Explanation

The formula p ∨ (p ∧ q) can be simplified using the distributive property of ∨ over ∧, which states that p ∨ (q ∧ r) = (p ∨ q) ∧ (p ∨ r).

Applying this property, we get:

p ∨ (p ∧ q) = (p ∨ p) ∧ (p ∨ q) = p ∧ (p ∨ q) = p

Explanation

Perplexed means پریشان

Puzzle means پریشان

Explanation

Elated means پرجوش

Thrilled means پرجوش

Overjoyed بے حد خوشی ہوئی۔

Explanation

¬p (read as "not p") represents the negation of statement p.

In this case, p is the statement "x + y = 3".

The negation of this statement would be "x + y ≠ 3", which means any situation where x + y is not equal to 3.

Explanation

The degree of a function is the highest power of the variable(s) in the function. In this case, the function is: f(x, y) = √x + √y / (x + y)

The highest power of x and y in the function is 1/2, which comes from the terms √x and √y.

Explanation

The given expression is: (x^3 + 2x^2y - xy - 2y^2) / (x + 2y)

To evaluate the limit, we can substitute x = 2 and y = 1: (2^3 + 2(2)^2(1) - 2(1) - 2(1)^2) / (2 + 2(1)) = (8 + 8 - 2 - 2) / 4 = 12 / 4 = 3.

Explanation

This is a alternating series with terms of the form x^n / n, which converges for -1 < x ≤ 1.

Explanation

The area of the circle with radius 10 cm is A₁ = π(10)² = 100π cm².

The area of the circle with radius 10.1 cm is A₂ = π(10.1)² = 102.01π cm².

The percentage change in area is:

((A₂ - A₁) / A₁) × 100%

= ((102.01π - 100π) / 100π) × 100%

= (2.01 / 100) × 100%

= 4%

So, the percentage change in the area is 4%.

Explanation

To evaluate the limit, we can start by rewriting the expression as:

lim (sin(2x) + a*sin(x)) / x^3

As x approaches 0, the sine functions can be approximated by their arguments, so we get:

lim (2x + a*x) / x^3 = lim (2 + a) / x^2

For this limit to exist, the numerator must be 0, so we set 2 + a = 0, which gives a = -2.

Explanation

The Cauchy Mean Value Theorem states that if functions f(x) and g(x) are continuous on the closed interval [a, b] and differentiable on the open interval (a, b), then there exists a point c in (a, b) such that: (f(b) - f(a))g'(c) = (g(b) - g(a))f'(c)

In this case, f(x) = x², g(x) = x², and the interval is [1, 2]. Plugging in the values, we get: (4 - 1)(2c) = (4 - 1)(2c)

Simplifying, we get: c = 14/9

So, the value of c is 14/9.