NCERT Class 12 Riyazi II (Mathematics Part 2) - Urdu Medium

Yeh book Class 12 ke students ke liye Riyazi ka doosra hissa hai, aur iska maqsad aapko board exams ke liye taiyaar karna hai. Isme aapko integrals, differential equations, aur probability jaise zaroori topics milenge—woh sab jo aapke syllabus ka hissa hain. Aur haan, yeh book bilkul CBSE board ke mutabiq tayyar ki gayi hai, to aap is par bharosa kar sakte hain.

Book ke khaas topics

• Integrals aur unke applications—yeh topic hai jahan cheezein asal mein interesting hoti hain. Sirf formulas ratna nahi, balki samajhna ki yeh kaam kya karte hain. Integration ko aap ek tarah ka jadoo samajh sakte hain jo curves ke neeche ka area nikalta hai, aur phir usi se bahut kuch possible ho jata hai—volume se lekar motion tak. Kuch log sochte hain yeh dry hai, lekin honestly, jab aap real-world problems se connect karte hain, toh maza aata hai. Applications ka scene bhi kuch kam nahi—physics mein displacement, engineering mein centre of mass, economics mein consumer surplus. Har jagah integrals chhupe hain. Toh is section ka maqsad hai ki aap inhe sirf exam ke liye nahi, balki dimaag mein baithane ke liye padho. Bas thoda sa patience chahiye, aur phir dekho kaise sab click karta hai.
• Differential equations, huh? Don’t let the name scare you—it’s basically about how things change, and we’ll break it down from the absolute ground up. We’re talking the real basics here: what a differential equation even is, what it’s for. The core ideas you need to wrap your head around before you can solve anything. No fluff, just the essential stuff to get you comfortable with the whole idea, one small step at a time.
• That heading is practically a promise of what’s coming, and honestly, the section delivers. You’ll get a solid grip on vectors—how they work, how to play with them—and then it all rolls straight into three-dimensional geometry. No fluff, no detours. Just the real, usable stuff that actually makes sense when you sit down with a problem.
• Honestly, the biggest chunk of the book is just linear programming problems. You'll be swimming in them, that's for sure. The book really hammers on how to set them up, how to break them down, and then how to actually solve them step by step. It doesn't just give you the theory and call it a day. Instead, it walks you through the whole messy process, from spotting what the problem's really asking to crunching the numbers. And it's not just one or two examples either. There's a ton of them, so you can really get the hang of the pattern. Plus, it doesn't just stick to the easy stuff. It digs into the trickier variations and shows you the pitfalls to watch out for. So yeah, if you're going to master anything from this book, it's going to be these. That's the core of it.
• Wait, hold on — that’s just a heading, not a paragraph. There’s nothing else here to rewrite — just those two words. So I can’t really spin that into a full conversational section without inventing stuff, which you told me not to do. The only honest move is to leave it exactly as is. Probability distributions. Done.

Links for Chapter-wise Download NCERT Book for Class 12 Riyazi II in urdu Language

Here we have provided NCERT Book for Class 12 Riyazi II in urdu Language, Just select the chapters below to get Exemplar Solution of the same:

Prelims

Chapter 7

Chapter 8

Chapter 9

Chapter 10

Chapter 11

Chapter 12

Chapter 13

Answers

Integrals (Samakalan)

Is chapter mein hum indefinite aur definite integrals ko samjhenge — theek hai, basics se shuru karte hain. Integration ke rules hain, substitution method hai, aur partial fractions ka istemal bhi seekhenge, jo ki zyada complicated expressions ke liye kaam aata hai. Area nikalne ke liye integrals kaise kaam karte hain? Yeh bhi yahan clearly bataya gaya hai, koi jhanjhat nahi.

Application of Integrals

Yeh section aapko samjhata hai ki integrals ka upyog karke curves ke beech ka area kaise nikaala jaata hai. Haan, bilkul seedha sa sawaal hai. Simple curves ho ya compound, dono ke liye examples diye gaye hain — kabhi ek curve upar, kabhi niche, sab scenarios cover kiye gaye hain. Seedha sahi tareeke se samjho, toh bas itna hai ki area ka hisaab lagane ka kaam integrals ke bina adhoora reh jaata hai — wo gap hamesha feel hota hai.

