https://pop.com.sgExplosive pop stories with broad entertainment appeal.en-usThu, 03 Sep 2026 10:48:56 -0400https://pop.com.sg/need-help-with-the-integral-int-frac-2-tan-x-3-5-sin-2-x-4-dx.htmlhttps://pop.com.sg/need-help-with-the-integral-int-frac-2-tan-x-3-5-sin-2-x-4-dx.htmlI'm having a problem resolving the following integral, spent almost all day trying. Any help would be appreciated. $$\int \frac{2\tan(x)+3}{5\sin^2(x)+4}\,dx$$...Thu, 03 Sep 2026 14:41:05 -0400https://pop.com.sg/why-cos-2-2x-frac-1-2-1-cos-4x.htmlhttps://pop.com.sg/why-cos-2-2x-frac-1-2-1-cos-4x.htmlWhy: $$\cos ^2(2x) = \frac{1}{2}(1+\cos (4x))$$ I don't understand this, how I must to multiply two trigonometric functions? Thanks a lot....Thu, 03 Sep 2026 14:16:12 -0400https://pop.com.sg/what-is-the-isometry-and-isometry-group.htmlhttps://pop.com.sg/what-is-the-isometry-and-isometry-group.htmlCan someone explain with some simple examples what is meant by the isometry and isometry group? I'm a student of physics and I often come across this term in texts of general relativity for various ...Thu, 03 Sep 2026 14:15:15 -0400https://pop.com.sg/volume-of-a-pencil-closed.htmlhttps://pop.com.sg/volume-of-a-pencil-closed.htmlA pencil sharpened at both ends is formed of a cylindrical barrel together with two conical points. The pencil measures 110mm from tip to tip, has radius $r$ mm, and the cylindrical barrel is $h$ m......Thu, 03 Sep 2026 14:05:20 -0400https://pop.com.sg/prove-that-the-limit-exists-and-then-find-its-limit.htmlhttps://pop.com.sg/prove-that-the-limit-exists-and-then-find-its-limit.htmlWe are given a function $h(x)$ which strictly positive. The function $h$ is defined on $\mathbb{R}$ and that satisfies the following:\ $$\lim_{x\to 0}(h(x)+\frac{1}{h(x)})=2$$ The question is to ......Thu, 03 Sep 2026 14:02:35 -0400https://pop.com.sg/cubic-equation-in-pocket-calculator.htmlhttps://pop.com.sg/cubic-equation-in-pocket-calculator.htmlIs it possible to input and automaticly get solutions for the cubic equation using Texas Instruments TI-30X IIS calculator?...Thu, 03 Sep 2026 13:56:10 -0400https://pop.com.sg/how-to-find-values-from-1-equation-with-3-variables.htmlhttps://pop.com.sg/how-to-find-values-from-1-equation-with-3-variables.htmlIs there any way to get something out of: $50x + 5y - 18z = 0$ ? I found this trying to solve this: \begin{equation} xyz+xyz+xyz=zzz \end{equation} in which xyz are not multiplying themselves. T......Thu, 03 Sep 2026 13:51:12 -0400https://pop.com.sg/how-can-i-find-the-hypotenuse-and-opposite-sides-of-triangle.htmlhttps://pop.com.sg/how-can-i-find-the-hypotenuse-and-opposite-sides-of-triangle.htmlBased on the given information how can I find the pixel values of the hypotenuse and opposite sides of the triangle?...Thu, 03 Sep 2026 13:49:53 -0400https://pop.com.sg/how-do-you-draw-a-direction-field-for-2x2-matrix.htmlhttps://pop.com.sg/how-do-you-draw-a-direction-field-for-2x2-matrix.htmlI understand $AX = X'$ and by doing so, you get both equation for derivative of $x_1 and $x_2$. When I make a $x_1$ and $x_2$ plot, I am confused regarding which derivative of $x_1$ or $x_2$ to ch......Thu, 03 Sep 2026 13:45:06 -0400https://pop.com.sg/how-do-we-decide-the-direction-of-vector-that-is-orthogonal-to-some-ot.htmlhttps://pop.com.sg/how-do-we-decide-the-direction-of-vector-that-is-orthogonal-to-some-ot.htmlassume that two vector given with relationship below: $$\vec{n}\cdot \vec{u} = 0 $$ Then $\vec{n}$ and $\vec{u}$ are orthogonal vectors. Assume that we now have the direction of $\vec{n}$ Then t......Thu, 03 Sep 2026 13:36:48 -0400https://pop.com.sg/what-is-a-formal-polynomial.htmlhttps://pop.com.sg/what-is-a-formal-polynomial.htmlI'm starting to study Field Theory by myself, the books don't say explicitly what a polynomial is, I mean, what the $x$ of $f(x)$ in $F[x]$ is? $x\in F$? When I take $f(\alpha)$ am I taking the ele......Thu, 03 Sep 2026 13:31:33 -0400https://pop.com.sg/is-dodecahedron-tesselation-somehow-possible.htmlhttps://pop.com.sg/is-dodecahedron-tesselation-somehow-possible.htmlIn this video (at 3:25) there is an animation of planets inside a dodecahedron matrix (or any data-structure that best fit this 3d mosaic). I tried reproducing it with 12 sided dices, or in Blender......Thu, 03 Sep 2026 13:28:44 -0400https://pop.com.sg/definite-integral-word-problem.htmlhttps://pop.com.sg/definite-integral-word-problem.htmlThe sales of a plastic widget were estimated to be: $P(t)  =  8000te ^{−0.5t}$ where $t$ is in weeks, and $P(t)$ is in units per