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Q21. The number of diagonals of a convex deodecagon (12-gon) is:
The formula for the number of diagonals in an n-sided polygon is n(n-3)/2. For n=12, it is 12(9)/2 = 54.
Q22. For which one of the following statements is the converse NOT true?
The converse is 'If a person has good qualifications, he gets employed.' This is objectively not always true (qualified people can be unemployed). The converses of the mathematical statements are absolute truths.
Q23. 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.
Q24. 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.
Q25. Suppose three meetings of a group of professors were arranged in Mumbai, Delhi and Chennai. Each professor attended exactly two meetings. 21 attended Mumbai, 27 attended Delhi and 30 attended Chennai. How many attended both Chennai and Delhi?
Let x, y, and z be the number of professors attending (Mumbai & Delhi), (Delhi & Chennai), and (Mumbai & Chennai) respectively. We have: x + z = 21, x + y = 27, and y + z = 30. Adding these gives 2(x + y + z) = 78, so x + y + z = 39. To find Delhi & Chennai (y), we calculate 39 - (x + z) = 39 - 21 = 18.
Q26. A woman starts shopping with Rs. X and Y paise, spends Rs. 3.50 and is left with Rs. 2Y and 2X paise. The amount she started with is:
Let the initial amount be (100X + Y) paise. She spends 350 paise. Remaining amount = (200Y + 2X) paise. Equation: 100X + Y - 350 = 200Y + 2X. Simplifying gives 98X - 199Y = 350. By substituting the options, if X=32 and Y=14, we get 98(32) - 199(14) = 3136 - 2786 = 350, which satisfies the equation.
Q27. Income tax is R% for the first 2 lakhs and (R+10)% for income exceeding 2 lakhs. If total tax is (R+5)% of annual income, what is the annual income?
Let income be I. Tax = 2(R/100) + (I-2)(R+10)/100 = I(R+5)/100. Solving: 2R + (I-2)(R+10) = I(R+5) => 2R + IR + 10I - 2R - 20 = IR + 5I => 5I = 20 => I = 4.
Q28. Abdul travels thrice the distance Catherine travels and twice the distance Binoy travels. Catherine's speed is 1/3 of Abdul's and 1/2 of Binoy's. Who reaches first?
Let Abdul's distance be 6D and speed be 3S. Then Catherine's distance is 2D and speed is S. Binoy's distance is 3D and speed is 2S. Time taken = Distance/Speed. Abdul: 6D/3S = 2D/S. Catherine: 2D/S. Binoy: 3D/(1.5S) = 2D/S. All reach at the same time.
Q29. The houses of three sisters lie in the same row, but the middle sister does not live in the middle house. In the morning, the shadow of the eldest sister's house falls on the youngest sister's house. What can be concluded for sure?
In the morning, shadows fall to the West. If the Eldest's shadow falls on the Youngest, the Eldest is directly East of the Youngest (Order: Eldest -> Youngest). Since the Middle sister cannot be in the middle house, the order of the three houses must be either (Middle, Eldest, Youngest) or (Eldest, Youngest, Middle). In both valid scenarios, either the Eldest or the Youngest occupies the middle house.
Q30. 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.