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MATH CAPSULE PART – II
Time, Speed & Distance
The Speed of a moving body is the Distance travelled by it in unit Time.
So
Distance travelled = Speed × Time
Total Time taken to cover some distance = Distance / Speed
Speed is either measured in Kilometer/ hour or meter/ second
To convert Kilometer/ hour in meter/second,
To convert meter/second in Kilometer/hour,
If a car covers a certain distance at x km/hr and an equal distance at y km/hr, the average speed of the whole journey
2xy/(x + y) km/hr
Speed and time are inversely proportional (when distance is constant)
Speed ∝1/ Time (When Distance is constant)
If the ratio of the speeds of A and B is a : b, then the ratio of the times taken by them to cover the same distance is 1/a:1/b or b : a Concept of Relative Speed: Case1: Two bodies are moving in opposite directions at speed V1 & V2 respectively. The relative speed is defined as
Vr =V1+V2 Case2: Two bodies are moving in same directions at speed V1 & V2 respectively. The relative speed is defined as
Vr =|V1–V2|
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Concept of Trains
The basic concept for train related problem is Speed = Distance / time. but we should kept in mind these discussed points below.
(i) When the train is crossing a moving object, the speed has to be taken as the relative speed of the train with respect to the object.
(ii) The distance to be covered when crossing an object, whenever trains crosses an object will be equal to:
Length of the train + Length of the object
NOTE- When train is crossing a stationary object (with length) like bridge, platform, and then its Length is added to the length of train to get required length.
When train is crossing a pole, tree, man etc.. then their length is neglect with respect to train, Here only length of train is considered.
Condition:
When Train crosses single object:
(Let the speed of train is st & length of train Lt)
1. Train Crosses a stationary object (without length like tree, man, pole etc..)
So time taken by train to cross the object =
=
2. Train Crosses a stationary object of Length L
So time taken by train to cross the object =
=
3. Train crosses a moving object of length L with speed sl in the same direction of train
So time taken by train to cross the object =
4. Train crosses a moving object of length L with Speed Sl in the opposite direction of train
So time taken by train to cross the object =
When two train crossing each other in both directions:
Let length of one train = L ; Length of Second train = L2 They are crossing each other in opposite direction in t1 sec and same direction in t2 sec respectively, Then, Speed of faster train = (L1 + L2) /2 [1/t1 + 1/t2]
Speed of slower train = (L1+ L2) / 2 [1/t1 – 1/t2]
If two trains (or bodies) start at the same time from points A and B towards each other and after crossing they take a and b sec in reaching B and A respectively, then:
(A's speed) : (B's speed) = (b : a)^1/2
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Boat & Streams
Downstream/Upstream:
In water, the direction along the stream is called downstream. and, the direction against the stream is called upstream.
If the speed of a boat in still water is u km/hr and the speed of the stream is v km/hr, then:
If the speed downstream is a km/hr and the speed upstream is b km/hr, then:
Speed in still water = 1
(a + b) km/hr. 2
Rate of stream = 1
(a - b) km/hr. 2
Time & Work
Concept
If A can do work in n days, then 1 day work of A = 1/n and vice versa.
If A is thrice as good a workman as B, then: Ratio of work done by A and B = 3 : 1.
Ratio of times taken by A and B to finish a work = 1 : 3. Man – Day – Work If M1 men can do W1 work in D1 days working H1 hours per day and M2 men can do W2 work in D2 days working H2 hours per day (when all men work at same rate)
If A can do a piece of work in p days and b can do in q days, then A and B together can complete the same work in
pq/(p+q) days
Speed downstream = (u + v) km/hr.
Speed upstream = (u - v) km/hr.
M1 D1 H1 / W1 = M2 D2 H2 / W2
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Pipe and Cistern
The pipe and cistern problem can be done on the concept of positive work and negative work. The pipe is used to fill tank or something reservoir etc. mainly pipe are two types
Inlet pipe: it is used to fill the tank.
Outlet pipe: it is used to empty the tank
So Inlet pipe work taken as Positive and Outlet pipe work taken as negative .
1. If a pipe can fill a tank in x hours, then:
part filled in 1 hour = 1
. x
2. If a pipe can empty a tank in y hours, then:
part emptied in 1 hour = 1
. y
3. If a pipe can fill a tank in x hours and another pipe can empty the full tank in y hours (where y > x), then on opening both the pipes, then
the net part filled in 1 hour =
1 -
1
. x y
4. If a pipe can fill a tank in x hours and another pipe can empty the full tank in y hours (where x > y), then on opening both the pipes, then
the net part emptied in 1 hour =
1 -
1
. y x
The same concept can be applied for one, two, three and more pipes.
