Sem VI-CDO-302 - MRU - Aptitude Handbook

Manav Rachna University
Career Development Centre
Handbook for
Quantitative Aptitude
Semester VI
Subject Name: PCE-IV
Subject Code: CDO-302
Semester: Jan-May’2020
Branch: CSE, ECE, ME
QAPD – I Semester IV
Student Name: …………………………
Roll Number: …………………………….
Section: …………………………………….
CONTENT
Chapter 1 - Permutation and Combinations
U Principle of Counting, Selection, Arrangement and Applications 1-8
N
I Chapter 2 - Probability
9-15
Bayes Theorem and Conditional Probability, Coins, Dices and Cards
T
1
Chapter 3 – Geometry
16-23
U Triangles, Quadrilaterals, Circle and Tangents theorem and Properties
N Chapter 4 - Mensuration – I (Areas & Formulae)

24-31
I Polygons, Types of Triangles & Quadrilaterals, Circle and Mixed Figures
T Chapter 5 - Mensuration – II (Surface Area & Volume)

32-40
Cubes & Cuboids, Cone, Cylinder, Sphere, Prism and Pyramid
2 Chapter 6 – Algebra
Linear & Quadratic Equations, Inequalities, Integral Solutions, Max & Min 41-50
value
CHAPTER - 1
PERMUTATION
&
COMBINATION
Handbook for Quantitative Aptitude (Sem VI) Page 1

PERMUTATION
The different arrangement of a given number or things by taking some or all at a time, are called Permutations.
a)All permutations (or arrangements) made with the letters of a, b, c by taking two at a time are(ab,ba,ac,ca,bc,cb)
b) All permutations made with the letters a,b,c taking all at a time are: (abc,acb,bac,bca,cab,cba)
Number of Permutations:
Number of all permutations of n things, taken r at a time, is given as:-
n
pr= n (n-1) (n-2)……………………(n-r+1) = n!/ (n-r)!
1.6P2 = (6*5) =30

7
2. P3 = (7*6*5) =210
Note: - Number of all permutation that is n things, we can take all at a time=n!
First, we find the number of vowels, here we have three vowels that is E,O,U. Then, we count the consonant that is two.
Now, we count the number of vowels as a single unit that is vowel E,O, and U count as a single unit and add it with
consonant. So, now we have 3 units.
(2 consonant+3 vowels as a single unit)=3*3 (Three vowel)
Important Results
1.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 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!)(p3!)……………….(pr!)]
Example 1 :-In how many different ways the words ‘HOUSE’ can be arranged?
Solution 1 :- The word HOUSE contains 5 letters. Therefore, 5P5=5!=120
Example 2:-In how many several ways the word KOLKATA can be prepared so that as vowel always comes together?
Solution 2:-Here, we have 3 vowels i.e. O, A and A.Here, we have 4 consonants.Now, we count vowels as a single unit.
(4 consonant+3 Vowels as a single unit) =5 *3 (Three Vowels)
5!*3! =720 ways
Example 3:-In how many several ways the word CAME can be arranged, that the vowels not come together?
Solution 3:-Here, we have 2 vowels that is A and E. Here, we have 2 consonants. We count the number of vowels as a
single unit i.e. vowel A and E.
(2 Consonant+2 vowels as a single unit)=3*2(Two Vowel) =12

Combination
Each of the different groups or selections which can be formed by taking some or all of a number of objects. In
combinations we can select only one item at time. It is based on selection or choice.
Number of Combination:-
The number of all combinations of n things, taken r at a time is:-
nC =n! /r! (n-r)! , If n=r then nC =0 and nC =1
r r 0
nC =nC
r (n-r)=1
Example:-Find the value of 16C ?13
Solution: - 16 C (16-13) =560
Difference b/w Permutation & Combination
Note:-
1. If order is important, PQ will be different from QP, PR will be different from RP and QR will be different from RQ.
2. If order is not important, PQ will be same as QP, PR will be same as RP and vice versa.
Hence, if order is important, question is related to Permutations. Otherwise, it is related to combinations.

Type 1 - Principles of Counting
Q1. A person wants to go from city A to city C via city B. There are 4 routes from A to B and 5 routes from B to C. In how
many ways can he travel from A to C?
A. 20 Ways B.25 Ways C.30 Ways D.35 Ways
Q2. In how many ways can 5 prizes be distributed among 4 boys when every boy can take one or more prizes ?
A. 1025 B.1024 C.1020 D.1021
Q3. A college offers 6 courses in the morning and 4 in the evening. Find the number of ways a student can select exactly
one course, either in the morning or in the evening?
A. 11 B.12 C.10 D.15
Q4. There are 3 questions in a question paper. If the questions have 4,3 and 2 solutions respectively, find the total
number of solutions?
A. 20 B.18 C.17 D.24
Q5. How many 3-digit numbers can be formed with the digits 1,4,7,8 and 9 if the digits are not repeated?
A. 60 B.64 C.65 D.70
Type 2 - Selection
Q6.A boy has 3 library cards and 8 books of his interest in the library. Of these 8, he does not want to borrow Chemistry
part-II unless Chemistry Part-I is also borrowed. In how many ways can he choose the three books to be borrowed?
A. 41 B.40 C.45 D.48
Q7.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 atleast one boy should be there?
A. 200 B.209 C.205 D.220
Q8.A box contains 1010 balls out of which 33 are red and rest is blue. In how many ways can a random sample of 66
balls be drawn from the bag so that at the most 22 red balls are included in the sample and no sample has all the 66
balls of the same color?
A. 166 B.165 C.168 D.170
Q9.How many groups of 6 persons cab be formed from 8 men and 7 women?
A. 5000 B.5005 C.5010 D.6000
Q10.There is a 7-digit telephone number with all different digits. If the digit at extreme right and extreme left are 5 and
6 respectively. Find how many such telephone numbers are possible?
A. 6720 B.6700 C.6708 D.6270
Q11.The number of ways in which a committee of 3 ladies and 4 gentlemen can be appointed from a group consisting
of 8 ladies and 7 gentlemen, if Mrs. X refuses to serve in a committee if Mr. Y is it’s member.
A. 2065 B.2165 C.2650 D.2560

Type 3 - Selection & Arrangement
Q12.There are 5 yellow, 4 green and 3 black balls in a bag. All the 12 balls are drawn one by one and arranged in a row.
Find out the number of different arrangements possible.
A. 27725 B.27720 C.22770 D.3000
Q13.A mixed doubles tennis game is to be played between two teams (Each team consists of 1 male and 1 female).There
are 4 married couples. No team is to consist of a husband and his wife. What is the maximum number of games that can
be played?
A. 48 B.46 C.40 D.84
Q14.In how many different ways can 4 books A, B, C and D be arranged one above another in a vertical order such that
the books A and B are never in continuous position?
A. 10 B.12 C.15 D.20
Q15.2 Men and 1 women board a bus in which 5 seats are vacant, one of these 5 seats is reserved for ladies. A woman
may or may not sit on the seat reserved for ladies. In how many different ways can the 5 seats be occupied by these
passengers?
A. 36 Ways B.30 Ways C.40 Ways D.35 Ways
Type 4 - Alphabets and Digits_
Q16.In how many different ways can the letters of the word ‘MATHEMATICS’ be arranged so that the vowels always
come together?
A. 120960 B.121960 C.125965 D.122960
Q17.In how many ways can the letters of the word ‘DIRECTOR ’be arranged so that the 3 vowels are never together?
A. 19,000 B.18,000 C.18,500 D.18,400
Q18.In how many different ways can the letters of word ‘JUDGE’ be arranged so that the vowels always come together?
A. 45 B.47 C.48 D.84
Q19.How many 3 letters computer passwords can be formed (no repetition allowed) with atleast 1 symmetric letter?
A. 12870 B.13500 C.18270 D.17820
Q20.How many different numbers can be formed from the digits 3,4,5,6 and 7 when repetitions is allowed?
A. 3908 B.3905 C.3900 D.3907
Q21.In how many ways,can the letters of the word ‘ASSASSINATION’ be arranged, so that all the S are together?
A. 150200 B.151201 C.151200 D.115200
Type 5 - Linear/Circular Arrangements
Q22. In how many ways can 3 men and 3 ladies be seated at around table such that no two men are seated together?
A. 13 B.12 C.15 D.18
Q23. In how many ways can 5 boys and 5 girls be seated at a round table so that no two girls may be together?
A. 2880 B.2550 C.2885 D.2775

Q24. In how many ways can 5 men and 2 women be arranged at a round table if two women are never together?
A. 470 B.480 C.420 D.400
Q25. 20 persons we invited to a party. In how many ways can they be seated in a round table such that two particular
persons sit on either side of the host?
A. 18!*2 B.17!*2 C.15! D.20!*2
Q26. How many necklaces of 10 beads each can be made from 20 beads of different colors?
A.19! / (10!)2 B.20! / (11!)2 C.15! / (12!)2 D.16! (10!)2
Q27. In how many different ways can five persons stand in a line for a group photograph?
A. 125 B.120 C.122 D.130
Type 6 - Identical Objects
Q28. Find the number of ways in which 12 students may be equally divided into three groups?
A. 3355 B.5775 C. 7755 D.8822
Q29.Find the number of ways in which 12 mangoes may be equally divided among 3 boys?
A. 57750 B.34650 C. 23450 D.12345
Q30. Find the number of ways of distributing 7 identical balls into three boxes so that no box is empty and each box
being large enough to accommodate all balls.
A. 15 B.73 C.37 D.7! /3!
Q31.In how many ways can a pack of 52 cards be divided equally among four players in order?
A. (52!)4 B.4*(13!) C.52! / (13!)4 D. None of these
Q32.In how many ways is it possible to make a selection by taking any number of all 20 fruits, namely 9 mangoes,7
oranges and 4 apples?
A. 199 B.20C3 C.20P3 D.399
Q33. A bag contains 4 mangoes and 5 oranges. In how many ways can I make a selection so as to take atleast one mango
and one orange?
A. 465 B.365 C.8 D.19
Q34.In how many ways a person can choose one or more beverages out of sodawater, Mirinda, whisky and coffee?
A. 16 B.4 C.15 D.4!
Type 7 - Applications
Q35. On a circle there are 9 points selected. How many triangles with edges in these points exist?
A. 80 B.84 C.82 D.91
Q36. There are 7 non-collinear points. How many triangles can be drawn by joining these points?
A. 32 B.33 C.35 D.40

Q37. How many lines can you draw using 3 non collinear (not in a single line) points A, B and C on a plane?
A. 2 B.3 C.4 D.5
Q38. In a Plane there are 37 straight lines, of which 13 pass through the point A and 11 pass through the point B.
Besides, no three lines pass through one point, no lines passes through both points A and B , and no two are parallel.
Find the number of points of intersection of the straight lines?
A. 525 B.555 C.545 D.535
Q39. 12 points lie on a circle. How many cyclic quadrilaterals can be drawn by using these points?
A. 470 B.450 C.495 D.475
Q40. If there are 15 dots on a circle, how many triangles can be formed?
A. 455 B.480 C.475 D.490
Q41. How many parallelograms will be formed if 7 parallel horizontal lines intersect 6 parallel vertical lines?
A. 215 B.315 C.115 D.415
Q42. A polygon 7 sides.How many diagonals can be formed?

