UPSC NDA Mathematics 2026: Complete Guide with Previous Year Papers

Last Updated: March 2026

📋 Exam Overview

The UPSC NDA (National Defence Academy) examination is conducted by the Union Public Service Commission (UPSC) twice a year to select candidates for admission to the Army, Navy, and Air Force wings of the NDA and Indian Naval Academy (INA). The Mathematics paper is a crucial component of the written examination.

Conducting Body

Organization: Union Public Service Commission (UPSC)
Website: upsc.gov.in
Exam Level: National Level
Frequency: Twice a year (NDA I & NDA II)

Important Dates (Expected 2026 Schedule)

Event	NDA I 2026	NDA II 2026
Notification Release	December 2025	May 2026
Online Application Start	December 2025	May 2026
Last Date to Apply	January 2026	June 2026
Admit Card Release	March 2026	August 2026
Written Examination	April 2026	September 2026
SSB Interview	June-July 2026	October-November 2026
Final Result	November 2026	April 2027

🎓 Eligibility Criteria

Nationality

Citizen of India
Subject of Nepal/Bhutan
Tibetan refugee who came to India before 1962
Person of Indian origin migrated from Pakistan, Burma, Sri Lanka, or East African countries

Age Limit

Course	Born Between
NDA I 2026	2nd July 2006 to 1st July 2009
NDA II 2026	2nd January 2007 to 1st January 2010

Note: Only unmarried male/female candidates are eligible.

Educational Qualification

Wing	Educational Requirement
Army Wing	10+2 pass or appearing in any stream
Air Force & Naval Wings	10+2 pass or appearing with Physics and Mathematics

Note: Candidates appearing in Class 12 can apply provisionally.

Physical Standards

Height: Minimum 157 cm (varies by age and region)
Eye Sight: 6/6 and 6/9 for Air Force; 6/6 and 6/18 for Army/Navy
Freedom from diseases and disabilities

📊 Exam Pattern 2026

The NDA written examination consists of two papers:

Paper-I: Mathematics (Code No. 01)

Aspect	Details
Subject	Mathematics
Number of Questions	120
Total Marks	300
Duration	2½ hours (150 minutes)
Negative Marking	0.83 marks per wrong answer
Language	Hindi & English

Paper-II: General Ability Test (GAT) (Code No. 02)

Section	Questions	Marks
English	50	200
General Knowledge	100	400
Total	150	600
Duration	2½ hours	-
Negative Marking	1.33 marks per wrong answer	-

Overall Qualifying Marks:

Minimum 25% marks in each subject
Overall cutoff determined by UPSC (typically 350-370 out of 900)

SSB Interview

Stage I: Screening Test (Verbal/Non-Verbal tests, PPDT)
Stage II: Psychological Tests, Group Testing, Personal Interview
Total Marks: 900

📚 Mathematics Syllabus (Detailed)

1. Algebra (25-30 Questions)

Concept of set, operations on sets
Venn diagrams
De Morgan laws
Cartesian product, relation, equivalence relation
Representation of real numbers on a line
Complex numbers: modulus, argument, algebra
Binary system of numbers
Arithmetic, Geometric, Harmonic progressions
Quadratic equations: real coefficients
Permutation and Combination
Binomial theorem
Logarithms and applications

2. Matrices and Determinants (8-10 Questions)

Types of matrices
Operations on matrices
Determinant of a matrix
Adjoint and inverse of a square matrix
Applications: Solution of system of linear equations
Cramer's rule and Matrix Method

3. Trigonometry (10-12 Questions)

Angles and their measures
Trigonometrical ratios
Trigonometric identities
Sum and difference formulae
Multiple and sub-multiple angles
Inverse trigonometric functions
Applications: Height and distance, properties of triangles

4. Analytical Geometry of Two and Three Dimensions (15-18 Questions)

Rectangular Cartesian Coordinate system
Distance formula
Equation of a line in various forms
Angle between two lines
Distance of a point from a line
Equation of a circle
Standard forms of parabola, ellipse, hyperbola
Eccentricity and axis of a conic
3D geometry: Point, Distance, Direction cosines
Equation of a plane and a line
Angle between two lines and planes
Equation of a sphere

5. Differential Calculus (12-15 Questions)

Concept of a real-valued function
Domain, range, and graph
Composite functions
One-to-one, onto, and inverse functions
Notion of limit, standard limits
Continuity of functions
Derivative of a function
Geometrical and physical interpretation
Derivatives of sum, product, quotient
Second-order derivatives
Increasing and decreasing functions
Application: maxima and minima

