Master NDA Mathematics: The Ultimate Collection of Past Papers

🎯 Prime Maths Academy

Master NDA Mathematics | Jai Hind!
The Ultimate Collection of Past Papers

Clearing the NDA Mathematics paper is never an accident — it is the result of focused, intelligent, and strategic practice. Transition from board exams to the competitive rigor of UPSC NDA with our meticulously curated archive of authentic past papers.

120+

Questions per Paper

2.5

Hours Time Limit

10+

Years of Papers

8

Core Topic Areas

Explore the Archive →

"As an educator, I have seen firsthand that clearing the NDA Mathematics paper is never an accident — it is the result of focused, intelligent, and strategic practice. The transition from standard Class XI and XII board exams to the competitive rigor of the UPSC NDA examination requires a massive shift from lengthy, step-by-step solving to rapid, objective, and analytical thinking."

Vinod Singh — Prime Maths

📚 The Archive

What You Will Find in Our NDA Mathematics Archive

Our dedicated repository is structured to streamline your revision process. Here's what makes this resource an essential part of your study toolkit.

📅

Year-Wise Collections

Access a meticulously organized vault of past NDA mathematics question papers, including recent Phase I and Phase II exams. Every paper is preserved in its original format for authentic practice.

🎯

Topic-Wise Segregation

Beyond full papers, dive into content categorized by crucial sections. This allows you to aggressively target your weakest areas and build mastery topic by topic.

Sets & Relations Algebra Trigonometry Calculus 2D Geometry 3D Geometry Probability Statistics

📋

Authentic Format

Every problem is presented exactly as it appeared on the official UPSC examinations, giving you a true sense of the required academic rigor and complexity demanded by the examiners.

⚡

Targeted Practice

Practicing with these structured materials helps you decode question patterns and understand exactly what the examiners demand in terms of conceptual clarity and shortcut application.

⚔️ Know Your Battlefield

The NDA Mathematics paper is notorious for its brutal time constraint. Familiarize yourself with the exact structure before you step in.

120

Questions

2.5 hrs

Duration

-0.83

Negative Marking

🧠 Core Topics

Master Every Critical Area

Our archive covers every topic that appears in the NDA Mathematics syllabus. Focus your preparation on high-weightage areas first.

Sets, Relations & Functions

High Weightage

Algebra

Very High Weightage

Trigonometry

High Weightage

Calculus

Very High Weightage

2D Geometry

Medium Weightage

3D Geometry

Medium Weightage

Probability

High Weightage

Statistics

Medium Weightage

🏆 Why Past Papers Work

The Real Benefits of Solving Past Papers

Dedicating consistent time to this archive will fundamentally transform your approach to the subject. Here's how.

01

🧘 Eliminate Exam Anxiety

The unknown is the biggest source of stress. By repeatedly exposing yourself to the authentic formatting and phrasing of NDA questions, the final UPSC exam will feel like just another practice session. Familiarity breeds confidence.

02

⏱️ Master Time Management & Accuracy

The NDA Mathematics paper is notorious for its brutal time constraint — 120 questions in just 2.5 hours, coupled with strict negative marking. Setting a timer while tackling these past papers will train your brain to optimize speed, teaching you exactly which questions to attempt immediately and which time-traps to skip.

03

🔍 Identify Recurring Concepts

While the exact numbers change, the core concepts and specific application tricks tested repeat year after year. Working through this collection will quickly reveal the high-weightage topics and recurring models you absolutely cannot afford to miss on exam day.

Your Journey to NDA Success Starts Here

Join thousands of aspirants who have transformed their preparation with authentic past papers. The path to mastering NDA Mathematics is not about luck — it's about practice, strategy, and consistency.

Start Practicing Today →

📍 Prime Maths Academy — Empowering Future Officers

Relations and Functions Advanced Problems for JEE Mains IIT Advanced Indian Statistical Institute and WBJEE

Prime Maths

25 Multiple Choice Questions (MCQs) on Relations and Functions for students of class XI and XII preparing for board examinations or JEE Mains, IIT Advanced, Indian Statistical Institute (B. Math & B.Stat) WBJEE or any other competitive entrance examination.

👨‍🏫 Author: Singh

📞 WA: +91-9038126497

Prime Maths

Advanced Algebra - Relations and Functions

Test your understanding of core concepts.

0

Done

25

Left

0%

Score

📝 Instructions

• This quiz contains 25 multiple choice questions.
• Select only one correct answer per question.
• Use the navigator to jump between questions.
• Submit when you are finished to see results.

