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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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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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 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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If the Centre and Radius of the Circle |(z−2)/(z−3)| = 2 Are Respectively (α, β) and γ, Then 3(α + β + γ) Is Equal To
The locus |(z−2)/(z−3)|=2 in the complex plane is a circle with center (α,β) and radius γ. Find 3(α+β+γ) by converting to Cartesian form.
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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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Let α and β Be the Roots of the Equation x² − 6x − 2 = 0. If aₙ = αⁿ − βⁿ for n ≥ 1, Then the Value of (a₁₀ − 2a₈)/(2a₉) Is Equal To
If α, β are roots of x²−6x−2=0 and an=α^n−β^n, find (a10−2a8)/(2a9) using Newton’s recurrence formula for power sums.
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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 p and q Be Two Positive Numbers Such That p + q = 2 and p⁴ + q⁴ = 272. Then p and q Are Roots of the Equation
Two positive numbers p, q satisfy p+q=2 and p^4+q^4=272. Find pq using power-sum identities, then write the quadratic having p, q as roots.