If the force \( \overrightarrow{\mathrm{F}_{1}}=(3,-1) \) acts at the point \( \mathrm{A}(1,2), \overrightarrow{\mathrm{F}_{2}} \) acts at the point \( \mathrm{B}(-1,1) \) and the two forces form a couple, then the algebraic measure of the moment of the couple \( = \) \( \qquad \) moment unit. (a) 5 (b) 2 (c) -5 (d) -2 In the opposite figure : A fine lamina of a uniform thickness and density in the form of a right-angled triangle at B and of weight \( 30 \mathrm{~kg} . \mathrm{wt} \). where \( \mathrm{AB}=9 \mathrm{~cm} ., \mathrm{BC}=6 \mathrm{~cm} \). the lamina is suspended in a pin at a small hole near to the vertex \( B \), a couple in its plane acted on it to make it in equilibrium when \( \overline{\mathrm{AB}} \) is horizontal, then the algebraic measure of the moment of the couple \( = \) \( \qquad \) kg.wt.cm. (a) 135 (b) -90 (c) -135 (d) 90

To find the moment of the couple formed by the forces \( \overrightarrow{\mathrm{F}_{1}} \) and \( \overrightarrow{\mathrm{F}_{2}} \), we need to first calculate the position vectors for both points A and B. The position vector from A to B is \( \overrightarrow{AB} = (-1-1, 1-2) = (-2, -1) \). Next, we compute the moment by taking the cross product of the position vector with one of the forces, which gives: \[ \mathrm{M} = \overrightarrow{AB} \times \overrightarrow{\mathrm{F}_{1}} = (-2, -1) \times (3, -1) = (-2)(-1) - (-1)(3) = 2 + 3 = 5 \quad \text{(moment unit)} \] Thus, the algebraic measure of the moment of the couple is **5**. For the lamina problem, considering it is in the shape of a right-angled triangle with sides \( \mathrm{AB} = 9 \mathrm{~cm} \) and \( \mathrm{BC} = 6 \mathrm{~cm} \), we can find the centroid, which acts at \( \frac{1}{3} \) of the height from the base BC and \( \frac{1}{3} \) of the base length from A. The weight \( W \) of the lamina is \( 30 \mathrm{~kg} \), and the moment about point B when the line AB is horizontal is given by \( \text{Moment} = W \cdot d \), where \( d \) is the distance from B to the centroid along the vertical axis. After calculating, we find \( d \) to be \( 90 \) kg.wt.cm. Thus, the algebraic measure of the moment of the couple is **90**.