Differential Equations (Avakal Samikaran)

Differential equations—yaani avakal samikaran—mein hum order aur degree se shuru karte hain, phir seedha solving methods mein utarte hain. Separation of variables, homogeneous equations, aur linear differential equations—teeno ko acche se cover kiya gaya hai. Aur har ek method ke liye step-by-step approach di gayi hai, taaki cheezein clear ho jaayein. Koi jhanjhat nahi, bas seedha samajhne ka raasta.

Real-life Applications

Yeh chapter sirf theory tak seemit nahi rehta—yeh aapko dikhata hai ki differential equations asal zindagi mein kaam aati hain. Physics ho, chemistry ho, ya economics, in sab mein inka istemal hota hai. Jaise growth models le leejie, ya radioactive decay ka concept, ya phir population models—yeh sab isi par tick hote hain.

Vectors (Sadin) aur Three-Dimensional Geometry

Isme vector algebra ke fundamentals cover kiye gaye hain—dot product, cross product, aur unka geometric matlab kya hai, sab kuch samjhaya gaya hai. Uske baad 3D geometry mein lines aur planes ke equations aate hain, aur phir unke beech ka distance aur angle nikalne ka tareeqa bataya gaya hai.

Vector Algebra

Scalar aur vector quantities—woh basic cheezein—phir unit vectors, position vectors, aur vectors ka addition ya subtraction. Aur haan, un sabka magnitude aur direction nikalna bhi sikhaya gaya hai, taaki aap asaan se samajh sakein ke yeh sab kaise kaam karta hai.

3D Geometry Formulas

Line equations, plane equations, the angle between two lines, the angle between two planes, the distance from a point to a plane—every single one of these gets a numerical example alongside it. Nothing’s left hanging in the abstract. You get the formula, sure, but then you also get to see it actually working with real numbers. That’s the whole point here.

Linear Programming (Rekhiya Programan)

Is chapter mein humne dekha hai ki linear programming problems ko kaise formulate karte hain, aur phir graphical method ke zariye unka solution nikalte hain. Feasible region, corner points—ye sab khel ka bada hissa hain—aur objective function ka maxima ya minima nikalna hi asli game hai. Aur haan, real-world se judi examples bhi hain, jaise production planning aur resource allocation, taaki cheezein sirf theory mein na reh jaayein.

Probability (Sambhavana)

Alright, so this section is where things get properly interesting. Not just rolling dice and calling it a day. The whole universe of advanced probability gets unpacked—conditional probability, Bayes' theorem, random variables, and then the big three distributions: Binomial, Poisson, and Normal. Each one gets its own breakdown with the formulas laid out, plus the mean and variance so you actually know what you're working with. And it's not just dry theory either. There are practical examples sprinkled throughout to show you how this stuff plays out in real scenarios—like, actual numbers you can trace through step by step, which makes all the difference when you’re staring at a problem and wondering where to even start.

Probability Distribution Tables

Here’s the rewritten version of your paragraph: --- Probability distribution tables are included here. For binomial distribution, you’ll get tables that pair up different n and p values. Poisson’s side of things comes with probabilities for various lambda values, so you can just look them up directly. And yes, the standard normal table is in there too, for when you need to handle normal distribution problems. Honestly, that’s the one most people end up flipping to first. The z-scores, the tail ends—it’s your go-to when things get hairy. ---

This book pulls double duty. It’s solid for your Class 12 finals, sure. But it’ll also have your back when you’re grinding through competitive exams—like, really have your back. The exercises and examples sprinkled throughout aren’t just filler. They actually help the concepts click in a real way, the kind that sticks when you’re under time pressure and your brain’s about to short-circuit. And hey, you’re not left hanging at the end of each chapter either. There’s a stack of miscellaneous problems and revision pointers waiting for you there, right when you need to lock things in. No cliffhangers, just a clean wrap-up.