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Q21. A notebook contains 100 statements. Statement n says 'This notebook contains n false statements'. Which statement is correct?
If statement 99 is true, there are 99 false statements and 1 true statement. This is consistent because if any other statement were true, it would contradict the count of false statements.
Q22. A boy holds one end of a rope of length l and the other end is fixed to a thin pole of radius r (r<<l). Keeping the rope taut, the boy goes around the pole causing the rope to get wound around the pole. Each round takes 10 s. What is the speed with which the boy approaches the pole?
With each full revolution around the pole, the rope wraps around the circumference of the pole, shortening the distance between the boy and the pole by 2*pi*r. Since one round takes 10 seconds, the speed of approach is Distance / Time = (2*pi*r) / 10 = pi*r / 5.
Q23. A wheel barrow with unit spacing between its wheels is pushed along a semi-circular path of mean radius 10. The difference between distances covered by the inner and outer wheels is:
With a unit spacing (1) and mean radius 10, the outer radius is 10.5 and the inner is 9.5. Distance for a semi-circle is πr. Difference = π(10.5) - π(9.5) = π(10.5 - 9.5) = π.
Q24. Two runners A and B start running from diametrically opposite points on a circular track in the same direction. If A runs at a constant speed of 8 km/h and B at a constant speed of 6 km/h and A catches up with B in 30 minutes, what is the length of the track?
A runs 2 km/h faster than B. In 30 minutes (0.5 hours), A closes a distance of 2 km/h * 0.5 h = 1 km. Since they started diametrically opposite, this 1 km represents exactly half the length of the circular track. Therefore, the full track is 2 km long.
Q25. A bicycle tube has a mean circumference of 200 cm and a circular cross-section diameter of 6 cm. What is the approximate volume of water required to fill it?
Volume of a torus = (Area of cross-section) * (Mean circumference). Radius of cross-section = 3 cm. Area = pi * 3^2 = 9pi. Volume = 9pi * 200 = 1800 pi.
Q26. A person purchases two chains from a jeweller, one weighing 18 g made of 22 carat gold and another weighing 22 g made of 18 carat gold. Which one of the following statements is correct?
Pure gold is 24 carats. Gold in chain 1: 18g * (22/24) = 16.5g. Gold in chain 2: 22g * (18/24) = 16.5g. Both chains contain exactly the same absolute mass of pure gold.
Q27. A train running at 36 km/h crosses a mark on the platform in 8 sec and takes 20 sec to cross the platform. What is the length of the platform?
Speed = 36 km/h = 10 m/s. Train length = Speed * time to cross mark = 10 * 8 = 80m. Total distance to cross platform = Speed * 20 = 200m. Platform length = 200m - 80m = 120m.
Q28. A 2 m long ladder is to reach a wall of height 1.75 m. The largest possible horizontal distance of the ladder from the wall could be:
The ladder forms the hypotenuse of a right-angled triangle. Using Pythagoras theorem: Horizontal Distance = sqrt(2^2 - 1.75^2) = sqrt(4 - 3.0625) = sqrt(0.9375). Since sqrt(1) is 1, sqrt(0.9375) must be slightly less than 1 meter.
Q29. A rectangular flask of length 11 cm, width 8 cm and height 20 cm has water filled up to height 5 cm. If 21 spherical marbles of radius 1 cm each are dropped in the flask, what would be the rise in water level?
The volume of a single spherical marble is (4/3)*pi*r^3 = (4/3)*(22/7)*(1)^3 = 88/21 cm^3. The volume of 21 marbles is 21 * (88/21) = 88 cm^3. This displaced volume causes the water to rise by a height 'h'. The volume of the displaced water column is Base Area * h = 11 * 8 * h = 88h. Equating the two: 88h = 88, so h = 1 cm.
Q30. There are small and large bacteria of the same species. If S is surface area and V is volume, then which of the following is correct?
For any geometric shape, Surface Area (S) scales with the square of the radius (r^2) while Volume (V) scales with the cube (r^3). Therefore, the surface-to-volume ratio (S/V) scales inversely with size (1/r). As an organism gets smaller, its surface-area-to-volume ratio increases.