week. How many widgets were sold in the first $9$ weeks? To sta......Thu, 03 Sep 2026 13:27:52 -0400https://pop.com.sg/what-is-the-difference-between-the-dot-product-and-the-scalar-projecti.htmlhttps://pop.com.sg/what-is-the-difference-between-the-dot-product-and-the-scalar-projecti.htmlI don't understand the difference between the dot product of two vectors and the scalar projection of a vector onto another one. To me it looks like they are both (geometrically) the length of the ...Thu, 03 Sep 2026 13:21:31 -0400https://pop.com.sg/multivariable-differential-equation.htmlhttps://pop.com.sg/multivariable-differential-equation.htmlGiven $u=f(2x-y)+g(x-2y)$, show that $$2 \frac{\mathrm{d}^2 u}{\mathrm{d}x^2} + 5 \frac{\mathrm{d}^2 u}{\mathrm{d}x\,\mathrm{d}y} + 2\frac{\mathrm{d}^2 u}{\mathrm{d}y^2} = 0.$$ I'm not even sure wh......Thu, 03 Sep 2026 13:18:42 -0400https://pop.com.sg/from-randn-to-bivariate-gaussian-distribution-image.htmlhttps://pop.com.sg/from-randn-to-bivariate-gaussian-distribution-image.htmlIn Matlab, I generated a bivariate Gaussian distribution with mean 50 and standard deviation 50: G=50*randn((100*100)/2,2)+50; histogram2(G(:,1),G(:,2)); The 3D histogram gives me the output I am ...Thu, 03 Sep 2026 13:17:00 -0400https://pop.com.sg/calculate-line-integral-with-respect-to-arc-length.htmlhttps://pop.com.sg/calculate-line-integral-with-respect-to-arc-length.htmlCalculate the line integral with respect to arc length of $\int_Czds$, where $C$ is parametrized by $γ(t) = (tcost,tsint,t), 0 ≤ t ≤ t_0.$ I calculate the norm of $γ'(t)$ and I get $\sqrt {2+t^2}$.......Thu, 03 Sep 2026 13:11:19 -0400https://pop.com.sg/how-do-i-simplify-frac-sin-2x-1-cos-2x.htmlhttps://pop.com.sg/how-do-i-simplify-frac-sin-2x-1-cos-2x.html$$\dfrac{\sin(2x)}{1-\cos (2x)}$$ How do I simplify given expression? My attempt: We know the double-angle identity for $\sin(2x)$ and $\cos(2x)$ as shown below $$\sin(2x) = 2\sin (x)\cos(x)$$......Thu, 03 Sep 2026 13:07:23 -0400https://pop.com.sg/is-infinity-the-reciprocal-of-zero-is-zero-the-reciprocal-of-infinity.htmlhttps://pop.com.sg/is-infinity-the-reciprocal-of-zero-is-zero-the-reciprocal-of-infinity.htmlIs infinity the reciprocal of zero? Is zero the reciprocal of infinity? It would make sense that they would be--they behave in a similar way (anything multiplied by zero or infinity results in zero......Thu, 03 Sep 2026 13:06:56 -0400https://pop.com.sg/when-is-there-arbitrary-variable-in-system.htmlhttps://pop.com.sg/when-is-there-arbitrary-variable-in-system.htmli am just stuck right now. what was the theorem again about the arbitrary variables in the system of equations? if i have 2 equations with 3 variables, then last variable will be arbitrary, right? ......Thu, 03 Sep 2026 13:05:34 -0400https://pop.com.sg/fraction-multiplication-with-square-root-as-the-numerator-closed.htmlhttps://pop.com.sg/fraction-multiplication-with-square-root-as-the-numerator-closed.htmlremoved - unclear question.......Thu, 03 Sep 2026 12:55:01 -0400https://pop.com.sg/grad-f-in-index-notation.htmlhttps://pop.com.sg/grad-f-in-index-notation.htmlIf $f(\mathbf{r})=\vert\mathbf{r}\vert^4$ How would you calculate $\operatorname{grad} f$ in index notation? I get that $\vert\mathbf{r}\vert^2=x_ix_i$ but how do I represent $f$ in index notation?...Thu, 03 Sep 2026 12:54:03 -0400https://pop.com.sg/mode-of-lognormal-distribution.htmlhttps://pop.com.sg/mode-of-lognormal-distribution.htmlSuppose $$y=e^{x}$$ where x is normal with mean mu and variance sigma. Then I see how to derive mode of f(y) (distribution of y), as we need to find the value y that makes $$f'(y)==0$$ However, why......Thu, 03 Sep 2026 12:44:14 -0400https://pop.com.sg/calculating-the-address-of-an-element-in-an-n-dimensional-array.htmlhttps://pop.com.sg/calculating-the-address-of-an-element-in-an-n-dimensional-array.htmlIn a single dimensional array the address of an element of an array say A[i] is calculated using the following formula Address of $A[i] =B+W * (i–L_B)$ where $B$ is the base address of the array, W......Thu, 03 Sep 2026 12:31:56 -0400https://pop.com.sg/decompose-a-real-symmetric-matrix.htmlhttps://pop.com.sg/decompose-a-real-symmetric-matrix.htmlProve that, without using induction, A real symmetric matrix $A$ can be decomposed as $A = Q^T \Lambda Q$, where $Q$ is an orthogonal matrix and $\Lambda$ is a diagonal matrix with eigenvalues of $......Thu, 03 Sep 2026 12:16:49 -0400https://pop.com.sg/total-probability-theorem-partition.htmlhttps://pop.com.sg/total-probability-theorem-partition.htmlIn