10. Prove that a finite set of points Z1, Z2, Zn cannot have any accumulation points.

Determine whether the following sets are linearly dependent or linearlyindependent. (a) {(; )( )} . -9} in Maxa(R) 4 (b) {(- ).(- 9G -)} in M2x2(R) 4 (c) {x 3 + 2x2, -x2 + 3x + 1, x3 – x2 + 2x – 1} in P3(R) (d) {x 3 – x, 2x2 + 4, -2x3 + 3x2 + 2x + 6} in P3(R) (e) {(1, -1, 2), (1, -2, 1), (1, 1, 4)} in R3 (f) {(1, –1, 2), (2, 0, 1), (–1, 2, –1)} in R3 {(÷ ) ( -) (; 0. 2 ).(- )} in Max2(R) (g) (m) {(-; ') (? ). -2)} in M2x2(R) () (x 4 - х3 + 5x2 - 8х + 6, -х4 + х3 - 5х2 + 5х - 3, x 4 + 3x2 – 3x + 5, 2x4 + 3x3 + 4x2 – x + 1, x3 – x + 2} in P4(R) ) (x4 - х3 + 5x2 - 8х + 6, -х4 + x3 - 5x2 + 5х - 3, х4 + 3x2 - 3х + 5, 2х4 + х3 + 4x2 + 8х} in P4(R)

Suppose a firm's technology is represented by the production function f(L, K) = min(L, 2K), where L is the quantity of labor and K is the quantity of capital. Consider the isoquant through input combination (L, K) = (4, 8). Which of the following statements is true? The isoquant through input combination (L, K) = (4, 8) has a corner at A input combination (L, K) = (4, 8). The isoquant through input combination (L, K) = (4, 8) has a corner at input combination (L, K) = (4, 2). %3D %3D The isoquant through input combination (L, K) = (4, 8) has a corner at C input combination (L, K) = (8, 8). The isoquant through input combination (L, K) = (4, 8) has a corner at input combination (L, K) = (8, 4).

7. Is this a linearly independent subset of R³ ? {(1, 4, 6), (1, 5, 8), (2, 1, 1), (0, 1, 0)} Justify your answer. —

3. Determine if the set Q n (0,1) is open, closed or neither. What is the boundary of the set? Also determine the accumulation points of this set. Explain your answers.

For which value of k is { (1, 2, 3), (1, 0, 1), (2, k, 8) } a linearly dependent set?

Suppose a and b are positive integers and 2021 a 2022 2022 b 2023 If s is the minimum value of a+b, what is the last digit of s? (A) 7 (B) 8 (C) 9 (D) 0

Suppose the product of x and y is 49 and both x and y are positive. What is the minimum possible sum of x and y?The minimal possible sum is

3. Express the following regions in the stated coordinate systems. You do not need to justify your answer but showing your though process may help with partial marks. Your answer should be a set in the following form: {(V1, V2, V3) : a ≤ v₁ ≤ b, f(v1) ≤ v2 ≤ g(v1), Ø(V1, V2) ≤ v3 ≤ (v₁, v₂)} where (v₁, V2, V3) are the three variables of the coordinate system in questions in any order, a, b R are constants, f, g are functions of v₁ and 6, are functions of v₁, V2. (a) The region outside the cylinder x² + y² ≤ r² and inside the sphere x² + y² + z² r< R. Express this in the cylindrical and spherical coordinate systems. - R² assuming (b) The region in the first octant (x, y, z ≥ 0) below the plane x + 2y + 3z = 6. Express this in the cylindrical, spherical and Cartesian coordinate system.

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