A. 16 B.10 C.14 D.20
Type 8 - Miscellaneous
Q43.How many 3-digit numbers can be formed from the digits 2,3,5,6,7 and 9,which are divisible by 5 and none of the
digits is repeated?
A. 22 B.20 C.25 D.28
Q44.There are 14 points in a plane, out of which 4 are collinear. Find the number of triangles made by these points?
A. 390 B.365 C.360 D.350
Q45.A question paper consists of 2 sections having respectively 3 and 5 questions. The following note is given on the
paper .It is not necessary to attempt the entire question. One question from each section is compulsory. In how many
ways, a candidate can select the question?
A. 217 B.220 C.218 D.271
Q46.A five-digit number divisible by 3 is to be formed by using the digits 0,1,2,3,4 and 5 without repetition,the total
number of ways by which this can be done is?
A. 220 B.216 C.226 D.261
Q47.A new flag is to be designed with 6 vertical strips using some or all of the color yellow,green,blue and red.Then,the
number of ways this can be made such that no two adjacent stripes have the same color is?
A. 972 B.970 C.927 D.900
Q48.How many numbers can be formed from 1, 2, 3, 4, 5(without repetition), when the digit at the unit’s place must be
greater than that in the ten’s place?
A. 65 B.62 C.60 D.70
Q49.Everybody in a room shakes hand with everybody else. The total no. of handshakes is 66.The total number of
persons in the room is?
A. 10 B.11 C.12 D.20

FAQs @ Placements
Q50. How many words with or without meaning, can be formed by using all the letters of the word, ‘ORANGE’, using
each letter exactly once?
A. 700 B.720 C.750 D.800
Q51. There are 28 stations between Ernakulam and Chennai. How many second class tickets have to be printed, so that
a passenger can travel from one station to any other station?
A. 800 B.820 C.850 D.870
Q52. A college has 10 basketball players. A 5-member team and a captain will be selected out of these 10 players. How
many different selections can be made?
A. 1300 B.1250 C.1260 D.1200
Q53. From a group of 5men and 4 women, 3 persons are to be selected to form a committee so that at least 2 men are
there are on the committee. In how many ways can it be done?
A. 20 B.50 C.65 D. 86
Q54. In how many different ways can 6 different balls be distributed to 4 different boxes, when each box can hold any
number of ball?
A. 4009 B.4095 C.4096 D.4090
Q55. If the letters of the word VERMA are arranged in all possible ways and these words are written out as in a
dictionary, then the rank of the word VERMA is :
A. 108 B.110 C.112 D.115
Q56. First, second and third prizes are to be awarded at an engineering fair in which 13 exhibits have been entered. In
how many different ways can the prizes be awarded?
A. 1736 B.1216 C.1716 D.1720
Q57. How much number of times will the digit ‘7' are written when listing the integers from 1 to 1000?
A. 243 B.300 C.450 D.350
Q58. Goldenrod and No Hope are in a horse race with 6 contestants. How many different arrangements of finishes are
there if No Hope always finishes before Goldenrod and if all of the horses finish the race?
A. 360 B.400 C.700 D.500
Q59. How many 5-digit positive integers exist the sum of whose digits are odd?
A. 36, 000 B.38, 000 C.45, 000 D.96, 000
Q60. A college has 10 basketball players. A 5-member team and a captain will be selected out of these 10 players.
How many different selections can be made?
A. 1260 B.1360 C.1560 D.1440

CHAPTER-2
PROBABILITY

Probability
Probability means the chances of happening or occurring of an event. The probability of an event tells that how likely
the event will happen. Probability is a chance of prediction. If the probability that an event will occur is “x”, then the
probability an event will not occur is “1-x”.
Definitions
1. An experiment is a situation involving chances or probability that leads to results called outcomes.
2. An outcome is the result of a single trial of an experiment.
3. An event is one or more outcomes of an experiment.
4. Probability is the measure of how likely an event is.
Random Experiment
An operation which produces an outcome is known as experiment. When an experiment is conducted repeatedly under
the same conditions the results cannot be unique but may be one of the various possible outcomes. Such an experiment
is called as Random Experiment.
Here, we cannot predict the outcomes. Tossing a coin is a random experiment.
Example:-
1. Rolling a die.
2. Drawing a card from a pack of cards.
3. Taking out a ball from a bag containing balls of different colors.
TRIAL
Performing a random experiment is called a Trial.
Sample Space:
The set of all possible outcomes of a random experiment is called as a Sample Space. Denoted by S.
Sample Space, S= {1, 2, 3, 4, 5, 6}
Event:
Any possible outcome or combination of outcome is called as Event. The every subset of the sample space, S is called as
an Event .Events are usually denoted by A,B,C,D,E,F. When a coin is tossed, getting a head or a tail is an event.
S= {H, T}, A= {H, B}, B= {T}
Here, events A and B are subsets of the sample space S.
Equally Likely Events:

Two or more events are said to be equally likely if each one of them has an equal chance of occurring. In tossing a coin,
getting a head and getting a tail are equally likely events.

Mutually Exclusive Events:
Two or more events are said to be mutually exclusive when the occurrence of anyone event excludes the occurrence of
the other event. Mutually Exclusive Events cannot occur simultaneously.
Example:
In throwing a die, Let be the event of getting an odd number and F be the event of getting an even number.
E={1,3,5}
F={2,4,6}
Exhaustive Events:
If two or more events together constitute the sample space S then, these events are said to be Exhaustive Events. In
throwing a die, the events of getting an odd number and the events of getting an even number together from the
sample space. So, they are Exhaustive Events.
Sure Events:
Since, S is a subset of S,S itself is an event and S is called Sure or Certain Event.
Impossible Event:
Let F be an event of getting more than two heads in tossing two coins simultaneously. F={}=∅.So,F is an
impossible event .
Favorable Outcomes:-
The outcomes corresponding to the desired event are called as the Favorable Outcomes. In rolling a die there
are 6 outcomes.
Note
1. The probability of sure event is1.That is, P(S)=1
2. The probability of impossible event is 0.That is P(∅)=0.

Type 1 - Coins
Q1.When tossing two coins once. What is the probability of heads on both the coins?
A. 1/4 B.1/8 C.1/8 D.1/16
Q2.A fair coin is tossed 100 times. The probability of getting head an odd number of times is?
A. 1/4 B.1/2 C.1/8 D.1/16
Q3.Three unbiased coins are tossed. What is the probability of getting atmost 2 heads?
A. 7/8 B.1/8 C.5/8 D.2/8
Type 2 - Cards
Q4.One card is randomly drawn from a pack of 52 cards. What is the probability that the card drawn is a face card?
A. 5/13 B.4/13 C.3/13 D.2/13
Q5. One card is randomly drawn from a pack of 52 cards. What is the probability that the card drawn is either a red card
or a king?
A. 7/13 B.5/13 C.6/13 D.2/13
Q6.From a set of 17 cards numbered 1,2,3……………………17 one is drawn at random. The probability that the number is
divisible by 3 or 7 is?
A. 6/17 B.5/17 C.8/17 D.7/17
Q7.In shuffling a pack of card 3 is accidently dropped then the chance that the missing card should be of different suit is?
A. 169/425 B.196/425 C.155/425 D.170/425
Type 3 - Dice
Q8.A dice is rolled twice. What is the probability of getting a sum equal to 9?
A. 2/9 B.1/9 C.3/9 D.5/9
Q9.A dice is rolled three times and sum of three numbers appearing on the uppermost face is 15.The chance that the
first roll was four is:-
A. 1/108 B.2/108 C.3/108 D.4/108
Q10.A six-faced dice is so biased that is twice as likely to show an even number when throw. It is thrown twice. The
probability that the sum of two numbers thrown is even is?
A. 4/9 B.5/9 C.7/9 D.7/9
Type 4 - With/Without Replacement

Q11. Given is a basket of fruits containing 4 oranges, 5 apples and 1 pears. We pick three fruits with replacement from
the basket. Find the probability of getting an orange and two apples.
A. 1/3 B.1/2 C.1/4 D.1/5
Q12. A jar contains 10 blue balls and 11 red balls. Two balls are drawn without replacement. What is the probability of
getting two red balls?
A. 11/42 B.10/42 C.9/42 D.13/42

Q13. We have a bag containing 4 yellow, 5 green and 6 orange candies. We draw two candies without replacement. Find
the probability of getting both candies green?
A. 4/21 B.2/21 C.5/21 D.6/21
Q14. A basket contains 3 apples and 7 oranges. A fruit is drawn and put back in the basket. This process is repeated 3
times. What is the probability that selected three fruits are orange?
A. 343/1000 B.353/1000 C.454/1000 D.654/1000
Q15.A bag contains 5 red balls and 7 blue balls. Two balls are drawn at random without replacement, then find the
probability of that one is red and the other is blue?
A. 33/66 B.35/66 C.34/66 D.37/66
Q16.A basket contains 5 black and 8 yellow balls.Four balls are drawn at random and not replaced.What is the
probability that they are of different colors alternatively.
A. 56/429 B.57/429 C.58/429 D.68/429
Type 5 - Bayes Theorem and Conditional Probability
Q17. The probability that a student is not a swimmer is 1/5. Then the probability that one of the five students, four are
swimmers is?
2
A. 5C4(1/5)(4/5)4 B.None of these C.(4/5)2(1/5) D.(3/5) (1/4)
Q18. I forgot the last digit of a 7-digit telephone number. If 1 randomly dial the final 3 digits after correctly dialling the
first four, then what is the chance of dialing the correct number?
A. 1/1000 B.1/1001 C.1/990 D.1/999
Q19. A man can hit a target once in 4 shots. If he fires 4 shots in succession, what is the probability that he will hit his
target?
A. 81/256 B.175/256 C.1/256 D.0
Q20. Dinesh speaks truth in 3/4 cases and Abhishek lies in 1/5 cases. What is the percentage of cases in which both
Dinesh and Abhishek contradict each other in stating a fact?
A. 60% B.40% C.35% D.50%
Q21. An anti-aircraft gun can fire four shots at a time. If the probabilities of the first, second, third and the last shot
hitting the enemy aircraft are 0.7, 0.6, 0.5 and 0.4, what is the probability that four shots aimed at an enemy aircraft will
bring the aircraft down?
A. 0.916 B.0.900 C.0.964 D.0.708
Q22. A and B play a game where each is asked to select a number from 1 to 5. If the two numbers match, both of them
win a prize. The probability that they will not win a prize in a single trial is:
A. 21/25 B.20/25 C.19/25 D.18/25
Q23. Two squares are chosen at random on a chessboard. What is the probability that they have a side in common?
A. 1/18 B.1/20 C.63/64 D.1/9

Q24.The probability that a man lives after 10 years is ¼ and that his wife is alive after 10 years is 1/3.The probability that
neither of them is alive after 10 years is?
A. 1/2 B.3/4 C.1/4 D.1/5
Q25.The are 100 students in a college class of which 36 are boys studying Statistics and 13 girls not studying Statistics .If
there are 55 girls in all, then the probability that a boy picked up at random is not studying Statistics is:-
A. 2/5 B.1/5 C.3/5 D.4/5
Q26.A bag contains 2 yellow,3 green and 2 blue balls. Two balls are drawn at random. What is the probability that none
of the balls drawn is blue?
A. 11/21 B.12/21 C.10/21 D.13/21
Q27.Out of 15 students studying in a class,7 are from Maharashtra,5 are from Karnataka and 3 are from Goa, four
students are to be selected at random. What are the chances that atleast one is from Karnataka?
A. 10/13 B.9/13 C.12/13 D.11/13
Q28. If the probability for A fail in an examination is 0.2 and that of B is 0.3,then the probability that either A or B fails is?
A. 0.44 B.0.55 C.0.42 D.0.40
Q29.Two dice and two coins are tossed. The probability that both the coins show head and the sum of the numbers
found on the two dice is a prime number is?
A. 4/48 B.3/48 C.5/48 D.7/48
Q30.A speaks truth in 60% of the cases and B in 80% of the cases. In what percentage of cases are they likely to
contradict each other, narrating the same incident?
A. 44% B.55% C.40% D.50%
Q31.From 4 children,2 women and 4 men,4 person are selected. The probability that there are exactly 2 children among
the selected persons is?
A. 9/12 B.7/12 C.5/12 D.11/12
Q32.The probability that a man can hit a target is 3/4.He tries 5 times. The probability that he will hit the target atleast
three times, is?
A. 459/512 B.450/512 C.495/512 D.359/512
Q33.The letters B, G, I, N, R is rearranged to form the word ‘BRING’. Find its probability?
A. 2/120 B.1/120 C.3/120 D.4/120
Q34. Four unit squares are chosen at random on a chessboard. What is the probability that three of them are of one
color and fourth is of opposite color?
A. 80/427 B.160/427 C.640/1281 D.320/1281
Q35. Two urns contain 5 white and 7 black balls and 3 white and 9 black balls respectively. One ball is transferred to the
second urn and then one ball is drawn from the second urn. Find the probability that the first ball transferred is black,
given that the ball drawn is black?
A. 13/23 B.14/23 C.11/23 D.10/23

