Important Questions Maths Class 12 2025-26 PDF with Solution

Mathematics is a crucial subject in CBSE Class 12, forming the foundation for higher education and careers in engineering, finance, data science, and research. Excelling in the subject requires conceptual clarity, strategic problem-solving, and consistent practice.

The CBSE Important Questions for Class 12 Mathematics help students focus on frequently asked, high-scoring questions from each chapter. These questions are carefully curated to align with the latest CBSE exam pattern, ensuring thorough preparation.

• By practising these important questions, students can:
• Strengthen their understanding of key concepts.
• Develop effective problem-solving strategies.
• Improve speed and accuracy in exams.

Below, we have provided the links to downloadable PDFs of chapter-wise important questions for class 12 Mathematics and those for different categories of marks.

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Some More Important Question Answers for Class 12 Maths (2025-26)

Question 1: If f(x) = 3x + 2, find f⁻¹(x) and verify that f(f⁻¹(x)) = f⁻¹(f(x)) = x.

Answer: Let y = f(x) = 3x + 2. To find the inverse, we interchange x and y and solve for y.
x = 3y + 2 ⇒ y = (x - 2)/3.
Therefore, f⁻¹(x) = (x - 2)/3.
Now verify:
f(f⁻¹(x)) = 3((x - 2)/3) + 2 = (x - 2) + 2 = x.
Also, f⁻¹(f(x)) = ((3x + 2) - 2)/3 = 3x/3 = x.
Hence verified.

Question 2: Simplify sin⁻¹(cos x) for x in [0, π/2].

Answer: We know cos x = sin(π/2 - x).
So sin⁻¹(cos x) = sin⁻¹(sin(π/2 - x)) = π/2 - x.
Therefore, sin⁻¹(cos x) = π/2 - x when x ∈ [0, π/2].

Question 3: If A = [[2, 3], [1, 4]] and B = [[1, 2], [3, 1]], find AB - BA.

Answer: First, multiply A and B.
AB = [[(2)(1)+(3)(3), (2)(2)+(3)(1)], [(1)(1)+(4)(3), (1)(2)+(4)(1)]].
AB = [[2+9, 4+3], [1+12, 2+4]] = [[11, 7], [13, 6]].
Now find BA.
BA = [[(1)(2)+(2)(1), (1)(3)+(2)(4)], [(3)(2)+(1)(1), (3)(3)+(1)(4)]].
BA = [[2+2, 3+8], [6+1, 9+4]] = [[4, 11], [7, 13]].
Now AB - BA = [[11-4, 7-11], [13-7, 6-13]] = [[7, -4], [6, -7]].

Question 4: Evaluate the determinant
|3 2 1|
|2 3 1|
|1 2 3|.

Answer: Expand along the first row.
= 3(3×3 - 1×2) - 2(2×3 - 1×1) + 1(2×2 - 3×1).
= 3(9 - 2) - 2(6 - 1) + (4 - 3).
= 3×7 - 2×5 + 1 = 21 - 10 + 1 = 12.

Question 5: If f(x) = x³ and g(x) = √x, find (f ∘ g)(x) and (g ∘ f)(x).

Answer: (f ∘ g)(x) = f(g(x)) = f(√x) = (√x)³ = x^(3/2).
(g ∘ f)(x) = g(f(x)) = g(x³) = √(x³) = x^(3/2).
Hence both compositions are equal and give x^(3/2).

Question 6 : Prove that the function f(x) = x² is continuous everywhere but not differentiable at x = 0.

Answer: For continuity,
Left hand limit at 0 = lim(x→0⁻) x² = 0,
Right hand limit at 0 = lim(x→0⁺) x² = 0,
and f(0) = 0.
So LHL = RHL = f(0). Therefore, f(x) is continuous at 0.
For differentiability,
f'(x) = 2x.
At x = 0, left derivative = lim(h→0⁻) (0² - h²)/h = 0, right derivative = lim(h→0⁺) (h² - 0)/h = 0.
Both are equal, so actually f(x) = x² is differentiable at 0 too (so differentiable everywhere). Correction noted: f(x)=|x| would have been non-differentiable, not x².

Question 7: Find equation of the tangent to y = x² + 3x + 2 at x = 1.

Answer: Slope = dy/dx = 2x + 3.
At x = 1, slope m = 2(1) + 3 = 5.
Point on curve = (1, 1² + 3×1 + 2) = (1, 6).
Equation of tangent: y - 6 = 5(x - 1) → y = 5x + 1.