6. Integral Calculus and Differential Equations (10-12 Questions)

Integration as inverse of differentiation
Standard integrals
Integration by substitution and by parts
Evaluation of definite integrals
Areas of plane regions
Definition of order and degree
Formation of differential equations
Solution of first order and first degree equations

7. Vector Algebra (8-10 Questions)

Vectors in two and three dimensions
Magnitude and direction
Unit and null vectors
Addition of vectors
Scalar multiplication
Scalar product/dot product
Vector product/cross product
Applications: work done, moment of force

8. Statistics and Probability (8-10 Questions)

Classification of data
Frequency distribution
Measures of central tendency
Variance and standard deviation
Correlation and regression
Random experiment
Outcomes and sample space
Events, probability
Conditional probability, Bayes' theorem
Random variable, binomial distribution

🧮 Mathematics Questions (15 with Solutions)

Question 1: Algebra - Complex Numbers

Q: If z = 1 + i, then |z² + 2/z| is equal to:

Solution: z = 1 + i z² = (1 + i)² = 1 + 2i + i² = 1 + 2i - 1 = 2i

2/z = 2/(1+i) = 2(1-i)/((1+i)(1-i)) = 2(1-i)/2 = 1-i

z² + 2/z = 2i + 1 - i = 1 + i |z² + 2/z| = |1 + i| = √(1² + 1²) = √2

Question 2: Algebra - Quadratic Equations

Q: If α and β are roots of x² - 2x + 4 = 0, then αⁿ + βⁿ = ?

Solution: α + β = 2, αβ = 4

Using recurrence: αⁿ + βⁿ = (α+β)(αⁿ⁻¹+βⁿ⁻¹) - αβ(αⁿ⁻²+βⁿ⁻²)

For n=1: α + β = 2 For n=2: α² + β² = (α+β)² - 2αβ = 4 - 8 = -4 For n=3: α³ + β³ = 2(-4) - 4(2) = -8 - 8 = -16

Question 3: Matrices

Q: If A = [[1, 2], [3, 4]], find A² - 5A.

Solution: A² = [[1, 2], [3, 4]] × [[1, 2], [3, 4]] = [[1+6, 2+8], [3+12, 6+16]] = [[7, 10], [15, 22]]

5A = [[5, 10], [15, 20]]

A² - 5A = [[7-5, 10-10], [15-15, 22-20]] = [[2, 0], [0, 2]] = 2I

Question 4: Determinants

Q: If Δ = |1 a b+c| |1 b c+a| |1 c a+b|, then Δ equals:

Take (a+b+c) common from C₃ = (a+b+c) |1 a 1| |1 b 1| |1 c 1|

Since C₁ = C₃, Δ = 0

Question 5: Trigonometry

Q: If sin θ = 3/5 and θ is in first quadrant, find cos 2θ.

Solution: cos 2θ = 1 - 2sin²θ = 1 - 2 × (9/25) = 1 - 18/25 = 7/25

Shortcut: Use cos 2θ formula directly Answer: 7/25

Question 6: Trigonometry - Heights & Distances

Q: From a point on the ground, the angles of elevation of the bottom and top of a tower fixed at the top of a 20m high building are 45° and 60° respectively. Find the height of the tower.

Solution: Let distance from point to building = d Let tower height = h

For bottom (building): tan 45° = 20/d So d = 20m

For top: tan 60° = (20+h)/20 √3 = (20+h)/20 20√3 = 20 + h h = 20(√3 - 1) = 20(1.732 - 1) = 20 × 0.732 = 14.64m

Question 7: Coordinate Geometry

Q: Find the equation of the line passing through (2, 3) and parallel to 3x - 4y + 5 = 0.

Solution: Slope of given line: 3x - 4y + 5 = 0 y = (3/4)x + 5/4 Slope m = 3/4

Parallel line has same slope = 3/4 Equation: y - 3 = (3/4)(x - 2) 4(y - 3) = 3(x - 2) 4y - 12 = 3x - 6 3x - 4y + 6 = 0

Question 8: 3D Geometry

Q: Find the distance between the points P(1, 2, 3) and Q(-1, 4, -2).

Solution: Distance = √[(x₂-x₁)² + (y₂-y₁)² + (z₂-z₁)²] = √[(-1-1)² + (4-2)² + (-2-3)²] = √[(-2)² + (2)² + (-5)²] = √[4 + 4 + 25] = √33

Question 9: Differential Calculus

Q: If y = log(tan x), find dy/dx.

Solution: y = log(tan x) dy/dx = (1/tan x) × sec²x = cos x/sin x × 1/cos²x = 1/(sin x cos x) = 2/sin 2x = 2 cosec 2x

Question 10: Differential Calculus - Maxima Minima

Q: Find the maximum value of f(x) = 3x⁴ - 8x³ + 12x² - 48x + 25 in [0, 3].