FAQs @ Placements
Q36. 1. If a number is chosen at random from the set {1, 2, 3…, and 100}, then the probability that the chosen number is
a perfect cube is:-
A. 1/25 B. 1/2 C. 4/13 D. 1/10
Q37. The probability that it is Friday and that a student is absent is 0.03. Since there are 5 school days in a week, the
probability that it is Friday is 0.2. What is the probability that a student is absent given that today is Friday?
A. 10% B. 15% C. 12% D. 13%
Q38. A person tosses an unbiased coin. When head turns up, he gets Rs.8 and tail turns up he loses Rs.4. If 3 coins are
tossed, what is probability that the gets a profit of Rs.12?
A. 3/8 B. 5/8 C. 3/4 D. 1/8
Q39. A number n is chosen from {2, 4, 6 … 48}. The probability that ‘n’ satisfies the equation (2x – 6) (3x + 12) (x – 6) (x –
10) = 0 is:-
A. 1/24 B. 1/12 C. 1/8 D. 1/6
Q40. If 10 letters are to be placed in 10 addressed envelopes, then what is the probability that at least one letter is
placed in wrong addressed envelope?
A. 1/10! B.1/9! C.1- (1/10!) D.9/10
Q41. Amit throws a biased coin on which the head appears in 65% of the situations. In a game involving this coin, if Amit
is paid Rs.15 per head and he has to pay Rs.20 for a tail, then in the long run, per game Amit makes an average……
A. Profit of Rs.2.25 B. Loss of Rs.2.25 C. Profit of Rs.2.75 D. Loss of Rs.2.75
Q42. From a box containing a dozen bulbs, of which exactly one half are good, and four bulbs are chosen at random to
fit into the four bulb holders in a room. The probability that the room gets lighted is…………
A. 2/3 B.1/3 C.33/44 D. 32/33
Q43. From a bag containing 6 pink and 8 orange balls, 8 balls are drawn at random. The probability that 5 of them are
pink and the rest are orange is…………
A. 16/143 B.19/143 C.17/143 D. 13/143
Q44. A problem is given to three students whose chances of solving it are 1/2, 1/3 and 1/4 respectively. What is the
probability that the problem will be solved?
A. 2/4 B.3/4 C.1/4 D.7/12
Q45. A man and his wife appear in an interview for two vacancies in the same post. The probability of husband's
selection is (1/7) and the probability of wife's selection is (1/5). What is the probability that only one of them is
selected?
A. 1/7 B.3/7 C.2/7 D.4/7

CHAPTER-3
GEOMETRY

FUNDAMENTAL CONCEPTS
Classification of angles:
Acute angle: An angle greater than 0° but less than 90° is called an acute angle
Right angle: An angle exactly equal to 90° is called a right angle
Obtuse angle: An angle greater than 90°but less than 180° is called an obtuse angle.
Straight angle: An angle exactly equal to 180° is called a straight angle.
Reflex angle: An angle greater than 180° but less than 360° is called a reflex angle.
Complete angle: An angle exactly equal to 360° is called a complete angle.
Complementary angles: If the sum of two angles equal to 90°, then the angles are complementary.
Supplementary angles: If the sum of two angles equal to 180°, then the angles are supplementary.
Adjacent angles: If two angles have a common hand and a common vertex then they are adjacent.
Basic properties of triangles:

1. Sum of all interior angles of a triangle = 180°
2. Largest angle is opposite to the longest side in a triangle or the longest side is opposite to the largest angle.
3. Smallest angle is opposite to the shortest side or the shortest side is opposite to the smallest angle.
4. Angles opposite to the equal sides are equal and the sides opposite to the equal angles are equal.
5. Sum of any two sides is greater than the third side.( Triangle inequality)
6. Difference of any two sides is lesser than the third side.
Exterior angles of a triangle: Any one of the three sides of a triangle is extended up to an outside point then the angle
formed by the extension line and one side of the triangle is called the exterior angle of the triangle. An exterior angle of
a triangle is equal to the sum of its opposite interior angles. Sum of all exterior angles = 360°
Geometrical centers of a triangle:

Circum center(S): The point of concurrency of the perpendicular bisectors of the three sides of a triangle is called the
Circum center of the triangle.
If S is the circum center and SA = SB = SC = R, where R is the circum radius.
In an acute angled triangle circum center lies inside the triangle.
In a right angled triangle, circum centre is the midpoint of the hypotenuse, hence the circum radius = 1/2 (hypotenuse).
In an obtuse angled triangle circum centre lies outside the triangle.
In center (I): it is the point of concurrency of all the angle bisectors of a triangle
If 'I' is the in centre and ID is a perpendicular drawn to the side BC.
ID is the in-radius of the triangle and that is denoted by 'r'.
Ortho center of a triangle (O): Point of concurrency of the three altitudes of a triangle is called the orthocenter and it is
denoted by ‘O’.
In an acute angled triangle ortho-center lies inside the triangle.
In a right angled triangle ortho center is the vertex where right angle formed.
In an obtuse angled triangle ortho center lies outside the triangle.
Centroid of a triangle (G): Centroid is the point of concurrency of all the medians.
The line segment joining one vertex and the midpoint of its opposite side is called median.
D, E and F are the mid points of the sides BC, AC and AB respectively.
AD, BE and CF are the medians.G is the Centroid.
In a right angled triangle the length of the median drawn to the hypotenuse is equal to half the length of the
hypotenuse.
Centroid divides medians in the ratio 2:1.
Internal angle bisector theorem: In ΔABC, AD is the angle bisector of ∠A, then AD/AC = BD/CD and BD * AC - AB * CD =
AD2
External angle bisector theorem: BE is the angle bisector of ∠CBD. Then BC/AB = CE/AE.
Apollonius theorem: In triangle ABC, AD is the median from vertex A to side BC.
AB2 + AC2 = 2(AD2 + BD2)

Regular polygons: In a polygon if all sides are equal in length and all the angles are equal in measure, then it is a regular
polygon.
Eg: square, equilateral triangle etc.
Trapezium: One pair of opposite sides are parallel.
Height of a trapezium is equal to the distance between the parallel sides.
Area = ½ * h(a+b), where a and b are the lengths of the parallel sides and h is distance between the parallel sides.
In a trapezium if the lengths of the nonparallel sides are equal, then it is an isosceles trapezium.
Parallelogram: Pair of opposite sides are parallel. i.e. AB || CD and BC || AD .
Height of a parallelogram is the distance between two parallel sides.
Diagonals bisect each other. i.e. M is the midpoint of AC and BD.
Δ AMB ≅ Δ CMD and ΔAMD ≅ ΔCMB.
Δ AMB, Δ CMD, ΔAMD and ΔCMB are all equal in area. i.e. Area of each triangle =1/4 * Area of the parallelogram
Area = b * h, where 'b' is base and 'h' is height.
Rhombus: All sides are equal
Diagonals bisect each other at 90°.
ΔAMB ≅ ΔCMD ≅ ΔAMD ≅ ΔCMB.
Area of each triangle = 1/4 * Area of the
Rhombus Area = 1/2 * (d1 * d2), where d1 and d2 are the diagonals.
Rectangle: Opposite sides are equal in length.
Opposite sides are parallel.
Diagonals are equal and bisect each other.
Four triangles formed by the intersection of two diagonals are all equal in area. i.e. area of each triangle = 1/4 * Area of
the rectangle.
Out of the above mentioned triangles, opposite triangles are congruent.
Area = length * breadth
Perimeter = 2 ( length + breadth)
Square: All sides are equal.All angles are right angles.
Diagonals are equal and bisect each other at 90°.
All the four triangles formed by the intersection of diagonals are congruent.
Area = side2 = a2
Perimeter = 4a
Length of diagonal = √2 * a
Circles: Circle is the collection of all points which are equidistant from a particular point on a plane. This particular point
is called the center of the circle and constant distance is the radius of the circle.
O – center r – radius, Diameter(d) = 2r
Area = π r2
Circumference = 2 πr
Angle around the centre = 360°
Area and arc length of a sector:
If the central angle of a sector = θ°, then; θ/3600 = arc length of the sector/ 2πr2 = area of the sector/ πr2
Area of sector = θ/360 * πr2
Arc length of the sector = θ/360* 2πr
Properties in a Circle:
1. Perpendicular from the center of a circle to a chord will bisect the chord. If two chords are intersecting in a circle,
then; PT * QT = RT * ST.
2. Angle inscribed in a semi circle is right angle.
3. Angle subtended at the center of a circle by an arc is double the angle subtended by it at any point on the remaining
part of the circumference.
4. Angles in the same segment of a circle are equal/Angles lie in the same arc are equal.
Tangents and secants to a circle: If a line touches the circle at exactly one point, then the line is a tangent to the circle.
AB is a tangent and T is the point of tangency.

There are infinitely many tangents are possible to a circle. Through one point on the circumference of a circle, there is
exactly one tangent is possible to a circle.
From an outside point, there are two different tangents are possible to a circle.
In the diagram PQ and PR are the tangents from an outside point P.
PQ = PR
Radius to the point of tangency is perpendicular to the tangent.
In the below diagram AB is a tangent and ACD is a secant to the same circle.
AB2 = AC * AD
If PBA and PDC are two secants to the same circle.
PB * PA = PC * PD
Cyclic quadrilateral: A quadrilateral inscribed in a circle is called Cyclic Quadrilateral.
Area of a cyclic quadrilateral = √((S-a)(S-b)(S-c)(S-d))
Where a, b, c and d are the four sides of the quadrilateral and S = a+b+c+d / 2
Opposite angles of a cyclic quadrilateral are supplementary. i.e. ∠A+∠C= ∠B+ ∠D=180°
Ptolemy's Theorem: If ABCD is a cyclic quadrilateral, then the product of the two diagonals is equal to the sum of the
product of the opposite sides. AC * BD = (AB * CD) + (AD * BC)
Results on Quadrilaterals:
i. The diagonals of a parallelogram bisect each other.
ii. Each diagonal of a parallelogram divides it into triangles of the same area.
iii. The diagonals of a rectangle are equal and bisect each other.
iv. The diagonals of a square are equal and bisect each other at right angles.
v. The diagonals of a rhombus are unequal and bisect each other at right angles.
vi. A parallelogram and a rectangle on the same base and between the same parallels are equal in area.
vii. Of all the parallelogram of given sides, the parallelogram which is a rectangle has the greatest area

Practice Exercises
Q1. The ratio of areas of a circle to that of a square, if the circle circumscribes the square is,
A. π:3 B. π:2 C. 2π:3 D. 2π:1
Q2. O is the centre of a circle with AC & BC being tangents originating from point c, which lies external to the circle. If
angle ACB is 50 degrees, then angle AOB in degrees is,
A. 80 B. 90 C.120 D. 130
Q3. In the figure above AC is the diameter of the circle. If angle ACB is 35 degrees then angle CAB is?
A. 35 B. 55 C. 65 D. 70
Q4. In the figure, AT is a tangent to the circle, with point of tangency at A. AB is a chord and angle
BAT is 59 degrees. Angle ACB in degrees will be,
A. 99 B. 111 C. 121 D. 131
Q5. In the above figure AD & BC are chords of the circle which intersect at
point E (not marked in the figure). If angle AED is 112 degrees & AD is parallel to BC, then angle ADE in degrees is,
A. 24 B. 34 C. 42 D. 43
Q6. The maximum number of tangents that can be drawn to two circles touching each other is,
A. 1 B. 2 C. 3 D. 4
Q7. In a circle of radius 13 cm, a chord AB is positioned at a distance of 5 cm, from its centre. Find the length of the
chord?
A. 12cm B. 18cm C. 20cm D. 24cm
Q8. Two circles with radius 5cm & 3cm are drawn on a plane paper such that the distance between their centers is 8 cm.
Which of the following is true?
A. Intersect at 2 points B. Touches each other externally C. Touches each other internally D. None of the above
Q9. In the figure AC & BD are not diameters of the circle. If Angle ABD is 55 degree then angle ACD is
A. 55 degree B. 52 degree C. 90 degrees D. Cannot be determined

Q10. ABCD is a cyclic quadrilateral. Side AD is extended to a point E. If angle ABC is 105 degree then angle CDE is,
A. 126 degrees B. 100 degrees C. 105 degrees D. 97.5 degrees
Q11. A circle has a diameter of 14cm.The length of the tangent drawn to it from a point outside the circle which is at a
distance of 25cm from the centre of the circle is,
A. 24cm B. 26cm C. 27cm D. 26.5cm
Q12. Find the centroid of the triangle whose vertices are: (3, -5), (-7, 4), (10, -2)
A. (2, -1) B. (-3, 1) C. (5, 2) D.(3, 4)
Q13. Find the third vertex of the triangle, if two of its vertices are at (-3, 1), (0, -2) and the centroid is at the origin
A. (2, 1) B. (3, 1) C. (3, 4) D. (5, 2)
Q14. Find the circum-centre of the triangle whose vertices are (-2, -3), (-1,0), (7, -6)
A. (4, 8) B. (1, 3) C. (2, -2) D. (3, -3)
Q15. The three vertices of a parallelogram are (1, 1), (4, 4), (4, 8). Find the fourth vertex
A. (-3, 1) B. (1, 5) C. (1, -3) D. (5, 1)
Q16. Find the centre of the circle passing through (5, -8), (2, -9), (2, 1)
A. (2, -4) B. (-3, 1) C. (5, 2) D. (3, 4)
Q17. If the points (1, 4) (r, -2), (-3, 16) are collinear. Find r
A. 7 B. 5 C. 6 D. 3
Q18. Find the area of the largest triangle that can be inscribed in a semicircle of diameter 6 cm?
A. 10 cm2 B. 9 cm2 C. 7.5 cm2 D. 8.25 cm2
Q19. Find the area of a cyclic quadrilateral whose sides are 8cm, 10cm, 12cm & 16 cm,
A. √15015 cm2 B. √30030 cm2 C.√10010 cm2 D. None of these
Q20. The perimeter of a semicircle of radius 14 cm is,