Permutation and Combination:
Factorial Notation:
Let n be a positive integer. Then, factorial n, denoted n! is defined as:
n! = n(n - 1)(n - 2) ... 3.2.1.
Examples: i. We define 0! = 1.
ii. 4! = (4 x 3 x 2 x 1) = 24. iii. 5! = (5 x 4 x 3 x 2 x 1) = 120.
Permutations:
The different arrangements of a given number of things by taking some or all at a time, are called permutations.
Examples:
i) All permutations (or arrangements) made with the letters a, b, c by taking two at a time are (ab, ba, ac, ca, bc, cb).
ii) All permutations made with the letters a, b, c taking all at a time are: ( abc, acb, bac, bca, cab, cba)
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Number of Permutations:
Number of all permutations of n things, taken r at a time, is given by:
nPr = n(n - 1)(n - 2) ... (n - r + 1) = n!
(n - r)!
Examples:
a. 6P2 = (6 x 5) = 30. b. 7P3 = (7 x 6 x 5) = 210. c. Number of all permutations of n things, taken all at a time = n!
An Important Result:
If there are n subjects of which p1 are alike of one kind; p2 are alike of another kind;p3 are alike of third kind and so on and pr are alike of rth kind, such that (p1 + p2 + ... pr) = n.
Then, number of permutations of these n objects is = n!
(p1!).(p2)!.....(pr!)
Combinations:
Each of the different groups or selections which can be formed by taking some or all of a number of objects is called a combination.
Examples:
1. Suppose we want to select two out of three boys A, B, C. Then, possible selections are AB, BC and CA. Note: AB and BA represent the same selection.
2. All the combinations formed by a, b, c taking ab, bc, ca. 3. The only combination that can be formed of three letters a, b, c taken all at a time is abc. 4. Various groups of 2 out of four persons A, B, C, D are:
AB, AC, AD, BC, BD, CD.
Note that ab ba are two different permutations but they represent the same combination.
Number of Combinations:
The number of all combinations of n things, taken r at a time is:
nCr = n!
= n(n - 1)(n - 2) ... to r factors
. (r!)(n - r)! r!
Note:
I) nCn = 1 and nC0 = 1. II) nCr = nC(n - r)
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Probability
Experiment: An operation which can produce some well-defined outcomes is called an experiment.
Random Experiment: An experiment in which all possible outcomes are know and the exact output cannot be predicted in advance, is called a random experiment.
Examples:
i. Rolling an unbiased dice. ii. Tossing a fair coin.
iii. Drawing a card from a pack of well-shuffled cards. iv. Picking up a ball of certain colour from a bag containing balls of different
colours.
Details:
i) When we throw a coin, then either a Head (H) or a Tail (T) appears. ii) A dice is a solid cube, having 6 faces, marked 1, 2, 3, 4, 5, 6 respectively. When we
throw a die, the outcome is the number that appears on its upper face. iii) A pack of cards has 52 cards.
It has 13 cards of each suit; name Spades, Clubs, Hearts and Diamonds. Cards of spades and clubs are black cards. Cards of hearts and diamonds are red cards. There are 4 honors of each unit.
These are Kings, Queens and Jacks. These are all called face cards.
Sample Space:
When we perform an experiment, then the set S of all possible outcomes is called the sample space.
Examples:
1. In tossing a coin, S = {H, T} 2. If two coins are tossed, the S = {HH, HT, TH, TT}. 3. In rolling a dice, we have, S = {1, 2, 3, 4, 5, 6}.
Event: Any subset of a sample space is called an event.
Probability of Occurrence of an Event:
Let S be the sample and let E be an event.
Then, E S.
P(E) = n(E)
. n(S)
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Results on Probability:
1) P(S) = 1 2) 0 P (E) 1 3) P( ) = 0 4) For any events A and B we have : P(A B) = P(A) + P(B) - P(A B) 5) If A denotes (not-A), then P(A) = 1 - P(A).
Mensuration
Rectangle
Area of rectangle = length (l) * breadth (b) = lb
Perimeter of rectangle = 2( l + b)
Where l= length of the rectangle, b= breadth of rectangle
Square
Area = a* a = a2
Perimeter = 4a
Where a= side of Square
Rhombus
Area = ½ * Product of Diagonals
Perimeter = 4* length of Side = 4l
Circle
Let r = radius, d = diameter of circle.