the Total Probability Theorem we assume that the sample space is partitioned into subsets. If we consider $B$ to be the sample space and $A_1$, $A_2$ to be the partition then the theorem says: $$\...Thu, 03 Sep 2026 12:12:54 -0400https://pop.com.sg/confused-about-limits-when-denominator-is-0.htmlhttps://pop.com.sg/confused-about-limits-when-denominator-is-0.htmlSo I thought that whenever the denominator of a function was $0$ then the limit does not exist. Until I read the arithmetic rules for limits of functions that states if $f$ and $g$ are functions, and ...Thu, 03 Sep 2026 12:08:53 -0400https://pop.com.sg/discontinuity-vs-not-continuity.htmlhttps://pop.com.sg/discontinuity-vs-not-continuity.htmlWhat is difference between Discontinuous and not continuous of a function $f$ at some point $x=a?$ I think there is not difference between the two. Is there some difference between these two relate......Thu, 03 Sep 2026 12:03:51 -0400https://pop.com.sg/how-to-factor-this-cubic-polynomial-closed.htmlhttps://pop.com.sg/how-to-factor-this-cubic-polynomial-closed.htmlI am asking for detailed steps how to factor the cubic polynomial $${x^3-3x+2}$$ The solution is $${(x-1)^2(x+2)}$$...Thu, 03 Sep 2026 11:57:11 -0400https://pop.com.sg/what-are-some-interesting-counterexamples-given-by-finite-topological-.htmlhttps://pop.com.sg/what-are-some-interesting-counterexamples-given-by-finite-topological-.htmlAccording to Wikipedia, 'finite topological spaces are often used to provide examples of interesting phenomena or counterexamples to plausible sounding conjectures.' I have been studying the book '...Thu, 03 Sep 2026 11:35:25 -0400https://pop.com.sg/prime-divisors-of-60.htmlhttps://pop.com.sg/prime-divisors-of-60.htmlHow to find all prime divisors of $60!$?, where $60!$ denotes the factorial of $60$, the product all integers from $1$ to $60$. Is there any solution to solve this? Thank you....Thu, 03 Sep 2026 11:31:40 -0400https://pop.com.sg/find-the-probability-that-a-randomly-chosen-applicant-who-was-approved.htmlhttps://pop.com.sg/find-the-probability-that-a-randomly-chosen-applicant-who-was-approved.htmlOf all the people applying for a certain job, $89\%$ are qualified. The personnel manager claims that she approves qualified people $61\%$ of the time and she approves an unqualified person $12\%$ ......Thu, 03 Sep 2026 11:30:30 -0400https://pop.com.sg/how-to-figure-of-the-laplace-transform-for-log-x.htmlhttps://pop.com.sg/how-to-figure-of-the-laplace-transform-for-log-x.htmlI was looking at a table of common Laplace transforms of functions when I came across the transform for $\log x$. Apparently, the transform is as follows: $$\mathcal{L} \left\{ \log x\right\}=-\f......Thu, 03 Sep 2026 11:29:05 -0400https://pop.com.sg/how-would-a-triangle-for-sin-90-degree-look.htmlhttps://pop.com.sg/how-would-a-triangle-for-sin-90-degree-look.htmlI am studying trigonometry in my school and learned that in a triangle the side opposite to angle theta should be taken as perpendicular side - hypotenuse remains the same and the third remaining s......Thu, 03 Sep 2026 11:28:03 -0400https://pop.com.sg/series-solution-to-y-039-039-xy-039-y-0.htmlhttps://pop.com.sg/series-solution-to-y-039-039-xy-039-y-0.htmlSo I'm learning to solve ODE's with series on my own using Boyce and DiPrima and exercise #3 is irking me...just looking for power series solutions around the ordinary point... $$y''-xy'-y=0$$ So I......Thu, 03 Sep 2026 11:27:44 -0400https://pop.com.sg/what-is-the-difference-between-lim-h-to0-frac-0-h-and-lim-h-to-infty-f.htmlhttps://pop.com.sg/what-is-the-difference-between-lim-h-to0-frac-0-h-and-lim-h-to-infty-f.htmlWhat is the difference (if any) between- $$\lim_{h\to0}\frac{0}{h} \text{ and } \lim_{h\to\infty}\frac{0}{h}$$ I argue that both must be $=0$ since the numerator is exactly $0$. But one fellow ...Thu, 03 Sep 2026 11:20:34 -0400https://pop.com.sg/compound-interest-with-regular-monthly-contributions-formula.htmlhttps://pop.com.sg/compound-interest-with-regular-monthly-contributions-formula.htmlI have discover formulas to build Investment Calculator by using compound interest (TVM) concept formula. The base of the formula Find Future Value: FV= P(1 +r/n) ^ (nt) + PMT * [(((1 + r/n) ^ (nt......Thu, 03 Sep 2026 11:16:19 -0400https://pop.com.sg/what-is-reform-calculus.htmlhttps://pop.com.sg/what-is-reform-calculus.htmlIn an answer to another question I asked, Isaac suggested a book that is the standard "reform calculus" book. In a comment, I asked what