Find the Coefficient of t⁴ in the Expansion of [(1 − t⁶)/(1 − t)]³
JEE Main 2019 PYQ: find the coefficient of t^4 in [(1-t^6)/(1-t)]^3 by rewriting it as (1-t^6)^3 (1-t)^-3 and using the negative-binomial expansion.
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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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If the Line ax + y = c Touches Both the Curves x² + y² = 1 and y² = 4√2x, Then |c| Is Equal To
Given the line ax+y=c is tangent to both the circle x²+y²=1 and the parabola y²=4√2x, find |c| by combining the circle’s tangent-distance condition with the parabola’s tangent-in-slope-form condition.
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The Equation of a Common Tangent to the Circle x² + y² − 6x = 0 and the Parabola y² = 4x Is
Find the equation of a common tangent to the circle x²+y²−6x=0 and the parabola y²=4x, by writing the parabola’s tangent in slope form and applying the circle’s tangent-distance condition.
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If A = [cos θ, i sin θ; i sin θ, cos θ], θ = π/24 and A⁵ = [a b; c d], Where i = √−1, Then Which One of the Following Is Not True
For A=[cosθ isinθ;isinθ cosθ] with θ=π/24 and A⁵=[a b;c d], determine which of four given statements about a,b,c,d is not true, by proving Aⁿ has the same cos(nθ)/isin(nθ) pattern.
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The Tangents to the Curve y = (x−2)² − 1 at Its Points of Intersection With the Line x − y = 3 Intersect at the Point
Find where the tangents to y=(x−2)²−1 at its points of intersection with the line x−y=3 meet, by finding the two intersection points, their slopes, and solving the tangent-line system.
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The Value of cos(π/2²)·cos(π/2³)⋯cos(π/2¹⁰)·sin(π/2¹⁰) Is
Find the value of the product cos(π/4)cos(π/8)···cos(π/1024)sin(π/1024) using the repeated identity sinθcosθ=½sin2θ to telescope it down.
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If a Circle C Passing Through the Point (4, 0) Touches the Circle x² + y² + 4x − 6y = 12 Externally at the Point (1, −1), Then the Radius of C Is
Circle C passes through (4,0) and touches x²+y²+4x−6y=12 externally at (1,−1). Find C’s radius using the tangent-point-lies-on-the-line-of-centres property.
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If the Length of the Perpendicular From the Point (β, 0, β), β ≠ 0, to the Line x/1 = (y−1)/0 = (z+1)/(−1) Is √(3/2), Then β Is Equal To
The perpendicular distance from (β,0,β) to the line x/1=(y−1)/0=(z+1)/(−1) is √(3/2). Find β using the cross-product distance formula.
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Let α and β Be Two Roots of the Equation x² + 2x + 2 = 0. Then α¹⁵ + β¹⁵ Is Equal To
If α, β are the roots of x²+2x+2=0, find α¹⁵+β¹⁵ by converting the complex roots to polar form and using De Moivre’s theorem.