Evaluate I = ∫₀^(π/4) log(1 + tan x) dx
Evaluate the definite integral of log(1+tan x) from 0 to π/4 using King’s property, which turns 1+tan x into 2/(1+tan x) under the substitution x → π/4 − x.
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If the Integral of t·f(t) From 0 to x Equals x²·f(x) and f(2) = 3, Then f(6) Equals
JEE Main 2025 PYQ: if ∫₀ˣ t·f(t)dt = x²·f(x) and f(2)=3, find f(6) by differentiating both sides to form and solve a differential equation for f(x).
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Aman and Bhuvan Throw a Pair of Dice Alternately. In Order to Win They Have to Get a Sum of 8. Find Their Respective Probabilities of Winning if Aman Starts the Game
Aman and Bhuvan throw a pair of dice alternately, and whoever first gets a sum of 8 wins. Find each player’s probability of winning if Aman starts, using an infinite geometric series.
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If y = mx + 4 Is a Tangent to Both the Parabolas y² = 4x and x² = 2by, Then b Is Equal To
JEE Main 2020 PYQ: the line y=mx+4 is tangent to both y²=4x and x²=2by. Find b by matching the tangency condition on the first parabola, then applying it on the second.
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A Spiral Is Made Up of Successive Semicircles With Centres Alternately at A and B, of Radii 0.5 cm, 1.0 cm, 1.5 cm, 2.0 cm, … What Is the Total Length of Such a Spiral Made Up of Thirteen Consecutive Semicircles
NCERT Class 10 Arithmetic Progressions problem: a spiral of 13 successive semicircles with radii increasing by 0.5cm each time. Find its total length using the AP sum formula.
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If S₂₀ = 30 and S₃₀ = 20 for an Arithmetic Progression, Find S₅₀
Given the sum of the first 20 terms of an AP is 30 and the sum of the first 30 terms is 20, find S50 by solving for the first term and common difference.
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Two Numbers k₁ and k₂ Are Randomly Chosen From the Set of Natural Numbers. Then the Probability That the Value of i^k₁ + i^k₂ (Where i = √−1) Is Non-Zero Is Equal To
JEE Main 2025 PYQ: k1 and k2 are chosen randomly from the natural numbers — find the probability that i^k1+i^k2 is non-zero, using the 4-cycle of powers of i.
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If One End of a Focal Chord of the Parabola y² = 16x Is at (1, 4), Then the Length of This Focal Chord Is
JEE Main 2019 PYQ: one end of a focal chord of the parabola y²=16x is at (1,4). Find the length of the chord by locating its other endpoint via substitution.
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Let f(x) = 5ˣ/(5ˣ + 5). Then the Sum of the Series f(1/20) + f(2/20) + f(3/20) + ⋯ + f(39/20) Is Equal To
JEE Main 2021 PYQ: for f(x)=5^x/(5^x+5), find the sum f(1/20)+f(2/20)+…+f(39/20) using the pairing identity f(x)+f(2-x)=1.
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Let the Centroid of an Equilateral Triangle ABC Be at the Origin and One of Its Sides Lie Along the Straight Line x + y = 3. If R and r Are the Radii of the Circumcircle and Incircle of Triangle ABC, Then R + r Is Equal To
JEE Main 2021 PYQ: an equilateral triangle has its centroid at the origin and one side along x+y=3. Find R+r using the R=2r relation for equilateral triangles.