Question Navigator

Progress 0%

Question 1

The number of reflexive relations on a set \(A\) of \(n\) elements is equal to

A\( 2^{n^2}\) B\( n^2\) C\( 2^{n(n-1)} \) D\( n^2-n\)

For the relation to be reflexive, \((a_i,a_i) \in A \times A \quad \forall a_i \in A\) So the from the remaining \(n^2 - n\) elements, any number of elements can be choosen to form a subset of \(A \times A\)

Question 2

Let \(X\) be a non-void set. If \(\rho_1\) and \(\rho_2\) be the transitive relations on \(X\), then [ \( \circ\) denotes composition ]

A \(\rho_1 \circ \rho_2\) is transitive relation B \(\rho_1 \circ \rho_2\) is not transitive relation C \(\rho_1 \circ \rho_2\) is equivalence relation D \(\rho_1 \circ \rho_2\) is not any relation on \(X\)

Let's define a set \(X = \{1, 2, 3, 4, 5\}\). Now, let's create two relations on \(X\), \(\rho_1\) and \(\rho_2\), both of which are transitive:Let \(\rho_1 = \{(1, 2), (3, 4)\}\) Let \(\rho_2 = \{(2, 3), (4, 5)\}\) (Note: These are trivially transitive). Now, let's find the composition \(\rho_1 \circ \rho_2\). By definition of composition, \((x, z) \in \rho_1 \circ \rho_2\) if there is some intermediate element \(y\) such that \((x, y) \in \rho_1\) and \((y, z) \in \rho_2\). So, the resulting composition is:\(\rho_1 \circ \rho_2 = \{(1, 3), (3, 5)\}\) which is clearly not transitive.

Question 3

If one root of \(x^2 + px - q^2 = 0\), \(p\) and \(q\) are real, be less than \(2\) and other be greater than \(2\), then

A \( 4+2p+q^2 > 0 \) B \( 4+2p+q^2 \\< 0 \) C \( 4+2p-q^2 > 0 \) D \( 4+2p-q^2 \\< 0 \)

Let \(y=x^2+px-q^2\) as \(x \rightarrow \pm \infty, y \rightarrow \infty\) thus the resulting parabola is open in the positive direction of \(y\)-axis. As \(2\), lies between the roots, \(y\) is less than \(0\) at \(x=2\).

Question 4

If \(R\) and \(Q\) are equivalence relations on set \(A\), then which of the following is not an equivalence relation

A\(R^{-1} \cap Q^{-1}\) B\(Q^{-1}\) C\(R \cup Q\) D\(R \cap Q\)

Let \( A=\{1,2,3\}, P =\{(1,1),(2,2),(3,3),(2,3),(3,2)\}\) and \(Q =\{(1,1),(2,2),(3,3),(1,2),(2,1)\}\). It is easy to see that both \(P \text{ and } Q\) are equivalence relation on \(A\) but \(P \cup Q\) is not an equivalence relation.

Question 5

Let \( \rho \) be a relation defined on set of natural numbers \( \mathbb{N} \), as \( \rho = \{(x, y) \in \mathbb{N} \times \mathbb{N} : 2x + y = 41\} \). Then domain A and range B are

A\( A \subset \{x \in \mathbb{N} : 1 \le x \le 20\} \) and \( B \subset \{y \in \mathbb{N} : 1 \le y \le 39\} \) B\( A = \{x \in \mathbb{N} : 1 \le x \le 15\} \) and \( B = \{y \in \mathbb{N} : 2 \le y \le 30\} \) C\( A \equiv \mathbb{N}, B \equiv \mathbb{Q} \) D\( A \equiv \mathbb{Q}, B \equiv \mathbb{Q} \)

See that \(y= 41 - 2x\) and for \(x \in \{1,2,...,20 \}\), \( y \in \{1,3,5,...,39\}\)

Question 6

In \(\mathbb{R}\), a relation \(\rho\) is defined as follows: \(\forall a,b \in \mathbb{R}\), \(a\rho b \quad \text{holds iff} \quad a^2 -4ab+3b^2=0 \)

A\(\rho\) is equivalence relation B\(\rho\) is only symmetric C\(\rho\) is only reflexive D\(\rho\) is only transitive

See that \(a^2-4ab+3b^2 =0 \implies a=b\) or \(a=3b\)

Question 7

Let \( f : \mathbb{R} \to \mathbb{R} \) be a function defined by \( f(x) = \frac{e^{|x|} - e^{-x}}{e^x + e^{-x}} \), then

A \( f \) is both one-one and onto B \( f \) is one-one but not onto C \( f \) is onto but not one-one D \( f \) is neither one-one nor onto

\Injectivity (One-One): For \( x < 0 \), \( |x| = -x \). $$ f(x) = \frac{e^{-x} - e^{-x}}{e^x + e^{-x}} = 0 $$ Since \( f(-1) = f(-2) = 0 \), multiple inputs yield the same output. Therefore, \( f \) is not one-one. 2. Surjectivity (Onto): For \( x \ge 0 \), \( |x| = x \). $$ f(x) = \frac{e^x - e^{-x}}{e^x + e^{-x}} $$ The range for \( x \ge 0 \) is \( [0, 1) \) ( use limits), the total range of \( f \) is \( [0, 1) \). Since the range \( [0, 1) \neq \) codomain \( \mathbb{R} \), \( f \) is not onto.

