Let α, β Be the Roots of the Equation x² − x + 2 = 0 With Im(α) > Im(β). Then α⁶ + α⁴ + β⁴ − 5α² Is Equal To
Given α, β are the roots of x²−x+2=0 with Im(α)>Im(β), find α⁶+α⁴+β⁴−5α² by repeatedly reducing powers using α²=α−2.
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Let P(4, 4√3) Be a Point on the Parabola y² = 4ax and PQ Be a Focal Chord of the Parabola. If M and N Are the Feet of Perpendiculars Drawn From P and Q on the Directrix, Then the Area of the Quadrilateral PQMN Is Equal To
P(4,4√3) lies on y²=4ax, and PQ is a focal chord. M, N are feet of perpendiculars from P, Q to the directrix. Find the area of quadrilateral PQMN using the focal-chord parameter relation.
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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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If a and b Are Unit Vectors, Then the Greatest Value of √3|a+b| + |a−b| Is
Find the greatest value of √3|a+b|+|a−b| where a and b are unit vectors, by writing both magnitudes in terms of the half-angle between them.
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Find the Positive Integer n for Which 2·2² + 3·2³ + 4·2⁴ + ⋯ + n·2ⁿ = 2ⁿ⁺¹⁰
Find the positive integer n satisfying 2·2²+3·2³+…+n·2ⁿ=2^(n+10), by summing the series via the standard shift-and-subtract technique for a sum of the form Σk·rᵏ.
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Let the Solution Curve y = y(x) of dy/dx − 3x⁵tan⁻¹(x³)y/(1+x⁶)^(3/2) = 2x·exp((x³−tan⁻¹x³)/√(1+x⁶)) Pass Through the Origin. Then y(1) Is Equal To
The solution curve y=y(x) of dy/dx − 3x⁵tan⁻¹(x³)y/(1+x⁶)^(3/2) = 2x·exp[(x³−tan⁻¹x³)/√(1+x⁶)] passes through the origin. Find y(1).
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For p > 0, the Vector v₂ = 2i + (p+1)j Is Obtained by Rotating v₁ = √3 p i + j by an Angle θ About the Origin in a Counterclockwise Direction. If tan θ = (α√3 − 2)/(4√3 + 3), Then the Value of α Is Equal To
For p>0, vector v2=2i+(p+1)j is obtained by rotating v1=√3pi+j by angle θ. Given tanθ=(α√3−2)/(4√3+3), find α using the rotation-preserves-length property.
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Let the Centre of a Circle C Be (α, β) and Its Radius r < 8. Let 3x + 4y = 24 and 3x − 4y = 32 Be Two Tangents and 4x + 3y = 1 Be a Normal to C. Then (α − β + r) Is Equal To
A circle with radius r<8 has 3x+4y=24 and 3x−4y=32 as tangents and 4x+3y=1 as a normal. Find α−β+r using the tangent-distance formula and the fact that a normal passes through the centre.
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The Solution of the Differential Equation cos x·dy = y(sin x − y)dx, for 0 < x < π/2, Is
Solve the differential equation cos x dy = y(sin x − y)dx for x in (0,π/2), by recognising it as a Bernoulli equation and substituting t=1/y.
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Let the Mean and Standard Deviation of Marks of Class A of 100 Students Be 40 and α, and of Class B of n Students Be 55 and 30 − α. If the Mean and Variance of the Combined Class of 100 + n Students Are 50 and 350, Then the Sum of Variances of Classes A and B Is
Class A (100 students, mean 40, SD α) and Class B (n students, mean 55, SD 30−α) combine to mean 50, variance 350. Find the sum of variances of A and B.