31. For two vectors \( \vec{A} \) and \( \vec{B} \), if \( \vec{A} \times \vec{B}=6 \hat{\imath}+2 \hat{j}+5 \hat{k}, \vec{A} \cdot \vec{B}=-13, \vec{A}+\vec{B}=2 \hat{i}+\hat{j}+2 \vec{k} \) then find: \( |\vec{A}|=3 \), the vectors \( \vec{A} \) and \( \vec{B} \) a) The two b) The area of the parallelogram and the triangle having \( \vec{A} \) and \( \vec{B} \) as their edge. c) A third vector having magnitude twice \( \vec{A} \) and direction perpendicular to both vectors.

Explain the difference between exponential growth and exponential decay.

Explain the difference between exponential growth and exponential decay.

[The most general such \( R \) is \( U \sqrt{A} \), where \( U \) is some isometry.] 6- Let \( x_{1}, \ldots, x_{n} \) be \( n \) vectors. Let \( G \) be a matrix with elements \( G_{i j}=\left(x_{i}, x_{j}\right) \). Show that \( G \) is a positive matrix. [ \( G \) is called the Gram matrix.]

75. Questions in which only two alternatives are possible is called A. Multiple choice questions B. Dichotomous questions C. Open ended questions D. Structured questions

68. Which step in the market research process allows conclusions to be drawn an recommendations to be formulated? A. Data collection B. Data analysis C. Qualitative analysis D. Quantitative analysis