Explanation

The hyperbolic tangent function tanh(y) is defined as:

tanh(y) = (e^y - e^(-y)) / (e^y + e^(-y))

Explanation

Ravenous means انتہائی بھوک یا لالچی

Voracious means ایک بہت بڑی بھوک یا کسی چیز کی خواہش کا ہونا

Explanation

Given n(n + 1)(n + 5):

1. n(n + 1) is always even (multiple of 2) since it's the product of two consecutive integers.

2. For divisibility by 3:

    If n is a multiple of 3, the expression is divisible by 3.

    If n = 3k + 1, then n + 5 = 3k + 6 is divisible by 3.

    If n = 3k + 2, then n + 1 = 3k + 3 is divisible by 3.

Given these cases, n(n + 1)(n + 5) is divisible by 3.

Explanation

Given the operation a * b = a + b + 1:

Let's find the identity element 'e' such that a * e = a.

a * e = a + e + 1 = a

This implies e + 1 = 0, so e = -1.

Now, to find the inverse 'x' of 'a' such that a * x = e = -1:

a * x = a + x + 1 = -1

This implies a + x = -2.

x = -2 - a

x = -a - 2

Explanation

Onslaught means a fierce or violent attack, which fits well with "by the enemy".

Onslaught means حملہ

Explanation

Given the function f(x) = 3x - 2

This is a linear function with a slope of 3. Since it's linear and defined for all real numbers x, its range is all real numbers.

As x varies from -∞ to ∞, f(x) also varies from -∞ to ∞.

Explanation

Sedulity means کسی کے کام میں تندہی یا مستقل کوشش

Carelessness means توجہ یا کوشش کا فقدان

Explanation

"Annihilate" means to completely destroy, which fits the context of weapons and global threat.

Annihilate means فنا

Explanation

Given y = acos(x) + bsin(x) + 4

First derivative: dy/dx = -asin(x) + bcos(x)

Second derivative: d²y/dx² = -acos(x) - bsin(x)

Notice that d²y/dx² = -(acos(x) + bsin(x))

Given y = acos(x) + bsin(x) + 4, we rearrange to find acos(x) + bsin(x) = y - 4

Substituting into the second derivative equation gives:

d²y/dx² = -(y - 4)

d²y/dx² = -y + 4

d²y/dx² + y = 4

The answer is d²y/dx² + y = 4.

Explanation

Let's directly evaluate the system of equations:

1x + 3y = 1

3x + λy = 3

For infinitely many solutions, these equations must represent the same line. Multiplying the first equation by 3 gives:

3x + 9y = 3

Comparing this with the second equation 3x + λy = 3, we see that λ must equal 9 for the equations to be identical.