There are 8 rice bags looking a like, seven of which have equal weight and one is slightly heavier. The weighing balance is of unlimited capacity. Using this balance, the minimum number of weights required to identify the heavier bag is
Which of the following assertions are correct?
(a) Adding 7 to each entry in a list adds 7 to the mean of the list.
(b) Adding 7 to each entry in a list adds 7 to the standard deviation of the list.
(c) Doubling each entry in a list doubles the means of the list.
(d) Doubling each entry of a list leaves the standard deviation of list unchanged
-
Solution
(a) Mean = \(\frac{a + b + c}{3}\)
According to each term
\(mean \; = \;\frac{\left ( a + 7 \right ) + \left ( b + 7 \right ) + \left ( c + 7 \right )}{3} = \frac{a + b + c}{3} + \frac{3 \times 7}{3}\) = mean + 7
(b) Adding a constant to each term does not change standard deviation.
(c) Mean = \(\frac{a + b + c}{3}\)
Doubling each term
\(mean \; = \; \frac{2a + 2b + 2c}{3} = \frac{2\left (a + b + c \right )}{3} \; = \; 2 \times mean\)
(d) Doubling each term changes the standard deviation.
The variable cost (V) of manufacturing a product varies according to the equation V = 4q, where q is the quantity produced. The fixed cost (F) of production of same product reduces with q according to the equation F = 100/q. How many units should be produced to minimize the total cost (V + F)?
-
Solution
Total Cost, T = 4q + 100/q
When total cost becomes minimum, first derivative of T becomes 0 and second derivative at the minimum point will be positive.
Differentiating T with respect to q and equating to 0,
4 - 100/q2 = 0 => q = +5 or -5.
Since, we can't have negative number of product, q = 5.
Taking second derivative, at q = 5 gives 200/125 = 8/5 > 0, and hence 5 is the minimum point.
An automobile plant contracted to buy shock absorbers from two supplies X and Y. X supplies 60% and Y supplies 40% of the shock absorbers. All shock absorbers are subjected to a quality test. The ones that pass the quality test are considered reliable of the X1s shock absorbers, 96% are reliable of the y’s shock absorbers, 72% are reliable. The probability that a randomly selected shock absorber,which is found to be reliable, is made by y is
A political party orders an arch for the entrance to the ground in which the annual convention is held. The profile of the arch follows the equation y = 2x – 0.1x2. Where y is the height of the arch in meters. The maximum possible height of arch is
-
Solution
Given:
Profile of each following equation,
y = 2x – 0.1x2
To find maximum possible height
take dy⁄dx = 0
dy⁄dx = 2 - (2 × 0.1x) = 2 - 0.2x = 0
∴ x = 10 m
Which of the following is the closest in the meaning to the word given below.
LATITUDE
Choose the most appropriate word from the options given below to complete the following sentences. It was her view that the country’s problems had been ________ by foreign technocrates, so that to invite them to come back would be counter-productive.
Choose the word from the options given below that is most nearly opposite is meaning to the given word.
F R E Q U E N C Y
Choose the most appropriate word from the option giving below to complete the following sentence. H is rather casual remarks on politics _______ his lack of seriousness about the subject.
The cost function for product in a firm is given by 5q2, where q is the amount of production. The firm can sell the product at a market price of ₹ 50 per unit. The number of units to be produced by the firm such that the profit is maximized is
-
Solution
Cost function for product = 5q2
Selling price = 50 × q\
∴ Profit = 50q – 5q2
P = 50q – 5q2
For maximum profit, dp⁄dq = 0
dp⁄dq = 50 - 2 × 5q
= 50 – 10q = 0
∴ q = 5