A. 44cm B. 72cm C. 88cm D. 48cm
Q21. Find the area of an equilateral triangle that has sides equal to 10 cm.
A. 43.3 cm2 B. 52.7 cm2 C. 60 cm2 D. 63.8 cm2
Q22. Triangle ABC, shown below, has an area of 15 mm 2. Side AC has a length of 6 mm and side AB has a length of 8
mm and angle BAC is obtuse. Find length of BC.
A. 12 mm B. 13.23 mm C. 15 mm D. 15.6 mm
Q23. If the largest angle in a triangle is 70degrees, what is least possible value of the smallest angle of the triangle?
A. 69 degrees B. 1 degrees C. 40 degrees D. 39 degrees
Q24. Find the number of non overlapping triangles which can be formed from the vertices of an octagon.
A. 56 B. 36 C. 16 D. 6

Q25. An order was placed for the supply of a carpet whose breadth was 6 m and length was 1.44 times the breadth.
What will be the cost of a carpet whose length and breadth are 40% more and 25% more respectively than the first
carpet. Given that the cost of carpet is Rs. 45 per sq m?
A. Rs. 3642.40 B. Rs. 3868.80 C. Rs. 4216.20 D. Rs. 4082.40
Q26. The perimeter of a square is double the perimeter of a rectangle. The area of the rectangle is 480 sq cm. Find the
area of the square.
A. 200 cm2 B. 72 cm2 C.162 cm2 D. Cannot be determined
Q27. A cube of side one meter length is cut into small cubes of side 10 cm each. How many such small cubes can be
obtained?
A. 10 B. 100 C. 1000 D. 10000
Q28. The volumes of two cones are in the ratio 1 : 10 and the radii of the cones are in the ratio 1 : 2. What is the ratio of
their heights?
A. 2 : 5 B. 1 : 5 C. 3 : 5 D. 4 : 5
Q29. A metallic sphere of radius 12 cm is melted and drawn into a wire, whose radius of cross section is 16 cm. What is
the length of the wire?
A. 6 cm B. 8 cm C. 9 cm D. None of these
Q30. The difference between the length and breadth of a rectangle is 23m. If it’s perimeter is 206m. Then its area is:
A. 2520 m2 B. 2420 m2 C. 2480 m2 D. 2529 m2
Q31. Find the length of the tangent from a point which is at a distance of 5cm from the center of the circle of
radius 3cm.
A. 4cm B. 3cm C. 2cm D. 1cm
Q32. A chord of length 6cm is drawn in a circle of radius 5cm. Calculate its distance from the center of the circle.
A. 1cm B. 4cm C. 8cm D. 10cm
Q33. How many bricks, each measuring 25cm x 11.25cm x 6cm, will be needed to build a wall of
8m x 6m x 22.5cm?
A. 5600 B. 6000 C. 6400 D. 7200
Q34. A boat having a length 3m and breadth 2m is floating on a lake. The boat sinks by 1cm when a man gets on it. The
mass of the man is:
A. 12 kg B. 60 kg C. 72 kg D. 96 kg
Q35. Find the area of the isosceles triangle whose perimeter is 20 cm and the ratio of equal side to unequal side is 3:4.
A. 8√5 cm2 B. 16√5 cm2 C. 12√5 cm2 D. 10√5 cm2
Q36. Find the circum radius of a triangle whose sides are 6cm,6cm and 8cm.
A. 7/√5 cm B. 8/√5 cm C. 9/√5 cm D. 11/√5 cm
Q37. In triangle ABC, BC is produced upto D. If angle ACD is 128 degrees, then find the angle of BAC, if length of the sides
AC and BC are equal
A. 69 degrees B. 59 degrees C. 64 degrees D. Cannot be determined
Q38. In a right angled triangle ABC, right angled at B, and ∠ACB=30. If AB= 5 units, find the measure of AC.
A. More than 10 B. 10 C. 8.6 D. 5.3

Q39. Triangle ABC is similar to EDF, AB = 5, ED = 22, and AC = 15 then find the measure of EF.
A. 40 B. 44 C. 63 D. 66
Q40. Let ABC be an equilateral triangle and AX, BY, CZ be the altitudes. Then the right statement out of the four given
responses is?
A. AX = BY = CZ B. AX ≠ BY ≠ CZ C. AX = BY ≠ CZ D. None of these

CHAPTER-4
MENSURATION – I
AREAS

Polygons:
A closed plane figure made up of several line segments that are joined together. The sides do not cross each other.
Exactly two sides meet at every vertex.
Types of Polygons
Regular - all angles are equal and all sides are the same length. Regular polygons are both equiangular and equilateral.
Equiangular - all angles are equal.
Equilateral - all sides are the same length.
Convex - a straight line drawn through a convex polygon crosses at most two sides. Every interior angle is
less than 180°.
Concave - you can draw at least one straight line through a concave polygon that crosses more than two
sides. At least one interior angle is more than 180°.
Polygon Formulas:
(N = # of sides and S = length from center to a corner)

1. Area of a regular polygon = (1/2) N sin(360°/N) S2
2. Sum of the interior angles of a polygon = (N - 2) x 180°
3. The number of diagonals in a polygon = 1/2 N(N-3)
4. The number of triangles (when you draw all the diagonals from one vertex) in a polygon = (N - 2)
Results on Triangles:
1. Sum of the angles of a triangle is 180°.
2. The sum of any two sides of a triangle is greater than the third side.
3. Pythagoras Theorem: In a right-angled triangle, (Hypotenuse)2 = (Base)2 + (Height)2.
4. The line joining the mid-point of a side of a triangle to the positive vertex is called the median.
5. The point where the three medians of a triangle meet, is called centroid. The centroid divided each of the medians in
the ratio 2 : 1.
6. In an isosceles triangle, the altitude from the vertex bisects the base.
7. The median of a triangle divides it into two triangles of the same area.
8. The area of the triangle formed by joining the mid-points of the sides of a given triangle is one-fourth of the area of
the given triangle.

Results on Quadrilaterals:
1. The diagonals of a parallelogram bisect each other.
2. Each diagonal of a parallelogram divides it into triangles of the same area.
3. The diagonals of a rectangle are equal and bisect each other.
4. The diagonals of a square are equal and bisect each other at right angles.
5. The diagonals of a rhombus are unequal and bisect each other at right angles.
6. A parallelogram and a rectangle on the same base and between the same parallels are equal in area.
7. Of all the parallelogram of given sides, the parallelogram which is a rectangle has the greatest area.
IMPORTANT FORMULAE
TRIANGLES:
1. Area of a triangle = ½ x Base x Height.
2. Area of a triangle = √𝑠(𝑠 − 𝑎)(𝑠 − 𝑏)(𝑠 − 𝑐)

where a, b, c are the sides of the triangle and s = ½ (a + b + c).
√3
3. Area of an equilateral triangle = × (𝑠𝑖𝑑𝑒 2 )
4
a
4. Radius of incircle of an equilateral triangle of side a = .
23
a
5. Radius of circumcircle of an equilateral triangle of side a = .
3
6. Radius of incircle of a triangle of area and semi-perimeter r = .
s
QUADRILATERAL:
1. Area of a rectangle = (Length x Breadth).
2. Perimeter of a rectangle = 2(Length + Breadth).
3. Area of a square = (side)2 = ½ (diagonal)2
4. Area of 4 walls of a room = 2 (Length + Breadth) x Height.
5. Area of parallelogram = (Base x Height).
6. Area of a rhombus = ½ x (Product of diagonals).
7. Area of a trapezium = ½ x (sum of parallel sides) x distance between them.
CIRCLE:
1. Area of a circle = 𝜋𝑅 2 , where R is the radius.
2. Circumference of a circle =2𝜋𝑅 .
2𝜋𝑅𝜃
3. Length of an arc = , where is the central angle.
360
𝜋𝑅2 𝜃
4. Area of sector = 360

Type 1 - Triangles
Q1. The perimeter of an equilateral △ is 72√3 meters. Find its height?
A. 63 metres B. 55 metres C. 40 metres D. 36 metres
Q2. One of the angles of a triangle is 60∘ and length of its two sides is 6 cm and 7 cm. The length of the third side of the
triangle is?
A. √43 B. 3 + √22 C. √43 𝑜𝑟 3 + √22 D. None
Q3. The altitude drawn to the base of an isosceles triangle is 8 cm and the perimeter is 32 cm. Find the area of the
triangle?
A. 50 B. 60 C. 70 D. 80
Q4. The base of a triangular field is three times its altitude. If the cost of cultivating the field at Rs. 24.68 per hectare be
Rs. 333.18, find its base and height?
A. B=900;H=300 B. B=300;H=900 C. B=600;H=700 D. B=500;H=900
Q5. Find the area of a right-angled triangle whose base is 12 cm and hypotenuse is 13 cm?
A. 30 B. 40 C. 50 D. 60
Type 2 - Quadrilaterals
Q6. How many tiles whose length and breadth are 12 cm and 5 cm respectively will be needed to fit in a rectangular
region whose length and breadth are respectively 144 cm and 100 cm?
A. 160 B. 240 C. 320 D. 450
Q7. A man walked diagonally across a square lot. Approximately, what was the percent saved by not walking along the
edges?
A. 30 B. 40 C. 50 D. 60
Q8. The diagonal of a rectangle is 17 cm long and its perimeter is 46 cm. Find the area of the rectangle?
A. 110 B. 120 C. 130 D. 140
Q9. A parallelogram has sides 30 m and 14 m and one of its diagonals is 40 m long. Then its area is?
A. 136 B. 236 C. 336 D. 436
Q10. The length of a rectangular hall is 5m more than its breadth. The area of the hall is 750 m. The length of the hall is?
A. 20 B. 25 C. 30 D. 35
Q11. One side of a rectangular field is 15 m and one of its diagonal is 17 m. Find the area of field?
A. 110 B. 120 C. 130 D. 140
Q12. The length of a rectangle is halved, while its breadth is tripled. What is the percentage change in area?
A. 20% B. 30% C. 40% D. 50%
Q13. The length of a rectangle is twice its breadth if its length is decreased by 5 cm and breadth is increased by 5 cm, the
area of the rectangle is increased by 75 cm2. Find the length of the rectangle?
A. 30 B. 40 C. 50 D. 60

Q14. The difference between the length and breadth of a rectangle is 23 m. If its perimeter is 206 m, then its area is?
A. 2520 B. 2420 C. 2320 D. 2620
Q15. An error 2% in excess is made while measuring the side of a square. The percentage of error in the calculated area
of the square is?
A. 1.04 B. 2.04 C. 3.04 D. 4.04
Q16. If the length of a certain rectangle is decreased by 4 cm and the width is increased by 3 cm, a square with the same
area as the original rectangle would result. Find the perimeter of the original rectangle?
A. 20 B. 30 C. 40 D. 50
Q17. The diagonal of a rectangle is cm and its area is 20 cm2. The perimeter of the rectangle must be?
A. 18 B. 28 C. 38 D. 48
Q18. The ratio between the perimeter and the breadth of a rectangle is 5 : 1. If the area of the rectangle is 216 cm 2,
what is the length of the rectangle?
A. 16 B. 18 C. 20 D. 22
Q19. If the length of the diagonal of a square is 20 cm, then its perimeter must be?
A. 40 cm B. 30 cm C. 10 cm D. 15 cm
Q20. The cost of fencing a square field @ Rs. 20 per metre is Rs.10.080. How much will it cost to lay a three meter wide
pavement along the fencing inside the field @ Rs. 50 per m2?
A. 53800 B. 43800 C. 83800 D. 73800
Q21. The ratio of the area of a square to that of the square drawn on its diagonal is?
A. 1:2 B. 2:3 C. 3:1 D. 4:1
Q22. The base of a parallelogram is twice its height. If the area of the parallelogram is 72 cm2, find its height?
A. 6 B. 7 C. 8 D. 9
Q23. The length of a rectangular plot is 20 metres more than its breadth. If the cost of fencing the plot @ Rs. 26.50 per
metre is Rs. 5300, what is the length of the plot in metres?
A. 20 B. 200 C. 300 D. 400
Q24. A room of 5m 44cm long and 3m 74cm broad is to be paved with square tiles. Find the least number of square tiles
required to cover the floor?
A. 136 B. 146 C. 166 D. 176
Q25. The ratios of areas of two squares, one having its diagonal double than the other is?
A. 2:1 B. 2:3 C. 3:1 D. 4:1
Q26. What is the least number of squares tiles required to pave the floor of a room 15.17 m long and 9.02 m broad?
A. 814 B. 714 C. 614 D. 713
Q27. A rectangular lawn 55 m by 35 m has two roads each 4 m wide running in the middle of it. One of the side is
parallel to the length and the other parallel to breadth. The cost of graveling the roads at 75 paise per m2 is?
A. Rs.58 B. Rs.158 C. Rs.258 D. Rs.358
Q28. The difference between two parallel sides of a trapezium is 4 cm and perpendicular distance between them is 19
cm. If the area of the trapezium is 475 cm2, find the lengths of the parallel sides?
A. 27 and 23 B. 24 and 23 C. 25 and 23 D. 22 and 23