Question 8: Find ∫(3x² + 4x + 2) dx.

Answer: Integrate term by term.
∫3x² dx = x³, ∫4x dx = 2x², ∫2 dx = 2x.
So final answer = x³ + 2x² + 2x + C.

Question 9: Find the area under the curve y = x² between x = 0 and x = 2.

Answer: Area = ∫(0 to 2) x² dx = [x³/3]₀² = (8/3) - 0 = 8/3 square units.

Question 10: Solve differential equation dy/dx = x + y.

Answer: Rewrite as dy/dx - y = x.
This is a linear equation in y.
Integrating factor = e^(-∫1 dx) = e^(-x).
Multiply both sides by e^(-x):
e^(-x) dy/dx - e^(-x) y = x e^(-x).
Left side = d/dx(y e^(-x)) = x e^(-x).
Integrate both sides:
y e^(-x) = ∫x e^(-x) dx = -(x+1)e^(-x) + C.
Multiply by e^x: y = -x - 1 + C e^x.

Question 11: Find the dot product of vectors a = 2i + 3j + k and b = i + 2j + 3k.

Answer: a·b = (2)(1) + (3)(2) + (1)(3) = 2 + 6 + 3 = 11.

Question 12: Find the angle between vectors a = 3i - 2j and b = i + 2j.

Answer: a·b = 3×1 + (-2)×2 = 3 - 4 = -1.
|a| = √(3² + (-2)²) = √13, |b| = √(1² + 2²) = √5.
cosθ = (-1)/(√13 × √5) = -1/√65.
θ = cos⁻¹(-1/√65).

Question 13: Find equation of line passing through (1, 2, 3) and parallel to vector 2i - j + k.

Answer: Vector form: r = (i + 2j + 3k) + t(2i - j + k). Cartesian form: (x - 1)/2 = (y - 2)/(-1) = (z - 3)/1.

Question 14: Find the shortest distance between lines
r₁ = (i + 2j + 3k) + t(2i - j + k) and r₂ = (2i + 3j + k) + s(i + j - k).

Answer: Direction vectors are a₁ = (2, -1, 1), a₂ = (1, 1, -1).
Distance formula: D = |(b₂ - b₁)·(a₁ × a₂)| / |a₁ × a₂|.
b₁ = (1,2,3), b₂ = (2,3,1).
Compute a₁ × a₂ = determinant |i j k; 2 -1 1; 1 1 -1| = i((-1)(-1)-1×1) - j(2(-1)-1×1) + k(2×1 - (-1)×1).
= i(1-1) - j(-2-1) + k(2+1) = 0i - j(-3) + k(3) = 3j + 3k.
So |a₁ × a₂| = √(3² + 3²) = 3√2.
(b₂ - b₁) = (1,1,-2).
Dot with (3,3,0) = (1×0 + 1×3 + (-2)×3) = 3 - 6 = -3.
So D = | -3 | / 3√2 = 1/√2.

Question 15: Solve the linear programming problem to maximize Z = 3x + 5y subject to
x + 2y ≤ 10, 3x + 2y ≤ 18, x, y ≥ 0.

Answer: Draw lines x + 2y = 10 and 3x + 2y = 18.
Find intersection points.
x + 2y = 10 ⇒ y = (10 - x)/2.
3x + 2y = 18 ⇒ y = (18 - 3x)/2.
Solving both: (10 - x)/2 = (18 - 3x)/2 ⇒ 10 - x = 18 - 3x ⇒ 2x = 8 ⇒ x = 4.
Then y = (10 - 4)/2 = 3.
So vertices: (0,0), (0,5), (4,3), (6,0).
Now compute Z at these:
(0,0):0; (0,5):25; (4,3):27; (6,0):18.
Maximum Z = 27 at (4,3).

Question 16: Two cards are drawn from a deck of 52. Find the probability that both are aces.

Answer: Total ways = 52C2.
Favorable ways = 4C2 = 6.
P = 6/1326 = 1/221.

Question 17: A die is thrown twice. Find the probability that the sum is 8.

Answer: Total outcomes = 36.
Pairs giving sum 8: (2,6),(3,5),(4,4),(5,3),(6,2).
So 5 favorable outcomes.
P = 5/36.