the phrase "reform calculus" means, and Isaac provided a lin......Thu, 03 Sep 2026 11:13:24 -0400https://pop.com.sg/book-recomendation-for-elliptic-curve-cryptography.htmlhttps://pop.com.sg/book-recomendation-for-elliptic-curve-cryptography.htmlI have already taken a course on Cryptography, The course focused mainly on the public-key cryptography based on the algebraic structure of elliptic curves over finite fields. Now, I would like to ...Thu, 03 Sep 2026 11:13:22 -0400https://pop.com.sg/is-division-allowed-in-rings-and-fields.htmlhttps://pop.com.sg/is-division-allowed-in-rings-and-fields.htmlIs division allowed in ring and field? The definition of ring I am using here does not require the presence of multiplicative inverse. I think in general, division is not a well-defined operation......Thu, 03 Sep 2026 11:08:07 -0400https://pop.com.sg/why-does-the-volume-element-in-spherical-polar-coordinates-contain-a-s.htmlhttps://pop.com.sg/why-does-the-volume-element-in-spherical-polar-coordinates-contain-a-s.htmlThe volume element in Cartesian coordinates is $dx dy dz$, the volume of a rectangular prism with side lengths being the length elements along the three rectangular axes. In spherical polar coordi......Thu, 03 Sep 2026 11:00:26 -0400https://pop.com.sg/triple-integral-in-a-sphere.htmlhttps://pop.com.sg/triple-integral-in-a-sphere.htmlHow can I integrate $\sqrt{x^2+y^2+z^2}$ over the region $x^2+y^2+z^2 \le 1$? The problem is I do not know how to set up the triple integral. I guess that I can integrate with respect to $x$ first......Thu, 03 Sep 2026 10:48:00 -0400https://pop.com.sg/prove-the-projection-transformation-is-linear.htmlhttps://pop.com.sg/prove-the-projection-transformation-is-linear.htmlLet $P_z : \mathbb R^3 \to \mathbb R^3$ be the function which orthogonally projects onto the $x$-$y$ linearity plane (that is, the set of vectors $(x, y, z) \in \mathbb R^3$ with $z = 0$). I......Thu, 03 Sep 2026 10:46:07 -0400https://pop.com.sg/why-the-column-space-of-a-matrix-is-useful.htmlhttps://pop.com.sg/why-the-column-space-of-a-matrix-is-useful.htmlI know what is the column space of a matrix: it is basically the subspace formed by the linear combinations of the columns (vectors) of a matrix. From wikipedia, we have the following nice picture: ...Thu, 03 Sep 2026 10:40:43 -0400https://pop.com.sg/line-integral-equals-zero.htmlhttps://pop.com.sg/line-integral-equals-zero.htmlCurrently going through some line integral problems and in the worked example provided, the line integral evaluated to become zero. The curve C that they have provided was a simple closed curve (i......Thu, 03 Sep 2026 10:35:37 -0400https://pop.com.sg/finding-int-x-xdx.htmlhttps://pop.com.sg/finding-int-x-xdx.htmlI'm trying to find $\int x^x \, dx$, but the only thing I know how to do is this: Let $u=x^x$. $$\begin{align} \int x^x \, dx&=\int u \, du\\[6pt] &=\frac{u^2}{2}\\[6pt] &=\dfrac{\lef......Thu, 03 Sep 2026 10:32:53 -0400https://pop.com.sg/duality-theorem-optimal-solution-of-both-primal-and-dual.htmlhttps://pop.com.sg/duality-theorem-optimal-solution-of-both-primal-and-dual.htmlThe Duality Theorem, in short, states that the optimal value of the Primal (P) and Dual (D) Linear Programs are the same if the solution, of either, is a basic feasible solution. My question is t......Thu, 03 Sep 2026 10:30:43 -0400https://pop.com.sg/what-does-it-mean-to-converge-in-law.htmlhttps://pop.com.sg/what-does-it-mean-to-converge-in-law.htmlIf $X_1, X_2, X_3, \cdots$ is a sequence of independent identically distributed random variables with $E[X_i] < \infty$ and $Var(X_i)< \infty$ such that the sequence $Y_n = 3 \frac{X_1 + X_2 + \...Thu, 03 Sep 2026 10:23:49 -0400https://pop.com.sg/how-to-approach-vector-space-of-matrices.htmlhttps://pop.com.sg/how-to-approach-vector-space-of-matrices.htmlThe following is from this Wikipedia page: Let $\mathbf{F}^{m\times n}$ denote the set of $m\times n$ matrices with entries in $\mathbf{F}$. Then $\mathbf{F}^{m\times n}$ is a vector space over $\...Thu, 03 Sep 2026 10:12:51 -0400https://pop.com.sg/find-how-many-terms-there-are-in-this-geometric-sequence-1-2-4-8-16777.htmlhttps://pop.com.sg/find-how-many-terms-there-are-in-this-geometric-sequence-1-2-4-8-16777.htmlFind how many terms there are in this geometric sequence: $-1, 2, -4, 8, ..., -16777216$ My attempt: $a_k=a.r^{k-1}$ And in this sequence: $a=-1$, $r=-2$ So $a_k=(-1){(-2)}^{k-1}$ $-16777216=(-1){......Thu, 03 Sep 2026 10:10:27 -0400https://pop.com.sg/construct-a-square-without-a-ruler-duplicate.htmlhttps://pop.com.sg/construct-a-square-without-a-ruler-duplicate.htmlHow can I construct a square using only a pencil and a compass, i.e. no ruler. Given: a sheet of paper with $2$ points marked on it, a pencil and a compass. Aim: plot $2$ vertices such that the $......Thu, 03 Sep 2026 10:09:17 -0400https://pop.com.sg/a-total-of-28-percent-of-american-males-smoke-cigarettes-7-smoke-cigar.htmlhttps://pop.com.sg/a-total-of-28-percent-of-american-males-smoke-cigarettes-7-smoke-cigar.htmlI was able to find out how many smoke neither cigars nor cigarettes as: $P(E \cup F)$ is the event that someone smokes a cigar OR a cigarette $P(E \cup F)^c$ is the event someone smokes neither ......Thu, 03 Sep 2026 10:09:15 -0400https://pop.com.sg/cof-ab-cof-a-cdot-cof-b.htmlhttps://pop.com.sg/cof-ab-cof-a-cdot-cof-b.htmlLet $A$ and $B$ be two $n\times n$ matrices and let cof$(A)$ and cof$(B)$ denote the cofactor matrix of A and B respectively. Then show that cof($AB$) = cof($A$)$\cdot$cof($B$)? I am trying to sh......Thu, 03 Sep 2026 10:07:39 -0400https://pop.com.sg/find-a-grammar-that-generates-this-palindrome-language.htmlhttps://pop.com.sg/find-a-grammar-that-generates-this-palindrome-language.htmlThis is a homework problem. The problem is: Find a grammar that generates this language: L = {wcw^R: w ∈ {a,b}+ } over alphabet Σ = {a, b, c}. I have tried many different transitions, but can't......Thu, 03 Sep 2026 10:06:50 -0400https://pop.com.sg/use-spherical-coordinates-to-evaluate-the-triple-integral.htmlhttps://pop.com.sg/use-spherical-coordinates-to-evaluate-the-triple-integral.htmlUse spherical coordinates to evaluate : $$ \int_{0}^{4} \int_{0}^{\sqrt{16 - x^2}} \int_{\sqrt{x^2 + y^2}}^{\sqrt{32 - (x^2 + y^2)}} \sqrt{x^2 + y^2 + z^2 \space } \, dz\,dy\,dx$$ Here is what I have ...Thu, 03 Sep 2026 10:06:15 -0400https://pop.com.sg/how-to-determine-coercive-functions.htmlhttps://pop.com.sg/how-to-determine-coercive-functions.htmlA continuous function $f(x)$ that is defined on $R^n$ is called coercive if $\lim\limits_{\Vert x \Vert \rightarrow \infty} f(x)=+ \infty$. I am finding it difficult to understand how the norm of ...Thu, 03 Sep 2026 09:57:50 -0400https://pop.com.sg/limit-as-x-approaches-infinity-for-y-sec-1-x.htmlhttps://pop.com.sg/limit-as-x-approaches-infinity-for-y-sec-1-x.htmlI'm a beginner and I'm using a basic graphing calculator. I understand I can input $sec^{(-1)}x$ as $cos^{(-1)}(1/x)$, but even as I'm looking at the graph, I don't get it. How do I determine the a......Thu, 03 Sep 2026 09:52:39 -0400https://pop.com.sg/laplace-transform-of-cosh-bt.htmlhttps://pop.com.sg/laplace-transform-of-cosh-bt.htmlSo, one of my homework assignments is to take the Laplace transform of a function such that $f(t)=\cosh{bt}$. I figured it would be equivalent to: $$\int^\infty_0{e^{-st}\frac{e^{bt}+e^{-bt}}{2}}\...Thu, 03 Sep 2026 09:42:43 -0400https://pop.com.sg/where-039-s-the-foolish-part-prove-that-0-0-2-duplicate.htmlhttps://pop.com.sg/where-039-s-the-foolish-part-prove-that-0-0-2-duplicate.htmlI've been across this on the web: \begin{align} \frac{0}{0} & = \frac{100-100}{100-100} \\ & = \frac{10^2-10^2}{10(10-10)} \\ & = \frac{(10-10)(10+10)}{(10)(10-10)} \\ & = \frac{(1......Thu, 03 Sep 2026 09:40:10 -0400https://pop.com.sg/how-to-find-length-of-vector-in-a-particular-direction.htmlhttps://pop.com.sg/how-to-find-length-of-vector-in-a-particular-direction.htmlHow do you solve this: How do I find how long the red line is?...Thu, 03 Sep 2026 09:38:42 -0400https://pop.com.sg/differential-equation-capacitor-closed.htmlhttps://pop.com.sg/differential-equation-capacitor-closed.htmlThe voltage of a capacitor can be described with the differential equation $ \frac {du} {dt} + \frac {1} {RC} u = 0$ where the voltage is u(t) at the time t. Solve the differential equation: Don't ...Thu, 03 Sep 2026 09:31:13 -0400https://pop.com.sg/alternate-translation-for-every-real-number-except-zero-has-a-multipli.htmlhttps://pop.com.sg/alternate-translation-for-every-real-number-except-zero-has-a-multipli.htmlA given text states, “Every real number except zero has a multiplicative inverse" (where mul- tiplicative inverse of a real number x is a real number y such that xy = 1). It offers the following ...Thu, 03 Sep 2026 09:15:09 -0400https://pop.com.sg/determine-slope-given-one-point-and-y-intercept.htmlhttps://pop.com.sg/determine-slope-given-one-point-and-y-intercept.htmlThe graph of the relation y = mx + 35 passes through the point (8,–77). Determine the value Of m. I understand how to find the y intercept when given slope and a point, but i can't seem to find the ...Thu, 03 Sep 2026 09:13:21 -0400https://pop.com.sg/prove-by-induction-2-n-c-n-0-c-n-1-cdots-c-n-n-duplicate.htmlhttps://pop.com.sg/prove-by-induction-2-n-c-n-0-c-n-1-cdots-c-n-n-duplicate.htmlThis is a question I came across in an old midterm and I'm not sure how to do it. Any help is appreciated. $$2^n = C(n,0) + C(n,1) + \cdots + C(n,n).