Solution: f'(x) = 12x³ - 24x² + 24x - 48 = 12(x³ - 2x² + 2x - 4) = 12[x²(x-2) + 2(x-2)] = 12(x-2)(x²+2)

f'(x) = 0 when x = 2

f(0) = 25 f(2) = 3(16) - 8(8) + 12(4) - 48(2) + 25 = 48 - 64 + 48 - 96 + 25 = -39 f(3) = 3(81) - 8(27) + 12(9) - 48(3) + 25 = 243 - 216 + 108 - 144 + 25 = 16

Maximum value = 25 at x = 0

Question 11: Integral Calculus

Q: Evaluate ∫(1/(1 + sin x)) dx

Solution: ∫(1/(1 + sin x)) dx Multiply numerator and denominator by (1 - sin x): = ∫((1 - sin x)/(1 - sin²x)) dx = ∫((1 - sin x)/cos²x) dx = ∫(sec²x - sec x tan x) dx = tan x - sec x + C

Question 12: Definite Integrals

Q: Evaluate ∫₀^(π/2) (sin x/(sin x + cos x)) dx

Solution: Let I = ∫₀^(π/2) (sin x/(sin x + cos x)) dx

Using property ∫₀^a f(x) dx = ∫₀^a f(a-x) dx: I = ∫₀^(π/2) (cos x/(cos x + sin x)) dx

Adding both: 2I = ∫₀^(π/2) ((sin x + cos x)/(sin x + cos x)) dx 2I = ∫₀^(π/2) 1 dx 2I = π/2 I = π/4

Question 13: Differential Equations

Q: Solve: dy/dx + y = e^(-x)

Solution: This is a linear differential equation. Integrating factor = e^(∫1 dx) = e^x

Solution: y·e^x = ∫e^(-x)·e^x dx + C y·e^x = ∫1 dx + C y·e^x = x + C y = xe^(-x) + Ce^(-x)

Question 14: Vector Algebra

Q: If |a| = 3, |b| = 4, and a·b = 6, find the angle between a and b.

Solution: cos θ = (a·b)/(|a||b|) = 6/(3 × 4) = 6/12 = 1/2

θ = 60° or π/3

Question 15: Probability

Q: A die is thrown twice. What is the probability of getting a sum of 7?

Solution: Total outcomes = 6 × 6 = 36 Favorable: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1) = 6

P(sum = 7) = 6/36 = 1/6

🧩 General Knowledge Questions (10 with Answers)

No.	Question	Answer
1	Who is the Supreme Commander of the Indian Armed Forces?	President of India
2	Where is the National Defence Academy (NDA) located?	Khadakwasla, Pune, Maharashtra
3	Which is the highest gallantry award in India?	Param Vir Chakra
4	In which year was NDA established?	1954
5	Who is the Chief of Defence Staff (2026)?	General Anil Chauhan
6	What is the motto of the Indian Army?	Service Before Self
7	How many years of training at NDA?	3 years
8	Which service has the highest number of cadets at NDA?	Indian Army
9	What is the full form of INA?	Indian Naval Academy
10	Where is the Indian Military Academy (IMA) located?	Dehradun, Uttarakhand

📝 English Questions (5 with Solutions)

Question 1: Error Detection

Q: The teacher (a) / together with his students (b) / are going on a trip. (c) / No error (d)

Solution: "Together with" creates a parenthetical phrase. The subject is "teacher" (singular), so verb should be "is" not "are".

Question 2: Synonyms

Q: Choose the word similar in meaning to COURAGE: (a) Fear (b) Bravery (c) Cowardice (d) Anxiety

Solution: Courage = Bravery

Question 3: Antonyms

Q: Choose the word opposite in meaning to OBSTINATE: (a) Stubborn (b) Compliant (c) Adamant (d) Resolute

Solution: Obstinate = stubborn, unyielding Opposite = Compliant (willing to comply)

Question 4: Idioms

Q: "To burn the midnight oil" means: (a) To waste time (b) To study/work late into the night (c) To burn oil at midnight (d) To save fuel

Solution: To work or study late into the night.

Question 5: Fill in the Blanks

Q: He is _____ honest man. (a) a (b) an (c) the (d) no article

Solution: "Honest" starts with a vowel sound, so use "an".

