Class 12 Maths NCERT Exemplar Questions

This page is for the ones slogging through Class 12 Maths NCERT Exemplar. You get the key questions here, with complete step-by-step solutions attached. Honestly, if you’re prepping for the CBSE board exam, these problems are pure gold. Not just helpful either—they’re the exact practice that gets you walking into that exam hall feeling like you’ve already seen the answers. Short ones, long ones, messy ones—they cover it all. So yeah, dig in.

Exemplar Questions ka Importance

• Exemplar questions — they push you way beyond the usual stuff. You’re not just ticking off basic problems—you’re wrestling with the tricky ones that actually make your brain work. That’s where the real practice happens, the kind that sticks.
• Exemplar questions are the real deal. They don’t just ask you to repeat a formula; they make you actually use it. The whole point is application. You’ll see concepts thrown into situations you weren’t expecting, which forces your brain to connect dots instead of just memorizing steps. That’s where the real learning happens—when you’re stuck, scratching your head. Suddenly it clicks. It’s not about rote practice; it’s about getting comfortable with the messy, unpredictable side of problem-solving. Honestly, if you only study past papers, you’re missing half the picture. These questions show you how the theory bends and twists in real exam scenarios. They’re your secret weapon, honestly. Don’t skip them.
• Exemplar Questions ka importance samajhna ho toh ek baat seedhi si hai—ye aapko board exams ke toughest sawalon ke liye taiyaar karte hain. Aur woh bhi asli tareeke se, sirf theory ratke nahi.

Here we have provided NCERT Exemplar Questionfor Class 12 Maths in hindi Language, Just select the chapters below to get Exemplar Solution of the same:

संबंध एव फलन

प्रतिलोम त्रिकोणमितीय फलन

आव्यूह

सारणिक

सांतत्य और अवकलनीयता

अवकलज के अनुप्रयोग

समाकल

स्माकलो के अनुप्रयोग

अवकल समीकरण

सदिश बीजगणित

त्रिविमयि ज्यामिति

रैखिक प्रोग्रामन

प्रायिकता

Class 12 Maths Exemplar: Chapter-wise Important Concepts

NCERT Exemplar’s Class 12 Maths section isn’t just about solving problems; it’s about really getting the concepts. For each chapter, you’ll find questions that start off simple and gradually climb to that advanced, brain-stretching level. Working through them honestly changes things—your conceptual clarity gets sharper with every attempt. You won’t just memorize formulas; you’ll start seeing why they work. And that’s where the real edge lies.

Chapter 1: Relations and Functions

Is chapter mein exemplar questions ka zyada tar jor relations ke types, functions ki composition, aur invertible functions par hi hota hai. Chalo ek example lete hain—

So, the exemplar question here asks whether the function f: R → R, given by f(x) = 1 + x², is one-one. We're about to show it's not. How? Simple—just plug in 1 and -1. You get f(1) = 2 and f(-1) = 2. Same output for two different inputs. That alone kills the one-one property, right there. No fancy algebra needed. The proof basically writes itself.

Chapter 2: Inverse Trigonometric Functions

Iska matlab hai ki is chapter mein aapko principal values, inverse trigonometric functions ki properties, aur unke domains se related questions milte hain. Exemplar problems mein usually expressions ko properties use karke simplify karna hota hai—woh thoda tricky ho sakta hai, par practice se clear ho jata hai.

Chapter 5: Continuity and Differentiability

Chapter 5 is a big deal in Class 12 Maths, no two ways about it. The Exemplar questions here are all about checking continuity and working through differentiability step by step. And yeah, you'll definitely run into some piecewise functions too—those are pretty much guaranteed to show up.