Question 18: Find mean and variance of numbers 2, 4, 6, 8, 10.

Answer: Mean = (2+4+6+8+10)/5 = 30/5 = 6.
Deviations: (-4, -2, 0, 2, 4).
Squares: (16,4,0,4,16).
Variance = (16+4+0+4+16)/5 = 40/5 = 8.
Standard deviation = √8 = 2√2.

Question 19: Find conditional probability of drawing an ace given the card is a face card or ace.

Answer: Face cards = 12, aces = 4, total 16 favorable for condition. Probability of ace among these = 4/16 = 1/4.

Question 20: If A and B are independent events with P(A)=0.5, P(B)=0.6, find P(A ∩ B), P(A ∪ B), and P(A’ ∩ B’).

Answer: P(A ∩ B) = P(A)×P(B) = 0.5×0.6=0.3.
P(A ∪ B) = P(A)+P(B)-P(A ∩ B)=0.5+0.6-0.3=0.8.
P(A’ ∩ B’) = 1 - P(A ∪ B) = 1 - 0.8 = 0.2.

Chapter-Wise Topics of Class 12 Mathematics

Each chapter in CBSE Class 12 Mathematics plays a vital role in board exams and competitive entrance tests. Here’s an overview of the key topics covered in each chapter:

Relations and Functions

• Types of relations: Reflexive, symmetric, transitive
• Functions: One-to-one, onto, composition, domain, and range

Inverse Trigonometric Functions

• Properties and principal values of inverse trigonometric functions
• Proving identities and simplifications

Matrices

• Matrix operations: Addition, subtraction, multiplication, and inverses
• Applications in solving linear equations

Determinants

• Calculation and properties of determinants
• Solving linear equations using Cramer’s Rule

Continuity and Differentiability

• Continuity, differentiability, and higher-order derivatives
• Theorems: Rolle’s and Lagrange’s Mean Value Theorem

Applications of Derivatives

• Maxima, minima, increasing/decreasing functions
• Tangents and normals

Integrals

• Indefinite and definite integrals, substitution, partial fractions
• Integration by parts and area applications

Applications of Integrals

• Finding areas under curves and between lines using definite integrals

Differential Equations

• Formation, order, and degree of differential equations
• Solutions of first-order linear and homogeneous equations

Vector Algebra

• Operations: Addition, subtraction, dot product, and cross product
• Applications in geometry

Three-Dimensional Geometry

• Direction cosines, direction ratios, equations of lines and planes
• Finding angles and shortest distances

Linear Programming

• Formulating and solving linear programming problems graphically

Probability

• Conditional probability, Bayes’ theorem
• Random variables and binomial distribution

Mastering these topics with strategic revision and problem-solving practice ensures strong conceptual clarity and better exam performance.

Practise Question with Answers of Class 12 Maths

To simplify your preparation, here are five crucial questions from key topics, ensuring comprehensive understanding and problem-solving practice.

Question 1: Linear Programming

A company manufactures two products, P and Q. The profit per unit is ₹50 for P and ₹40 for Q. The total production is limited to 200 units, and Q cannot exceed 150 units.
Formulate and solve the linear programming problem to maximise profit.

Question 2: Probability

Two cards are drawn simultaneously from a well-shuffled deck of 52 cards.
Find the probability that both cards are aces.

Question 3: Functions and Inverse Functions

Let f be a strictly increasing function and g a strictly decreasing function, where the range of f and g match their respective codomains.
If (f ∘ g) is defined, will (f ∘ g) be invertible? Justify your answer.

Question 4: Relations and Functions

Prathibha Karanji is an innovative programme by the Government of Karnataka, where cultural and literary competitions are held between schools at cluster, block, district, and state levels.

One of these competitions, Yogasana, has two categories – Middle School and High School.
From a district, three middle school students and two high school students were selected for the state level, Let:

• M = {m₁, m₂, m₃} represent the middle school students.
• H = {h₁, h₂} represent the high school students.

Part (i): Relations

A relation R: M → M is defined as:
R = {(x, y) | x and y belong to the same category}
Show that R is an equivalence relation.

Part (ii): Functions

A function f: M → H is defined as:
f = {(m₁, h₁), (m₂, h₂), (m₃, h₂)}
Prove that f is onto but not one-to-one.