Q29. A rectangular plot measuring 90 meters by 50 meters is to be enclosed by wire fencing. If the poles of the fence are
kept 5 meters apart, how many poles will be needed?
A. 65 m B. 45 m C. 55 m D. 56 m
Q30. The area of a rectangle is 460 m2. If the length is 15% more than the breadth, what is the breadth of the
rectangular field?
A. 20 m B. 30 m C. 40 m D. 50 m
Q31. 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 m2, then what is the width of the road?
A. 2.91 m B. 3 m C. 5.82 m D. None of these
Q32. The diagonal of the floor of a rectangular closet is 7 feet. The shorter side of the closet is 4 feet. What is the area of
the closet in square feet?
A. 5 ¼ B. 13 ½ C. 27 D. 37
Q33. A parallelogram has sides 30 m and 14 m and one of its diagonals is 40 m long. Then its area is?
A. 136 B. 236 C. 336 D. 436
Q34. Then, AB + BC = 2x metres. If the diagonal of a rectangle is 17 cm long and its perimeter is 46 cm. Find the area of
the rectangle?
A. 110 B. 120 C. 130 D. 140
Q35. The area of the square formed on the diagonal of a rectangle as its side is 108 1/3 % more than the area of the
rectangle. If the perimeter of the rectangle is 28 units, find the difference between the sides of the rectangle?
A. 8 B. 12 C. 6 D. 2
Q36. The length of a rectangular floor is more than its breadth by 200%. If Rs. 324 is required to paint the floor at the
rate of Rs. 3 per m2, then what would be the length of the floor?
A. 27 m B. 24 m C. 18 m D. 21 m
Q37. An order was placed for the supply of a carpet whose breadth was 6 m and length was 1.44 times the breadth.
What be the cost of a carpet whose length and breadth are 40% more and 25% more respectively than the first carpet.
Given that the ratio of carpet is Rs. 45 per m2?
A. Rs. 3642.40 B. Rs. 3868.80 C. Rs. 4216.20 D. Rs. 4082.40
Q38. The area of a square is 4096 cm2. Find the ratio of the breadth and the length of a rectangle whose length is twice
the side of the square and breadth is 24 cm less than the side of the square?
A. 18 : 5 B. 7 : 16 C. 5 : 14 D. None of these
Q39. A room is half as long again as it is broad. The cost of carpeting the at Rs. 5 per m2 is Rs. 270 and the cost of
papering the four walls at Rs. 10 per m2 is Rs. 1720. If a door and 2 windows occupy 8 m2, find the dimensions of the
room?
A. l=18; b=6;H=6 B. l=6; b=5;H=18 C. l=6;b=18;H=15 D. l=5;b=18;H=18
Q40. In measuring the sides of a rectangle, one side is taken 5% in excess and the other 4% in deficit. Find the error
percent in the area?
A. 0 .7% B. 0.8% C. 0.9% D. 0.3%

Type 3 - Circle
Q41. A chord AB of a circle of radius 5.25 cm makes an angle of 60∘ at the centre of the circle. Find the area of the major
segment?
A. 168 cm2 B. 100 cm2 C. 84 cm2 D. 70 cm2
Q42. A wooden door wedge is in the shape of a sector of a circle of radius 10 cm with angle 24∘ and constant thickness 3
cm. Find the volume of wood used in making the wedge?
A. 62.83 cm3 B. 57.88 cm3 C. 41.22 cm3 D. 333.15 cm3
Q43. Three circles of radius 3.5 cm are placed in such a way that each circle touches the other two. The area of the
portion enclosed by the circles is?
A. 1.967 B. 1.867 C. 1.767 D. 1.567
Q44. The inner circumference of a circular race track, 14 m wide, is 440 m. Find radius of the outer circle
A. 44 B. 22 C. 33 D. 84
Q45. A wheel makes 1000 revolutions in covering a distance of 88 km. Find the radius of the wheel?
A. 14 B. 13 C. 12 D. 11
Q46. The area of a circular field is 13.86 hectares. Find the cost of fencing it at the rate of Rs. 4.40 per metre?
A. 2808 B. 3808 C. 4808 D. 5808
Q47. The diameter of the driving wheel of a bus is 140 cm. How many revolutions per minute must the wheel make in
order to keep a speed of 66 kmph?
A. 150 B. 250 C. 350 D. 550
Q48. Find the length of a rope by which a cow must be tethered in order that it may be able to graze an area of 9856
m2?
A. 56 m B. 16 m C. 14 m D. 76 m
Q49. The area of a circle of radius 5 is numerically what percent its circumference?
A. 150% B. 250% C. 350% D. 450%
Q50. The sector of a circle has radius of 21 cm and central angle is 135°. Find its perimeter?
A. 91.5 cm B. 93.5 cm C. 94.5 cm D. 92.5 cm
Type 4 – Polygons
Q51. How many sides are there in a convex polygon having 27 diagonals?
A. 6 B. 9 C. 15 D. 12
Q52. Exterior angle of a regular polygon is 1/3 rd of interior angle. How many sides are there?
A. 6 B. 7 C. 8 D. 9
Q53. ABCDE is regular pentagon, diagonal AD divides angle CDE in two parts. Find the ratio of angles ADE to angle ADC?
A. 2:1 B. 2:3 C. 3:2 D. 1:2
Q54. The difference between exterior angle of (n-1) sided regular polygon and exterior angle of (n+2) regular polygon is
6 degree, what is n?
A. 13 B. 10 C. 15 D. 12

Q55. Ratio of interior angle to exterior angle of a regular polygon is 7:2, what is the number of sides?
A. 8 B. 10 C. 15 D. 9
Q56. The ratio of number of sides of two regular polygons is 1:2, ratio of measures of their interior angles is 3:4, what is
the number of sides?
A. 1:2 B. 3:4 C. 3:8 D. 2:1
Type 4 - Mixed
Q57. Four equal sized maximum circular plates are cut off from a square paper sheet of area 784 cm2. The circumference
of each plate is?
A. 22 cm B. 44 cm C. 66 cm D. 88 cm
Q58. Five concentric squares are given. If the area of the circle inside the smallest square is 77 square units and the
distance between the corresponding corners of consecutive squares is 1.5 units, find the difference in the areas of the
outermost and innermost squares?
A. 1254 sq units B. 1008 sq units C. 877 sq units D. 240 sq units
Q59. The area of the largest circle that can be drawn inside a rectangle with sides 18 cm by 14 cm is?
A. 49 B. 154 C. 378 D. 1078
Q60. A wire can be bent in the form of a circle of radius 56 cm. If it is bent in the form of a square, then its area will be?
A. 7744 B. 8844 C. 5544 D. 4444
Q61. Find the ratio of the areas of the in circle and circum circle of a square?
A. 1:1 B. 1:2 C. 1:3 D. 1:4
Q62. A circular swimming pool is surrounded by a concrete wall 4 ft wide. If the area of the concrete wall surrounding
11
the pool is 𝑡ℎ that of the pool, then the radius of the pool is?
25
A. 10 ft B. 20 ft C. 30 ft D. 40 ft
Q63. There are two circles of different radii. The area of a square is 784 cm2 and its side is twice the radius of the larger
circle. The radius of the larger circle is seven - third that of the smaller circle. Find the circumference of the smaller
circle?
A. 6∏ cm B. 8∏ cm C. 12∏ cm D. 16∏ cm
Q64. A 3 by 4 rectangle is inscribed in circle. What is the circumference of the circle?

A. 2.5π B. 3π C. 5π D. 4π
Q65. The length of a rectangle is two - fifths of the radius of a circle. The radius of the circle is equal to the side of the
square, whose area is 1225 square units. What is the area (in square units) of the rectangle if the rectangle if the
breadth is 10 units?
A. 140 B. 156 C. 175 D. 214
Q66. The perimeter of a circle and a square field are equal. What is the diameter of the circle field if the area of the
square field is 484 m2?
A. 14 m B. 21 m C. 28 m D. None of these

CHAPTER – 5
MENSURATION – II
(Surface Area & Volume)

Lateral Surface Area: The surface area is the area that describes the material that will be used to cover a geometric
solid. When we determine the surface areas of a geometric solid we take the sum of the area for each geometric form
within the solid.
Total Surface Area: Area of lateral surface + base area
Volume: The volume tells us something about the capacity of a figure
Vertex: The common vertex of the triangular faces of a pyramid is called the vertex of the pyramid.
Slant height: The slant height of regular right-pyramid is the line segment joining the vertex to the mid-point of anyone
of the sides of the base.
Lateral faces: The side of a pyramid is known as its lateral faces. If the base of a pyramid is a polygon of n sides then it
has n lateral faces, each one of which is a triangle and 2n edges.
CUBOID:
Let length = l, breadth = b and height = h units. Then
Volume = (l x b x h) cubic units.
Surface area = 2(lb + bh + lh) sq. units.
Diagonal = l2 + b2 + h2 units.
CUBE:
Let each edge of a cube be of length a. Then,
Volume = a3 cubic units.
Surface area = 6a2 sq. units.
Diagonal = 3a units.
CYLINDER:
Let radius of base = r and Height (or length) = h. Then,
Volume = (𝜋𝑟 2 h) cubic units.
Curved surface area = (2𝜋𝑟ℎ) sq. units.
Total surface area = 2𝜋𝑟(ℎ + 𝑟) sq. units.
CONE:
Let radius of base = r and Height = h. Then,
Slant height, l = h2 + r2 units.
1
Volume = 𝜋𝑟 2 ℎ cubic units.
3
Curved surface area = 𝜋𝑟𝑙 sq. units.
Total surface area =(𝜋𝑟𝑙 + 𝜋𝑟 2 ) sq. units.

SPHERE:
Let the radius of the sphere be r. Then,
4
Volume = 3 𝜋𝑟 3 cubic units.
Surface area =4𝜋𝑟 2 sq. units.
HEMISPHERE:
Let the radius of a hemisphere be r. Then,
2
Volume = 3 𝜋𝑟 3 cubic units.
Curved surface area =2𝜋𝑟 2 sq. units.
Total surface area = 3𝜋𝑟 2 sq. units.
Note: 1 litre = 1000 cm3.
PRISM:
A prism is a solid, whose side faces are parallelograms and whose ends (or bases) are congruent parallel rectilinear
figures.
Volume of prism = Area of base × Height.

Lateral Surface area = Perimeter of base × Height.
Total surface area = Lateral surface area + 2(Area of base)
PYRAMID:
A Pyramid Is A Polyhedron Whose Base Is A Polygon Of Any Number Of Sides And Whose Other Faces Are Triangle With
A Common Vertex.
Volume of pyramid = 1/3 (Area of base) × height
Lateral Surface area = 1/2 (Perimeter of base) × Slant height.
Total surface area = Area of base + lateral surface area.