• Let’s take a closer look at this one. We’ve got f(x) = |x – 1| + |x – 2|, and we need to check continuity at x = 1 and x = 2. Straight off, both pieces—those absolute value terms—are continuous everywhere on their own. No jumps, no breaks. But that doesn’t mean we can just assume the sum behaves nicely at those exact points. We’ve got to actually verify it. So here’s the thing. For continuity at a point, three conditions must hold: the function is defined there, the limit exists as x approaches that point. The limit equals the function’s value. Let’s run through it for x = 1 first. Compute f(1): that’s |1 – 1| + |1 – 2|, which is 0 + 1, so f(1) = 1. Now check the left-hand limit as x creeps up to 1 from below. When x is just under 1, x – 1 is negative, so |x – 1| becomes -(x – 1) = 1 – x. Meanwhile, x – 2 is still negative, so |x – 2| = 2 – x. That gives us (1 – x) + (2 – x) = 3 – 2x. Let x → 1⁻, and you get 3 – 2(1) = 1. Now the right-hand limit, from above. For x slightly greater than 1 but less than 2, x – 1 is positive, so |x – 1| = x – 1, and x – 2 is still negative, so |x – 2| = 2 – x. Sum those: (x – 1) + (2 – x) = 1 — interesting—it’s a constant. Let x → 1⁺, and the limit is 1. Both one-sided limits agree, and they match f(1) = 1. So yes, continuous at x = 1. No drama there. Now let’s do x = 2. First, f(2) = |2 – 1| + |2 – 2| = 1 + 0 = 1. For the left-hand limit, x approaches 2 from below, so we’re in that middle zone where x is between 1 and 2. Same as before: |x – 1| = x – 1, and |x – 2| = 2 – x. The sum is still 1, regardless of x. So the left-hand limit is 1. For the right-hand limit, x > 2. Now both terms are positive: |x – 1| = x – 1, and |x – 2| = x – 2. Sum that up: (x – 1) + (x – 2) = 2x – 3. Let x → 2⁺, and you get 2(2) – 3 = 1. Both one-sided limits hit 1, and f(2) is also 1. So it’s continuous at x = 2 as well. Here’s the kicker: even though the expression changes its algebraic form depending on which side you’re on—like switching gears at x = 1 and again at x = 2—the function still manages to be continuous at both points. The graph would just have these “corners” or kinks, but no actual breaks. So yeah, f is continuous at x = 1 and x = 2. Both check out fine.
• Alright, here's the thing. We're going to calculate both the left-hand and right-hand limits, and we're doing that for each of the two points. Simple enough, right?

Chapter 6: Application of Derivatives

Chapter 6 isn't just about theory—these exemplar problems drag calculus right into the real world. You’ll see rates of change popping up in daily scenarios, functions flipping between increasing and decreasing, tangents and normals doing their thing. Those classic maxima-minima puzzles where you’re hunting for the biggest or smallest value. It’s all about applying derivatives to actual situations, not just staring at formulas.

Here’s a problem for you. We’ve got a 28-meter wire, and we’re cutting it into two pieces. One piece gets bent into a square, the other into a circle. The goal? Figure out how to split that wire so the combined area of both shapes is as small as possible. Classic optimization stuff, right? So here’s the plan. We’ll let x be the length of one piece, which means the other piece is 28 minus x. Then we write down the area of the square in terms of x, and the area of the circle in terms of the leftover wire. Add them up. We’ve got our total area as a function of x. Then we take the derivative, set it to zero, and hunt for those critical points. That’s where the minimum’s hiding. A little calculus magic, and we’re done.

Exemplar Questions Solve Karne ke Fayde

Exemplar questions ko apni routine mein shaamil karna—chahe woh hafte mein do ya teen baar hi kyun na ho—aapke liye kaafi kuch badal sakta hai. Pehli baat toh yeh ki aapka concept clarity ekdum next level par chala jaata hai. Sirf theory padhna aur yeh sochna ki "haan, main samajh gaya" kaafi nahi hai, kyunki jab tak aap khud question solve karke dekhte nahi, tab tak pata hi nahi chalta ki asli problem kahan hai. Exemplar ke sawaal aise hote hain jo aapko alag angle se sochne par majboor kar dete hain, aur isi wajah se aapke andar ek aisi problem-solving skill develop hoti hai jo exams mein kaam aati hai—chahe woh board ho ya competitive. Aur sabse badi baat, regular practice se speed aur accuracy dono mein faayda hota hai. Jo log sirf routine questions karte hain, unse aap obviously ek step aage rahoge, kyunki exemplar mein woh tricky twists hote hain jo aapko mentally tayyar karte hain. Toh haan, agar aap serious hain toh exemplar ko skip karna apne saath unfair karna hai, aur yeh woh cheez hai jo aapko revision ke time bhi bahut help karti hai.

