If x = −4 Is a Root of |x 2 3; 1 x 1; 3 2 x| = 0, Then Find the Sum of the Other Two Roots
Given x=−4 satisfies the determinant equation |x 2 3; 1 x 1; 3 2 x|=0, expand it into a cubic and find the sum of the remaining two roots.
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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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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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There Are Two Bags. One Bag Contains Six Green and Three Red Balls, the Second Bag Contains Five Green and Four Red Balls. One Ball Is Transferred From the First Bag to the Second, Then One Ball Is Drawn From the Second Bag. Find the Probability That It Is a Red Ball
Bag 1 has 6 green, 3 red balls; Bag 2 has 5 green, 4 red. One ball moves from Bag 1 to Bag 2, then one is drawn from Bag 2. Find the probability it’s red.
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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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Show That the Lines (x+1)/3 = (y+3)/5 = (z+5)/7 and (x−2)/1 = (y−4)/3 = (z−6)/5 Intersect
Prove that the lines (x+1)/3=(y+3)/5=(z+5)/7 and (x−2)/1=(y−4)/3=(z−6)/5 intersect, by finding parameter values that satisfy all three coordinate equations.
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If S = {z ∈ C : |z − i| = |z + i| = |z − 1|}, Then n(S) Is
Find the number of complex numbers z satisfying |z−i|=|z+i|=|z−1| simultaneously, by recognising z as the circumcenter of the triangle formed by i, −i, and 1.
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Let ABCD Be a Parallelogram Such That AB = q, AD = p and ∠BAD Is an Acute Angle. If r Is the Vector That Coincides With the Altitude Directed From Vertex B to Side AD, Then r Is Given By
In parallelogram ABCD with AB=q, AD=p, and acute ∠BAD, find the vector r for the altitude from vertex B to side AD, using vector projection.
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Find the Value of √(12 + √(12 + √(12 + ⋯)))
Find the value of the infinite nested radical √(12+√(12+√(12+…))) by setting it equal to x and squaring, reducing it to a quadratic equation.
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Let A and B Be Two Finite Sets With m and n Elements. The Total Number of Subsets of A Is 56 More Than the Total Number of Subsets of B. Then the Distance of the Point P(m, n) From Q(−2, −3) Is
Sets A, B have m, n elements, with A’s subset count 56 more than B’s. Find m and n via prime factorisation of 2^m−2^n=56, then the distance from P(m,n) to Q(−2,−3).