Observe the pattern of powers of 10 modulo 7: , , , , , . Any power where n is a multiple of 10 modulo 6 (like 10, 100, 1000) results in . Therefore, every term in this sequence reduces to , which is 4. There are 10 terms in the given sequence if we interpret the final term as concluding the logical base-10 progression up to 10 iterations. . .
Q. 57. Three statements followed by two conclusions numbered I and II are given below…
Statements:
I. Mostly pets are cats.
II. Only cats are dogs.
III. No cats are rabbits.
Conclusions:
I. Some pets are rabbits.
II. All pets being rabbits is not a possibility.
Choose the correct option:
(a) Both conclusion I and conclusion II follow.
(b) Only conclusion I follows
(c) Only conclusion II follows
(d) Neither conclusion I nor conclusion II follows
Answer:
(c) Only conclusion II follows
Explanation:
Statement I means “Some pets are cats”. Statement III means “No cats are rabbits”. Therefore, there is no positive connection to deduce that “Some pets are rabbits” (Conclusion I is false). Because some pets are definitely cats, and no cats can be rabbits, it is a certainty that those specific pets can never be rabbits. Thus, “All pets being rabbits is not a possibility” is absolutely true (Conclusion II follows).
Q. 58. The length of candle B is 3 times the length of candle A. The speed of burning of candle B is 4 times the speed of burning of candle A. A party begins with the lighting of these two candles. The party ends when the heights of these candles become equal. If the numerical value of length (in metre) of the burnt away portion of candle A and the speed (in metre per hour) of its burning are same, how long, in hours, did the party continue?
(a) 1/2
(b) 1
(c) 1 1/4
(d) 1 1/2
Answer:
(b) 1
Explanation:
Let length of A be and speed of A be . Thus length of B is and speed of B is . Let the time elapsed be $t$. When heights are equal: . The problem states the burnt portion of A (St) is numerically equal to its speed (S). Therefore, , which implies hour.
Q. 59. A cube of dimensions 8x8x8 is painted with 3 different colours such that opposite faces have the same colour. The number of unit cubes that have exactly 2 faces painted and with more than 1 colour is:
(a) 32
(b) 48
(c) 60
(d) 72
Answer:
(d) 72
Explanation:
Unit cubes with exactly 2 faces painted are located on the 12 edges of the main cube (excluding the 8 corners). An 8x8x8 cube has such cubes per edge. edge cubes. Because opposite faces share the same color, any two adjacent faces meeting at an edge must have different colors. Thus, every single one of the 72 edge cubes is painted with more than 1 color.
Q. 60. A king ordered that a golden crown be made for him from 8 kg of gold and 2 kg of silver. The goldsmith took away some amount of gold and replaced it by an equal amount of silver and the crown, when made, weighed 10 kg. Archimedes knew that under water gold lost 1/20th of its weight, while silver lost 1/10th. When the crown was weighed under water, it was 9.25 kg. How much gold was stolen by the goldsmith?
(a) 3 kg
(b) 2 kg
(c) 1 kg
(d) 0.5 kg
Answer:
(a) 3 kg
Explanation:
Let the actual amount of gold in the final crown be $G$ and silver be $S$. We know . The weight lost in water is . The formula for weight loss is . Substituting $S = 10 – G$, we get . Multiply by 20 to clear the denominator: . The crown was supposed to have 8 kg of gold but only has 5 kg. Therefore, of gold was stolen.
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