Area = π * radius2 = π r2 = ¼π d2
Circumference of Circle = 2πr
Radius r= d/2
Cylinder
Volume of cylinder = πr2h
Total surface area of cylinder = 2πr(r + h)
Curved Surface Area = 2πrh
Where r = radius of base h= height of cylinder
Sphere
Volume of sphere= 4/3 π r3 = 1/6 π d3
Surface Area of sphere = 4 π r2 = πd2
Where r = radius of sphere d = diameter of sphere
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Hemisphere
Volume of hemisphere = 2/3 π r3
Surface area of hemisphere = 3π r2
Where r= radius of hemisphere
Cone
Volume of right circular cone = 1/3 π r2h
Area of base of a cone = π r2
Curved Surface Area of Cone = π r l
Total Surface area of cone = πr(r+l)
Where r= radius of base, l = lateral height of cone , h = height of cone
Lateral height of Cone l = {(h2+r2)}1/2
Area of Sector of Circle = π r2 * θ/ 360
Where θ = measure of the angle of the sector, r = radius of sector
Length of an arc = 2π r* θ/360
Cube
Volume of Cube = l * l * l = l3
Length of Diagonal of Cube = √3 l
Where l= side of cube
Cuboid
Volume of cuboid = l *b* h
Length of Diagonal of Cuboid = (l2+b2+h2)1/2
(Where l = length b= breadth h = height)
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Concept Clearing Quiz
1. Amit walks at 14 km/hr instead of 10 km/hr, he would have walked 20 km more. The actual distance travelled by him is: A) 45 B) 50 C) 55 D) 60 E) None of these
2. A train can travel 50% faster than a car. Both start from point A at the same time and reach point B 75 kms away from A at the same time. On the way, however, the train lost about 12.5 minutes while stopping at the stations. The speed of the car is:: A) 100 km/hr B) 105 km/hr C) 150 km/hr D) 200 km/hr E) None of these
3. Excluding stoppages, the speed of a bus is 54 kmph and including stoppages, it is 45 kmph. For how many minutes does the bus stop per hour?: A) 10 B) 15 C) 20 D) 25 E) None of these
4. In a flight of 600 km, an airplane was slowed down due to bad weather. Its average speed for the trip was reduced by 200 km/hr and the time of flight increased by 30 minutes. The duration of the flight is:: A) 45 minutes B) 50 minutes C) 55 minutes D) 60 minutes E) None of these
5. A man on tour travels first 160 km at 64 km/hr and the next 160 km at 80 km/hr. The average speed in km/hour for the first 320 km of the tour is: A) 35.11 B) 55.71 C) 71.11 D) 66.67 E) None of these
6. Prashant is travelling on his cycle and has calculated to reach point A at 2 P.M. if he travels at 10 kmph, he will reach there at 12 noon if he travels at 15 kmph. At what speed must he travel to reach A at 1 P.M.? A) 10 kmph B) 15 kmph C) 20 kmph D) 25 kmph E) None of these
7. It takes eight hours for a 600 km journey, if 120 km is done by train and the rest by car. It takes 20 minutes more, if 200 km is done by train and the rest by car. The ratio of the speed of the train to that of the cars is: A) 3:4 B) 4:5
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C) 7:9 D) 8:11 E) None of these
8. A train overtakes two persons walking along a railway track. The first one walks at 4.5 km/hr. The other one walks at 5.4 km/hr. The train needs 8.4 and 8.5 seconds respectively to overtake them. What is the speed of the train if both the persons are walking in the same direction as the train?: A) 80 kmph B) 81 kmph C) 85kmph D) 90kmph E) None of these
9. Two, trains, one from Kolkata to Delhi and the other from Delhi to Kolkata, start simultaneously. After they meet, the trains reach their destinations after 9 hours and 16 hours respectively. The ratio of their speeds is: A) 4:3 B) 3:4 C) 2:3 D) 3:2 E) None of these
10. Two trains of equal lengths take 10 seconds and 15 seconds respectively to cross a telegraph post. If the length of each train be 120 meters, in what time (in seconds) will they cross each other travelling in opposite direction? A) 10 B) 12 C) 30 D) 25 E) None of these
11. A boat running upstream takes 8 hours 48 minutes to cover a certain distance, while it takes 4 hours to cover the same distance running downstream. What is the ratio between the speed of the boat and speed of the water current respectively? A) 8:3 B) 3:8 C) 7:9 D) 9:12 E) None of these
12. A boatman goes 2 km against the current of the stream in 1 hour and goes 1 km along the current in 10 minutes. How long will it take to go 5 km in stationary water? A) 1hour B) 1 hour 15 minutes C) 2 hour D) 2hour 15 minutes E) None of these
13. Speed of a boat in standing water is 9 kmph and the speed of the stream is 1.5 kmph. A man rows to a place at a distance of 105 km and comes back to the starting point. The total time taken by him is? A) 20 hour B) 21 hour C) 23 hour D) 24 hour E) None of these