$$ Prove this statement is true for all $n \ge ......Thu, 03 Sep 2026 09:12:26 -0400https://pop.com.sg/what-is-the-derivative-of-xf-x-closed.htmlhttps://pop.com.sg/what-is-the-derivative-of-xf-x-closed.htmlIf we have a function $f(x)$ what would the value of the following expression be? $$\frac{d}{dx}[xf(x)]$$...Thu, 03 Sep 2026 09:12:04 -0400https://pop.com.sg/how-to-find-bn-for-limit-comparison.htmlhttps://pop.com.sg/how-to-find-bn-for-limit-comparison.htmlI am asking how to do the Limit comparison test for this: $$\lim {An\over Bn} =L$$ You choose $B_n$ yourselve, but how do you choose it? Example: $$A_n = \frac{3n^2 + 5n + 1}{\sqrt{(n^5 + 5 )}}$......Thu, 03 Sep 2026 09:07:09 -0400https://pop.com.sg/modeling-with-markov-chains-and-one-step-analysis.htmlhttps://pop.com.sg/modeling-with-markov-chains-and-one-step-analysis.htmlI have set up the following model: Let $X_n$ be the number of heads in the $n$-th toss and $P(X_0=0)=1$. I can calculate the transition matrix $P$. Define $$ T=\min\{n\geq 0\mid X_n=5\}. $$ Then ......Thu, 03 Sep 2026 08:56:40 -0400https://pop.com.sg/how-to-solve-this-indefinite-integral.htmlhttps://pop.com.sg/how-to-solve-this-indefinite-integral.htmlI'm trying to understand how to solve this indefinite integral using the power rule. $$ \begin{eqnarray} \int \frac{x}{\sqrt{x}} dx &=& \int \frac{x^1}{x^\frac{1}{2}} dx \end{eqnarray} $$ ...Thu, 03 Sep 2026 08:54:31 -0400https://pop.com.sg/why-do-some-people-state-that-039-zero-is-not-a-number-039.htmlhttps://pop.com.sg/why-do-some-people-state-that-039-zero-is-not-a-number-039.htmlEvery now and then I read about people who wonder whether zero is a number. It never occurred to me to question this, so I checked the Wikipedia page which, when talking about the Rules of Brahmagu......Thu, 03 Sep 2026 08:40:47 -0400https://pop.com.sg/sum-of-geometric-series-formula-duplicate.htmlhttps://pop.com.sg/sum-of-geometric-series-formula-duplicate.htmlI just need the formula for the sum of geometric series when each element in the series has the value $1/2^{j+1}$, where $j = 0, 1, 2, \ldots, n$. Please help. Someone told me it is: $$S = 2 - \f......Thu, 03 Sep 2026 08:27:51 -0400https://pop.com.sg/how-far-can-one-see-over-the-ocean.htmlhttps://pop.com.sg/how-far-can-one-see-over-the-ocean.htmlSince Earth is a sphere, one has only a limited visibility radius. How far is that, actually? This Q&A was inspired by this question, about whether or not Legolas can see the 24km distant Ride......Thu, 03 Sep 2026 08:26:54 -0400https://pop.com.sg/understanding-the-bockstein-homomorphism-in-group-cohomology.htmlhttps://pop.com.sg/understanding-the-bockstein-homomorphism-in-group-cohomology.htmlLet $k$ be an algebraically closed field of characteristic $p>0$. Let $G$ be a finite group. In group cohomology, the Bockstein homomorphism is the connecting homomorphism $$\beta:H^n(G,\math......Thu, 03 Sep 2026 08:23:37 -0400https://pop.com.sg/why-is-a-circle-not-simply-connected.htmlhttps://pop.com.sg/why-is-a-circle-not-simply-connected.htmlTo be simply-connected means to be path-connected and able to continuously shrink a closed curve while remaining in the domain. According to wikipedia, a circle is not simply connected, but a dis......Thu, 03 Sep 2026 08:05:46 -0400https://pop.com.sg/how-do-i-go-about-simplifying-this-complex-radical.htmlhttps://pop.com.sg/how-do-i-go-about-simplifying-this-complex-radical.htmlI'm having a difficult time trying to simplify the radical below. When I type the radical to the left into my calculator, I arrive at the correct answer to the right. But I cannot seem to figure ou......Thu, 03 Sep 2026 08:04:12 -0400https://pop.com.sg/is-closedness-a-proper-word.htmlhttps://pop.com.sg/is-closedness-a-proper-word.htmlIn one of my papers I had to prove a list of properties of a set, say, $S=\{a,b,c\}$. Among them we have a fact that $S$ is downward closed with respect to a binary relation $R$. I found it awkward......Thu, 03 Sep 2026 08:03:58 -0400https://pop.com.sg/what-does-it-mean-when-f-a-is-defined-closed.htmlhttps://pop.com.sg/what-does-it-mean-when-f-a-is-defined-closed.htmlI