Express the complex quotient (3 + 4i) / (1 - 2i) in standard rectangular form a + bi.
Complex Numbers And Induction
Question · Easy
Find the modulus and principal argument of the complex number z = -1 - i√3, where Arg(z) ∈ (-π, π].
Which of the following is the polar form r(cos θ + i sin θ) of the complex number z = 2√3 - 2i for θ ∈ (-π, π]?
If z₁ = 2(cos(π/4) + i sin(π/4)) and z₂ = 3(cos(π/12) + i sin(π/12)), find the product z₁z₂ in rectangular form a + bi.
Question · Medium
Using De Moivre's theorem, evaluate (1 + i)^8.
Simplify the expression (cos θ + i sin θ)^5 / (cos θ - i sin θ)^3.
Find the square roots of the complex number 3 - 4i.
Question · Hard
What is the exact value of (cos(π/6) + i sin(π/6))^6?
In the complex Argand plane, what geometric figure is represented by the equation |z - (2 + 3i)| = 5?
Find real numbers x and y such that (x + yi)(1 + 2i) = 7 + 4i.
Find the three distinct roots of the equation z^3 = 8i in rectangular form.
In a proof by mathematical induction for the summation identity 1 + 2 + 3 + ... + n = n(n + 1)/2, after assuming the statement holds for n = k, what term must be added to both sides of the inductive hypothesis?
When proving by mathematical induction that 7^n - 1 is divisible by 6 for all integers n ≥ 1, which algebraic decomposition correctly demonstrates that 7^(k+1) - 1 is divisible by 6 given the induction hypothesis 7^k - 1 = 6m?
Using mathematical induction or summation formulas, find the sum S_n = 1·2 + 2·3 + 3·4 + ... + n(n + 1).
Find the smallest positive integer n such that (1 + i√3)^n is a purely real number.
Find the distance between the two points A(1, -2, 4) and B(4, 2, -8) in three-dimensional Cartesian space.
Analytical Geometry 3d Vectors
A straight line in space makes angles α = 60° with the positive x-axis and β = 45° with the positive y-axis. If it makes an acute angle γ with the positive z-axis, find γ.
Given vectors a = 2i - j + 3k and b = 4i + 2j - pk, determine the value of the scalar p such that a and b are perpendicular.
Find the unit vector in the direction of the vector v = 3i - 2j + 6k.
Compute the cross product a × b of the vectors a = i + 2j - k and b = 2i - j + k.
Find the area of the parallelogram formed by the adjacent vectors a = 2i + j - k and b = -i + 2j + 3k.
What are the direction cosines (l, m, n) of the vector v = 2i - 3j + 6k?
Find the Cartesian equation of the plane passing through the point P(1, 2, -3) with normal vector n = 3i - j + 2k.
Find the perpendicular distance from the point P(2, 3, -1) to the plane 2x - 2y + z + 12 = 0.
Find the acute angle between the two planes 2x - y + z = 6 and x + y + 2z = 3.
Which of the following is the symmetric Cartesian equation of the line passing through point A(2, -1, 3) and parallel to vector d = 3i + 4j - 5k?
Find the value of k such that the line (x - 1)/2 = (y + 2)/3 = (z - 4)/k is parallel to the plane 2x + 4y - z = 5.
Calculate the scalar triple product a · (b × c) for vectors a = i + j + k, b = 2i - j + 3k, and c = 3i + 2j - k.
Find the coordinates of the midpoint of the line segment joining points P(3, -5, 7) and Q(-1, 3, 1).
Which of the following is the Cartesian equation of the straight line passing through the point (1, 2, 3) and perpendicular to the plane 4x - 2y + 5z = 7?
In how many different ways can 5 distinct mathematics textbooks be arranged side by side on a shelf?
Permutations Combinations Conics
A committee of 4 members is to be selected from a group of 7 men and 5 women. In how many ways can the committee be formed if it must contain exactly 2 men and 2 women?
How many distinct 4-digit numbers can be formed using the digits 1, 2, 3, 4, 5, 6 if each digit can be used at most once in each number?
Find the number of distinct permutations of all the letters in the word 'MATHEMATICS'.
In how many distinct ways can 6 people be seated around a round dining table?
Using the binomial theorem, determine the coefficient multiplying x³ in (2x - 1)⁵.
Find the coordinates of the focus and the equation of the directrix for the parabola y² = 12x.
Determine the vertex and focus of the parabola (x - 2)² = 8(y + 1).
Find the eccentricity e of the ellipse x²/25 + y²/16 = 1.
Find the coordinates of the foci of the ellipse x²/9 + y²/25 = 1.
Find the length of the latus rectum of the ellipse 4x² + 9y² = 36.
Find the equations of the asymptotes of the hyperbola x²/16 - y²/9 = 1.
Determine the eccentricity of the hyperbola x²/9 - y²/16 = 1.
Find the coordinates of the foci of the hyperbola y²/36 - x²/64 = 1.
From a standard deck of 52 playing cards, in how many ways can a 5-card hand containing exactly 3 aces be dealt?
Determine the amplitude and period of the trigonometric function y = -3 sin(4x - π/2) + 1.
Trigonometric Functions And Differentiation
Find the exact principal value of arcsin(-√3/2).
For acute angles A and B with tan A = 3/4 and tan B = 1/7, evaluate tan(A + B).
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