NIMCET 2022 — Mathematics PYQ

Step-by-step Derivations 
Step 1: Define Variables and Set up Equations 
Let $h$ be the height of the tower. Let $D$ be the foot of the tower. Let $A$, $B$, and $C$ be the three collinear points on the road leading to the foot of the tower. The angles of elevation from $A$, $B$, and $C$ are $30^{\circ }$, $45^{\circ }$, and $60^{\circ }$, respectively. Let $CD=x$, $BD=y$, and $AD=z$. From the properties of right-angled triangles, the following relationships are established: For point $C$: $\tan (60^{\circ })=\frac{h}{x}⟹h=x\tan (60^{\circ })=x\sqrt{3}$. For point $B$: $\tan (45^{\circ })=\frac{h}{y}⟹h=y\tan (45^{\circ })=y(1)=y$. For point $A$: $\tan (30^{\circ })=\frac{h}{z}⟹h=z\tan (30^{\circ })=\frac{z}{\sqrt{3}}$. [1]  
Step 2: Express Distances in Terms of Height 
From the equations in Step 1, the distances from the foot of the tower to the points are expressed in terms of $h$: $x=\frac{h}{\sqrt{3}}$. $y=h$. $z=h\sqrt{3}$. 
Step 3: Calculate AB and BC 
The distances $AB$ and $BC$ are calculated as follows: $BC=BD-CD=y-x=h-\frac{h}{\sqrt{3}}=h\left(1-\frac{1}{\sqrt{3}}\right)=h\left(\frac{\sqrt{3}-1}{\sqrt{3}}\right)$. $AB=AD-BD=z-y=h\sqrt{3}-h=h(\sqrt{3}-1)$. 
Step 4: Determine the Ratio AB:BC 
The ratio of $AB$ to $BC$ is determined by dividing the expression for $AB$ by the expression for $BC$: $\frac{AB}{BC}=\frac{h(\sqrt{3}-1)}{h\left(\frac{\sqrt{3}-1}{\sqrt{3}}\right)}=\frac{1}{\frac{1}{\sqrt{3}}}=\sqrt{3}$. Therefore, the ratio $AB:BC=\sqrt{3}:1$. 
Final Answer 
The final answer is \boxed{\text{√3: 1}}.