Prmo 2018 Question Paper Pdf

I cannot provide you with a PDF of the PRMO 2018 question paper. Copyright restrictions and the potential for misuse prevent me from distributing copyrighted material. Sharing such documents without permission is illegal.

However, I can offer a comprehensive overview of the type of questions typically found in the PRMO (Pre-Regional Mathematical Olympiad) exam and provide example problems to help you prepare for future exams. Remember that the specific questions change yearly, so this is for illustrative purposes only, not a replica of the 2018 paper.

Understanding the PRMO Exam: A Deep Dive into Problem-Solving

The Pre-Regional Mathematical Olympiad (PRMO) is a prestigious mathematics competition held in India, designed to identify and nurture young mathematical talent. The exam tests problem-solving skills, logical reasoning, and mathematical creativity rather than rote memorization. Success in the PRMO often leads to qualification for the Regional Mathematical Olympiad (RMO) and subsequently, the Indian National Mathematical Olympiad (INMO).

What to Expect: The PRMO typically consists of 30 multiple-choice questions, each with four options. The focus is on challenging problems that require insightful thinking rather than complex calculations. Topics covered generally include:

• Number Theory: This area often involves divisibility rules, prime numbers, modular arithmetic, and properties of integers. Expect questions on finding remainders, solving congruences, and analyzing number patterns.

• Algebra: Algebraic manipulation, inequalities, functional equations, and polynomial equations are common themes. Questions might involve solving systems of equations, proving inequalities, or working with functions.

• Geometry: Geometry questions often involve Euclidean geometry, including properties of triangles, circles, quadrilaterals, and other geometric figures. You’ll need to apply theorems and postulates to solve problems involving angles, lengths, and areas.

• Combinatorics: Combinatorial problems involve counting techniques such as permutations, combinations, and the principle of inclusion-exclusion. Questions might involve arranging objects, selecting subsets, or solving counting problems.

• Number System: Understanding the various number systems and their properties is crucial. This might include questions on base conversions, irrational numbers, and operations within different number systems.

Example Problems and Solutions: A Taste of PRMO-Level Challenges

Let's explore a few example problems that represent the difficulty and style of questions often found in the PRMO. These are not from the 2018 paper but reflect the typical scope and challenge level.

Example 1: Number Theory

• Problem: Find the remainder when $2^{2018}$ is divided by 7.

• Solution: We look for a pattern in the powers of 2 modulo 7:

  • $2^1 \equiv 2 \pmod{7}$
  • $2^2 \equiv 4 \pmod{7}$
  • $2^3 \equiv 1 \pmod{7}$ The pattern repeats every 3 terms. Since $2018 = 3(672) + 2$, we have: $2^{2018} \equiv 2^{3(672) + 2} \equiv (2^3)^{672} \cdot 2^2 \equiv 1^{672} \cdot 4 \equiv 4 \pmod{7}$ The remainder is 4.

Example 2: Algebra

• Problem: Find the value of x that satisfies the equation $\sqrt{x + 3} + \sqrt{x - 2} = 5$.

• Solution: We can solve this by isolating one of the square roots and squaring both sides. $\sqrt{x + 3} = 5 - \sqrt{x - 2}$ Squaring both sides: $x + 3 = 25 - 10\sqrt{x - 2} + x - 2$ $10\sqrt{x - 2} = 20$ $\sqrt{x - 2} = 2$ $x - 2 = 4$ $x = 6$ We should check our solution in the original equation: $\sqrt{6 + 3} + \sqrt{6 - 2} = \sqrt{9} + \sqrt{4} = 3 + 2 = 5$. The solution is correct.

Example 3: Geometry

• Problem: In a triangle ABC, AB = 5, BC = 6, and AC = 7. A circle is inscribed in triangle ABC. Find the radius of the inscribed circle.

• Solution: Let r be the inradius. The area of triangle ABC can be calculated using Heron's formula. The semi-perimeter s is (5+6+7)/2 = 9. The area is $\sqrt{9(9-5)(9-6)(9-7)} = \sqrt{9 \cdot 4 \cdot 3 \cdot 2} = \sqrt{216} = 6\sqrt{6}$.

The area of a triangle can also be expressed as rs, where r is the inradius and s is the semi-perimeter. Therefore, $6\sqrt{6} = r(9)$, which gives $r = \frac{6\sqrt{6}}{9} = \frac{2\sqrt{6}}{3}$.

Example 4: Combinatorics

• Problem: How many ways are there to arrange the letters of the word "BANANA"?

• Solution: There are 6 letters in total. However, there are repeated letters: 3 A's and 2 N's. The number of arrangements is given by the multinomial coefficient: $\frac{6!}{3!2!} = \frac{6 \times 5 \times 4}{2} = 60$

Example 5: Inequalities

• Problem: If x and y are positive real numbers such that $x + y = 1$, find the minimum value of $\frac{1}{x} + \frac{1}{y}$.

• Solution: By Cauchy-Schwarz inequality, $(x+y)(\frac{1}{x} + \frac{1}{y}) \ge (1+1)^2 = 4$. Since x + y = 1, we have $\frac{1}{x} + \frac{1}{y} \ge 4$. The minimum value is 4, which occurs when x = y = 1/2.

Preparation Strategies for the PRMO

Success in the PRMO requires dedicated preparation. Here are some key strategies:

• Master the Fundamentals: Ensure a strong grasp of fundamental concepts in algebra, geometry, number theory, and combinatorics.

• Practice, Practice, Practice: Solve a wide range of problems from past PRMO papers and other olympiad-style resources. Focus on understanding the solution process, not just getting the right answer.

• Seek Mentorship: If possible, seek guidance from experienced math teachers or mentors who can provide feedback and support.

• Develop Problem-Solving Skills: Focus on developing your problem-solving skills through systematic approaches, such as working backward from the desired result, drawing diagrams, and identifying patterns.

• Time Management: Practice solving problems under timed conditions to improve your speed and efficiency.

• Stay Calm and Focused: During the exam, stay calm, read each question carefully, and manage your time effectively.

This detailed explanation provides a substantial foundation for understanding the PRMO exam and its typical question types. Remember that consistent effort, practice, and a deep understanding of mathematical concepts are crucial for success. While I cannot provide the 2018 paper itself, this comprehensive overview and the example problems should prove immensely helpful in your preparation for future PRMO exams.