Merging man & maths https://www.mathcity.org/ Thu, 04 Jun 2026 02:43:30 +0000 FeedCreator 1.8 https://www.mathcity.org/_media/logo.svg https://www.mathcity.org/ https://www.mathcity.org/fsc-part2-ptb/definitions Definitions: FSc Part 2 (Mathematics): PTB On this page, all the definitions of “Calculus and Analytic Geometry, MATHEMATICS 12” (Mathematics FSc Part 2 or HSSC-II), Punjab Textbook Board (PTB) Lahore, Pakistan are given. We are very thankful to $A=x^2$$f:X\to Y$$X$$f:X\to Y$$y$$Y$$y=ax+b$$x$$y$$f(x)=2x-6$$p(x) = {a_n}{x^n} + {a_{n - 1}}{x^{n - 1}} + {a_{n - 2}}{x^{n - 2}} + ... + {a_1}x + {a_0}$${a_0},\,{a_1},\,{a_2},...,{a_n}$$f(x)=ax+b$$X$$I:X\to X$$X$$Y$$C:X \rightarrow Y$$C(x)=a$$x \in X$$… anonymous@undisclosed.example.com (Anonymous) Sun, 20 Oct 2024 18:16:52 +0000 https://www.mathcity.org/fsc-part2-ptb/definitions-muzzammil-subhan Definitions: Mathematics 12: PTB by Muzzammil Subhan Definitions from Calculus and Analytic Geometry, MATHEMATICS 12, published by Punjab Textbook Board (PTB) Lahore, Pakistan. We are very thankful to Muzzammil Subhan for his valuable contribution. Download or view PDF for all definitions. Samples is given below$P(x)=a_0 x^0+a_1 x^1+a_2 x^2+\ldots . .+a_{n-1} x^{n-1}+a_n x^n$$n \in W$$a_0, a_1, a_2, \ldots, a_n \in R$$f(x)=a x+b$$a, b \in R$$a \neq 0$$f(x)=x$$f(x)=c$$c \in R$$\frac{P(x)}{Q(x)}$… anonymous@undisclosed.example.com (Anonymous) Sun, 20 Oct 2024 18:16:28 +0000 https://www.mathcity.org/fsc-part2-ptb/important-questions/unit-05-linear-inequalities-and-linear-programming Unit 05: Linear Inequalities and Linear Programming Here is the list of important questions. * Graph the solution region of $2x+y \geq 2$ --- BSIC Gujranwala (2016) * Graph the feasible region subject to the following constraint: --- BSIC Gujranwala (2016)$2x-3y \leq 6$$2x+3y \leq 12$$x \geq 0$$y \geq 0$$2x+y\geq 2$$x+2y\leq10$$x\geq0,y\geq0$$2x+3y\leq 12$$z=x+3y$$2x+5y\leq30$$5x+4y\leq20$$x\geq0$$y\geq0$$x+2y\leq 14$$3x+4y\leq 36$$2x+y\leq 10$$x\geq0, y\geq0$$f(x)=2x+5y$$-x\leq8$$-y\leq… anonymous@undisclosed.example.com (Anonymous) Sun, 07 Feb 2021 16:47:57 +0000 https://www.mathcity.org/fsc-part2-ptb/subjective-mathematics-m-shahbaz Subjective Mathematics 12th by Muhammad Shahbaz [Subjective Mathematics 12th by Muhammad Shahbaz] These handwritten notes are sent by Muhammad Shahbaz. We are very thankful to him for sending these notes. These notes are based on the book “Calculus and Analytic Geometry, MATHEMATICS 12 (Mathematics FSc/ICS Part 2 or HSSC-II), Punjab Textbook Board (PTB) Lahore, Pakistan.'' There are total seven (7) units in this book. anonymous@undisclosed.example.com (Anonymous) Tue, 23 Sep 2025 19:59:04 +0000 https://www.mathcity.org/fsc-part2-ptb/important-questions/unit-03-integration Unit 03: Integration Here is the list of important questions. * Evaluate $\int \frac{1}{\sqrt{x}(\sqrt{x}+1)}dx$ --- BSIC Gujranwala (2016) * Find $\int \frac{1}{1+ cosx}dx$ --- BSIC Gujranwala (2016) * Evaluate $\int \frac{1}{x \ln x}dx$ --- BSIC Gujranwala (2016) * Find $\int x \ln x dx$ --- BSIC Gujranwala (2016) * Evaluate $\int e^{2x}(-sinx+2cosx)dx$ --- BSIC Gujranwala (2016)$\int^2_1(x^2+1)dx$$\int^{\frac{\pi}{4}}_0 \sec x(\sec x+\tan x)dx$$\sin y cosec x \frac{dy}{d… anonymous@undisclosed.example.com (Anonymous) Sun, 07 Feb 2021 16:47:55 +0000