Type 1 - Cubes & Cuboids
Q1. The dimensions of a room are 10m x 7m x 5m. There are 2 doors and 3 windows in the room. The dimensions of the
doors are 1m x 3m. One window is of size 2m x 1.5m and the other 2 windows are of size 1m x 1.5m. The cost of painting
the walls at Rs. 3 per m2 is?
A. Rs.174 B. Rs.274 C. Rs.374 D. Rs.474
Q2. A room is half as long again as it is broad. The cost of carpeting at Rs. 5 per m2 is Rs. 270 and the cost of papering the
four walls at Rs. 10 per m2 is Rs. 1720. If a door and 2 windows occupy 8 m2, find the dimensions of the room?
A. b=6;l=18;h=6 B. b=5;l=6;h=18 C. b=18;l=6; h=15 D. b=18; l=5; h=18
Q3. How many cubes of 3 cm edge can be cut out of a cube of 18 cm edge?
A. 36 B. 232 C. 216 D. 484
Q4. A rectangular block 6 cm by 12 cm by 15 cm is cut up into an exact number of equal cubes. Find the least possible
number of cubes?
A. 30 B. 40 C. 10 D. 20
Q5. A hall is 15 m long and 12 m broad. If the sum of the areas of the floor and the ceiling is equal to the sum of the
areas of four walls, the volume of the hall is?
A. 720 B. 900 C. 1200 D. 1800
Q6. Three solid cubes of sides 1 cm, 6 cm and 8 cm are melted to form a new cube. Find the surface area of the cube so
formed?
A. 486 B. 586 C. 686 D. 786
Q7. A cistern 6m long and 4 m wide contains water up to a depth of 1 m 25 cm. The total area of the wet surface is?
A. 49 B. 50 C. 53.5 D. 55
Q8. The volume of a wall, 5 times as high as it is broad and 8 times as long as it is high, is 12.8 m3. Find the breadth of
the wall?
A. 40 cm B. 30 cm C. 20 cm D. 10 cm
Q9. A cistern of capacity 8000 litres measures externally 3.3 m by 2.6 m by 1.1 m and its walls are 5 cm thick. The
thickness of the bottom is?
A. 90 cm B. 1 dm C. 1 m D. 1.1 cm
Q10. How many bricks each measuring 25cm by 11.25cm by 6 cm, will be needed to build a wall of 8m by 6m by 22.5m?
A. 5600 B. 600 C. 6400 D. 7200
Q11. In a shower, 5 cm of rain falls. The volume of water that falls on 1.5 hectares of ground is?
A. 75 m3 B. 750 m3 C. 7500 m3 D. 75000 m3
Q12. The size of the wooden block is 5 cm x 10 cm x 20 cm. How many such blocks will be required to construct a solid
wooden cube of minimum size?
A. 6 B. 8 C. 12 D. 16
Q13. A large cube is formed from the material obtained by melting three smaller cubes of sides 3 cm, 4 cm and 5 cm.
What is the ratio of the total surface areas of the smaller cubes and the large cube?
A. 2 : 1 B. 3 : 2 C. 25 : 18 D. 27 : 20

Q14. The height of the wall is 6 times its width and length of the wall is 7 times its height .if the volume of the wall be
16128 m3.Its widths is?
A. 4 m B. 5 m C. 6 m D. 7 m
Q15. The cost of the paint is Rs.36.50 per kg. If 1 kg of paint covers 16 ft2, how much will it cost to paint outside of a
cube having 8 feet each side?
A. Rs.962 B. Rs.672 C. Rs.546 D. Rs.876
Q16. Water flows into a tank 200 m x 160 m through a rectangular pipe of 1.5 m x 1.25 m @ 20 kmph . In what time (in
minutes) will the water rise by 2 meters?
A. 92min B. 93min C. 95min D. 96min
Q17. A boat having a length 3 m and breadth 2 m is floating on a lake. The boat sinks by 1 cm when a man gets on it. The
mass of the man is?
A. 12 kg B. 60 kg C. 72 kg D. 96 kg
Type 2 - Cylinder
Q18. The curved surface area of a cylindrical pillar is 264 m2 and its volume is 924 m3. Find the ratio of its diameter to its
height?
A. 3 : 7 B. 7 : 3 C. 6 : 7 D. 7 : 6
Q19. A hollow iron pipe is 21 cm long and its external diameter is 8 cm. If the thickness of the pipe is 1 cm and iron
weight, then the weight of the pipe is?
A. 3.6 kg B. 3.696 kg C. 36 kg D. 36.9 kg
Q20. There is a cylinder circumscribing the hemisphere such that their bases are common. Find the ratio of their
volumes?
A. 3/2 B. 5/2 C. 7/2 D. 9/2
Q21. How many iron rods, each of length 7 m and diameter 2 cm can be made out of 0.88 m3 of iron ?
A. 500 B. 600 C. 400 D. 300
Q22. A single pipe of diameter x has to be replaced by six pipes of diameters 12 cm each. The pipes are used to covey
some liquid in a laboratory. If the speed/flow of the liquid is maintained the same then the value of x is?
A. 14.69 cm B. 29.39 cm C. 18.65 cm D. 22.21 cm
Q23. A square sheet of paper is converted into a cylinder by rolling it along its length. What is the ratio of the base
radius to the side of the square?
A. 1: 2𝜋 B. 2𝜋 ∶ 1 C. 𝜋: 1 D. 1: 𝜋
Type 3 – Cone
Q24. A right triangle with sides 3 cm, 4 cm and 5 cm is rotated the side of 3 cm to form a cone. The volume of the cone
so formed is?
A. 12 pi cm3 B. 15 pi cm3 C. 16 pi cm3 D. 20 pi cm3

Q25. Two cones have their heights in the ratio 1:3 and the radii of their bases in the ratio 3:1. Find the ratio of their
volumes?
A. 3:1 B. 2:1 C. 4:1 D. 5:1
Q26. The slant height of a right circular cone is 10 m and its height is 8 m. Find the area of its curved surface?
A. 40 pi m2 B. 50 pi m2 C. 60 pi m2 D. 70 pi m2
Q27. The radius of base and height of a cone are increased by 10%. Find the percentage increase in volume?
A. 33.5% B. 33.1% C. 32.1% D. 53.1%
Q28. What is the total surface area of a right circular cone of height 14 cm and base radius 7 cm?
A. 344.35 cm2 B. 462 cm2 C. 498.35 cm2 D. None of these
Q29. Two cylindrical buckets have their diameters in the ratio 3:1 and their heights are as 1: 3. Find the ratio of their
volumes?
A. 2:1 B. 3:1 C. 4:1 D. 5:1
Q30. A Conical tent was erected by the army at a base camp with height 3 m and base diameter 10 m. If every person
requires 3.92 m3 air, then how many persons can be seated in that tent approximately?
A. 20 B. 19 C. 17 D. 22
Q31. Baskin n Robbin’s biggest ice-cream cardboard cone has height of 24 cm and slant height of 25 cm. One sheet of
cardboard makes 8 such cones. What is area of the sheet?
A. 4400 cm2 B. 6236 cm2 C. 6000 cm2 D. 550 cm2
Type 4 - Sphere
Q32. If a solid sphere of radius 10 cm is molded into 8 spherical solid balls of equal radius, then surface area of each ball
(in sq.cm) is?
A. 100 π B. 101/π C. 99 π/12 D. 54/13π
Q33. When Jaya divided surface area of a sphere by the sphere’s volume, she got the answer as 1/18 cm. What is the
radius of the sphere?
A. 24 cm B. 6 cm C. 54 cm D. 4.5 cm
Q34. Sam has a solid metal ball with diameter 6cm. He melts it and uses the material for making a solid cylinder. If the
diameter of the cylinder is same as the ball, what would its height be?
A. 4 cm B. 4.5 cm C. 6 cm D. 8 cm
Q35. The area of the base of a right circular cone is 154cm2 and its height is 14cm. Taking π = 22/7, the curved surface of
the cone is?
A. 154 x 5 cm2 B. 154 x 7 cm2 C. 11 cm2 D. 5324 cm2
Q36. Three solid metallic balls of diameter 3 cm, 4 cm and 5 cm are melted into a solid sphere. The radius of new sphere
is?
A. 6 cm B. 4 cm C. 3 cm D. 5 cm
Q37. A solid rubber sphere weighs 6 kg when its diameter is 6 cm. Using the same material, a hollow sphere is made
with outer diameter 18 cm and inner diameter 12 cm. What is its weight?
A. 114 kg B. 96 kg C. 72.64 kg D. 64 33 kg

Q38. A sphere with surface area (792/7) m2 has a volume?
A. 7 cm2 B. 792 cm3 C. 792/7 cm3 D. 792π/7 cm3
Q39. Volume of a sphere with radius r is obtained by multiplying its surface area by r and a constant. Find the value of
constant?
A. 3 B. 1/3 C. 4/3 D. 1
Q40. A sphere of maximum volume is cut out from a solid hemisphere of radius r, The ratio of the volume of the
hemisphere to that of the cut out sphere is?
A. 3 : 2 B. 4 : 1 C. 4 : 3 D. 7 : 4
Q41. The capacities of two hemispherical vessels are 6.4 litres and 21.6 litres. The areas of inner curved surfaces of the
vessels will be in the ratio of?
A. 2–√3 :3–√2 B. 2 : 3 C. 4 : 9 D. 16 : 81
Type 5 - Prism and Pyramid
Q42. The base of a right prism is an equilateral triangle of area 173 cm2 and the volume of the prism is 10380 cm3. The
area of the lateral surface of the prism is (use √3=1.73) ?
A. 1200 cm2 B. 2400 cm2 C. 3600 cm2 D. 4380 cm2
Q43. If the slant height of a right pyramid with square base is 4 m and the total slant surface area of the pyramid is 12
m2, then the ratio of total slant surface area and area of the base is?
A. 16 : 3 B. 24 : 5 C. 32 : 9 D. 12 : 3
Q44. The length of each edge of a regular tetrahedron is 12 cm. The area (in cm2) of the total surface of the tetrahedron
is?
A. 288√3 B. 144√2 C. 108√3 D. 144√3
Q45. A right pyramid stands on a square base of side 16 cm and its height is 15 cm. The area of its slant surface is (in
cm2)?
A. 514 B. 544 C. 344 D. 444
Q46. The base of right, prism is an equilateral triangle. If the lateral surface area and volume is 120 cm2, 40√3 cm3
respectively them the side of base of the prism is?
A. 4 cm B. 5 cm C. 7 cm D. 40 cm
Q47. Each edge of a regular tetrahedron is 4 cm. Its volume (in cm3) is?
A. (16√3)/3 B. 16√3 C. (16√2)/3 D. 16√2
Q48. The base of a right prism is a quadrilateral ABCD, given that AB = 9 cm, BC = 14 cm, CD = 13 cm,
DA = 12 cm and ∠DAB=90°. If the volume of the prism be 2070 cm3, then the area of lateral surface is?
A. 720cm2 B. 810 cm2 C. 1260 cm2 D. 2070 cm2
Q49. The total surface area of a regular triangular pyramid with each edges of length 1 cm is?
A. 2 √2 cm2 B. √3 cm2 C. 4 cm2 D. 4√3 cm2
Q50. Base of a right pyramid is a square of side 10 cm. If the height of the pyramid is 12 cm, then its total surface area
is?
A. 360 cm2 B. 400 cm2 C. 460 cm2 D. 260 cm2

Q51. The base of a right prism is a trapezium whose lengths of two parallel sides are 10 cm and 6 cm and distance
between them is 5 cm. If the height of the prism is 8 cm, its volume is?
A. 300 cm3 B. 300.5 cm3 C. 320 cm3 D. 310 cm3
Type 6 – Mixed
Q52. A cone and a hemisphere have equal bases and equal volumes. Find the ratio of their heights?
A. 1:21:2 B. 2:12:1 C. 3:13:1 D. 3:43:4
Q53. A cone and sphere have the same radius of 12 cm. Find the height of the cone if the cone and sphere have the
same volume?
A. 18 cm B. 24 cm C. 36 cm D. 48 cm
Q54. A solid wooden toy is in the shape of a right circular cone mounted on a hemisphere. If the radius of the
hemisphere is 4.2 cm and the total height of the toy is 10.2 cm, find the volume of the wooden toy?
A. 104.22 cm3 B. 162.08 cm3 C. 427.56 cm3 D. 266.11 cm3
Q55. A metallic sheet is of rectangular shape with dimensions 48 m x 36 m. From each of its corners, a square is cut off
so as to make an open box. If the length of the square is 8m, the volume of the box (in m3) is?
A. 4830 B. 5120 C. 6420 D. 8960
Q56. 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 m2, then what is the width of the road?
A. 2.91 m B. 3 m C. 5.82 m D. None of these
Q57. 66 cubic centimeters of silver is drawn into a wire 1 mm in diameter. The length of the wire in metres will be?
A. 84 B. 90 C. 168 D. 336
Q58. A cylindrical rod of iron, whose height is equal to its radius, is melted and cast into spherical balls whose radius is
half the radius of the rod. Find the number of balls?
A. 3 B. 4 C. 5 D. 6
Q59. Find the ratio of the surfaces of the inscribed and circumscribed spheres about a cube?
A. 1:2 B. 1:3 C. 1:4 D. 1:5
Q60. What part of the volume of a cube is the pyramid whose base is the base of then cube and whose vertex is the
center of the cube?
A. 4:1 B. 5:1 C. 6:1 D. 7:1
Q61. What is the ratio between the volumes of a cylinder and cone of the same height and of the same diameter?
A. 2:1 B. 3:1 C. 4:1 D. 5:1
Q62. A rectangular block has the dimensions 5 cm x 6 cm x7 cm it is dropped into a cylindrical vessel of radius 6 cm and
height 10 cm. If the level of the fluid in the cylinder rises by 4 cm, what portion of the block is immersed in the fluid?
A. 22/7 x 24/35 B. 22/7 x 36 x 4 C. 22/7 x 36/5 D. 22/7 x 37/21
Q63. A 5 cubic centimeter cube is painted on all its side. If it is sliced into 1 cm3 cubes, how many 1 cm3 cubes will have
exactly one of their sides painted?
A. 9 B. 61 C. 98 D. 54

Q64. Raju has a cone, a hemisphere and a cylinder. They have the same height. Raju immerses them completely in a
bucket full of water. What will be ratio of volume of cylinder to cone to hemisphere, if they also have same bases?
A. 1:2:3 B. 3:1:2 C. 2:1:3 D. 1:1:1
Q65. 2 cm thick 14 metal plates are kept exactly one above the other. On the top plate a hemisphere of diameter 6cm is
kept. This just covers the top plate. What is the volume of the entire object?
A. 360π cm3 B. 144π cm3 C. 81 cm3 D. 270π cm3
Q66. A metallic hemisphere is melted and recast in the shape of a cone with the same base radius (R) as that of the
hemisphere. If H is the height of the cone, then ?
A. H = 2R B. H = 3R C. H = 3–√R D. H=4R
Q67. A hollow sphere of internal and external diameters 4 cm and 8 cm respectively, melted into a cone of base
diameter 8 cm. The height of the cone is?
A. 12 cm B. 14 cm C. 15 cm D. 18 cm

CHAPTER – 6
ALGEBRA

POLYNOMIALS
Linear Equation in two variables:
Quadratic Equation:- Equation in one variable with degree 2 in form ax2+bx+c=0 is a quadratic equation. Where a,b,c
are integers and a can’t be zero.

