S is a set of 10 distinct primes and A is the set of products of two or more elements of S. Find n(P) where P={(x,y): x∈S, y∈A, x divides y}.
If the midpoint of a chord of the ellipse x²/9+y²/4=1 is (√2, 4/3), and the chord’s length is 2√α/3, then α is: (1) 18 (2) 22 (3) 26 (4) 20.
Step-by-step solution to the integral of cos(x)/[(4+sin²x)(5−4cos²x)] dx, using a sine substitution and partial fractions to reduce it to two arctangent terms.
Consider f:R→R defined by f(x)=2x/√(1+9x²). If (f∘f∘⋯∘f, 10 times)(x)=2^10x/√(1+9αx²), find the value of √(3α+1).
The sum of the series 1+6+9(1²+2²+3²)/7+12(1²+2²+3²+4²)/9+15(1²+…+5²)/11+… up to 15 terms is: (1)7820 (2)7520 (3)7830 (4)7510.
If four distinct points with position vectors a,b,c,d are coplanar, which scalar triple product identity must hold? A worked derivation from the coplanarity condition.
Let E1: x²/9+y²/4=1. Ellipses Ei share E1’s centre and eccentricity, with minor axis of Ei equal to major axis of Ei+1. Find (5/π)ΣAi.
If the domain of log5(18x−x²−77) is (α,β) and the domain of log(x−1)[(2x²+3x−2)/(x²−3x−4)] is (γ,δ), then α²+β²+γ² is: A)195 B)179 C)186 D)174.
A JEE Main 2025 binomial coefficient summation simplified by rewriting the ratio of consecutive coefficients, then splitting into two telescoping sums.
A JEE Main 2025 definite integral solved using the King’s property trick for limits symmetric about zero, avoiding a direct, messier computation.