12th Maths Important Questions: Chapter-wise with Answers (Samacheer Kalvi)

Exam pattern and marks split#

The Maths paper in the Tamil Nadu HSC (+2) public exam is a 90-mark theory paper plus 10 marks of internal assessment awarded by the school. The pattern in the answer box is the one used in recent public question papers; the March 2026 Maths exam was held on 17 March 2026. The March 2027 HSC Maths public exam is scheduled for 29 March 2027. Check the latest DGE model paper before your exam in case of changes.

How to read the split:

• Part IV carries 35 marks, the largest share. Every question has an A or B choice, so for each chapter you should be fully ready with at least one long-answer type.
• Parts II and III together carry 35 marks. You choose 7 of 10 in each, but Q.30 and Q.40 are compulsory, so a small part of every chapter matters.
• Part I (20 marks) spans the whole syllabus and often tests a property or a short calculation rather than a long method.

The Samacheer Kalvi 12th Maths textbook is in two volumes. No new textbook has been issued for Class 12 in 2026-27 (new books came only for Classes 1 to 3); state officials have said revised books for Classes 4 to 12 are planned for 2027-28.

12th maths important questions: Volume 1#

These are our own textbook-style questions with answers, not copies from any guide or past paper. In the exam, show every step.

Chapter 1: Applications of Matrices and Determinants

1. (2 marks) Find the inverse of the matrix with rows (2, 3) and (1, 2). Answer: determinant = 4 − 3 = 1, so the inverse is the adjoint: rows (2, −3) and (−1, 2).
2. (3 marks) Find the rank of the matrix with rows (1, 2, 3), (2, 4, 6) and (1, 1, 1). Answer: the second row is twice the first, and rows 1 and 3 are independent, so the rank is 2.
3. (3 marks) Solve 2x + y = 5, x − y = 1 by Cramer's rule. Answer: Δ = −3, Δx = −6, Δy = −3, so x = 2 and y = 1.
4. (5 marks) Solve x + 2y + z = 7, x + 3z = 11, 2x − 3y = 1 by the matrix inversion method or Gaussian elimination. Answer: x = 2, y = 1, z = 3. Substitute back into all three equations to check.

Also practise: consistency of a system using the rank method, and properties of the adjoint.

Chapter 2: Complex Numbers

1. (2 marks) Simplify i¹⁹⁸³. Answer: 1983 leaves remainder 3 when divided by 4, so i¹⁹⁸³ = i³ = −i.
2. (2 marks) Find the modulus and principal argument of 1 + i√3. Answer: modulus 2, argument π/3.
3. (3 marks) Show that (1 + i)⁴ = −4. Answer: (1 + i)² = 2i, and (2i)² = −4.
4. (3 marks) If ω is a complex cube root of unity, find (1 + ω − ω²)(1 − ω + ω²). Answer: since 1 + ω + ω² = 0, the factors are −2ω² and −2ω; the product is 4ω³ = 4.
5. (5 marks) Use De Moivre's theorem to find (cos π/6 + i sin π/6)⁶. Answer: cos π + i sin π = −1. Also practise finding the nth roots of a complex number and the locus of z from a modulus condition (for example, the modulus of z − 2 equal to 3 is a circle with centre (2, 0) and radius 3).

Chapter 3: Theory of Equations

1. (2 marks) Form the cubic equation whose roots are 1, 2 and 3. Answer: x³ − 6x² + 11x − 6 = 0.
2. (2 marks) If 2 + i is a root of a quadratic with real coefficients, find the equation. Answer: the other root is 2 − i, so x² − 4x + 5 = 0.
3. (3 marks) If α, β, γ are the roots of x³ − 2x² + 3x − 4 = 0, find α² + β² + γ². Answer: (Σα)² − 2Σαβ = 4 − 6 = −2.
4. (3 marks) Show that x⁴ + 3x² + 5 = 0 has no real roots. Answer: the left side is always at least 5, so it can never be zero (Descartes' rule of signs gives the same result).
5. (5 marks) Solve x³ − 9x² + 26x − 24 = 0 given that the roots are in AP. Answer: take a − d, a, a + d; 3a = 9 gives a = 3; the product 3(9 − d²) = 24 gives d = ±1. The roots are 2, 3, 4.