Question 8

Let \( f : \mathbb{R} \to \mathbb{R} \) be a function defined by \( f(x) = 2026x^3+2024x+2023 \), then

A \( f \) is both one-one and onto B \( f \) is one-one but not onto C \( f \) is onto but not one-one D \( f \) is neither one-one nor onto

It is easy to see that \(f\) is injective. For surjectivity, think about the nature of roots of a cubic equation.

Question 9

Let \(u + v + w = 3\), \(u, v, w \in \mathbb{R}\) and \(f(x) = ux^2 + vx + w\) be such that \(f(x + y) = f(x) + f(y) + xy\), \(\forall x, y \in \mathbb{R}\). Then \(v\) is equal to

A\(\frac{5}{2}\) B\(\frac{1}{2}\) C\(\frac{1}{\sqrt{2}}\) D\(3\)

Start by substituting \(f(x) = ux^2 + vx + w\) directly into the given equation \(f(x + y) = f(x) + f(y) + xy\). Expand the left side, and then compare the coefficients of the \(xy\) terms and the constant terms on both sides. This will reveal the values of \(u\) and \(w\) first, which you can then use with the condition \(u + v + w = 3\) to find \(v\).

Question 10

Let \(X\) and \(Y\) be two non-empty sets. Let \(f:X \rightarrow Y\) be a function. For \(A \subset X\) and \(B \subset Y\) , define \(f(A)=\{f(x):x \in A\}\) and \(f^{-1}(B)=\{x \in X : f(x) \in B \}\), then

A\(f(f^{-1}(B))=B\) B\(f(f^{-1}(B)) \subset B\) C\(f^{-1}(f(A))=A\) D\(f^{-1}(f(A)) \subset A\)

Collection of all pre-images will be mapped into \(B\). Not every element in \(Y\) has a pre-image. Additional exercise, what will happen if the function is surfective?

Question 11

Range of the function \(f(x)= \frac{x^2+x+2}{x^2+x+1}\ \: x \in \mathbb{R}\)

A\((1,\infty)\) B\((1, 1.5]\) C\((1,\frac{7}{3}]\) D\([1,\frac{7}{3}]\)

Rewrite the function as \(f(x)=1+\frac{1}{x^2+x+1}\)

Question 12

Let \(X=\{v,i,n,o,d\}\) and \(Y=\{p,m\}\). The number of onto ( surjective) functions from \(X\) to \(Y\) is

A\(32\) B\(28 \) C\(30\) D\(36\)

To find the number of onto functions, first calculate the total number of unrestricted mappings from \(\{v, i, n, o, d\}\) to \(\{p, m\}\), which is \(2^5\). Then, simply subtract the cases where the mapping is not onto—specifically, the exact number of scenarios where all \(5\) elements map exclusively to just a single output in the codomain.

Question 13

Find the natural number \(a\) for which \(\sum_{k=1}^{n} f(a+k) = 16(2^n-1)\) where the function \(f\) satisfies the relation \(f(x+y)=f(x)f(y)\) for all natural numbers \(x,y\) and further , \(f(1)=2\).

A\(11\) B\(5\) C\(7\) D\(3\)

Hint not yet added.

Question 14

The domain of the definition of the function \(f(x)=\frac{1}{4-x^2} + log_{10} (x^3-x)\) is

A \( (-1,0) \cup (1,2) \cup (3,\infty) \) B \( (-2,-1) \cup (-1,0) \cup (2,\infty) \) C \( (-1,0) \cup (1,2) \cup (2,\infty) \) D \( (1,2) \cup (2,\infty)\)

Critical points of \(x^3-x\) are \(-1,0\) and \(1\)

Question 15

Let \(f(x)= a^x \quad (a > 0) \) be written as \(f(x)=f_1(x)+f_2(x)\), where \(f_1(x)\) is an even function and \(f_2(x)\) is an odd function. Then \(f_1(x+y)+f_1(x-y)\) equals

A\(2f_1(x+y)f_2(x-y)\) B\(2f_1(x+y)f_1(x-y)\) C\(2f_1(x)f_2(x)\) D\(2f_1(x)f_1(x)\)

see that \(f(x)=a^x=f_1(x)+f_2(x)\) and \(\f(-x)=a^{-x}=f_1(x)-f_2(x))

Question 16

Let \(g(x) = 1 + x - [x]\) and \(f(x) = \begin{cases} -1, & x < 0 \\ 0, & x = 0 \\ 1, & x > 0 \end{cases}\), then for all \(x\), \(f[g(x)]\) is equal to

