If z Is a Complex Number Such That |z| ≥ 2, Then the Minimum Value of |z + 1/2| Is
Given |z|≥2, find the minimum value of |z+1/2| by writing z in polar form and minimising the distance from z to the point −1/2.
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For a Positive Constant a, Find dy/dx Where y = a^(t + 1/t) and x = (t + 1/t)^a
Find dy/dx where y=a^(t+1/t) and x=(t+1/t)^a, by treating u=t+1/t as an intermediate variable so the parameter t’s own derivative cancels out.
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If the Vectors a = i − j + 2k, b = 2i + 4j + k and c = λi + j + μk Are Mutually Orthogonal, Then (λ, μ) =
Given a=i−j+2k, b=2i+4j+k, and c=λi+j+μk are mutually orthogonal, find (λ,μ) by setting the dot products a·c and b·c to zero.
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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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If Three Positive Real Numbers a, b, c Are in AP Such That abc = 4, Then the Minimum Value of b Is
Given positive reals a,b,c in AP with abc=4, find the minimum value of b using the AM-GM inequality on a, b, c.
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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 Sum of the Series 1/(1·2) − 1/(2·3) + 1/(3·4) − ⋯ up to Infinity Is
Find the sum of the alternating series 1/(1·2)−1/(2·3)+1/(3·4)−… to infinity, by splitting each term via partial fractions and matching against the log(1+x) expansion.
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Let z Be a Complex Number Such That the Real Part of (z − 2i)/(z + 2i) Is Zero. Then the Maximum Value of |z − (6 + 8i)| Is Equal To
Given the real part of (z−2i)/(z+2i) is zero, show z traces a circle of radius 2 about the origin, then find the maximum distance from that circle to (6,8).
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If A + B + C = 180°, Prove That sin 2A + sin 2B − sin 2C = 4 cos A cos B sin C
Prove that sin2A+sin2B−sin2C=4cosA cosB sinC when A+B+C=180°, using the sum-to-product formula twice along with the supplementary-angle identities.
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The Number of Complex Numbers z Satisfying |z| = 1 and |z/z̄ + z̄/z| = 1 Is
Find how many complex numbers z satisfy both |z|=1 and |z/z̄+z̄/z|=1 simultaneously, by reducing the condition to x²−y²=±1/2 on the unit circle.