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Q11. Tiwari and Deo take 2 hours to do a job. Tiwari and Hari take 3 hours. Deo and Hari take 6 hours. Which statement is incorrect?
Work rates: T+D=1/2, T+H=1/3, D+H=1/6. Adding them: 2(T+D+H)=1. So T+D+H=1/2. Since T+D=1/2, H must be 0 (no work). Rates are: T=1/3 (3 hrs), D=1/6 (6 hrs), H=0. Thus, Hari being the fastest is incorrect.
Q12. How many digits are there in 3^16 when it is expressed in the decimal form?
We can estimate this using logarithms: log10(3^16) = 16 * log10(3) ≈ 16 * 0.4771 = 7.6336. Since the characteristic is 7, the number of digits is 7 + 1 = 8.
Q13. Write d=1 degree, r=1 radian and g=1 grad (100 grad = right angle). Then which of the following is true?
Comparing the sizes: 1 radian ≈ 57.3°, 1 grad = 0.9°, and 1 degree = 1°. So, 1° > 0.9°, and 57.3° is the largest. In the first quadrant, cosine decreases as the angle increases. Therefore, cos(1 rad) < cos(1 grad) < cos(1°).
Q14. 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.
Q15. The statement: 'The father of my son is the only child of your parents' is true in how many relations?
'Father of my son' is either the speaker (if male) or the speaker's husband. 'Only child of your parents' is the listener. This identifies a unique husband-wife relationship dynamic.
Q16. 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.
Q17. For which values of A and B is sin A = cot B?
Evaluate the options: If A=0 and B=pi/2, then sin(0) = 0, and cot(pi/2) = cos(pi/2)/sin(pi/2) = 0/1 = 0. Both sides equal 0, making the equation true.
Q18. What is the volume of soil in an open pit of size 2m x 2m x 10cm?
This is a logical trick question. Because it is an 'open pit', the soil has already been excavated and removed. Therefore, the volume of soil currently inside the pit is zero.
Q19. Four small squares of side x are cut out of a square of side 12 cm to make a tray by folding the edges. What is the value of x so that the tray has the maximum volume?
The volume V of the tray is Length * Width * Height = (12 - 2x)(12 - 2x)x. To maximize volume, we find the derivative dV/dx and set it to zero, which yields x = 2 and x = 6. Since x = 6 results in a volume of zero, the maximum volume occurs when x = 2.
Q20. The difference between the squares of the ages of a father and son is 899. The age of the father when the son was born was:
F^2 - S^2 = 899 => (F-S)(F+S) = 31 * 29 or 899 * 1. F-S is the age of the father at the son's birth. If F-S = 29 and F+S = 31, then F=30, S=1. (F-S)=29 is a valid age.