• Clear concept ban jaata hai. Aur phir complex problems bhi aasani se solve ho jaati hain. Matlab, sirf ratna nahi, samajhna seekhte ho.
• Board exam mein aane wale questions ka pattern samajh aata hai, aur practice bhi ho jaati hai. Kya hi baat hai — paper mein kya aayega, kaise aayega, woh sab ek baar mein clear ho jata hai. Yehi toh asli fayda hai iska.
• Time management is the real deal here. You genuinely learn how to crack tough questions when the clock’s ticking against you. That pressure? It becomes your training ground, so you’re not just solving problems—you’re solving them fast, within the limit, no panic. And that’s a skill that sticks with you long after the exam hall.
• Honestly, this is where the real magic happens. Once you've wrestled with these exemplar problems, walking into the exam hall feels completely different. You're not just hoping for the best anymore—you actually know you've seen the tough stuff before, and that alone changes your whole mindset. No more second-guessing yourself. You just sit down, look at the paper, and think, "Okay, I've got this." It's a genuine boost, the kind that shows up in your marks.

Tips for Solving Exemplar Problems

Agar aap exemplar problems mein atak jaate hain, toh ghabraane ki zaroorat nahi hai—bas in steps ko follow karein. Pehle problem ko dhyaan se padhein, har line ko samjhein, aur jo diya hai use alag se likh lein. Phir sochiye ki aapko kya find karna hai; ye clarity aadha solve kar deti hai. Agar phir bhi rukawat aaye, toh concepts ko revise karein, koi shortcut ya formula yaad karne ki koshish na karein. Thoda break lein, phir wapas aake try karein—kabhi kabhi dimaag ko rest dene se hi raasta nikal aata hai. Aur haan, answer ko verify karna mat bhoolen, kyunki chhote mistakes bhi bade problem ban jaate hain. Bas practice karte rahein, har baar ye process repeat karein, aur dheere dheere aapki speed aur accuracy dono badh jaayengi.

1. Read the question first—properly. Don’t just skim it. Figure out what it’s actually asking before you even think about solving it. That’s half the battle, honestly.
2. Jot down every related formula and theorem in your notebook. Seriously, it makes a world of difference. You’ll thank yourself later when you’re staring at a tricky problem and the answer’s right there on the page.
3. Try writing out every step as you go. Don't just jump straight to the final answer.
4. Agar koi problem atak jaye aur solution samajh na aaye, toh seedha answer key kholo. Usse method dekho, samjho ke kaise approach kiya hai. Phir usse band karke dobara khud se solve karo. Bas copy mat karo—pattern pakdo, phir apne dimaag se nikalne ki koshish karo.
5. Look, consistency beats cramming every single time. So here's the deal: carve out a little time each week and work through at least 10 to 15 exemplar problems—no exceptions, no excuses. That's your baseline. Some weeks you'll feel like doing more, and that's great, but never let that number slip below ten. It sounds simple, but you'd be surprised how many students skip this and then wonder why the exam feels like a foreign language. Make it a habit, not a chore. Set a reminder on your phone, grab a friend to keep you accountable, or just reward yourself with a snack afterwards—whatever works. The point is, show up weekly, grind through those problems. Watch your confidence climb.

Important Chapters for Board Exams

Okay, so for board exams, you can't just skim everything. Some chapters have exemplar questions that are an absolute must-do. Honestly, those are the ones that'll really test your understanding. So, make sure you're giving these specific chapters some serious extra attention:

• Calculus (Chapters 5, 6, 7, 8) is where the real heavy lifting happens. You’ve got derivatives, integrals, and differential equations all piled up in there, and guess what? It carries the biggest weightage in the exam, bar none. So if you’re short on time, this is the chapter block you simply can't skip—those marks are just sitting there, waiting for you to grab them.
• Important Chapters for Board Exams [PARA] Algebra (Chapters 3, 4): Matrices aur Determinants se sure-shot questions aate hain.
• Vectors and 3D Geometry, that’s Chapters 10 and 11 for you. Honestly, the numerical problems from these chapters in the exemplar? They’re the real deal. Don’t skip them, because the board exam loves pulling straight from that material.

So here’s the thing—before you even glance at those exemplar problems, make sure you’ve truly gone through your NCERT textbook. Like, properly gone through it. Get your basics solid first, then and only then dive into the advanced stuff. And when you do sit down with those questions, don’t just memorize the answers. That’s a trap. Really dig into the reasoning behind each solution, figure out why it works, and you’ll be set. Best of luck with your prep—you’ve got this!