Maths

If the Boolean expression (p ∧ q) ○ (p ⊕ q) is a tautology, then ○ and ⊕ are respectively by:

Let S₁, S₂, and S₃ be three sets defined as
S₁ = { z∈C : |z−1| ≤ √2}
S₂ = {z∈C : Re((1-i)z) ≥ 1}
S₃ = {z∈C : Im(z) ≤ 1}
Then the set S₁ ∩ S₂ ∩ S₃

The number of solutions of the equations |cot x| = cot x + 1/sin x in the interval [0, 2π] is ______.

The number of times the digit 3 will be written when listing the integers from 1 to 1000 is ______.

Let f(x) and g(x) be two functions satisfying f(x²) + g(4 - x) = 4x³ and g(4 - x) + g(x) = 0, the value of ∫[-4 to 4] f(x²) dx is ______.

A square ABCD has all its vertices on the curve x²y² = 1. The midpoints of its sides also lie on the same curve. Then, the square of area of ABCD is ______.

The equation of the planes parallel to the plane x - 2y + 2z - 3 = 0 which are at unit distance from the point (1,2,3) is ax + by + cz + d = 0. If (b-d) = K(c-a), then the positive value of K is ______.

Let z₁, z₂ be the roots of the equation z² + az + 12 = 0 and z₁, z₂ form an equilateral triangle with origin. Then, The value of |a| is ______.

The missing value in the following figure is ______. (A diagram with numbers in a circular pattern is shown).

The mean age of 25 teachers in a school is 40 years. A teacher retires at the age of 60 years and a new teacher is appointed in his place. If the mean age of the teachers in this school now is 39 years, then the age (in years) of the newly appointed teacher is: ______.