Chapter 4: Inverse Trigonometric Functions

1. (2 marks) Find the principal values of sin⁻¹(−1/2), cos⁻¹(−1/2) and tan⁻¹(−√3). Answer: −π/6, 2π/3 and −π/3.
2. (2 marks) Find the domain of sin⁻¹(2x − 1). Answer: −1 ≤ 2x − 1 ≤ 1, so 0 ≤ x ≤ 1.
3. (3 marks) Find sin(cos⁻¹(3/5)). Answer: 4/5.
4. (5 marks) Prove that tan⁻¹(1/2) + tan⁻¹(1/3) = π/4. Answer: (1/2 + 1/3) ÷ (1 − 1/6) = 1, and the product 1/6 is less than 1, so the sum is tan⁻¹1 = π/4.

Chapter 5: Two Dimensional Analytical Geometry-II

1. (2 marks) Find the centre and radius of x² + y² − 6x + 8y − 11 = 0. Answer: centre (3, −4), radius √(9 + 16 + 11) = 6.
2. (2 marks) Find the focus, directrix and length of latus rectum of y² = 12x. Answer: focus (3, 0), directrix x = −3, latus rectum 12.
3. (3 marks) Find the eccentricity and foci of x²/25 + y²/9 = 1. Answer: c² = 25 − 9 = 16, so e = 4/5 and the foci are (±4, 0).
4. (3 marks) Find the eccentricity of x²/16 − y²/9 = 1. Answer: c² = 16 + 9 = 25, so e = 5/4.
5. (5 marks) A parabolic arch is 20 m wide at the base and 5 m high at the centre. Find its height 4 m from the centre. Answer: with the vertex at the origin, x² = −20y; at x = 4, y = −0.8, so the height is 5 − 0.8 = 4.2 m.

Chapter 6: Applications of Vector Algebra

1. (3 marks) Find the volume of the parallelepiped with edges a = (1, 2, 3), b = (2, −1, 1) and c = (3, 1, −2). Answer: the scalar triple product is 1(2 − 1) − 2(−4 − 3) + 3(2 + 3) = 30, so the volume is 30 cubic units.
2. (3 marks) Find the distance of the point (1, 2, 3) from the plane 2x − y + 2z + 3 = 0. Answer: (2 − 2 + 6 + 3) ÷ √(4 + 1 + 4) = 9 ÷ 3 = 3 units.
3. (5 marks) Practise: the vector and Cartesian equations of a plane through three points, the shortest distance between two skew lines, and the angle between a line and a plane. A scalar triple product of zero means the three vectors are coplanar.

12th maths important questions: Volume 2#

Chapter 7: Applications of Differential Calculus

1. (2 marks) Verify the mean value theorem for f(x) = x² on [1, 3]. Answer: f′(c) = 2c must equal (9 − 1) ÷ 2 = 4, so c = 2, which lies in (1, 3).
2. (3 marks) Find the equations of the tangent and normal to y = x² at (2, 4). Answer: slope 4; tangent y = 4x − 4; normal x + 4y − 18 = 0.
3. (3 marks) Evaluate the limit of (1 − cos x)/x² as x → 0 using L'Hôpital's rule. Answer: 1/2.
4. (5 marks) Find the local maximum and minimum of f(x) = x³ − 3x + 2. Answer: f′(x) = 3x² − 3 = 0 at x = ±1; local maximum f(−1) = 4, local minimum f(1) = 0.
5. (5 marks) Show that among rectangles of perimeter 40 cm, the square has the greatest area. Answer: area = x(20 − x) is greatest at x = 10, giving a 10 cm square of area 100 cm².

Chapter 8: Differentials and Partial Derivatives

1. (2 marks) Use linear approximation to find √36.1. Answer: 6 + 0.1 × 1/12 ≈ 6.0083.
2. (2 marks) If y = x³, find dy when x = 2 and dx = 0.1. Answer: dy = 3x² dx = 1.2.
3. (3 marks) If u = x²y + 3xy², find ∂u/∂x and ∂u/∂y. Answer: 2xy + 3y² and x² + 6xy.
4. (5 marks) Verify Euler's theorem for f(x, y) = x³ + y³. Answer: f is homogeneous of degree 3, and x(3x²) + y(3y²) = 3(x³ + y³) = 3f.

