Is f(x)=(2^x−2^−x)/(2^x+2^−x) One-One, Onto, Both or Neither?
The function f:(−∞,∞)→(−∞,1), defined by f(x)=(2^x−2^−x)/(2^x+2^−x), is: neither one-one nor onto, onto but not one-one, both, or one-one but not onto?
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Find Vector d Perpendicular to Both b and c, With d·a = 21
Given vectors a=4i+5j−k, b=i−4j+5k, c=3i+j−k, find vector d perpendicular to both b and c such that d·a=21, by solving a system of three linear equations.
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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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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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Sum to 10 Terms of the Series r/(1+r²+r⁴), JEE Main 2023
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.