For equal roots, the discriminant must be zero: k^2 - 4*1*8 = 0, solving gives k = 4.

Using the product of the roots, c = 3 * (-2) = -6.

The condition for equal roots is given by the discriminant b^2 - 4ac = 0.

The sum of the roots is 3 + 4 = 7, hence m = -7.

For equal roots, the discriminant must be zero: p^2 - 4q = 0, hence p^2 = 4q.

Using Vieta's formulas, sum of roots = 1 + 2 = 3 and product of roots = 1*2 = 2. Thus, p = 2.

If A · B = 0, then the vectors are orthogonal.

A · B = 2x - y + 3z = 10.

Mode is the value that appears most frequently. Here, 50, 60, 70, and 80 all appear twice, but 50 is the smallest.

Mean = (10 + 20 + 30 + 40 + x) / 5 = 30. Solving gives x = 50.

Arranging the scores: 50, 60, 70, 80, 90, 100, 110. Median = 80 (4th value).

All scores occur only once, so there is no mode.

Arranging the scores: 60, 70, 80, 90, 100. Median = 80 (middle value).

Median = 80 (middle value when arranged: 60, 70, 80, 90, 100).

Perimeter = a + b + c = 7 + 24 + 25 = 56 cm.

A triangle with sides in the ratio 3:4:5 is a right triangle, as it satisfies the Pythagorean theorem.

A standard deviation of 0 indicates that all values in the data set are the same.

Variance = (Standard Deviation)^2 = 3^2 = 9.

Variance = (Standard Deviation)² = 4² = 16.

Variance is the square of the standard deviation. Variance = 5² = 25.

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.

The 4th term a_4 = S_4 - S_3 = (3(4^2) + 2(4)) - (3(3^2) + 2(3)) = (48 + 8) - (27 + 6) = 56 - 33 = 23.

The 5th term can be found using a_n = S_n - S_(n-1). Calculate S_5 and S_4, then find a_5 = S_5 - S_4 = (5(5^2) + 3(5)) - (5(4^2) + 3(4)) = 38.

The sum of the roots is given by -b/a = 3. Here, -(-3)/1 = 3, so p can be any value.

The complement of A, denoted A', includes all elements in U that are not in A. Thus, A' = {3, 4, 5}.

Standard deviation = √variance = √16 = 4.

Cosine of angle = (A · B) / (|A| |B|) = (1*2 + 2*1) / (√5 * √5) = 4/5, angle = cos^(-1)(4/5).

cos(θ) = (a · b) / (|a| |b|). a · b = 1*3 + 2*4 = 11, |a| = √(1^2 + 2^2) = √5, |b| = √(3^2 + 4^2) = 5. Thus, cos(θ) = 11 / (√5 * 5) = 11 / (5√5), θ = 60 degrees.

a + b = (2 + 1, -1 + 3) = (3, 2)