Chapter 9: Applications of Integration

1. (2 marks) Evaluate the integral of x³ from −1 to 1. Answer: 0, because x³ is an odd function.
2. (3 marks) Evaluate the integral of x eˣ from 0 to 1. Answer: [x eˣ − eˣ] from 0 to 1 = 0 − (−1) = 1.
3. (3 marks) Evaluate the integral of sin⁴x from 0 to π/2. Answer: (3/4)(1/2)(π/2) = 3π/16.
4. (5 marks) Find the area enclosed by y = x² and y = x. Answer: the integral of (x − x²) from 0 to 1 = 1/2 − 1/3 = 1/6 sq. units.
5. (5 marks) Find the volume generated when the region under y = x from x = 0 to x = 2 is rotated about the x-axis. Answer: π × (integral of x² from 0 to 2) = 8π/3 cubic units, the volume of a cone of radius 2 and height 2.

Chapter 10: Ordinary Differential Equations

1. (2 marks) Find the order and degree of (d²y/dx²)³ + (dy/dx)² + y = 0. Answer: order 2, degree 3.
2. (2 marks) Form the differential equation of y = Ae²ˣ + Be⁻²ˣ. Answer: y″ − 4y = 0.
3. (3 marks) Solve dy/dx = y/x. Answer: separating variables, y = Cx.
4. (5 marks) Solve dy/dx + y = eˣ. Answer: integrating factor eˣ gives y eˣ = e²ˣ/2 + C, so y = eˣ/2 + Ce⁻ˣ.
5. (5 marks) A population grows at a rate proportional to its size and doubles in 50 years. In how many years will it triple? Answer: t = 50 × ln 3/ln 2 ≈ 79.2 years.

Chapter 11: Probability Distributions

1. (2 marks) If P(X = x) = cx for x = 1, 2, 3, 4, find c. Answer: c(1 + 2 + 3 + 4) = 1, so c = 1/10.
2. (3 marks) For the same distribution, find the mean. Answer: Σx × cx = (1 + 4 + 9 + 16)/10 = 3.
3. (3 marks) A fair coin is tossed 4 times. Find the probability of exactly 2 heads, and the mean and variance of the number of heads. Answer: ⁴C₂(1/2)⁴ = 3/8; mean np = 2; variance npq = 1.
4. (5 marks) If f(x) = kx² for 0 ≤ x ≤ 3 (and 0 elsewhere) is a probability density function, find k and P(X ≤ 1). Answer: k × 9 = 1, so k = 1/9; P(X ≤ 1) = (1/9)(1/3) = 1/27.

Chapter 12: Discrete Mathematics

1. (2 marks) On the set of integers, a ∗ b = a + b − 1. Find the identity element and the inverse of a. Answer: a + e − 1 = a gives e = 1; a + a′ − 1 = 1 gives a′ = 2 − a.
2. (2 marks) Find the inverse of 3 in Z₇ under multiplication modulo 7. Answer: 3 × 5 = 15 ≡ 1, so the inverse is 5.
3. (3 marks) Using a truth table, show that (p ∧ q) → p is a tautology.
4. (5 marks) Verify De Morgan's law ¬(p ∨ q) ≡ ¬p ∧ ¬q with a truth table, and check whether a ∗ b = a + b − 1 is commutative and associative on the integers (both are true).

Repeated questions#

DGE does not publish lists of repeated questions or chapter weightage. In practice, most public exam questions are built on the textbook's worked examples and exercise problems, sometimes with changed numbers, and the same kinds of long answers return year after year: solving systems of equations, De Moivre's theorem and roots, conics in real-life settings, vector equations of planes and lines, maxima and minima, area and volume by integration, linear differential equations and probability density functions.

So treat the textbook as your 12th samacheer kalvi maths guide: work through every example and exercise yourself, and use a solutions book only to check. Past public papers, explained in our guide to the 12th public question paper, show how textbook ideas are framed in the exam.

How to practise#

1. Volume 1 first, then Volume 2, but revise both every week so the early chapters are not forgotten by March.
2. Prepare the A and B option for every Part IV question type. Since all seven 5-mark questions must be answered, being comfortable with both options gives you a real choice.
3. Make a theorem and formula sheet for conics, vectors and integration properties.
4. Write a full 3-hour paper every week from January, and mark it against the 20 + 14 + 21 + 35 split.
5. Practise the compulsory-question style. Q.30 and Q.40 can come from any chapter, so keep short answers from all 12 chapters fresh.
6. Link Maths with Physics. Calculus and vectors are used throughout our 12th physics guide, so revising them together saves time.
7. Get doubts cleared early. If a chapter still feels shaky, join live classes for Class 12 Maths with a teacher rather than leaving it for the last month.

When the exam is over, our page on the TN 12th result explains how to check marks and what to do next. For all Tamil Nadu State Board subjects, see the Samacheer Kalvi hub.