5. The beam AB shown in the Fig-5 supports a uniform load of intensity 3000 N/m acting over half the length of the beam. The beam rests on a soil layer that produces a uniformly distributed pressure load over the entire length. Draw the shear-force and bending-moment diagrams for this beam. (20 %).

7. In $\mathbb{R}^4$, let $w_1 = (1, 0, 1, 0)$, $w_2 = (0, 1, 0, 1)$, $w_3 = (1, 0, 0, 0)$. Prove that $\{w_1, w_2, w_3\}$ is linearly independent. Use the Gram-Schmidt process to compute the orthogonal vectors $v_1, v_2$ and $v_3$ such that $span\{w_1, w_2, w_3\} = span\{v_1, v_2, v_3\}$ and then normalize these vectors to obtain an orthonormal set. (15%)

5. The beam AB shown in the Fig-5 supports a uniform load of intensity 3000 N/m acting over half the length of the beam. The beam rests on a soil layer that produces a uniformly distributed pressure load over the entire length. Draw the shear-force and bending-moment diagrams for this beam. (20 %).

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