Variance is the square of the standard deviation. Therefore, variance = 7² = 49.

Variance is the square of the standard deviation. Therefore, variance = 3² = 9.

If all squares are rectangles, it implies that there are rectangles that are not squares, thus some rectangles are not squares.

The conclusion 'Some mammals are cats' is valid because if all cats are mammals, then it follows that at least some mammals must be cats.

The conclusion 'Some mammals are cats' is valid because all cats being mammals implies that there are mammals that are cats.

Since all cats are mammals and some mammals are not dogs, it follows that some cats are not dogs.

Stress = Young's modulus × Strain = 150 GPa × 0.01 = 150 MPa.

According to Hooke's Law, if stress is doubled, strain also doubles for elastic materials.

The sum S of an infinite GP is given by S = a / (1 - r). Here, 20 = a / (1 - 1/4) = a / (3/4). Thus, a = 20 * (3/4) = 15.

The sum of an infinite GP is given by S = a / (1 - r). Here, S = 10 and r = 1/3. Thus, 10 = a / (1 - 1/3) = a / (2/3) => a = 10 * (2/3) = 20.

Let the son's age be x. Then father's age = 4x. According to the problem, x + 4x = 50, which gives 5x = 50, so x = 10.

A triangle with one angle measuring 90 degrees is classified as a right triangle, as it adheres to the property of having one angle equal to 90 degrees.

In an equilateral triangle, each angle is equal, so each angle = 180/3 = 60 degrees.

The third angle can be found by subtracting the sum of the other two angles from 180 degrees: 180 - (50 + 70) = 60 degrees.

A triangle with one angle measuring 90 degrees is defined as a right triangle.

Let the two-digit number be 10a + b, where a is the tens digit and b is the units digit. We have a + b = 9 and 10a + b = 4(a + b). Solving these gives a = 4 and b = 5, so the number is 45.

The factors of 12 are 1, 2, 3, 4, 6, and 12. Their sum is 28.

Using the formula for the sum of a geometric series, S_n = a(1 - r^n) / (1 - r), we can solve for r. Here, S_5 = 1(1 - r^5) / (1 - r) = 31, leading to r = 3.

The sum S_5 = 5/2 * (2a + 4d). Here, 50 = 5/2 * (2a + 8), solving gives a = 10.

The average of the first n terms is given by S_n/n. Here, S_5 = 50, so the average = 50/5 = 10.

The sum of the first n natural numbers is given by the formula n(n + 1)/2 = 210. Solving n(n + 1) = 420, we find n = 14 or n = 15. Thus, n = 15.

The sum of the first n natural numbers is given by the formula n(n + 1)/2 = 255. Solving n(n + 1) = 510, we find n = 16.

The sum of the first n natural numbers is given by the formula n(n + 1)/2. Setting this equal to 2550, we have n(n + 1) = 5100. Solving the quadratic equation n^2 + n - 5100 = 0 gives n = 50.

The sum of the first n natural numbers is given by the formula n(n + 1)/2. Setting this equal to 5050, we have n(n + 1)/2 = 5050. Solving for n gives n = 100.

n(n + 1)/2 = 55, solving gives n = 10.

Using the formula S_n = a(1 - r^n) / (1 - r), we have 63 = 3(1 - r^4) / (1 - r). Solving gives r = 2.

The sum S_n can be negative if the common ratio r is negative and the first term a is also negative.

As n approaches infinity and |r| < 1, r^n approaches 0, thus S_n approaches a/(1-r).

Using the formula S_n = a(1 - r^n) / (1 - r), we have 81 = 3(1 - r^n) / (1 - r). Solving gives r = 3.

When r = 1, S_n = a(1 - 1^n)/(1 - 1) is indeterminate, but the sum of n terms is na.