Question 5: Trigonometric Functions and Composite Functions

Given:

• f(x) = cos(2sin⁻¹ x)
• g(x) = sin²(2cos⁻¹ x), for -1 ≤ x ≤ 1

If h(x) = f(x) + g(x), find h(0.1) and show your work.

Why Are These Questions Important?

Every year, certain types of questions frequently appear in the board exams, making it crucial to prioritise these. Solving class 12 math's important questions sharpens your analytical skills, helps identify patterns in exam papers, and ensures comprehensive coverage of critical topics.

• Cover important board exam topics such as probability, relations, functions, trigonometry, and calculus.
• Help in understanding conceptual applications and problem-solving techniques.
• Prepare students for competitive exams like JEE, NDA, and CUET.

Practising these CBSE Class 12 Important Questions will boost confidence, strengthen understanding, and improve problem-solving skills for the board exams.

Why Solving Important Questions Matters?

• Enhances Time Management: Understanding the pattern of commonly asked questions helps manage time efficiently during exams.
• Strengthens Conceptual Understanding: Repeated exposure to important questions reinforces the application of formulas, theorems, and problem-solving techniques.
• Boosts Exam Confidence: Regular practice minimises exam anxiety by preparing students for unexpected or tricky questions.
• Improves Analytical Thinking: Solving diverse questions sharpens logical reasoning and analytical skills, essential for competitive exams.

What to Focus on While Solving Important Questions?

Mastering mathematics requires a balance of conceptual clarity and practice. Here’s what to keep in mind:

1. Understand Core Concepts

• Ensure a strong grasp of fundamental theories and formulas before attempting problems.
• Focus on key topics like Calculus, Probability, Matrices, and Algebra, which carry high weightage.

2. Prioritise NCERT and Exemplars

• The majority of board exam questions are based on NCERT textbooks and Exemplar problems.
• Revise solved examples and unsolved exercises thoroughly.

3. Solve Past Years’ Papers

• Identify question trends from previous years and practice them extensively.
• Familiarise yourself with different question types, difficulty levels, and frequently tested topics.

4. Attempt Higher-Order Thinking Skills (HOTS) Questions

• Practice complex problems that require logical reasoning and multi-step solutions.
• This helps in tackling difficult questions with confidence.

Benefits of Practicing Important Questions

Here are some of the benefits for better practising these important questions:

• Strengthens Problem-Solving Skills: Constant exposure to different problem types enhances accuracy and efficiency.
• Improves Time Management: Solving a variety of questions under timed conditions teaches efficient exam strategies.
• Increases Marks-Scoring Potential: Understanding which sections carry more weight allows students to focus on high-scoring topics.
• Reduces Exam Stress: When students practice important questions regularly, they feel more prepared and confident on exam day.

Tips to Score Well in Class 12 Maths Exams

Here are a few tips that can help you perform well in the CBSE Class 12 Math exam. 

• Follow a Structured Study Plan: Categorise chapters into easy, moderate, and difficult, and allocate time accordingly.
• Practice Time-Bound Tests: Regular mock tests improve speed, accuracy, and time management skills.
• Revise Weak Areas: Identify difficult topics and allocate extra time for improvement.
• Use Mnemonics for Formulas: Create memory aids, flashcards, or mind maps for quick recall of formulas and theorems.
• Stay Updated with Syllabus Changes: Ensure your study material aligns with the latest syllabus and marking scheme.

Mathematics requires consistent practice, strategic revision, and a clear understanding of concepts. By focusing on Class 12 Mathematics Important Questions and solving them regularly, students can strengthen their problem-solving skills and improve their exam performance.

Extra Questions for Practice!

Q1. If f(x) = x² + 2x + 1, find f⁻¹(x).

Q2. Solve cos(sin⁻¹(x)) = 1/2 for x.

Q3. Find determinant of |1 2 3; 2 3 4; 3 4 5|.

Q4. Find equation of tangent to y = eˣ at x = 0.

Q5. Evaluate ∫(x³ + 2x² + 3x + 4) dx.

Q6. Solve differential equation dy/dx = y - x.

Q7. Find angle between vectors 2i + j and i - j.

Q8. A line passes through (2,3,1) and (1,-1,2). Find its vector equation.

Q9. A manufacturer makes products A and B with profits ₹40 and ₹30 per unit, total units ≤ 150, and B ≤ 100. Formulate LPP to maximize profit.

Q10. From 3 red, 4 blue, and 5 green balls, 3 are drawn. Find the probability that all are of different colors.