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14. A man rows to a place 48 km distant and come back in 14 hours. He finds that he can row 4 km with the stream in the same time as 3 km against the stream. The rate of the stream is: A) 2km/hr B) 3 km/hr C) 2.5 km/hr D) 4 km/hr E) None of these
15. A, B and C can do a piece of work in 20, 30 and 60 days respectively. In how many days can A do the work if he is assisted by B and C on every third day: A) 15 days B) 20 days C) 25 days D) 30 days E) None of these
16. A is thrice as good as workman as B and therefore is able to finish a job in 80 days less than B. Working together, they can do it in? A) 20 days B) 25 days C) 30 days D) 40 days E) None of these
17. A and B can complete a work in 15 days and 10 days respectively. They started doing the work together but after 2 days B had to leave and A alone completed the remaining work. The whole work was completed in : A) 10 days B) 12 days C) 20 days D) 25 days E) None of these
18. A and B together can do a piece of work in 30 days. A having worked for 16 days, B finishes the remaining work alone in 44 days. In how many days shall B finish the whole work alone? A) 40 days B) 50 days C) 55 days D) 60 days E) None of these
19. A machine A can print one lakh books in 8 hours, machine B can print the same number of books in 10 hours while machine C can print them in 12 hours. All the machines are started at 9 A.M. while machine A is closed at 11 A.M. and the remaining two machines complete work. Approximately at what time will the work (to print one lakh books) be finished? A) 1:05 PM B) 1:30 PM C) 11:35 AM D) 2 PM E) None of these
20. In how many different ways can the letters of the word 'MATHEMATICS' be arranged so that the vowels always come together?
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A) 124045 B) 20890 C) 133156 D) 120960 E) None of these
21. How many 4-letter words with can be formed out of the letters of the word, 'LOGARITHMS', if repetition of letters is not allowed? A) 400 B) 4050 C) 5040 D) 5773 E) None of these
22. In a group of 6 boys and 4 girls, four children are to be selected. In how many different ways can they be selected such that at least one boy should be there? A) 156 B) 209 C) 193 D) 245 E) None of these
23. In a bag, there are 8 red, 7 blue and 6 green balls. One ball is picked up randomly. What is the probability that it is neither red nor green? A) 3/91 B) 1/3 C) 3/7 D) 7/15 E) None of these
24. One card is drawn at random from a pack of 52 cards. What is the probability that the card drawn is a face card (Jack, Queen and King only)? A) 3/13 B) 1/13 C) 7/52 D) 9/13 E) None of these
25. Three taps A,B and C can fill a tank in 20,30and 40 minutes respectively. All the taps are opened simultaneously and after 5 minutes tap A was closed and then after 6 minutes tab B was closed .At the moment a leak developed which can empty the full tank in 60 minutes. What is the total time taken for the completely full? A) 15 minutes B) 24 minutes C) 30 minutes D) 48 minutes E) None of these
26. There are three taps A, B, and C. A takes thrice as much time as B and C together to fill the tank. B takes twice as much time as A and C to fill the tank. In how much time can the Tap C fill the tank individually, if they would require 10 hours to fill the tank, when opened simultaneously? A) 12 hour B) 48 hour C) 60 hour D) 24 hour E) None of these
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27. Three pipes A, B and C can fill a tank from empty to full in 30 minutes, 20 minutes, and 10 minutes respectively. When the tank is empty, all the three pipes are opened. A, B and C discharge chemical solutions P,Q and R respectively. What is the proportion of the solution R in the liquid in the tank after 3 minutes? A) 5/11 B) 7/11 C) 9/11 D) 3/11 E) None of these
28. The length of a rectangular plot is 20 meters more than its breadth. If the cost of fencing the plot at 26.50 per meter is Rs. 5300, what is the length of the plot in meters? A) 40 B) 50 C) 60 D) Data inadequate E) None of these
29. A Blanket, when washed, was found to have lost 20% of its length and 10% of its breadth. The percentage of decrease in area is? A) 25 B) 28 C) 35 D) 40 E) None of these
30. A rectangular park 60 m long and 40 m wide has two concrete crossroads running in the middle of the park and rest of the park has been used as a lawn. If the area of the lawn is 2109 sq. m, then what is the width of the road?? A) 3m B) 2m C) 1m D) 5m E) None of these