Consider a rectangular cardboard box of height 3, breadth 4 and length 10 units. There is a lizard in one corner $A$ of the box and an insect in the corner $B$ which is farthest from $A$. The length of the shortest path between the lizard and the insect along the surface of the box is
(a) $\sqrt{5^{2} + 10^{2}}$
(b) $\sqrt{7^{2} + 10^{2}}$
(c) $4 + \sqrt{3^{2} + 10^{2}}$
(d) $3 + \sqrt{4^{2} + 10^{2}}$

A room is in the shape of a rectangular box. The shortest path along the surface from one corner $A$ to the opposite corner $B$ has length $\sqrt{29}$ (given the relevant dimensions are $5$ and $2$). Find this shortest distance.

Two persons, both of height $h$, are standing at a distance of $h$ from each other. The shadow of one person cast by a vertical lamp-post placed between the two persons is double the length of the shadow of the other. If the sum of the lengths of the shadows is $h$, then the height of the lamp post is
(A) $\frac{\sqrt{3}}{2}h$
(B) $2h$
(C) $\left(\frac{1 + \sqrt{2}}{2}\right)h$
(D) $\left(\frac{\sqrt{3} + 1}{2\sqrt{2}}\right)h$.

Between 12 noon and 1 PM, there are two instants when the hour hand and the minute hand of a clock are at right angles. The difference in minutes between these two instants is:
(A) $32 \frac { 8 } { 11 }$
(B) $30 \frac { 8 } { 11 }$
(C) $32 \frac { 5 } { 11 }$
(D) $30 \frac { 5 } { 11 }$.

A straight road has walls on both sides of height 8 feet and 4 feet respectively. Two ladders are placed from the top of one wall to the foot of the other as in the figure below. What is the height (in feet) of the maximum clearance $x$ below the ladders?
(A) 3
(B) $2 \sqrt { 2 }$
(C) $\frac { 8 } { 3 }$
(D) $2 \sqrt { 3 }$

Two ships are approaching a port along straight routes at constant velocities. Initially, the two ships and the port formed an equilateral triangle. After the second ship travelled 80 km, the triangle became right-angled. When the first ship reaches the port, the second ship was still 120 km from the port. Find the initial distance of the ships from the port.
(A) 240 km
(B) 300 km
(C) 360 km
(D) 180 km

If two vertical poles 20 m and 80 m high stand apart on a horizontal plane, then the height (in m) of the point of intersection of the lines joining the top of each pole to the foot of other is
(1) 16
(2) 18
(3) 50
(4) 15

Two vertical poles of height, $20 m$ and $80 m$ stand apart on a horizontal plane. The height (in meters) of the point of intersection of the lines joining the top of each pole to the foot of the other, from this horizontal plane is:
(1) 16
(2) 12
(3) 18
(4) 15

In a shooting game, a player must avoid obstacles to shoot a target. A rectangular coordinate system is set up on the game screen with the lower left corner $O$ of the screen as the origin, the lower edge of the screen as the $x$-axis, and the left edge of the screen as the $y$-axis. The target is placed at point $P ( 12,10 )$. There are two walls in the screen (wall thickness is negligible), one wall extends horizontally from point $A ( 10,5 )$ to point $B ( 15,5 )$, and another wall extends horizontally from point $C ( 0,6 )$ to point $D ( 9,6 )$, as shown in the schematic diagram on the right. If a player at point $Q$ can shoot the target at point $P$ in a straight line without being blocked by the two walls, which of the following options could be the coordinates of point $Q$?
(1) $( 6,3 )$
(2) $( 7,3 )$
(3) $( 8,5 )$
(4) $( 9,1 )$
(5) $( 9,2 )$