MAT Exam 2023 Paper

Concept:Use $\tan \theta = \frac{\text{opposite}}{\text{adjacent}}$ by taking the horizontal distance between the wall and the tree as the base.Explanation:Let the horizontal distance between the wall and the tree be $x$.Let the height of the tree be $h$.The tree is taller than the wall, so the vertical difference between their tops is $h - 24$.From the bottom of the tree, the top of the wall is seen at an angle of elevation $30^{\circ}$.Therefore, $\tan 30^{\circ} = \frac{24}{x}$.This gives $x = 24\sqrt{3}$. From the top of the tree, the top of the wall is seen at an angle of depression $60^{\circ}$.So, $\tan 60^{\circ} = \frac{h-24}{x}$.Substitute $x = 24\sqrt{3}$:$$\sqrt{3} = \frac{h-24}{24\sqrt{3}}$$$$h - 24 = 24\sqrt{3} \times \sqrt{3} = 72$$$$h = 96 \text{ m}$$Also, $x = 24\sqrt{3} = 24 \times 1.732 = 41.56$ m.Thus, the required distance is $41.56$ m and the height of the tree is $96$ m.Answer:Option B: $41.56$ m and $96$ m.