Twelve straight lines are drawn in a plane such that no two of them are parallel and no three of them are concurrent. A circle is now drown in the same plane such that all the points of intersection of all the lines lie inside the circle. What is the number of non-overlapping regions in to which the circle is divided?
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Solution
One line divides the whole space into two and two lines into 4.
If no two lines are parallel and no three are concurrent,then 3rd line can cut the existing lines at most two places or three extra spaces are added. Thus 3 lines divide the whole space into 4 +3 =7.One more line can intersect at most 3 points and will provide additional 4 spaces. The nth line can increase the region by k if and only if it divides k of the old regions and it divides k (regions) if and only if it intersects the existing lines at at most k – 1 points.
1 line = 2 region
2 lines = 2 + 2 = 4 regions = 1+ 1
3 lines = 2+ 2+ 3= 7 regions =1 + 1 +2
4 lines = 2+ 2+ 3+ 4= 11 regions = 1 +1 +2 +3
n lines = 2+ 2+ 3 + 4+ ...n regions
= 1 + 1 + 2 + ... n –1 regions
= Sn + 1
where, Sn is sum of n natural numbers. n= 12 ⇒ number of non-overlapping regions inside the circle (as all intersects are inside the circle, the total regions are inside the circle) = 1 + S12= 1 + 78 = 79.
A, B and C starts running at the same time and from the same point around a circular track of 70 m radius A and B run clockwise and C counter clockwise. If A meets C every 88 seconds and B meets C every 110 s, then after how much seconds does A meets B?
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Solution
Considering relative speed, as A meets C every 88 seconds then if C is at a constant point then A covers the circular track in each 88 seconds same for B also which covers the distance in 110 seconds.
One afternoon a person left home at 2:45 pm according to the watch at home. Here ached office at 3:10 pm according to the watch at office. He stayed therefor 25 min and moved towards home at a speed twice than previous and reached home at 4 pm according to the watch at home. Find out by how much time office watch is either fast or late compared to watch at home?
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Solution
Total travel time including rest time according to watch at home from 2 : 35 pm to 4 : 00 pm = 1 h 25 min.Total travel time excluding rest time= 1 h 25 min – 25 min
= 1 h 60 min
The ratio of time for going to office and coming from office = 2 : 1 and total time taken only for travelling = 60 min ⇒ 3x = 60 min be 2x = 40 min.
He took 40 min to go to office. Thus the time while here ached at office must be 2.35 + 40 min = 3 : 15.
But the time at office watch was 3 :10. Hence the office watch was 5 min late.
Mr Das decide to walk down the escalator of a mall. He found that if he walks down 26 steps, he requires 30 seconds to reach the bottom. However, if the steps down 34 stair he would only require 18 seconds to get to the bottom. If the time is measured from the moment the top step begins to descend to the time he steps off the last step at the bottom, find out the height of the stair way insteps.
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Solution
Total number of steps elapsed in both the case is same as speed of escalator is constant. Hence according to question
26 + 30x = 34 + 18x
⇒ x = 2⁄3 steps
Thus, required number of steps= 26 + 30x
26 + 30x × 2⁄3 = 46 steps
Honeybees, unlike many other varieties of bees (such as bumblebees and wasps), are able to live through the winter. 30000 to 40000 bees within a honeybee hive could not individually move about in cold winter temperatures.But when ‘clustering together in a dense ball’, the bees generate heat by constantly moving their body parts. The cluster also moves slowly about the hive, eating honey stored in the combs. This nutrition, in addition to the heat generated by the cluster, enables the honeybee to survive the cold winter months.From the four options, find the statement which best describes the above passage.
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Solution
Honeybees, unlike many other varieties of bees, are able to live through the winter by clustering together in a dense ball for body warmth.Main ideas are unlike other bees, honey bees form cluster in winter to gain body warmth for survival. The numbers, how they eat, how they move are secondary ideas according to the passage.
Given a number with 1998 digits which is divisible by 9.Let x be the sum of its digits and y be the sum of digits of x and z be the sum of digits of y. Find z.
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Solution
Sum of the digits of a number with any number of digits,which is divisible by 9 is 9. Working in options wages.Thus, x = 9. And hence y= 9 and z = 9.
If there are 51 rupees consisting of coins of 1 rupees, 50 paisa and 25 paisa. Number of 50 paisa coin is double that of number of 1 rupee coin and four times that of 25 paisa,then what is the number of 50 paisa coins in the collection?
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Solution
Choose the most appropriate word from the options given below to complete the following sentence.Female sparrows and immatures are …….., while the typical adult male sparrow is ……… by its markings : a black bib,a gray cap, and white lines trailing down from the month.
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Solution
The word nondescript and the phrase easily recognized make for just the sort of contrast that lends coherence to the sentence as a whole.
Which of the following options is the closest in meaning to the word below?
Frugality
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Solution
Frugal means saving, not wasteful.
In the following question, a word is given below which precedes four more words. Pick the word from the four options which is most nearly opposite in meaning to the given word.
Distend
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Solution
Distend means to expand or stretch or swell.Hence,the opposite meaning is diminish.