Explanation

(a) Communism is a political and economic system in which the means of production, distribution and exchange are commonly owned and equally distributed. It is based on principle of for each according to his capacity and to each according to his needs. It is the last stage of socialism.

(b)(i) The means of production, distribution and exchange are commonly owned.

(ii) It is a classless society

(iii) Opposition not tolerated

(iv) The economy is collectively planned

(v) State withers away

(vi) Full employment

(vii) Decisions are taken by consensus.

Correct Answer: D. protein

Correct Answer: A. Customs Authority

Correct Answer: B. 252°

Explanation

Let 0 = angle of the minor sector

angle of the major sector = 360 - \(\theta\)(angle at a point)

2 \(\pi r\) = 54 + 126(i.e circumference of minor and major arc)

2\(\pi r = 180^o\)

r = \(\frac{180}{2\pi}\) = \(\frac{90}{\pi}\)

Lenght of ninor arc

= \(\frac{\theta}{360} \times 2 \pi r\)

54 = \(\frac{\theta}{360} \times 3 \pi r\)

\(\theta = \frac{360 \times 54}{2 \pi r}\)

but r = \(\frac{90}{\pi}\) substituting \(\frac{90}{\pi}\) for r

\(\theta = \frac{360 \times 54 \times \pi}{2 \times \pi \times 90}\)

\(\theta = 2 \times 54 = 108^o\)

angle of the major sector = 360 - 108^o

= 252^o

Explanation

(a) \(\int_{1} ^{2} \frac{x}{\sqrt{5 - x^{2}}} \mathrm {d} x\)

Let \(u^{2} = 5 - x^{2}\).

\(2u \mathrm {d} u = - 2x \mathrm {d} x \implies u \mathrm {d} u = - x \mathrm {d} x\)

\(\int \frac{x}{\sqrt{5 - x^{2}}} \mathrm {d} x = \int \frac{- u}{\sqrt{u^{2}}} \mathrm {d} u\)

= \(- \int \mathrm {d} u \)

= \(-[5 - x^{2}]_{1} ^{2}\)

= \((- (5 - 2^{2}) - (- (5 - 1^{2})))\)

= \(- 1 + 4 = 3\)

(b)(i) \(\begin{vmatrix} 2 & -3 & 1 \\ 0 & 1 & -2 \\ 1 & 2 & -3 \end{vmatrix}\)

= \(2(-3 + 4) + 3(0 + 2) + 1(0 - 1) = 7\)

(ii) \(2x - 3y + z = 10\)

\(0x + y - 2z = -7\)

\(x + 2y - 3z = -9\)

Let A be the matrix of coefficients.

\(A = \begin{pmatrix} 2 & -3 & 1 \\ 0 & 1 & -2 \\ 1 & 2 & -3 \end{pmatrix}\)

We write the set of equations in matrix form, i.e Ax = B.

\(\begin{pmatrix} 2 & -3 & 1 \\ 0 & 1 & -2 \\ 1 & 2 & -3 \end{pmatrix} \begin{pmatrix} x \\ y \\ z \end{pmatrix} = \begin{pmatrix} 10 \\ -7 \\ -9 \end{pmatrix}\).

We find the inverse of A, A\(^{-1}\).

Determinant of A = 7.

Co-factors:

\(A_{11} = + \begin{vmatrix} 1 & -2 \\ 2 & -3 \end{vmatrix} = +(-3 + 4) = 1\)

\(A_{12} = - \begin{vmatrix} 0 & -2 \\ 1 & 2 \end{vmatrix} = -(0 + 2) = -2\)

\(A_{13} = + \begin{vmatrix} 0 & 1 \\ 1 & 2 \end{vmatrix} = +(0 - 1) = -1\)

\(A_{21} = - \begin{vmatrix} -3 & 1 \\ 2 & -3 \end{vmatrix} = -(9 - 2) = -7\)

\(A_{22} = + \begin{vmatrix} 2 & 1 \\ 1 & -3 \end{vmatrix} = +(-6 - 1) = -7\)

\(A_{23} = - \begin{vmatrix} 2 & -3 \\ 1 & 2 \end{vmatrix} = -(4 - (-3)) = -7\)

\(A_{31} = + \begin{vmatrix} -3 & 1 \\ 1 & -2 \end{vmatrix} = +(6 - 1) = 5\)

\(A_{32} = - \begin{vmatrix} 2 & 1 \\ 0 & -2 \end{vmatrix} = -(-4 - 0) = 4\)

\(A_{33} = + \begin{vmatrix} 2 & -3 \\ 0 & 1 \end{vmatrix} = +(2 - 0) = 2\)

\(\therefore C = \begin{pmatrix} 1 & -2 & -1 \\ -7 & -7 & -7 \\ 5 & 4 & 2 \end{pmatrix}\)

\(\text{adj A} = C^{T} = \begin{pmatrix} 1 & -7 & 5 \\ -2 & -7 & 4 \\ -1 & -7 & 2 \end{pmatrix}\)

\(A^{-1} = \frac{\text{adj A}}{|A|} = \frac{1}{7} \begin{pmatrix} 1 & -7 & 5 \\ -2 & -7 & 4 \\ -1 & -7 & 2 \end{pmatrix}\)

\(x = A^{-1} b = \frac{1}{7} \begin{pmatrix} 1 & -7 & 5 \\ -2 & -7 & 4 \\ -1 & -7 & 2 \end{pmatrix} \begin{pmatrix} 10 \\ -7 \\ -9 \end{pmatrix}\)

= \(\frac{1}{7} \begin{pmatrix} 14 \\ -7 \\ 21 \end{pmatrix}\)

\(\begin{pmatrix} x \\ y \\ z \end{pmatrix} = \begin{pmatrix} 2 \\ -1 \\ 3 \end{pmatrix}\)

\(x = 2; y = -1 ; z = 3\)

Correct Answer: A. introduction-selection-breeding

Correct Answer: A. retract

Correct Answer: C. 4.4 cm

Explanation

Length = \(\frac{\theta}{360}\) x 2\(\pi\)r

= \(\frac{72}{360} \times 2 \frac{22}{7} \times \frac{0.5}{3.5}\)

= 4.4

Correct Answer: C. commercialization

Correct Answer: A. land