INEQUALITIES (WAVY CURVE METHOD)
The wavy curve method (also called the method of intervals) is a strategy used to solve inequalities of the form f(x)/g(x)
>0 or <0(>=0,<=0). The method uses the fact that f(x)/g(x) can only change sign at its zeroes and vertical asymptotes, so
we can use the roots of f(x) and g(x) to sketch a graph of the function over different intervals. There are some basic
rules. These are:-
Rule 1- Coefficient of x must be 1.
Rule 2- Power of factors should be odd. For even power ignore the factors.
Rule 3- Expression is in multiplication or division follow the same rules.
Rule 4- Started the curve from right end with positive sign, then alternatively negative & positive sign for the remaining
factor’s values.
Basic example-
3𝑥−𝑥 2
𝒔𝒐𝒍𝒗𝒆 𝒇𝒐𝒓 𝒙 ∶ (𝑥+4)2
≥ 0.
−𝑥(𝑥−3)
Step 1- Factor the polynomials: (𝑥+4)2
≥0
Step 2- Make the coefficient of the variable of all factors positive. Multiply/divide both sides of the inequality by -1 to
𝑥(𝑥−3)
remove the minus sign (remember that in doing so the inequality would reverse): (𝑥+4)2
≤0
Step 3- Find the roots and asymptotes of the inequality by equating each factor to 0:
x=0 and x-3=0 => x=3.
x+4=0 => x=-4.
Step 4- Plot the points on the number line. Now, start with the largest factor, i.e. 3. Initially, a curve from the positive
region of the number line should intersect that point (here 3). Now, look at the power of the respective factors. If it is
odd, then we have to change the path of the curve from their respective roots. If it is even, continue in the same region.
Here, the curve would change its path at 0 and 3 because their factors are odd powers. However, at 4, it would not
change its direction since its factor has an even power.

Now, if the inequality is either >= or <= 0, then we have to consider those values of x at which the inequality is equal to
0. However, as a rule of the wavy curve method, we should exclude the root of the factor in the denominator (here -4) in
our solution set.
So, our final answer is 𝑥 ∈ [0,3] .
INTEGRAL SOLUTIONS
Integral solutions for the given type of equation- x1+x2+⋯+xr=n.
Case-1: Positive integral solutions.
Let us understand the concept from an example:
X1 + X2 + X3= 8.
To solve this, imagine that there are 8 identical objects placed next to each other with gaps separating them.8 objects
have 7 gaps between them. Now, I can select 2 gaps from among the 7 in 7C2 ways. These selected gaps will hold the plus
signs of the given equation. Now, the number of objects to the left of the first plus sign, the number of objects between
the two plus signs and the number of objects to the right of the second plus sign will be the values of X1, X2 and
X3 respectively.
Therefore, number of positive integral solutions of equation x1+x2+⋯+xr=n

= Number of ways in which n identical balls can be distributed into r distinct boxes where each box must contain at least
one ball= (n-1)C(r-1)
Case-2: Number of Non-negative integral solutions
We will continue with our previous equation. The number of non-negative integral solutions will be different from
number of positive integral solutions as the value of variables can be 0 as well.
We will substitute the variables in the question such that this case would become similar to previous case. In previous
case, (X1, X2, X3) >= 1. In this case, (X1, X2, X3) >= 0. Therefore, (X1+1, X2+1, X3+1) >= 1. Substitute X1+1=Y1, X2+1=Y2 and
X3+1=Y3 in the given equation such that
(X1+1) + (X2+1) +(X3+1) = 11
=> Y1+Y2+Y3=11.
Now this case becomes similar to previous one and number of solutions is 10C2.
Therefore, Number of non-negative integral solutions of equation x1+x2+⋯+xr=n
= Number of ways in which n identical balls can be distributed into r distinct boxes where one or more boxes can be
empty.= (n+r-1)C(r-1)
Case-3: Constraints on the variables.
Consider following equation- A+B+C = 13, where, 1=< (A,B,C) <=6.
To solve this, replace A,B,C with P,Q,R such that P= 6-A, Q=6-B and R=6-C. Then, (6-P)+(6-Q)+(6-R)=13 which implies
P+Q+R=5. As A ranges from 1 to 6, P ranges from 0 to 5. Hence, the problem reduces to finding the non-negative
solutions of P+Q+R = 5. The number of non-negative solutions is 7C2 = 21.
Another way is to use the following concept. If the linear equation is x1 + x2 +..+ xr= n and 0<= (x1, x2 .… xr ) <=p then the
problem can be reduced to finding the exponent of xnin the expression ( 1 + x + x2 + x3..+ xp )r.
Example 1: Find the number of positive integral solutions of |x| + |y| = 10.
Solution 1: Let |x|= a and |y|=b. First find the positive integral solution of a+b = 10.
Number of non-zero integral solutions= 10-1 C 2-1 = 9. Now for each solution (a1, b1), the values of (x,y)= (a1, b1), (-a1, b1),
(a1, -b1) and (-a1, -b1). So total number of non-zero integral solutions= 4×9 =36.
Example 2: Find the number of positive integral for a,b,c and d such that their sum is not more than 15.
Solution 2: a + b + c + d < 15.
a + b + c + d = 14,13,12,11,10,9,8,7,6,5,4.(Since we need to find positive integral solutions, sum of 4 variables cannot be
less than 4)
Total no of positive solution = 13C3 + 12C3+ 11C3 … 3C3
= 286+220+165+120+84+56+35+20+10+4+1=1001.
Maximum and Minimum Values

It has been observed that the graph of f(x) = ax2 + bx + c extends upwardly or downwardly in all cases accordingly to a >
0 or a < 0. Now, when graph extends upwardly, then the vertex determines the minimum of f(x) i.e. when a > 0, then at x
= b/2a, f(x) attains its minimum equal to (–D/4a).
In the other case, when the graph extends downwardly, then the vertex determines the maximum of f(x), i.e., when a <
0, then at x = –b/2a, f(x) attains its maximum equal to (–D/4a). These same concepts are applicable in numerical which
demand the computation of maximum or minimum values of a function.
Some key Points
When a > 0, then maximum of f(x) does not exist and when a < 0, then minimum of f(x) does not exist. This happens
because of R being the domain of the function f(x) = ax2 + bx + c.
The given quadratic expression is
f(x) = ax2 + bx + c
If the Quadratic is in the form
y = a(x-h)2 + k
In such a form of equation ‘k’ is the value at the vertex.

Here ‘k’ gives us the maximum or the minimum value of the function accordingly as ‘a’ is positive or negative.
Coefficient of x2 Max value Min value

a>0 +infinity (4ac-b2)/4a
2
a<0 (4ac-b )/4a -infinity
Concepts: If the sum of the positive numbers are constant than their product is maximum when all numbers are equal.
Suppose that x, y, z . . . w are n positive variables and that c is a constant, then If x + y + z + ... + w = c, the value of xyz ...
w is greatest when x = y = z = ... = w = c/n.
For, ax+by=c ,then (xm)*(yn ) is maximum when ax/m =by/n.
For, xm*yn*zp= constant, then ax+by+cz is minimum when ax/m=by/n=cz/p.
Now, we discuss some of the illustrations based on these concepts:

Example 1: Find the greatest value of [(x+2)/ (2x2 + 3x + 6)] for real values of x?
Solution 1: The given function is [(x+2)/ (2x2 + 3x + 6)]
Clearly, the function (2x2 + 3x + 6) is a quadratic function with a = 2 > 0 and so it will have its minimum at x = –b/2a = –
3/4 and minimum value is
–D/4a = 9-48/ 4.2 = 39/8
Example 2: Find the maximum or minimum value of x2 + x + 1.
Solution 2: Comparing the given equation with the general form ax2 + bx + c
We get, a = 1. Since the value of a > 0 so, we will get a minimum value.
The minimum value is given by c-b2/4a = 1-12/4.1 = 3/4.
Illustration: Find the maximum or minimum value of -2(x-1)2 + 3.
Solution: As discussed above, this equation is of the form a(x-h)2 + k.
Here a = -2 and k = 3.
Now ‘a’ is negative which implies that the equation will have a maximum value. The maximum value in this case is given
by k = 3.

Type 1 – Linear & Quadratic Equations
1
Q1. Find the maximum value of the expression 𝑥 2 +5𝑥+10
.
A. 15/2 B. 1 C. 4/15 D. 1/3
Q2. Find the maximum value of the expression: x2+8x+20.

A. 4 B. 2 C. 29 D. None of These
Q3. Find the minimum value of the expression: p+(1/p); p>0 .

A. 1 B. 0 C. 2 D. Depend upon value of P
Q4. If the product of the roots of the equation x2-3(2a+4)x+a2+18a+81=0 is unity, then a can take the values as:-
A. 3,-6 B. 10,-8 C. -10,-8 D. -10,-6
Q5. For the equation 2(a+3)=4(a+2)- 48 ,the value of a will be:

A. -3/2 B. -3 C.-2 D. 1
Q6. The expression a2 + ab + b2 is _____________for a<0 , b<0 is?

A. <=0 B. <0 C. >0 D. =0
Q7. If the roots of equation x2+bx+c=0 differ by 2, then which of the following is true?
A. a2 c2=4(1+c) B. 4b+c=1 C. c2= 4 + b D. b2= 4(c+1)
Q 8. If f(x)=(x+2) and g(x) = (4x+5), and h(x) is defined as h(x) = f(x) g(x), then sum of roots of h(x) will be
A. 3/4 B. 13/4 C. (-13)/4 D. (-3)/4
Q9. If equation x2+bx+12=0 gives 2 as its one of the roots and x2+bx+q=0 gives equal roots then the value of b is?
A. 49/4 B.-8 C. 4 D.25/2
Q10. If the roots of the equation (a2+b2)x2-2(ac + bd)x + (c2+d2) = 0 are equal then which of the following is true?
A. ab=cd B. ad = bc C. ad=√bc D. ab = √cd
Q11. For what value of c the quadratic equation x2 -(c+6)x+2(2c-1) = 0 has sum of the roots as half of their product?
A. 5 B. -4 C. 7 D. 3
Q12. Two numbers a and b are such that the quadratic equation ax2+3x+2b = 0 has -6 as the sum and the product of the
roots. Find (a+b).
A. 2 B. -1 C. 1 D. -2
Q13. If a and b are the roots of the quadratic equation 5y2-7y+1 = 0 then find the value of (1/α) +(1/β) .
A. 7/25 B. -7 C. (-7)/25 D. 7
Q14. If a =√((7+4√3)) , what will be the value of (a+1/a) ?

