G.P. of Increasing Positive Terms, a1a5=28 and a2+a4=29, Find a6
Let a1,a2,a3,… be a G.P. of increasing positive terms. If a1a5=28 and a2+a4=29, then a6 is equal to: (A) 628 (B) 812 (C) 526 (D) 784.
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Seven Digit Numbers With Digit Sum 11, Using Only 1, 2 and 3
Let P be the set of seven digit numbers with sum of their digits equal to 11, formed using only digits 1, 2 and 3. Find the number of elements in P.
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Domain of log5(18x−x²−77) Is (α,β), Find α²+β²+γ²
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.
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5 Letters Chosen From Alphabet, Arranged in Order, Middle Letter M
A JEE Main 2025 problem where fixing alphabetical order turns an arrangement question into a pure selection (combinations) problem, split around a fixed middle letter.
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4 Boys, 3 Girls Seated: Girls Together, B1 and B2 Not Adjacent
A JEE Main 2025 seating-arrangement problem solved with the block method for ‘girls together’, then placing two specific boys into non-adjacent gaps.
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If Sn=1/2+1/6+1/12+… to n Terms, A.P. Sum is √(2026·S2025)
A JEE Main 2025 problem combining a telescoping series with an arithmetic progression, solved by simplifying Sn first, then using it to pin down the AP.
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Distance From C to Angle Bisector of OA, OB is 9/√2, Find a
Three points given by position vectors, a distance from an angle bisector — solved fastest by switching to plain coordinate geometry. JEE Main 2025.
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If I(m,n)=∫x^(m-1)(1-x)^(n-1)dx, Find I(9,14)+I(10,13)
A beta-function-style JEE Main 2025 integral solved without ever evaluating the integral itself — just substitution, addition, and pattern matching.
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If Σr²(³⁰Cᵣ)²/(³⁰Cᵣ₋₁)=α·2²⁹ From r=1 to 30, Find α
A JEE Main 2025 binomial coefficient summation simplified by rewriting the ratio of consecutive coefficients, then splitting into two telescoping sums.
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Complex Roots of x²-(3-2i)x-(2i-2)=0, Find αγ+βδ
A quadratic with complex coefficients and complex roots is easier factorised directly than approached through Vieta’s formulas. JEE Main 2025, solved.