If a rhombus has one side and one diagonal equal to √3 em, find length of its other diagonal.
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Solution
Diagonals bisect at 90°
∴\(\sqrt{3}^{2}=\left ( \frac{\sqrt{3}}{2} \right )^{2}+x^{2}\)
3 = 3⁄4 + X2 ∴ BD = 3
Circular pizza with negligible thickness is cut into x pieces by 4 straight line cuts. What is maximum and minimum value of x respectively?
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Solution
People eating in a certain cafeteria are either faculty members or students and the number of faculty members is 15% of total number of people in cafeteria. After some of the students leave, total number of persons remaining in cafeteria is 50% of the original total. The number of students who left is what fractional part of original number of students?
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Solution
Let total number of people originally = 100 ⇒ 15 faculty members and 85 students. Let x students leave.
∴ strength now = 50 which includes 15 faculty members i.e. 35 studehts
∴ 50 students left.
∴ Required fraction = \(\frac{50}{85}=\frac{10}{17}\)
Three machines can individually do a certain job in 8, 10, 12 hours respectively. Then the least possible part of job that can be done in one hour by two of machines together is what fraction of the greatest possible part of job that can be done in the same time by two machines working together?
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Solution
Least work is done when two less efficient machines work together.
∴ Least work done in 1 hour
=\(\frac{1}{10}+\frac{1}{12}=\frac{11}{60}\)
Maximum work done in 1 hour is when two most efficient machines work together.
∴ Maximum work done = \(\frac{1}{8}+\frac{1}{10}=\frac{9}{40}\)
∴ Required fraction = \(\frac{\frac{11}{60}}{\frac{9}{40}}=\frac{11}{60}\times \frac{40}{9}\)
A cistern has a capacity of 40,000 Liters. Initially it is half full. Every day a certain quantity of water is used but at the end of day (4⁄3) of used quantity is replaced into cistern. If amount of water used is same every day, then what is the total quantity of water used for refilling till cistern is full?
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Solution
For every' 'X' liters consume , \(\frac{4x}{3}\) liters is replace which means a net increase in volume by 1⁄3 liters.
Thus for net increase of 1 litre, Quantity of water to be added is 4 litres. Hence when total increase has to 20,000 liters, the amount of water to be actually refilled is 4 times 20,000 = 80,000 Liters.
a # b = a + b + ab
If for any ‘a’, there is a number ‘c’ such that a # c = a, then c is
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Solution
a # c = a ∴ a + c + ac = a ∴C + ac = 0
∴ c(l + a) = 0 ∴ c = 0 or a = -1
A supply of water will last for 150 days if 7.5 gallons leak off every day, but only for 100 days if 15 gallons leak off daily. What is total quantity of supply of water in gallons?
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Solution
Let daily consumption be x gallons.
Leak + consumption = capacity.
∴ 150(x + 7.5) = 100(x + 15) ∴ x = 7.5
∴ Capacity 100(7.5 + 15) = 2250 gallons
If p < q, then
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Solution
Consider (A) p2 < q2
P = -3 q = -2 .∴ p < q But p2 > q2
So (A)is not always true.
(B) p2>q2
Consider p = 2, q = 3 .'. p < q and also p2 < q2
∴ (8) is not always true.
(C) p < q2
Consider p = X q = ~ :. p < q But p > q2
So (C) is not always true.
(D) p3 < q3
For all possible values of p and q i.e. fractions, integers (positive or negative) if p < q, then p3 < q3
(E) p2 > q
Consider p = 2 q = 9 ∴ p < q and p2 < q So (E) is not always true.
2 – 3m? 5
Of the following symbols, which can be substituted for question mark in the above expression to make a true statement for all values of m such that -1 < m ≤ 2?
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Solution
m lies between -1 and 2 with 2 inclusive.
Largest value of m possible is 2.
∴ Putting m = 2, we get 2 - 3m = 2 - 6 = -4.
Putting m = -1, we get 2 - (-3) = 2 + 3 = 5.
But m is actually > -1.
∴ If m is > -1, then 2 - 3m < 5
The average height of 22 trees in a forest is 15 feet. The height of the tallest tree exceeds that of the shortest tree by 3 feet 4 inches. If these two trees are excluded, the average height drops by 2 inches.What is the height of the tallest tree?
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Solution
Total height of all 22 trees is
22 x 15 = 330 feet = 3960 inches.
Excluding the tallest and the shortest trees, the total height of remaining 20 trees is 20 x 14 feet 10 inches = 3560 inches.
∴ Height of the tallest and shortest tree is 400 inches. Since the difference in the height is 3 feet 4 inches
i.e. 40 inches. Tallest tree is 220 inches and shortest tree is 180 inches.
∴ Tallest tree is 18 feet 4 inches.