pi/6)$. I have been trying to calculate this question. My current working out is as follows: $\sin\theta = 3/4$ $\pi/2 < \theta < ......Fri, 14 Aug 2026 18:30:31 -0400https://pop.com.sg/what-is-the-cardinality.htmlhttps://pop.com.sg/what-is-the-cardinality.htmlWhat is the cardinality of the set A,B where My answer was 4 for set A and 6 for set B. When i looked at the text book for answer it is not provided in the text book and now i'm wondering what the ...Fri, 14 Aug 2026 18:28:17 -0400https://pop.com.sg/origin-of-the-terminology-connected-algebra.htmlhttps://pop.com.sg/origin-of-the-terminology-connected-algebra.htmlI was wondering what is the origin of the word "connected" for a connected algebra ? To be more precise, why is a graded $R$-algebra $A_{\ast}$ with an augmentation $A_{\ast} \to R$ that restricts......Fri, 14 Aug 2026 18:14:16 -0400https://pop.com.sg/geometry-book-recommendation.htmlhttps://pop.com.sg/geometry-book-recommendation.htmlCan someone recommend a good basic book on Geometry? Let me be more specific on what I am looking for. I'd like a book that starts with Euclid's definitions and postulates and goes on from there to ...Fri, 14 Aug 2026 18:13:32 -0400https://pop.com.sg/proof-of-theorem-the-principle-of-mathematical-induction.htmlhttps://pop.com.sg/proof-of-theorem-the-principle-of-mathematical-induction.htmlI have a question from my textbook and wanted to make sure that I understood it. I have marked in a green box the question that I have....Fri, 14 Aug 2026 18:13:27 -0400https://pop.com.sg/how-to-use-stars-and-bars-for-solving-number-of-integral-solutions-to-.htmlhttps://pop.com.sg/how-to-use-stars-and-bars-for-solving-number-of-integral-solutions-to-.htmlHow to use the stars and bars method? Say I want to find number of combinations I can get with $x_1+x_2+x_3+x_4=22$, where $x_i\in\mathbb{N}$. Is this the correct time to apply the method? This......Fri, 14 Aug 2026 18:12:02 -0400https://pop.com.sg/iff-if-and-only-if-vs-tfae-the-following-are-equivalent.htmlhttps://pop.com.sg/iff-if-and-only-if-vs-tfae-the-following-are-equivalent.htmlIf $P$ and $Q$ are statements, $P \iff Q$ and The following are equivalent: $(\text{i}) \ P$ $(\text{ii}) \ Q$ Is there a difference between the two? I ask because formulations of certain theor......Fri, 14 Aug 2026 18:03:39 -0400https://pop.com.sg/finding-extrema-non-critical-points-and-saddle-points-given-a-contour-.htmlhttps://pop.com.sg/finding-extrema-non-critical-points-and-saddle-points-given-a-contour-.htmlI'm trying to interpret the contour plot as like a mountain, where all the segregated areas are plains that are higher when it proceeds inwards. What is the significance of the numbers on x-axis an......Fri, 14 Aug 2026 18:03:01 -0400https://pop.com.sg/regular-octagons-and-diagonals-proof.htmlhttps://pop.com.sg/regular-octagons-and-diagonals-proof.htmlDefinition: A diagonal of a octagon is a line segment connecting any two non-adjacent vertices. Prove: Every vertex of the regular octagon will produce 2 diagonals that are parallel to at ...Fri, 14 Aug 2026 17:56:57 -0400https://pop.com.sg/logarithmic-differentiation-when-to-use.htmlhttps://pop.com.sg/logarithmic-differentiation-when-to-use.htmlSorry if this is an ignorant or uninformed question, but I would like to know when I can (or should use) logarithmic differentiation. I haven't taken calculus in a while so I'm quite rusty. So, le......Fri, 14 Aug 2026 17:52:51 -0400https://pop.com.sg/proof-of-additivity-of-domain-for-definite-integrals.htmlhttps://pop.com.sg/proof-of-additivity-of-domain-for-definite-integrals.htmlI would like to prove the following theorem: Theorem If $c \in (a,b)$ and $f$ is integrable on $[a,c]$ and $[c,b]$, then $f$ is integrable on $[a,b]$ and $$\int_{a}^{b}f = \int_{a}^{c}f + ......Fri, 14 Aug 2026 17:51:45 -0400https://pop.com.sg/finding-which-sets-are-subspaces-of-r3.htmlhttps://pop.com.sg/finding-which-sets-are-subspaces-of-r3.htmlHello. I have attached an image of the question I am having trouble with. I thought that it was 1,2 and 6 that were subspaces of $\mathbb R^3$. Here is my working: Rearranged equation ---> $x+y......Fri, 14 Aug 2026 17:50:09 -0400https://pop.com.sg/what-is-the-name-of-the-4d-shape-that-is-equivalent-to-a-cylinder-exce.htmlhttps://pop.com.sg/what-is-the-name-of-the-4d-shape-that-is-equivalent-to-a-cylinder-exce.htmlSorry if my question is poorly explained. Basically I want the name of the shape that always looks like a sphere in 3 dimensions when fed into 3d space standing up in the 4th dimension; just like h......Fri, 14 Aug 2026 17:45:09 -0400https://pop.com.sg/sign-language-and-deaf-mathematicians.htmlhttps://pop.com.sg/sign-language-and-deaf-mathematicians.htmlSomething I've often wondered (and I suppose this goes for all kinds of technical terminology, not just that of mathematics) is what kind of sign language exists for practising professional ...Fri, 14 Aug 2026 17:35:38 -0400https://pop.com.sg/the-meaning-of-spacecraft-attitude-represented-in-quaternion.htmlhttps://pop.com.sg/the-meaning-of-spacecraft-attitude-represented-in-quaternion.htmlI am reading the following paper about the attitude control of aircraft: http://ieeexplore.ieee.org/stamp/stamp.jsp?arnumber=1271671 The quaternion represents the relative orientation of two ...Fri, 14 Aug 2026 17:26:58 -0400https://pop.com.sg/using-rouche-039-s-theorem-to-find-no-of-roots-of-p-z-2z-4-2z-3-2z-2-7.htmlhttps://pop.com.sg/using-rouche-039-s-theorem-to-find-no-of-roots-of-p-z-2z-4-2z-3-2z-2-7.htmlLet $p(z)=2z^4-2z^3+2z^2+7$. I'm trying to determine the number of roots of $p(z)$ that lie inside $\{z\in\mathbb{C}:|z|<2\}$ I used Rouche's theorem and I'm trying to find find $f(z)$ such tha......Fri, 14 Aug 2026 17:22:00 -0400https://pop.com.sg/what-is-sine-of-a-real-number.htmlhttps://pop.com.sg/what-is-sine-of-a-real-number.htmlI never understand what the trigonometric function sine is.. We had a table that has values of sine for different angles, we by hearted it and applied to some problems and there ends the matter. T......Fri, 14 Aug 2026 17:10:26 -0400https://pop.com.sg/repeated-roots-of-a-polynomial.htmlhttps://pop.com.sg/repeated-roots-of-a-polynomial.htmlWhenever a part of the graph of a polynomial is in the form of a parabola whose vertex touches the x axis we conclude that a root is repeated at that point. Why is a root said to be repeated at that ...Fri, 14 Aug 2026 17:08:10 -0400https://pop.com.sg/what-is-sin-2-x-equal-to.htmlhttps://pop.com.sg/what-is-sin-2-x-equal-to.htmlLet's take the sine of $30^\circ$ which is one-half. If you take $\sin^2(30^\circ)$, would that just be the sine of $900$? Or would it be equal to one-quarter, or would it be equal to something ...Fri, 14 Aug 2026 16:58:41 -0400https://pop.com.sg/sphere-eversion-and-inversion.htmlhttps://pop.com.sg/sphere-eversion-and-inversion.htmlthe following things are asked because of a research I am doing for a short novel that I am writing. It contains only very little mathematics, however I want it to be correct. Do I understand corr......Fri, 14 Aug 2026 16:56:58 -0400https://pop.com.sg/show-that-int-0-1-times-0-1-frac-dx-dy-1-xy-frac-pi-2-6.htmlhttps://pop.com.sg/show-that-int-0-1-times-0-1-frac-dx-dy-1-xy-frac-pi-2-6.html$$ \int_{[0,1] \times [0,1]} \frac{dx \, dy}{1-xy}=\frac{\pi^2}{6}, $$ there is a hint for substitution given but I can't seem to get anywhere with it. $$ \begin{cases} x = \frac{u-v}{\sqrt{2}} \\ ......Fri, 14 Aug 2026 16:48:18 -0400https://pop.com.sg/questions-concerning-smallest-fraction-between-two-given-fractions.htmlhttps://pop.com.sg/questions-concerning-smallest-fraction-between-two-given-fractions.htmlI recently encountered this on a practice test Find the smallest positive integer $n$ such that there exists an integer $m$ satisfying $0.33 < \frac{m}{n} < \frac{1}{3}$ My answer: $$\begin{...Fri, 14 Aug 2026 16:19:07 -0400https://pop.com.sg/how-to-solve-this-simple-algebraic-equation-wolfram-symbolab-both-stum.htmlhttps://pop.com.sg/how-to-solve-this-simple-algebraic-equation-wolfram-symbolab-both-stum.htmlI need to solve the following equation for a hobby coding project I'm working on and I can't figure out the solution. Wolfram and SymboLab solvers are both stumped. $$0 = d - (r^{p-1} * (p-1) * c)......Fri, 14 Aug 2026 16:09:35 -0400https://pop.com.sg/discriminant-of-elliptic-curves.htmlhttps://pop.com.sg/discriminant-of-elliptic-curves.htmlIn the study of elliptic curves, specifically in Weierstrass form, you have the equation $E : y^2 = x^3 +ax +b$. However I have found the discriminant comes in two different forms: $\Delta = ......Fri, 14 Aug 2026 15:29:58 -0400https://pop.com.sg/two-independent-events-a-and-b-find-b-given-only-a-and-a-cup-b.htmlhttps://pop.com.sg/two-independent-events-a-and-b-find-b-given-only-a-and-a-cup-b.htmlSorry to be a burden but my maths exam is tomorrow and I have no idea how to do this at all. Here is the exact question : The two independent events $A$ and $B$ are such that $P(A) = 0.2$, $P(A\cu......Fri, 14 Aug 2026 15:25:23 -0400https://pop.com.sg/find-the-3-angles-of-the-hyperbolic-triangle.htmlhttps://pop.com.sg/find-the-3-angles-of-the-hyperbolic-triangle.htmlA(0,5) B(0,2) C(4,2) In Euclidean geometry the three points given are the vertices of a right-angled triangle. Find the three angles of the hyperbolic triangle with vertices A,B,C. Find the hyperb......Fri, 14 Aug 2026 15:22:41 -0400https://pop.com.sg/definition-of-minor-for-a-general-m-times-n-matrix.htmlhttps://pop.com.sg/definition-of-minor-for-a-general-m-times-n-matrix.htmlFollowing is the definition of minor given in the text : For a matrix $A$, consider its $k × k$ submatrix, obtained by taking $k$ rows and $k$ columns. The determinant of this matrix is calle......Fri, 14 Aug 2026 15:20:51 -0400https://pop.com.sg/what-is-a-null-set.htmlhttps://pop.com.sg/what-is-a-null-set.htmlI am very confused with null sets. I get that a set which has no elements will be called a null set but I am not getting the examples given below. Please help me by explaining how $P,Q,R$ are all......Fri, 14 Aug 2026 15:15:26 -0400https://pop.com.sg/absolute-value-graph-sketching-x-1-1.htmlhttps://pop.com.sg/absolute-value-graph-sketching-x-1-1.htmlWhere would you start if you were told to plot: $$||x-1|-1|$$ I looked at just $f(x) = |x-1|$ and noticed that the two equations are: $\pm (x-1)$ for $x \geq 1$ and $x < 1$. Extrapolating then......Fri, 14 Aug 2026 15:12:11 -0400https://pop.com.sg/how-can-i-prove-that-the-angles-of-the-tetrahedral-structure-is-109-5-.htmlhttps://pop.com.sg/how-can-i-prove-that-the-angles-of-the-tetrahedral-structure-is-109-5-.htmlHow can I prove that the dihedral angles of the regular tetrahedral structure is $109.5^\circ$ with calculus or any other technique. I could do it with geometry by considering a cube....Fri, 14 Aug 2026 15:08:55 -0400https://pop.com.sg/what-is-the-vertex-of-y-x-x.htmlhttps://pop.com.sg/what-is-the-vertex-of-y-x-x.htmlFor $x > 0$, the graph takes on a parabola-like shape. Is there a method to algebraically and not graphically determine the vertex of the graph? Desmos rendering of $x^x$...Fri, 14 Aug 2026 14:44:53 -0400https://pop.com.sg/construct-triangle-given-angle-and-centroid.htmlhttps://pop.com.sg/construct-triangle-given-angle-and-centroid.htmlI am stumped by Euclidea problem 8.11: From a triangle are given angle A and the centroid G Construct the points B and C. Please only a hint...Fri, 14 Aug 2026 14:44:42 -0400https://pop.com.sg/what-is-physical-interpretation-of-dot-product-duplicate.htmlhttps://pop.com.sg/what-is-physical-interpretation-of-dot-product-duplicate.htmlConsider two vectors $V_1$ and $V_2$ in $\mathbb{R}^3$. When we take their dot product we get a real number. How is that number related to the vectors? Is there any way we can visualize it?...Fri, 14 Aug 2026 14:43:37 -0400https://pop.com.sg/solve-this-logical-puzzle-only-using-logical-equivalencies-is-there-a-.htmlhttps://pop.com.sg/solve-this-logical-puzzle-only-using-logical-equivalencies-is-there-a-.htmlHere's a puzzle: Four friends have been identified as suspects for an authorized access into a computer system. They have made statements to the investigating authorities. Alice said "Carlos......Fri, 14 Aug 2026 14:31:11 -0400https://pop.com.sg/how-to-prove-that-the-altitudes-of-the-triangle-are-concurrent.htmlhttps://pop.com.sg/how-to-prove-that-the-altitudes-of-the-triangle-are-concurrent.htmlGiven a triangle of sides $A$ $B$ $C$ .how to prove that altitudes from sides $A$ $B$ $C$ are concurrent . There is a theorem called ceva's theorem.but i dont know how to use that theorem in this p......Fri, 14 Aug 2026 14:25:07 -0400https://pop.com.sg/the-position-of-a-ladder-leaning-against-a-wall-and-touching-a-box-und.htmlhttps://pop.com.sg/the-position-of-a-ladder-leaning-against-a-wall-and-touching-a-box-und.htmlI was reading a newspaper and there was a little math riddle, I thought "how funny, that's gonna be easy, let's do it" and here am I... The problem goes as follow : in a barn, there is a 1 meter c......Fri, 14 Aug 2026 14:23:58 -0400https://pop.com.sg/what-defines-a-chaotic-system.htmlhttps://pop.com.sg/what-defines-a-chaotic-system.htmlSuppose I have an equation of motion of some sort. All I know about chaotic systems is that small changes in initial conditions will increase the outcome at $t$ by exceedingly large amounts. My que......Fri, 14 Aug 2026 14:19:22 -0400https://pop.com.sg/solving-the-equation-11x-2-6000x-27500-0-preferably-without-the-quadra.htmlhttps://pop.com.sg/solving-the-equation-11x-2-6000x-27500-0-preferably-without-the-quadra.htmlI obtained this form while solving an aptitude question. $$\frac{3000}{x-50} + \frac{3000}{x+50} = 11$$ I changed it into quadratic equation $$11x^2 -6000x - 27500 =0$$ but I don't know how to ......Fri, 14 Aug 2026 14:17:14 -0400https://pop.com.sg/simply-explain-the-general-number-field-sieve.htmlhttps://pop.com.sg/simply-explain-the-general-number-field-sieve.htmlAs a beginner to the world of integer factorization, my idea of factoring an integer is to generate a large list of prime numbers below this number and to repeatedly try to divide the integer by th......Fri, 14 Aug 2026 14:16:59 -0400https://pop.com.sg/how-does-one-orient-a-simplicial-complex.htmlhttps://pop.com.sg/how-does-one-orient-a-simplicial-complex.htmlI have a simplicial complex, built out of hyper-tetrahedra (5-cells) with the topology of $S_{4}$ and I would like to assign an ordering to it's vertices (some couple thousand), so that I can apply a ...Fri, 14 Aug 2026 14:13:30 -0400https://pop.com.sg/finding-the-area-of-an-ellipse-using-the-formula.htmlhttps://pop.com.sg/finding-the-area-of-an-ellipse-using-the-formula.htmlI'm preparing for the ACTM Sate contest, and I stumbled across this problem. I've never learned how to decipher equations like this, so could someone