A. 7 B. 4 C. 3 D. 2
Q15. If the roots of the equation (a2+b2)x2 - 2b(a+c)x + (b2+c2) = 0 are equal then a,b,c are in.
A. AP B. GP C. HP D. Cannot be said
Q16. If a and b are the roots of the equation ax^2+bx+c = 0 then the equation whose roots are (a + 1/β) and (b + 1/α) is?
A. abx2+b(c+a)x + (c+a)2= 0 B. (c+a)x2+b(c+a)x+ac = 0 C. cax2+b(c+a)x+(c+a)2= 0 D.cax2+b(c+a)x+c(c+a)2=0

Q17. If x2+ax+b leaves the same remainder 5 when divided by x-1 or x+1 then the value of a and b are respectively
A. 0 and 4 B. 3 and 0 C. 0and 3 D. 4 and 0
Q18. Find all the values od b for which the equation x2–bx+1 = 0 does not possess real roots.
A. -1<b<1 B. 0<b<2 C. -2<b<2 D. -1.9<b<1.9
Directions (Q19 to Q21): Read the data given below and solve the question based on given data.
If a and b are roots of the equation x2+x-7=0, than find the following.
Q19. a2 + b2 =?
A. 10 B. 15 C. 5 D. 18
Q20. a3 + b3=?
A. 22 B. -22 C. 44 D. 36
Q21. For what value of c in the equation the roots of the equation would be opposite in signs?
A. (0,4) B. (-4,0) C. (0,3) D. (-4,4)
Q22. The set of real values of x for which the expression x2-9x+20 is negative is represented by
A. 4<x<5 B. 4<x<5 C. x<4 or x>5 D. -4<x<5
Q23. The expression x2+kx+9 becomes positive for what values of k(given that x is real)?
A. k<6 B. k>6 C. |k|<6 D. |k|>6
Q24. If 9(a-2) ÷ 3(a+4) = 81(a-11), then find the value of 3(a-8) + 3(a-6).
A. 972 B. 2916 C. 810 D. 2268
Q25. Find the number of solutions of a3+2(a+1)=a4, given that n is a natural number less than 100.
A. 0 B. 1 C. 2 D. 3
Q26. The number of positive integral of x that satisfy x3 -32x-5x2+64 =0 is /are?

A. 4 B. 5 C. 6 D. More than 6
Q27. Find the positive integral of x that satisfy the equation: x3 - 32x - 5x2+ 64=0.
A. 5 B. 6 C. 7 D. 8
Q28. If a,b,c are positive integers , such that (1/a)+(1/b)+(1/c) = 29/72 and c<b<a<60. How many set of (a,b,c) exist?
A. 3 B. 4 C. 5 D. 6
Type 2 - Inequalities
Directions (Q29 to Q35): Solve the given inequalities for x.
Q29. x2+5x+6>0
A. x>2 or x<-3 B. x<-2 C. x>-2 or x<-3 D. x<-2 or x>3
Q30. (x+3)(x-2)(x-1)>0
A. x>2 or -3<x<1 B. x<2 or -3<x<1 C. x>2 or 3<x<-1 D. None of these
(𝑥+3)(𝑥−2)
Q31. > 0.
(𝑥−1)
A. x>2 B. -3<x<1 C. x>2 or -3<x<1 D. None of these

Q32. (x-7)(x+4)6(x+5)>0.
A. x>7 or x<-5 B. x>7 or x<-5 except x=-4 C. x>7 D. x<-5
Q33. (x-1)99(x+2)100(x-3)101(x+4)102 <0

A. [1,3] B. [-1,3] C. (1,3] D. (1,3)
2𝑥−3
Q34. < −3
𝑥+4
A. (-inf, -4) B. (-9/5,+inf) C. x>-4 D. (-4,-9/5)
3𝑥 2 +7𝑥−6
Q35. <0
𝑥 2 −9𝑥+8
A. 1<x<8 or -3<x<2/3 B. 1<x<8 C. 3<x<2/3 D. None of these
Q36. Find the integer solution for (3x2-7x-6)(x2-5x+4)<0

A. 1 B. 2 C. 3 D. No solution
𝑥−3
Q37. Find the different integral solution for 𝑥+2 < 0
A. 3 B. 4 C. 1 D. 2
Q38. Find the interval of x for the given inequality: x2+x<x3+1.

A. (-inf,+inf) B. x>-1 C. x>-1 except x=1 D. (-1,1)
Type 3 –Integral Solutions

Q39. Find the positive integral solution of 𝑥 + 𝑦 + 𝑧 = 10.
A. 15 B. 26 C. 32 D. 36
Q40. Find the non-negative integral solution of 𝑥 + 𝑦 + 𝑧 = 10 𝑤ℎ𝑒𝑟𝑒,

𝑥 ≥ 2, 𝑦 ≥ 1 𝑎𝑛𝑑 𝑧 ≥ 1 .
A. 25 B. 24 C. 28 D. None of these
Q41. Find the positive and non-negative integral solution of 3𝑥 + 𝑦 + 𝑧 = 12.

A. 15,35 B. 15,30 C. 10,23 D. 12, 35
Q42. How to find the number of all the positive integral solutions of 5x+7y=100.
A. 5,5 B. 5,10 C. 10,12 D. 5,15
Q43. How do you find the number of integral solutions of the equation 2x + 3y = 763?
A. 0 B. 1 C. 45 D. Infinite
Type 4 – Maximum & Minimum Values

Q44. Find the maximum or minimum value of f(x) = 2x2 + 3x – 5.
A. Max 49/8 B. Min -49/8 C. Max -49/8 D. Min 49/8
Q45. Find the range of f(x) = -x2 + 4x – 5.

A. (-inf,+inf) B. (-inf,-1) C. (-inf,-1] D. None of these
2𝑥+5
Q46. Find maximum value of the given expression: 2𝑥 2 +6𝑥+7
A. [-inf,-1/5] B. [1,+inf] C. [-1/5,1] D. None of these

Q47. If g(x)= min(-x+3, 2+7x), then find the maximum value of g(x).
A. -23/8 B. 22 C. 23/5 D. 23/8
Q48. If f(x)=max (x2-2, x+2), then find minimum value of f(x).

A. -11/4 B. 17/4 C. -17/4 D. None of these
Q49. Find max and min value of f(x)=16 − |−𝑥 + 3|.

A. 16 and infinity B. 16 and –infinity C. infinity and 16 D. 16 and 3
Q50. Sum of the numbers are given (x+y+z)=9. Find the maximum value of the product (x*y*z)?
Q51. Sum of the numbers are given 2x+3y=15. Find the maximum value of the product (x2*y3)?
A. 27 B. 81 C. 105 D. 243
Q52. Product of the numbers is given x*y2=27. Find the minimum value of sum (32x+y).
Q53. If a, b and c are positive variables and a + b + c = 12, find maximum value of (a + 1) × (b + 2) × c.
A. 25 B. 27 C. 125 D. 130
Q54. For what value of a do the roots of the equation 2x2+6x+a=0, satisfy the conditions (α/β) + (β/α)<2.
A. a<0 or a>9/2 B. a>0 C. -1<a<0 D. -1<a<1
Q55. For what value of b and c would the equation x2+bx+c=0 have roots equal to b and c.
A. (0,0) B. (1,-2) C. (1,2) D. Both (a) and (b)
Q56. The sum of a fraction and its reciprocal equals 85/18. Find the fraction.
A. 2/6 B. 2/3 C. 2/9 D. 4/9
Q57. If both the roots of the quadratic equation ax2+bx+c=0 lie in the interval (0,3), then a lies in
A. (1,3) B. (-1,-3) C. (-√121/91,-√8) D. None of these
Q58. If the common factor of (ax2+bx+c) and (bx2+ax+c) is (x+2) then

A. a=b, or a+b+c=0 B. a=c, or a+b+c=0 C. a=b=c D. b=c, a+b+c=0
Q59. If p=2(2/3)+2(1/3) then which of the following is true?

A. p3-6p-6=0 B. p3-6p+6=0 C. p3+6p-6=0 D. p3+6p+6=0
Q60. If f(x)=x2+2x-5 and g(x)=5x+30 , then the roots of the quadratic equation g[ f(x) ] will be
A. -1,-1 B. 2,-1 C. -1+√2 , -1-√2 D. 1,2
Q61. If one of the quadratic equation ax2+bx+c=0 is three times the other, find the the relationship between a, b and c.
A. 3b2 = 16ac B. b2= 4ac C. (a+b)2= 4b D. (a2+c2)/ac=b/2
Q62. If x2-3x+2 is a factor of x4–ax2+b-0 then the values of a and b are

A. -5, -4 B. 5, 4 C. -5, 4 D. 5, -4
Q63. Value of the expression (x2-x+1)/(x-1) cannot lie between

A. 1,3 B. -1, -3 C. 1, -3 D. -1, 2
Q64. The value of p satisfying log3(p2+4p+12)= 2 are
A. 1, -3 B. -1, -3 C. -4, 2 D. -4, -2
Q65. If q, r>0 then roots of the equation x2+qx-r=0 are

A. Both negative B. Both positive C. Of opposite sign but equal D. Of opposite sign unequal
Q66. If two quadratic equation ax2+ax+3=0 and x2+x+b=0 have a common root x=1 then which of the following
statement hold true. (a) a+b= -3.5 (b) ab= 3 (c) a/b = 3/4 (d) a-b= -0.5
A. a,b,c B. b,c,d C. a,c,d D. a,b,d
Q67. If the expression ax2+bx+c is equals to 4 when x=0 leaves a remainder 4 when divided by x+1 and a remainder 6
when divided by x+2, then the values of a,b and c are respectively
A. 1,1,4 B. 2,2,4 C. 3,3,4 D. 4,4,4
Q68. If p and q are the roots of the equation x2–px+q=0 , then

A. p=1,q=-2 B. p=0,q=1 C. p=-2,q=0 D. p=-2,q=1
Q69. Sum of the real roots of the equation x2+5|x|+6 = 0

A. Equals to 5 B. Equals to 10 C. Equals to -5 D. None of these
Q70. For what values of p would the equation x2+2(p-1)x+p+5=0 possess at least one positive root?
A. p<=-1 B. p<1 C. p>-1 D. p>=1
Q71. If a, b ɶ {1,2,3,4}, then the number of equations of the form ax^2+bx+1=0 having real roots is
A.10 B. 7 C. 6 D. 12
Q72. If a2+b2+c2=1, then which of the following cannot be a value of (ab + bc + ca)?
A. 0 B. 1/2 C. -1/4 D. -1
Q73. If the roots of the equation x3–ax2+bx-1080= 0 are in the ratio 2: 4: 5, find the value of the coefficient of x2?
A. 33 B. 66 C. -33 D. 99
Q74. If a + 2b + 3c = 9 find maximum value of a × b × c. (Given a, b and c are positive).

A. 27 B. 9/2 C. 9/4 D. None of these
Q75. Davji shop sells samosas in boxes of different sizes. The samosas are priced at Rs. 2 per piece up to 200 samosas.
For every additional 20 samosas, the price of the whole lot goes down by 10 paisa per samosa. What should be the
maximum size of the box that would maximize the revenue?
A. 250 B. 300 C. 275 D. 187
Q76. Find the maximum and/or minimum value of (x^2 - x + 1)/(x^2 + x + 1) for real values of x.
A. Max 3 B. Min 1/3 C. Max 3 and Min 1/3 D. None of these
Q77. If 2a + 5b = 7; find the maximum value of (a+1)2 * (b+2)3.

A. 195 * 33/58 B. 194 * 33/58 C. 195 * 33/56 D. None of these
Q78. Find the least value of 3x + 4y + 5z for positive values of x and y, subject to the condition xyz = 6.
A. 6×(45)1/3 B. 6×(45)1/2 C. 3×(45)1/3 D. None of these

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Manav Rachna University

Career Development Centre

N Chapter 4 - Mensuration – I (Areas & Formulae)

T Chapter 5 - Mensuration – II (Surface Area & Volume)

Handbook for Quantitative Aptitude (Sem VI) Page 1

1.6P2 = (6*5) =30

(2 Consonant+2 vowels as a single unit)=3*2(Two Vowel) =12

Handbook for Quantitative Aptitude (Sem VI) Page 2

Difference b/w Permutation & Combination

Hence, if order is important, question is related to Permutations. Otherwise, it is related to combinations.

Handbook for Quantitative Aptitude (Sem VI) Page 3

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Type 4 - Alphabets and Digits_

Type 5 - Linear/Circular Arrangements

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Type 6 - Identical Objects

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Q42. A polygon 7 sides.How many diagonals can be formed?

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Here, events A and B are subsets of the sample space S.

Equally Likely Events:

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Type 4 - With/Without Replacement

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Type 5 - Bayes Theorem and Conditional Probability

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Basic properties of triangles:

Geometrical centers of a triangle:

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A. 99 B. 111 C. 121 D. 131

A. 55 degree B. 52 degree C. 90 degrees D. Cannot be determined

Handbook for Quantitative Aptitude (Sem VI) Page 20

Q20. The perimeter of a semicircle of radius 14 cm is,

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Equiangular - all angles are equal.

Equilateral - all sides are the same length.

(N = # of sides and S = length from center to a corner)

Handbook for Quantitative Aptitude (Sem VI) Page 25

1. Area of a triangle = ½ x Base x Height.

2. Area of a triangle = √𝑠(𝑠 − 𝑎)(𝑠 − 𝑏)(𝑠 − 𝑐)

Handbook for Quantitative Aptitude (Sem VI) Page 26

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Q64. A 3 by 4 rectangle is inscribed in circle. What is the circumference of the circle?

Handbook for Quantitative Aptitude (Sem VI) Page 31