Explanation

Given A = {x | x ∈ P ∧ 5 ≤ x ≤ 13}, where P is the set of prime numbers:

The prime numbers between 5 and 13 are 5, 7, 11, and 13.

So, A = {5, 7, 11, 13}, which has 4 elements.

The power set P(A) has 2^n elements, where n is the number of elements in A:

2^4 = 16

Explanation

Given [ ± x] = x:

This implies the absolute value or a similar concept where ±x results in a positive x.

Explanation

Use integration by parts:
Let $u=x$, $dv=\cos (x)dx$

Then:

$$\int x\cos (x)dx=x\sin (x)-\int \sin (x)dx=x\sin (x)+\cos (x)$$

Evaluate from $0$ to π/2

$$= left( frac{pi}{2} cdot 1 + cosleft( frac{pi}{2} right) right) - left( 0 + cos(0) right) = frac{pi}{2} - 1$$

π/2​−1

Explanation

Given 5x - 8 = -23:

Adding 8 to both sides gives:

5x = -15

Dividing both sides by 5 gives:

x = -15/5

x = -3

Explanation

Given y = a²:

In logarithmic form, this can be written as:

logₐ(y) = 2

Explanation

In the expression x√a, 'a' is called the radicand.

The radical symbol (✓) indicates a root, and the number or expression inside it (in this case, 'a') is the radicand.

Explanation

Given (2x, x + y) = (6, 2):

From the first component, 2x = 6, so x = 3.

From the second component, x + y = 2. Substituting x = 3:

3 + y = 2

y = -1

Explanation

(-2)⁴ means (-2) × (-2) × (-2) × (-2) = 16

-2⁴ means -(2⁴) = -16

So:

(-2)⁴ > -2⁴

Explanation

Given the expression (A and x∈B) represents the condition where x belongs to both A and B:

This directly implies the intersection of sets A and B.

Explanation

Given cosec(90 - θ):

Using trigonometric identities, cosec(90 - θ) = sec(θ).