Express tan⁻¹(cos x/(1 − sin x)), Where x ∈ (−3π/2, π/2), in Its Simplest Form
Simplify tan⁻¹(cosx/(1−sinx)) for x∈(−3π/2,π/2) using half-angle substitutions to reveal it equals tan(π/4+x/2), matching the principal range exactly.
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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.
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Two Sets A and B Are as Under: A = {(a,b) ∈ R×R : |a−5| < 1 and |b−5| < 1}, B = {(a,b) ∈ R×R : 4(a−6)² + 9(b−5)² ≤ 36}. Then
Set A is an open square defined by |a−5|<1 and |b−5|<1; set B is a filled ellipse. Determine whether A⊂B, B⊂A, or neither, by checking the square's corners against the ellipse.
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Let α, β Be the Roots of x² − ax − b = 0 With Im(α) < Im(β) and Pₙ = αⁿ − βⁿ. If P₃ = −5√7 i, P₄ = −3√7 i, P₅ = 11√7 i and P₆ = 45√7 i, Then |α⁴ + β⁴| Is Equal To
Given Pn=α^n−β^n for roots of x²−ax−b=0, and values of P3 through P6, find a and b using a linear recurrence, then compute |α⁴+β⁴|.
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If a, b, c Are Three Mutually Perpendicular Vectors of the Same Magnitude, Prove That a + b + c Is Equally Inclined With a, b and c
Prove that a+b+c makes equal angles with each of a, b, c when a, b, c are three mutually perpendicular vectors of the same magnitude, using dot products.
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Let P Be the Point on the Parabola y² = 8x Which Is at a Minimum Distance From the Centre C of the Circle x² + (y+6)² = 1. Then the Equation of the Circle Passing Through C and Having Its Centre at P Is
Find point P on the parabola y²=8x at minimum distance from the centre C of x²+(y+6)²=1, then find the equation of the circle through C centred at P.
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Let the Straight Lines AB: x + y − 2 = 0 and AC: 3y − x = 2 Intersect the x-axis at B and C Respectively. If P Is the Orthocentre of Triangle ABC, Then the Area of Triangle CPB Is
Lines AB: x+y−2=0 and AC: 3y−x=2 meet the x-axis at B and C. Find the area of triangle CPB, where P is the orthocentre of triangle ABC.
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Let a Straight Line L Pass Through P(2, −1, 3) and Be Perpendicular to the Lines (x−1)/2 = (y+1)/1 = (z−3)/(−2) and (x−3)/1 = (y−2)/3 = (z+2)/4. If L Intersects the yz-Plane at Q, Then the Distance Between P and Q Is
A line L through P(2,−1,3) is perpendicular to two given lines. Find where L meets the yz-plane and the distance PQ, using the cross product for L’s direction.
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Let an Ellipse With Centre (1, 0) and Latus Rectum of Length 1/2 Have Its Major Axis Along the x-axis. If Its Minor Axis Subtends an Angle of 60° at the Foci, Then the Square of the Sum of the Lengths of Its Minor and Major Axes Is Equal To
An ellipse centred at (1,0) has latus rectum 1/2 and its minor axis subtends 60° at the foci. Find the square of the sum of its major and minor axis lengths.
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A Person Has Undertaken a Construction Job. The Probabilities Are 0.65 That There Will Be a Strike, 0.80 That the Job Will Be Completed on Time if There Is No Strike, and 0.32 if There Is a Strike. Find the Probability the Job Will Be Completed on Time
A construction job has 0.65 probability of a strike, with different completion probabilities depending on whether there’s a strike. Find the overall probability of on-time completion using total probability.