am in Calculus I and my calculus jargon is lacking so please if you can explain, for lack of better terms, explain it simply....Thu, 03 Sep 2026 07:55:28 -0400https://pop.com.sg/can-a-function-have-an-x-input-with-two-y-outputs.htmlhttps://pop.com.sg/can-a-function-have-an-x-input-with-two-y-outputs.htmlFor example f(4) = 3, 2 f(5) = 4 f(6) = 10 Please forgive me if my example is atrocious, hopefully it gets my point across....Thu, 03 Sep 2026 07:52:31 -0400https://pop.com.sg/which-greek-letter-is-commonly-used-to-represent-a-count.htmlhttps://pop.com.sg/which-greek-letter-is-commonly-used-to-represent-a-count.htmlWhich Greek letter is commonly used to represent a count? For example, the Greek letter sigma ($\Sigma$) is commonly used to represent a sum....Thu, 03 Sep 2026 07:50:32 -0400https://pop.com.sg/odds-of-winning-at-minesweeper-with-perfect-play.htmlhttps://pop.com.sg/odds-of-winning-at-minesweeper-with-perfect-play.htmlHow would someone go about doing this? Assume that the first "click" will never be a bomb, and that the number of mines and the area are both known. Rather hoping there is a clever way to do this, ......Thu, 03 Sep 2026 07:45:03 -0400https://pop.com.sg/applying-difference-of-cubes-to-cube-roots.htmlhttps://pop.com.sg/applying-difference-of-cubes-to-cube-roots.htmlI am stumped as to why this application of the difference of cubes is valid... I am rationalizing the denominator. I don't understand the reasoning of why the difference of cubes formula is appli......Thu, 03 Sep 2026 07:44:01 -0400https://pop.com.sg/why-does-the-dot-product-between-two-unit-vectors-equal-the-cosine-on-.htmlhttps://pop.com.sg/why-does-the-dot-product-between-two-unit-vectors-equal-the-cosine-on-.htmlI'm confused about why does the dot product between two unit vectors equal the cosine on the angle between them Thank you....Thu, 03 Sep 2026 07:32:14 -0400https://pop.com.sg/how-can-i-check-that-a-hexagon-intersects-a-circle.htmlhttps://pop.com.sg/how-can-i-check-that-a-hexagon-intersects-a-circle.htmlIf I know the center positions and radius of a circle and hexagon, how can I determine whether are they touches or intersects each other? Update: Also the orientation of the hexagon is know:...Thu, 03 Sep 2026 07:23:34 -0400https://pop.com.sg/calculating-the-volume-of-a-hemisphere-given-only-the-height.htmlhttps://pop.com.sg/calculating-the-volume-of-a-hemisphere-given-only-the-height.htmlSo I have a math problem that I'm trying to solve with regards to hemispheres but am just confused about this part of the question If I have a hemisphere with radius $r$, then Volume $= \frac23......Thu, 03 Sep 2026 07:21:37 -0400https://pop.com.sg/how-to-intuitively-see-that-the-text-volume-of-a-pyramid-1-3-times-tex.htmlhttps://pop.com.sg/how-to-intuitively-see-that-the-text-volume-of-a-pyramid-1-3-times-tex.htmlI'm interested to know if anyone can point me to a non-calculus way of seeing that the $\text{volume of a pyramid} = \frac{1}{3}\times(\text{area of base})\times(\text{height})$. Yes, I've googled....Thu, 03 Sep 2026 07:20:56 -0400https://pop.com.sg/square-root-of-1-2-sqrt3-2.htmlhttps://pop.com.sg/square-root-of-1-2-sqrt3-2.htmlPlaying with Maple, I noticed that it gives the square root of $c = 1+\frac{\sqrt3}{2}$ as equal to $a = \frac{1}{2}+\frac{\sqrt3}{2}$. Indeed it checks out. But I got curious: how can I find that......Thu, 03 Sep 2026 07:15:50 -0400https://pop.com.sg/the-price-is-right-optimal-play.htmlhttps://pop.com.sg/the-price-is-right-optimal-play.htmlThe following situation happened on the Price is Right and I was curious about the optimal response. The rules are: A contestant rolls a wheel with 5 cent increments from 5 - 100 (20 numbers total)......Thu, 03 Sep 2026 07:13:03 -0400https://pop.com.sg/given-that-log-ab-log-ba-and-that-a-b-ne-1-and-a-ne-b-find-b-in-terms-.htmlhttps://pop.com.sg/given-that-log-ab-log-ba-and-that-a-b-ne-1-and-a-ne-b-find-b-in-terms-.htmlGiven that $\log_ba=\log_ab$, and that $a,b \ne 1$ and $a \ne b$, find $b$ in terms of $a$. I tried to solve this problem using changing the base of the first part of the equation: $$ \frac{\log......Thu, 03 Sep 2026 07:12:39 -0400https://pop.com.sg/what-are-expressions-in-mathematics.htmlhttps://pop.com.sg/what-are-expressions-in-mathematics.htmlLike algebraic expressions are logarithmic, Experimental, trigonometric, differential, etc., expressions also there? I am not referring to functions, but just expressions or equations. Are their ...Thu, 03 Sep 2026 07:08:55 -0400https://pop.com.sg/find-a-unit-vector-that-maximizes-the-directional-derivative-at-a-poin.htmlhttps://pop.com.sg/find-a-unit-vector-that-maximizes-the-directional-derivative-at-a-poin.htmlGiven the function $f(x,y)=g(2x+y), g'(3)=3$ Find the coordinates of a unit vector $v=(v_1,v_2)$ that finds the maximum directional derivative $D_ uf(1,1)$ of f in the point $(1,1)$. I know that I ...Thu, 03 Sep 2026 07:08:45 -0400https://pop.com.sg/probability-of-random-item-selection.htmlhttps://pop.com.sg/probability-of-random-item-selection.htmlIn a 15-piece batch there are 5 black balls and the 10 remaining balls are white. 