Roots of x²−√2x−√3=0, Pn=αⁿ−βⁿ, Find (11√3−10√2)P10+(11√2+10)P11−11P12
Let α, β (α>β) be roots of x²−√2x−√3=0 and Pn=αⁿ−βⁿ. Find (11√3−10√2)P10+(11√2+10)P11−11P12 using the Newton recurrence for power sums.
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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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The Sum to 10 Terms of the Series 1/(1+1²+1⁴) + 2/(1+2²+2⁴) + 3/(1+3²+3⁴) + … Is
The sum to 10 terms of the series 1/(1+1²+1⁴) + 2/(1+2²+2⁴) + 3/(1+3²+3⁴) + … is (A) 56/111 (B) 58/111 (C) 55/111 (D) 50/111.
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How to Find the Sum of 7 + 77 + 777 + … Up to n Terms
A step-by-step derivation of the sum of the series 7 + 77 + 777 + … up to n terms, using the repunit-pattern trick to write each term as 7(10^n-1)/9.
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Triangle PQR, Points A, B, C Divide Sides in Ratio 1:2, Find Area(PQR)/Area(ABC)
Let PQR be a triangle. Points A, B, C are on sides QR, RP, PQ such that QA/AR = RB/BP = PC/CQ = 1/2. Find Area(PQR)/Area(ABC) using position vectors.
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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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Let A, B, C Have Position Vectors OA = √3i + j, OB = i + √3j and OC = ai + (1−a)j. If the Distance of C From the Line Bisecting the Angle Between OA and OB Is 9/√2, the Sum of All Possible Values of a Is
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.