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.

Explanation

The given inequality is 3 < x < 7, which can be rewritten as |x-5|<2.

Explanation

To solve the inequality x−3/x+7 < 0, we need to find where the expression is negative.

We first determine the critical points by setting the numerator and denominator equal to zero:

x−3 = 0 gives x = 3, and x+7 = 0 gives x = −7.

Now, we create a sign chart with these critical points and test intervals to see where the expression is negative:

Test x = −8 (a number less than -7): −8−3/-8+7 = −11/-1 ​= 11>0 (positive)

Test x = 0 (a number between -7 and 3): 0−3/0+7 = −3/7 <0 (negative)

Test x = 4 (a number greater than 3): 4−3/ 4+7 = 1/11 >0 (positive)

From this, we see the expression is negative when x is between -7 and 3, so the solution is (−7,3).