3 balls are selected randomly. The batch is discarded if among the chosen ones at least one selected ball is white.......Thu, 03 Sep 2026 07:04:11 -0400https://pop.com.sg/find-the-length-of-parametric-curves-simple.htmlhttps://pop.com.sg/find-the-length-of-parametric-curves-simple.htmlso I have the following parametric curve: Now I'm trying to figure out how I should tackle this, I know the formula for finding lengths is $\int[\sqrt{g'(x)^2+f'(x)^2}]$ or when its a single ...Thu, 03 Sep 2026 07:00:25 -0400https://pop.com.sg/calculating-a-chromatic-polynomial.htmlhttps://pop.com.sg/calculating-a-chromatic-polynomial.htmlI am going through some questions in the "Bondy,Murty - Graph Theory with applications" book, and I have stumbled upon the following question: calculate the Chromatic Polynomial of the following ......Thu, 03 Sep 2026 06:56:25 -0400https://pop.com.sg/remainder-of-an-integer-divided-by-5-is-the-same-as-the-remainder-of-t.htmlhttps://pop.com.sg/remainder-of-an-integer-divided-by-5-is-the-same-as-the-remainder-of-t.htmlwe have been told in arithmetic that the remainder of an integer divided by 5 is the same as the remainder of the division of the rightmost digit by 5 how to use modular properties to prove this?...Thu, 03 Sep 2026 06:55:40 -0400https://pop.com.sg/understanding-the-difference-between-span-and-basis.htmlhttps://pop.com.sg/understanding-the-difference-between-span-and-basis.htmlI've been reading a bit around MSE and I've stumbled upon some similar questions as mine. However, most of them do not have a concrete explanation to what I'm looking for. I understand that the Sp......Thu, 03 Sep 2026 06:55:39 -0400https://pop.com.sg/proof-by-induction-sequence-of-integers.htmlhttps://pop.com.sg/proof-by-induction-sequence-of-integers.htmlSuppose a sequence of integers $a_1$, $a_2$, ... is defined as: $$a_1 = 3$$ $$a_2 = 6$$ $$a_n = 5a_{n-1} - 6a_{n-2} + 2$$ for all $n\ge3$ $\mathbf {Prove}$ $\mathbf {S(n)}$: $a_n = 1 + 2^{n-1} + ......Thu, 03 Sep 2026 06:38:00 -0400https://pop.com.sg/check-if-line-intersects-with-circles-perimeter.htmlhttps://pop.com.sg/check-if-line-intersects-with-circles-perimeter.htmlSay I have this circle here. What would be an algorithm to check if lines like the green or blue lines intersect the edge of the circle, but not the red line. So lets say if(amountOfPointsHit(l......Thu, 03 Sep 2026 06:20:04 -0400https://pop.com.sg/parameterization-of-a-part-of-a-sphere.htmlhttps://pop.com.sg/parameterization-of-a-part-of-a-sphere.htmlI have to parametrize $D=\{x^2+y^2+z^2\le 25,y\le -4\}$. I can see the I have to parametrize 2 surfaces : ($S_1$) the intersection between the plane $z=-4$ and the sphere: ($x^2+z^2\le 9$) ($S_2......Thu, 03 Sep 2026 06:12:38 -0400https://pop.com.sg/computing-the-expectation-of-the-square-of-a-random-variable-text-e-x-.htmlhttps://pop.com.sg/computing-the-expectation-of-the-square-of-a-random-variable-text-e-x-.htmlWhat is the rule for computing $ \text{E}[X^{2}] $, where $ \text{E} $ is the expectation operator and $ X $ is a random variable? Let $ S $ be a sample space, and let $ p(x) $ denote the probabil......Thu, 03 Sep 2026 06:07:16 -0400https://pop.com.sg/how-do-i-find-the-lateral-surface-area-of-an-elliptical-cone-not-a-fru.htmlhttps://pop.com.sg/how-do-i-find-the-lateral-surface-area-of-an-elliptical-cone-not-a-fru.htmlThe formula for the lateral surface area of a right circular cone is $\pi*r*l$ where $r$ is the circle's radius. But ellipse has a semi-major axis as well as a semi-minor one. Which one of them sho......Thu, 03 Sep 2026 05:49:57 -0400https://pop.com.sg/farmers-pen-part-a-part-b.htmlhttps://pop.com.sg/farmers-pen-part-a-part-b.htmlA) A rectangular pen is built with one side against a barn, 200 meters of fencing are used for the other three sides of the pen. What dimensions maximize the area of the pen? B) A rancher plans to......Thu, 03 Sep 2026 05:39:29 -0400