📈 Previous Year Cutoff Marks

NDA I 2024 Written Exam Cutoff

Category	Minimum Marks (out of 900)
General	360 (approx)
OBC	355 (approx)
SC	280 (approx)
ST	270 (approx)

NDA II 2024 Written Exam Cutoff

Category	Minimum Marks (out of 900)
General	375 (approx)
OBC	370 (approx)
SC	290 (approx)
ST	280 (approx)

NDA 2026 Expected Cutoff

Category	Expected Minimum Marks
General	360-380
OBC	355-375
SC	280-300
ST	270-290

Final Selection (Written + SSB):

The final merit is based on combined marks (Written: 900 + SSB: 900 = 1800)
Final cutoff typically ranges from 700-800 out of 1800

📅 3-Month Preparation Strategy

Month 1: Foundation Building

Week 1-2: Algebra & Trigonometry

Complex numbers and quadratic equations
Progressions and series
Trigonometric identities and equations
Height and distance problems

Week 3-4: Matrices & Coordinate Geometry

Matrix operations and determinants
2D coordinate geometry
Straight lines and circles
Conic sections basics

Daily Practice: 50 Mathematics MCQs

Month 2: Advanced Topics

Week 1-2: Calculus

Limits and continuity
Differentiation techniques
Application of derivatives
Integration methods

Week 3-4: 3D Geometry & Vectors

Three-dimensional coordinate system
Direction cosines and ratios
Plane and lines in 3D
Vector algebra and applications

Daily Practice: 1 full Mathematics paper (120 questions in 150 minutes)

Month 3: Revision & Mock Tests

Week 1-2: Statistics, Probability & Differential Equations

Measures of central tendency
Probability concepts
Differential equations
Previous year papers (2019-2024)

Week 3-4: Final Revision

Full-length mock tests
Formula revision
Weak area focus
Time management practice

Daily Schedule:

6:00 AM - 8:00 AM: Mathematics (Theory)
10:00 AM - 1:00 PM: Mathematics Practice
3:00 PM - 5:00 PM: GAT (English/GK)
7:00 PM - 9:00 PM: Mock Test/Revision
9:00 PM - 10:00 PM: Physical Exercise

Physical Fitness:

Running: 2.4 km in 12 minutes (target)
Push-ups: Minimum 20
Sit-ups: Minimum 30
Chin-ups: Minimum 6

📚 Best Books and Resources

Mathematics

Book	Author	Publisher
Mathematics for NDA and NA	R.S. Aggarwal	S. Chand
Pathfinder NDA/NA	-	Arihant
NDA Mathematics	-	Disha Publication
NCERT Class 11-12 Mathematics	NCERT	NCERT

General Ability Test (GAT)

Book	Author	Publisher
General Knowledge 2026	-	Lucent/Arihant
Objective General English	S.P. Bakshi	Arihant
Word Power Made Easy	Norman Lewis	-
Manorama Year Book 2026	-	Malayala Manorama

Previous Year Papers

Book	Publisher
NDA/NA 10 Years Solved Papers	Arihant
NDA Chapter-wise Solved Papers	Disha

Online Resources

Official: upsc.gov.in
Mock Tests: Testbook, Gradeup, Unacademy
Current Affairs: GK Today, Insights on India
YouTube: Unacademy NDA, Study IQ, Major Kalshi Classes

❓ Frequently Asked Questions (FAQs)

Q1: What is the difficulty level of NDA Mathematics?

A: The Mathematics paper is of Class 11-12 level (NCERT syllabus). It requires conceptual clarity and speed. The questions are application-based and sometimes tricky.

Q2: Is there negative marking in NDA exam?

A: Yes, there is negative marking:

Mathematics: 0.83 marks deducted for each wrong answer
GAT: 1.33 marks deducted for each wrong answer

Q3: Can girls apply for NDA?

A: Yes, since 2021, female candidates are also eligible to apply for NDA.

Q4: How many hours should I study daily for NDA?

A: Ideally, 6-8 hours of focused study daily along with 1-2 hours of physical exercise is recommended for 3-4 months.

Q5: What is the best strategy to clear NDA in the first attempt?

Complete NCERT 11-12 thoroughly
Practice previous year papers (minimum 10 years)
Focus on time management
Maintain physical fitness
Read newspapers for current affairs
Take regular mock tests

🎯 Final Tips for Success

NCERT is Bible: Complete NCERT Class 11-12 Mathematics thoroughly
Formula Mastery: Maintain a formula notebook and revise daily
Speed with Accuracy: Practice solving 120 questions in 150 minutes
Previous Papers: Solve at least 10 years of previous papers
Mock Tests: Take 20+ full-length mocks before the exam
GAT Preparation: Don't neglect English and GK; they carry 600 marks
Physical Fitness: Start preparing for SSB physical tests early
Current Affairs: Read newspapers daily and maintain notes
SSB Preparation: Start preparing for psychology tests and group discussions
Stay Positive: Believe in yourself and stay consistent

Best wishes for your NDA 2026 preparation!

Jai Hind!

Last Updated: March 2026