5-Digit Numbers >50000, Digits 0-7, First+Last Digit ≤8
How many 5-digit numbers above 50,000, built from digits 0-7 with repetition, have first-plus-last digit at most 8? JEE Main 2025, fully solved.
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Group A: 7 Boys 3 Girls, Group B: 6 Boys 5 Girls — Picnic Selection
A JEE Main 2025 combinatorics problem: invite 4 boys and 4 girls for a picnic with 5 from one group and 3 from another. Solved case by case.
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If I = ∫96x²cos²x/(1+eˣ)dx = π(απ²+β), Find (α+β)²
A JEE Main 2025 definite integral solved using the King’s property trick for limits symmetric about zero, avoiding a direct, messier computation.
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Words From KANPUR in Dictionary Order: 440th Word
All arrangements of the letters of KANPUR are sorted like a dictionary. Which word sits at position 440? A JEE Main 2025 permutations shortcut, solved.
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Value of (sin 70°)(cot 10° cot 70° − 1)
A short JEE Main 2025 trigonometry problem solved by converting cotangents to cosines and sines, then applying the cos(A+B) identity.
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6-Letter Words From MATHS Where Every Letter Appears Twice
How many 6-letter words can be formed from MATHS if every letter used must appear at least twice? A JEE Main 2025 permutations problem, solved fully.
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Position Vectors 4p+q-3r, -5p+q+2r, 2p-q+2r: Find α+2β+5γ
A JEE Main 2025 vector geometry problem solved using the Euler line property that the centroid divides the orthocenter-circumcenter segment 2:1.
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Two Ellipses, Eccentricity 1/√3: Area of Quadrilateral ABCD
Two ellipses with the same eccentricity, a given latus rectum product, and a known focal distance intersect at four points. A JEE Main 2025 area problem, solved.
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Unit Vector Perpendicular to b, c at Angles cos⁻¹(-1/3) and π/3
Find alpha when a unit vector perpendicular to two given vectors also satisfies two angle conditions. A JEE Main 2025 3D vector problem, solved fully.
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3-Digit Numbers Divisible by 2 and 3, Not 4 and 9
How many 3-digit numbers are divisible by both 2 and 3, but not by 4 or 9? Solved using LCMs and arithmetic progressions, JEE Main 2025 style.