Class 11th

Let there be three independent events E₁, E₂ and E₃. The probability that only E₁ occurs is α, only E₂ occurs is β and only E₃ occurs is γ. Let 'p' denote the probability of none of events occurs that satisfies the equations (α - 2β)p = αβ and (β - 3γ)p = 2βγ. All the given probabilities are assumed to lie in the interval (0, 1). Then, (Probability of occurrence of E₁) / (Probability of occurrence of E₃) is equal to.

If (2021)³⁷⁶² is divided by 17, then the remainder is .

The sum of possible values of x for tan^-1(x + 1) + cot^-1(1/(x - 1)) = tan^-1(8/31) is:

The line 2x - y + 1 = 0 is a tangent to the circle at the point (2, 5) and the centre of the circle lies on x - 2y = 4. Then, the radius of the circle is:

The value of 4 + 1 / (5 + 1 / (4 + 1 / (5 + 1 / (4 + .∞)))) is:

If the fourth term in the expansion of (x + xˡᵒᵍ²ˣ)⁷ is 4480, then the value of x where x ∈ N is equal to:

Choose the incorrect statement about the two circles whose equations are given below:
x² + y² - 10x - 10y + 41 = 0 and x² + y² - 16x - 10y + 80 = 0

In a triangle PQR, the co-ordinates of the points P and Q are (-2, 4) and (4, -2) respectively. If the equation of the perpendicular bisector of PR is 2x - y + 2 = 0, then the centre of the circumcircle of the ΔPQR is:

Suppose you have taken a dilute solution of oleic acid in such a way that its concentration becomes 0.01cm³ of oleic acid per cm³ of the solution. Then you make a thin film of this solution (monomolecular thickness) of area 4cm² by considering 100 spherical drops of radius ³√(3/(40π)) * 10⁻³ cm. Then the thickness of oleic acid will be x * 10⁻¹⁴ m. Where x is ______.

In a school, there are three types of games to be played. Some of the students play two types of games, but none play all the three games. Which Venn diagrams can justify the above statement?