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Q1. For a certain regular solid, Faces + Vertices = Edges + 2. For three such distinct objects, what is the total value of Faces + Vertices - Edges?
For one object, F + V - E = 2. For three distinct, non-touching objects, the total sum is 2 + 2 + 2 = 6.
Q2. A polynomial f(x) divided by x-5, x-3, or x-2 leaves a remainder of 1. Which of the following could be the polynomial?
The polynomial is of the form k(x-5)(x-3)(x-2) + 1. Expanding (x-5)(x-3)(x-2) gives x^3 - 10x^2 + 31x - 30. Adding the remainder 1 results in x^3 - 10x^2 + 31x - 29.
Q3. In ponds where there is an overabundance of aquatic plants like water hyacinth, the fish population is found to be low, the reason being:
When invasive plants like water hyacinth overgrow and eventually die, their decomposition by aerobic bacteria consumes massive amounts of dissolved oxygen from the water (increasing the Biological Oxygen Demand). This lack of oxygen suffocates the fish population.
Q4. 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.
Q5. 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.
Q6. 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.
Q7. Moriarty stole the treasure and hid it in one of ten pillars. The note handed to Sherlock read: 'The clue is hidden in this statement'. Which pillar is it?
The statement 'The clue is hidden in this statement' consists of exactly 9 words. In logic puzzles of this type, the count of the key indicator (words) often points to the solution. Therefore, it is the 9th pillar (IX).
Q8. If a plant with green leaves is kept in a dark room with only green light ON, which one of the following would we observe?
Green leaves appear green because they reflect green light while absorbing other wavelengths (red/blue) for photosynthesis. In a dark room illuminated only by green light, the plant will reflect almost all the light back, making it appear brighter than the non-reflective surroundings.
Q9. A frog hops exactly 1 meter at a time. What is the least number of hops required to reach a point 10 cm away?
The frog can hop 1m forward and then 1m back at an angle such that the final displacement is only 10cm. This is possible using the triangle inequality where two sides are 1m and the third is 0.1m.
Q10. 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.