This is an IIT-JEE problem that has been bothering me: Find the number of solutions to the equation sin x = x/10 for x ∈ ℝ. I tried solving it analytically but it seems impossible due to the transcendental nature. My teacher suggested a graphical approach. Could someone explain how to count the intersections methodically? I know that both functions are odd, so the number of positive solutions will be symmetric to negative ones, with x = 0 always being a solution.

I'm preparing for IIT-JEE and I need to understand the classic proof that √2 is irrational. I've seen the proof but I have some doubts about the logic. The proof assumes √2 = p/q in lowest terms, then squares both sides to get 2q² = p². Then it argues that p must be even, so p = 2k, and substituting gives q² = 2k², so q is also even, contradicting the assumption that p/q is in lowest terms. My question: Why must p be even if p² is even? And why does this proof not work for √4 (which is rational)?

I'm studying linear algebra and I need to understand how to compute the rank of a matrix. I know the definition: The rank of a matrix is the dimension of its column space (or row space). But practically, I've been told to convert the matrix to row echelon form and count the number of non-zero rows. However, I'm confused about: 1. Does the rank depend on whether I use row echelon form or reduced row echelon form? 2. What is the rank of a zero matrix? An identity matrix? 3. How do I find the rank of: A = 1 & 2 & 3 \ 2 & 4 & 6 \ 3 & 6 & 9 4. What does it mean for a matrix to have full rank?

I am taking an introductory Real Analysis course and we are studying the Cauchy criterion for convergence. I understand the definition: A sequence (aₙ) is Cauchy if for every ε > 0, there exists N ∈ ℕ such that for all m, n ≥ N, |aₙ - a_m| < ε. We are told that in ℝ, every Cauchy sequence converges (completeness property). But how do we actually prove this from the definition? I know it requires the supremum axiom or the nested interval property, but I need to see the full proof. Also, why does this property fail in ℚ?

Euclid's proof that there are infinitely many primes is famous, but I want to understand it deeply. Assume p₁, p₂, \ldots, pₙ are all the primes. Consider N = p₁ p₂ ⋯ pₙ + 1. Then N is either prime or has a prime factor not in our list, contradiction. My question: Could N itself be divisible by one of the existing primes? Euclid says no because dividing N by any pᵢ leaves remainder 1. But is that rigorous enough? And are there other proofs of infinitude of primes?

I'm studying A-Level calculus and I can state the Mean Value Theorem: If f is continuous on [a,b] and differentiable on (a,b), then there exists c ∈ (a,b) such that: f'(c) = f(b) - f(a)/b - a But I am having trouble building intuition for what this really means geometrically. Why is this theorem so important in analysis? Can someone provide a clear geometric explanation with a diagram description?