explain what these equations mean and how to so......Fri, 14 Aug 2026 14:13:14 -0400https://pop.com.sg/easiest-way-to-solve-this-integral-closed.htmlhttps://pop.com.sg/easiest-way-to-solve-this-integral-closed.htmlI was solving this problem from a calculus textbook and I got stuck at this particular problem. I tried to put it into Integral Calculator after I was unable to solve it, but now I wonder if there ......Fri, 14 Aug 2026 14:11:52 -0400https://pop.com.sg/trace-and-the-coefficients-of-the-characteristic-polynomial-of-a-matri.htmlhttps://pop.com.sg/trace-and-the-coefficients-of-the-characteristic-polynomial-of-a-matri.htmlLet $A\in M(\mathbb F)_{n \times n}$ Prove that the trace of A is minus the coefficient of $\lambda ^{n-1}$ in the characteristic polynomial of A. I had several ideas to approach this problem - ......Fri, 14 Aug 2026 14:07:18 -0400https://pop.com.sg/how-to-show-this-sequence-is-decreasing.htmlhttps://pop.com.sg/how-to-show-this-sequence-is-decreasing.htmlI was given a practice question in my math lecture today, it was to prove that the sequence below was decreasing: $a_n = \frac{2^n}{(n+1)!}$ using the condition that a sequence is decreasing if $......Fri, 14 Aug 2026 13:54:08 -0400https://pop.com.sg/probability-x-will-happen-out-of-two-tries.htmlhttps://pop.com.sg/probability-x-will-happen-out-of-two-tries.htmlSay you grab a card out a deck, check it, and put it back. If the probability of getting a non-club card is $\frac{3}{4}$, what is the probability it will happen at least once out of two tries? Y......Fri, 14 Aug 2026 13:49:51 -0400https://pop.com.sg/orthogonal-projection-onto-the-unit-simplex.htmlhttps://pop.com.sg/orthogonal-projection-onto-the-unit-simplex.htmlThe Unit Simplex is defined by: $$ \mathcal{S} = \left\{ x \in \mathbb{{R}^{n}} \mid x \succeq 0, \, \boldsymbol{1}^{T} x = 1 \right\} $$ Orthogonal Projection onto the Unit Simplex is defined by......Fri, 14 Aug 2026 13:48:20 -0400https://pop.com.sg/what-is-the-equation-of-a-half-sine-wave.htmlhttps://pop.com.sg/what-is-the-equation-of-a-half-sine-wave.htmlBasically, I am trying to generate a half sine wave using excel. The waveform that I am looking for is a wave which has a positive component from $0$ to $\pi$ and a zero component from $\pi$ to $2\......Fri, 14 Aug 2026 13:40:11 -0400https://pop.com.sg/find-the-partial-sum-of-the-series.htmlhttps://pop.com.sg/find-the-partial-sum-of-the-series.htmlCase: $$\sum _{n=1}^{\infty}(-1)^{n+1}(2n-1)$$ Question: $$S_n =\ ?$$ My partial solution and investigation: $$S_n = \sum _{k=1}^n (-1)^{n+1}(2n-1) = 1 -3+5-7+9-11+\cdots+(-1)^{n+1}(2n-1)$$ I see......Fri, 14 Aug 2026 13:38:23 -0400https://pop.com.sg/you-roll-a-6-sided-die-what-is-your-p-prime-closed.htmlhttps://pop.com.sg/you-roll-a-6-sided-die-what-is-your-p-prime-closed.htmlyou roll a 6 sided die. whqat is P(prime)? write your answer as a percentage...Fri, 14 Aug 2026 13:34:05 -0400https://pop.com.sg/proving-tan-20-circ-cdot-tan-30-circ-cdot-tan-40-circ-tan-10-circ.htmlhttps://pop.com.sg/proving-tan-20-circ-cdot-tan-30-circ-cdot-tan-40-circ-tan-10-circ.htmlhow to prove that : $\tan20^{\circ}.\tan30^{\circ}.\tan40^{\circ}=\tan10^{\circ}$? I know how to prove $ \frac{\tan 20^{0}\cdot\tan 30^{0}}{\tan 10^{0}}=\tan 50^{0}, $ in this way: $ \tan{20^......Fri, 14 Aug 2026 13:32:02 -0400https://pop.com.sg/determining-a-branch-of-logarithm.htmlhttps://pop.com.sg/determining-a-branch-of-logarithm.htmlThe question I have is that what is the explicit mapping that takes the value $-i \pi/2$ at $-i$ where the mapping is a branch of the logarithm in the slit plane $\mathbb{C}- [0,\infty)$? I'm famil......Fri, 14 Aug 2026 13:24:20 -0400https://pop.com.sg/is-it-possible-in-a-group-of-seven-people-for-each-to-be-friends-with-.htmlhttps://pop.com.sg/is-it-possible-in-a-group-of-seven-people-for-each-to-be-friends-with-.htmlIs it possible in a group of seven people for each to be friends with exactly 3 others? I know that the sum of degrees of vertices in a graph must be even....Fri, 14 Aug 2026 13:20:39 -0400https://pop.com.sg/how-to-find-the-average-distance-from-a-point-to-a-all-the-points-on-a.htmlhttps://pop.com.sg/how-to-find-the-average-distance-from-a-point-to-a-all-the-points-on-a.htmlSuppose point $C$ is directly above point $B$ of $\overline{AB}$. $|\overline{AB}| =2$, and the distance from $C$ to $B$ is $1$. How can I find the average distance from point $C$ to every point o......Fri, 14 Aug 2026 13:19:16 -0400https://pop.com.sg/proof-of-the-ham-sandwich-theorem.htmlhttps://pop.com.sg/proof-of-the-ham-sandwich-theorem.htmlI have doubts about the proof of the Ham-Sandwich theorem descibed on planetmath (http://planetmath.org/proofofhamsandwichtheorem) and wikipedia (http://en.wikipedia.org/wiki/Ham_sandwich_theorem):......Fri, 14 Aug 2026 13:18:31 -0400https://pop.com.sg/can-a-039-closed-039-curve-be-a-function.htmlhttps://pop.com.sg/can-a-039-closed-039-curve-be-a-function.htmlFor a 'closed' curve (I don't know if there is such in mathematics called closed curve, but I mean a curve which is 'closed') e.g.1)It is well-known unit circle with equation$$x^2+y^2=1$$e.g.2)It i......Fri, 14 Aug 2026 13:12:36 -0400https://pop.com.sg/how-can-i-find-the-volume-of-intersection-between-two-cubes.htmlhttps://pop.com.sg/how-can-i-find-the-volume-of-intersection-between-two-cubes.htmlHow can I find the volume of the intersection of two cubes aligned to the axis? The cubes can have different sizes and positions. (I add a picture to show a simple example case of two cubes) After ......Fri, 14 Aug 2026 12:39:00 -0400https://pop.com.sg/fundamental-solution-of-heat-equation.htmlhttps://pop.com.sg/fundamental-solution-of-heat-equation.htmlFor every bounded function $u_0\in C(\mathbb R)$ the function $$u(t,x)=\frac{1}{\sqrt{4\pi t}}\int_{-\infty}^{\infty}e^{-\frac{-(x-y)^2}{4t}}u_0(y)dy$$ is a solution of $\dfrac{\partial v}{\partial......Fri, 14 Aug 2026 12:34:58 -0400https://pop.com.sg/giving-proof-by-counterexample.htmlhttps://pop.com.sg/giving-proof-by-counterexample.htmlI just started learning college mathematics and one of the things I don't like is giving proofs by counterexamples. My question is how is disproving by giving counterexample is seen by advanced '...Fri, 14 Aug 2026 12:31:24 -0400https://pop.com.sg/is-it-possible-to-convert-a-one-to-many-function-to-a-one-to-one-funct.htmlhttps://pop.com.sg/is-it-possible-to-convert-a-one-to-many-function-to-a-one-to-one-funct.htmlIs it possible to convert a one-to-many function to a one-to-one function. The input is going to be unique and hence the out of the function should also be unique. Repeating output would only creat......Fri, 14 Aug 2026 12:30:14 -0400https://pop.com.sg/minewsweeper-numbers-of-mines-to-board-size-probability-of-wins.htmlhttps://pop.com.sg/minewsweeper-numbers-of-mines-to-board-size-probability-of-wins.htmlIn a typical minesweeper setup, we have: Beginner: 10 mines in 9x9 Intermediate: 40 mines in 16x16 Expert: 99 mines in 16x30 That means mines occupy ~12.35%, ~15.63%, and ~20.63% of the board, ...Fri, 14 Aug 2026 12:18:08 -0400https://pop.com.sg/what-is-a-intersection-b-039.htmlhttps://pop.com.sg/what-is-a-intersection-b-039.htmlI was answering the questions in my book and came over the question: What is A intersection B complement? I thought the answer would be shading everything EXCEPT the middle part where A and B inte......Fri, 14 Aug 2026 12:13:46 -0400https://pop.com.sg/find-a-and-b-so-that-f-is-differentiable-everywhere.htmlhttps://pop.com.sg/find-a-and-b-so-that-f-is-differentiable-everywhere.htmlThe piecewise function is $$f(x)=\begin{cases} ax^3 & x\leq 2 \\ x^2+b, & x>2 \end{cases} $$ First I plug in the 2 into both functions and set them equal to each other $$ 2^2+b=a(2)^3......Fri, 14 Aug 2026 12:03:01 -0400https://pop.com.sg/is-this-batman-equation-for-real-closed.htmlhttps://pop.com.sg/is-this-batman-equation-for-real-closed.htmlHardOCP has an image with an equation which apparently draws the Batman logo. Is this for real? Batman Equation in text form: \begin{align} &\left(\left(\frac x7\right)^2\sqrt{\frac{||x|-3|}{......Fri, 14 Aug 2026 11:56:44 -0400https://pop.com.sg/how-to-split-large-numbers-easily-for-factoring-the-middle-term.htmlhttps://pop.com.sg/how-to-split-large-numbers-easily-for-factoring-the-middle-term.htmlI was doing a problem when I got suddenly stuck with a middle term factorisation. The equation is as follows: $$x^4-289x^2+14400=0.$$ I managed to somehow solve it, but it took a lot of time. Can you ...Fri, 14 Aug 2026 11:52:07 -0400https://pop.com.sg/vertical-asymptotes-of-a-given-non-rational-radical-funtion.htmlhttps://pop.com.sg/vertical-asymptotes-of-a-given-non-rational-radical-funtion.htmlWe have that $f$ is a function $f(x) = x\sqrt{x+4}$. Hence, $f'(x) = \dfrac{3x+8}{2\sqrt{x+4}}$. Then, $\lim_{x \to -4^+}f'(x) = -\infty$. This means that $f$ has a vertical slope at $f(-4)$. I......Fri, 14 Aug 2026 11:47:43 -0400https://pop.com.sg/how-do-i-find-the-mode-when-my-data-points-have-decimals.htmlhttps://pop.com.sg/how-do-i-find-the-mode-when-my-data-points-have-decimals.htmlMy data has numbers such as 70.35, 70.39, 70.1. None of the numbers are exactly the same. How do I find the mode?...Fri, 14 Aug 2026 11:33:25 -0400https://pop.com.sg/is-it-necessary-for-the-imaginary-axis-to-be-perpendicular-to-the-real.htmlhttps://pop.com.sg/is-it-necessary-for-the-imaginary-axis-to-be-perpendicular-to-the-real.htmlThe Real number line is in one dimension. If you want to map a complex number, you would have to add a second dimension to that number line- the Imaginary-axis. The Imaginary-axis is always ...Fri, 14 Aug 2026 11:32:36 -0400https://pop.com.sg/what-is-the-definition-of-binary-sequence.htmlhttps://pop.com.sg/what-is-the-definition-of-binary-sequence.htmlCan I write an infinite binary sequence like so: ...0111001001, ...10010 because I saw some people write infinite binary set from left to right like so: 1011000... , 101111... But I was not sure ......Fri, 14 Aug 2026 11:28:24 -0400https://pop.com.sg/find-critical-and-saddle-points-for-a-multivariable-function.htmlhttps://pop.com.sg/find-critical-and-saddle-points-for-a-multivariable-function.htmlThe function is $f(x, y) = x^2 + 2y^2 + 2xy - 4y + 15$ here's what I did, $$f_x = 2x + 2y$$ $$f_y = 4y + 2x - 4$$ $$f_{xx} = 2$$ $$f_{yy} = 4$$ $$f_{xy} = 2$$ Putting $f_x = 0$ and $f_y = 0$, w......Fri, 14 Aug 2026 11:05:06 -0400https://pop.com.sg/when-is-the-equicorrelation-matrix-positive-definite.htmlhttps://pop.com.sg/when-is-the-equicorrelation-matrix-positive-definite.htmlShow that the matrix, $$ A= \begin{bmatrix} 1 & \rho & \dots & \rho \\ \rho & 1 & \dots & \rho \\ \vdots & \rho & \ddots & \vdots \\ \rho & \dot......Fri, 14 Aug 2026 11:04:35 -0400https://pop.com.sg/alternate-definition-of-a-derivative-question.htmlhttps://pop.com.sg/alternate-definition-of-a-derivative-question.htmlThe alternate form of the derivative is $$f'(a)=\lim_\limits{x \to a}\frac{f(x)-f(a)}{x-a}$$ Can this be rewritten to this and still be true? $$f'(a)=\lim_\limits{x \to a}\frac{f(a)-f(x)}{a......Fri, 14 Aug 2026 10:48:50 -0400https://pop.com.sg/formula-for-calculating-total-yearly-payments-with-yearly-escalation-o.htmlhttps://pop.com.sg/formula-for-calculating-total-yearly-payments-with-yearly-escalation-o.htmlMy knowledge on math/terminology is pretty terrible so seeking some help with a formula (which I'll be converting to javascript to be used in a calculator). I'm trying to calculate the total amoun......Fri, 14 Aug 2026 10:18:46 -0400https://pop.com.sg/finding-sum-of-factors-of-a-number-using-prime-factorization.htmlhttps://pop.com.sg/finding-sum-of-factors-of-a-number-using-prime-factorization.htmlGiven a number, there is an algorithm described here to find it's sum and number of factors. For example, let us take the number $1225$ : It's factors are $1, 5, 7, 25, 35, 49, 175, 245, 1225 $ an......Fri, 14 Aug 2026 10:17:21 -0400https://pop.com.sg/how-to-integrate-with-respect-to-x-2.htmlhttps://pop.com.sg/how-to-integrate-with-respect-to-x-2.html$$\int x\,dx^2$$ I'm having trouble comprehending this question. I can perform substitution, but when I do I come out wrong. Here's how I've happened to handle this: $\int x\,dx^2$ $u=x^2$ $du=......Fri, 14 Aug 2026 10:01:32 -0400https://pop.com.sg/which-day-of-the-week-will-be-100-days-from-now.htmlhttps://pop.com.sg/which-day-of-the-week-will-be-100-days-from-now.htmlIf today is Wednesday, what day of the week will it be $100$ days from now? Going forward $1,8,15,\dots,92,99$ days all result in Wednesday. So the $100$th day will be Thursday. But the solution ......Fri, 14 Aug 2026 09:49:16 -0400https://pop.com.sg/notation-for-eigenvalues.htmlhttps://pop.com.sg/notation-for-eigenvalues.htmlIs there a specific notation for eigenvalues? specifically, I'd like to write: $$m\equiv \text{smallest eigenvaue of }H$$ I've seen some sources write this as: $H\succeq mI$, where "$\succeq0$" mea......Fri, 14 Aug 2026 09:48:50 -0400https://pop.com.sg/what-is-the-mean-of-outcomes-of-six-faced-fair-die-closed.htmlhttps://pop.com.sg/what-is-the-mean-of-outcomes-of-six-faced-fair-die-closed.htmlA six faced fair die is tossed a large number of times (tending to infinity). What is the mean of outcomes of the die? My try: let the die be tossed $n$ times then the mean of outcomes $$=\frac{......Fri, 14 Aug 2026 09:37:45 -0400https://pop.com.sg/can-a-function-be-of-class-c1-even-if-its-partial-derivatives-are-not-.htmlhttps://pop.com.sg/can-a-function-be-of-class-c1-even-if-its-partial-derivatives-are-not-.htmlI know that if all partial derivatives of a function f exist and are continuous then the function is said to be of class C1 (continuously differentiable). However, I was not able to find whether ......Fri, 14 Aug 2026 09:30:59 -0400https://pop.com.sg/visualizing-the-4th-dimension.htmlhttps://pop.com.sg/visualizing-the-4th-dimension.htmlIn a freshers lecture of 3-D geometry, our teacher said that 3-D objects can be viewed as projections of 4-D objects. How does this helps us visualize 4-D objects? I searched that we can at least see ...Fri, 14 Aug 2026 09:29:19 -0400https://pop.com.sg/deval-function-in-matlab.htmlhttps://pop.com.sg/deval-function-in-matlab.htmlI can't understand how the deval function works in matlab, it's hard for me to understand the code, are there any books or descriptions of numerical methods where I can read about numerical algorit......Fri, 14 Aug 2026 09:29:05 -0400https://pop.com.sg/how-to-find-the-closed-form-to-the-fibonacci-numbers-duplicate.htmlhttps://pop.com.sg/how-to-find-the-closed-form-to-the-fibonacci-numbers-duplicate.htmlPossible Duplicate: Prove this formula for the Fibonacci Sequence How to find the closed form to the fibonacci numbers? I have seen is possible calculate the fibonacci numbers without recursi......Fri, 14 Aug 2026 09:17:00 -0400