A\(x\) B\(1\) C\(f(x)\) D\(g(x)\)

Note that \(x-[x] = \{x\}\) ( fractional part of \(x\) )

Question 17

Let \(\mathbb{N}\) be the set of natural numbers and two functions \(f\) and \(g\) defined as \(f,g:\mathbb{N} \rightarrow \mathbb{N}\) such that \(f(x) = \begin{cases} \frac{n+1}{2}, & \text{ if n is odd} \\ \frac{n}{2}, & \text{ if n is even} \end{cases}\) and \(g(n)=n-(-1)^n\). Then \(f\circ g\) is

Aone-one but not onto Bonto but not one-one Cboth one-one and onto Dneither one-one nor onto

Start by evaluating the composite function for the first couple of natural numbers. What exactly are the outputs for \(n = 1\) and \(n = 2\) when you calculate \((f \circ g)(n)\)? If you choose an even natural number for your initial input, say \(n = 2k\), what does the composite function \((f \circ g)(2k)\) ultimately simplify to?

Question 18

If \(f\) is an even function defined on the interval \((-5, 5)\), the four real values of \(x\) satisfying the equation: \(f(x) = f\left(\frac{x+1}{x+2}\right)\) are

A\(x = \frac{-1 \pm \sqrt{7}}{2} \quad \text{and} \quad x = \frac{-3 \pm \sqrt{5}}{2}\) B\(x = \frac{-1 \pm \sqrt{5}}{2} \quad \text{and} \quad x = \frac{-1 \pm \sqrt{5}}{2}\) C\(x = \frac{-2 \pm \sqrt{5}}{2} \quad \text{and} \quad x = \frac{-3 \pm \sqrt{5}}{2}\) D\(x = \frac{-1 \pm \sqrt{5}}{2} \quad \text{and} \quad x = \frac{-3 \pm \sqrt{5}}{2}\)

Since the function \(f\) is an even function, the condition \(f(a) = f(b)\) implies that either \(a = b\) or \(a = -b\).

Question 19

Let \(f:X \rightarrow X\) be such that \(f(f(x))=x\), for all \(x \in X\) and \( X \subset R \), then

A\(f\) is one-to-one B\(f\) is onto C\(f\) is one-to-one not onto D\(f\) is onto but not one-to-one

Hint not yet added.

Question 20

Let \( f : R \rightarrow R \) be such that \( f \) is injective and \( f(x)f(y) = f(x+y) \) for \( \forall x, y \in R \). If \( f(x) \), \( f(y) \), \( f(z) \) are in G.P., then \( x, y, z \) are in

AAP always and \(f(0)=1\) BGP always and \(f(0)=0\) CAP depending on the value of \( x, y, z \) DGP depending on the value of \( x, y, z \)

Hint not yet added.

Question 21

The domain of definition of \( f(x) = \sqrt{\frac{1 - |x|}{2 - |x|}} \) is

A\( (-\infty, -1) \cup (2, \infty) \) B\( [-1, 1] \cup (2, \infty) \cup (-\infty, -2) \) C\( (-\infty, 1) \cup (2, \infty) \) D\( [-1, 1] \cup (2, \infty) \)

Hint not yet added.

Question 22

Let \( f(x) = ax^2 + bx + c \), \( g(x) = px^2 + qx + r \) such that \( f(1) = g(1) \), \( f(2) = g(2) \) and \( f(3) - g(3) = 2 \). Then \( f(4) - g(4) \) is

A4 B5 C6 D7

Hint not yet added.

Question 23

Let \( R \) be the set of real numbers and the functions \( f : R \rightarrow R \) and \( g : R \rightarrow R \) be defined by \( f(x) = x^2 + 2x - 3 \) and \( g(x) = x + 1 \). Then, the value of \( x \) for which \( f(g(x)) = g(f(x)) \) is

A\( -1 \) B\( 0 \) C\( 1 \) D\( 2 \)

Hint not yet added.

Question 24

The minimum value of the function \(f(x)=2|x-1|+|x-2|\) is

A\( 0 \) B\( 1 \) C\( 2 \) D\( 3 \)

Sketch the graph.

Question 25

Let \(\mathrm{A}=\{-3,-2,-1,0,1,2,3\}\) and \(R\) be a relation on \(A\) defined by \(x R y\) if and only if \(2 x-y \in\{0,1\}\). Let \(l\) be the number of elements in \(R\). Let \(m\) and \(n\) be the minimum number of elements required to be added to \(R\) to make it a reflexive and symmetric relation, respectively. Then \(l+\mathrm{m}+ \mathrm{n}\) is equal to :-

A\(18\) B\(15\) C\(16\) D\(17\)

Hint not yet added.

Prime Maths - Excellence in Mathematical Education

© 2026 Prime Maths | Created by Vinod Singh