 ## Aptitude

 Question 1

In a population of N families, 50% of the families have three children, 30% of the families have two children and the remaining families have one child. What is the probability that a randomly picked child belongs to a family with two children?

 A 3/23 B 6/23 C 3/10 D 3/5
Question 1 Explanation:
Let us consider total no. of families = 100
In that 50% of families having 3 children
i.e., 50×3 = 150 (No. of children)
→ Likes that 30% have 2 children = 30×2 = 60
→ 20% have 1 children = 20×1 = 20
Probability of choosing a children the family have two children
= 60/(150+60+20) = 60/230 = 6/23
 Question 2

In a class of 200 students, 125 students have taken Programming Language course, 85 students have taken Data Structures course, 65 students have taken Computer Organization course; 50 students have taken both Programming Language and Data Structures, 35 students have taken both Data Structures and Computer Organization; 30 students have taken both Data Structures and Computer Organization, 15 students have taken all the three courses.How many students have not taken any of the three courses?

 A 15 B 20 C 25 D 35
Question 2 Explanation:
n = 200
PL = 125
DS = 85
CO = 65
PL & DS = 50
DS & CO = 35
CO & PL = 30
PL & DS & CO = 15 ⇒ Not taken any course = 200 - (60 +15+15+35+15+15+20)
= 200 - 175
= 25
 Question 3

Two straight lines are drawn perpendicular to each other in X-Y plane. If α and β are the acute angles the straight lines make with the X-axis, then α+β is ______.

 A 60° B 120° C 180° D 90°
Question 3 Explanation:
Two st. line perpendicular to each other in X-Y, if α & β are acute angles then α + β = ____ We know a + α + β = 180
α + β = 180 - 90
α + β = 90
 Question 4

The total revenue of a company during 2014-2018 is shown in the bar graph. If the total expenditure of the company in each year is 500 million rupees, then the aggregate profit or loss (in percentage) on the total expenditure of the company during 2014-2018 is ______. A 20% profit B 20% loss C 16.67% loss D 16.67% profit
Question 4 Explanation:
Bar-graph:
2014-2018
Total expenditure = 500 million/year = 500 × 5 = 2500 million
Revenue total (from the graph)
= 400 + 500 + 600 + 700 + 800
= 3000 million
Profit = 3000 - 2500 = 500
⟹ 500/2500 = 20% profit
 Question 5

The figure below shows an annular ring with outer and inner radii as b and a, respectively. The annular space has been painted in the form of blue colour circles touching the outer and inner periphery of annular space. If maximum n number of circles can be painted, then the unpainted area available in annular space is ______. A π[(b2 - a2) + π/4(b - a)2] B π[(b2 - a2) - π/4(b - a)2] C π[(b2 - a2) - n(b - a)2] D π[(b2 - a2) + n(b - a)2]
Question 5 Explanation: Area of non-painted between 2 circles area of part 1 = πb2 - πa2 = π(b2 - a2)
Radius of painted circles = b-a/2
Area of painted circle = π(b-a/2)2
For n circles = nπ(b-a/2)2
Non-painted area = π[b2 - a2- n/4(b - a)2]
 Question 6

Goods and Services Tax (GST) is an indirect tax introduced in India in 2017 that is imposed on the supply of goods and services, and it subsumes all indirect taxes except few. It is a destination-based tax imposed on goods and services used, and it is not imposed at the point of origin from where goods come. GST also has a few components specific to state governments, central government and Union Territories (UTs).

Which one of the following statements can be inferred from the given passage?

 A GST includes all indirect taxes. B GST is imposed at the point of usage of goods and services. C GST does not have a component specific to UT. D GST is imposed on the production of goods and services.
Question 6 Explanation:
GST is imposed at the point of usage of goods and services.
 Question 7

Select the word that fits the analogy:

` Cook : Cook :: Fly : _____  `
 A Flyer B Flying C Flew D Flighter
Question 7 Explanation:
Cook-cook - Noun-verb relation
Fly-Flyer
 Question 8

The drawn of the 21st century witnessed the melting glaciers oscillating between giving too much and too little to billions of people who depend on them for fresh water. The UN climate report estimates that without deep cuts to man-made emissions, at least 30% of the northern hemisphere’s surface permafrost could melt by the end of the century. Given this situation of imminent global exodus of billions of people displaced by rising seas, nation-states need to rethink their carbon footprint for political concerns, if not for environmental ones.

Which one of the following statements can be inferred from the given passage?

 A Nation-states are responsible for providing fresh water to billions of people. B Billions of people are affected by melting glaciers. C Nation-states do not have environmental concerns. D Billions of people are responsible for man-made emissions.
Question 8 Explanation:
Billions of people are affected by melting glaciers.
 Question 9

There are multiple routes to reach from node 1 to node 2, as shown in the network. The cost of travel on an edge between two nodes is given in rupees. Nodes ‘a’, ‘b’, ‘c’, ‘d’, ‘e’, and ‘f’ are toll booths. The toll price at toll booths marked ‘a’ and ‘e’ is Rs.200, and is Rs.100 for the other toll booths. Which is the cheapest route from node 1 to node 2?

 A 1-f-e-2 B 1-f-b-2 C 1-b-2 D 1-a-c-2
Question 9 Explanation: 1 - f - e - 2 ─ 100 + 100 + 200 = 400
1 - f - b - 2 ─ 100 + 0 + 200 = 300 ⇾ shortest [Answer]
1 - b - 2 ─ 300 + 200 = 500
1 - a - c ─ 2 - 200 + 100 + 100 = 400
 Question 10

Raman is confident of speaking English _____ six months as he has been practicing regularly_____the last three weeks.

 A for, in B during, for C for, since D within, for
 Question 11

If P = 3, R = 27, T = 243, then Q+S = ______.

 A 80 B 110 C 40 D 90
Question 11 Explanation:
P=3, R=27, T=243, Q+5=?
P=31, Q=32, R=33, S=34, T=35
Q+S = 32 + 34 = 9+81 = 90
 Question 12

His knowledge of the subject was excellent but his classroom performance was ______.

 A good B praiseworthy C extremely poor D desirable
Question 12 Explanation:
A Contrast is indicated in the above sentence.
 Question 13
The _________ is too high for it to be considered _________.
 A fair / fare B faer / fair C fare / fare D fare / fair
Question 13 Explanation:
Fare: The money paid for a journey. Fair: treating people equally without favouritism or discrimination (or) without cheating (or) trying to achieve unjust advantage The fare is too high for it to be considered fair.
 Question 14
A function y(x) is defined in the interval [0, 1] on the x-axis as Which one of the following is the area under the curve for the interval [0, 1] on the x-axis?
 A 5/6 B 6/5 C 13/6 D 6/13
Question 14 Explanation: We know that area of a rectangle = length * breadth = L * B
Area of rectangle (1) = L*B = (1/3) * 2 = (2/3)
Area of rectangle (2) = L*B = (3/4 - 1/3 ) * 3 = (5/4 )
Area of rectangle (3) = L*B = (1 - 3/4) * 1 = 1/4
Therefore Total area = Area(1+2+3) = (2/3) + (5/4) + (1/4) = 26/12 = 13/6
 Question 15
Let r be a root of the equation x2 + 2x + 6 = 0. Then the value of the expression (r + 2)(r + 3)(r + 4)(r + 5) is
 A 51 B -51 C 126 D -126
Question 15 Explanation: Question 16
Given below are four statements.
Statement 1: All students are inquisitive.
Statement 2: Some students are inquisitive.
Statement 3: No student is inquisitive.
Statement 4: Some students are not inquisitive.
From the given four statements, find the two statements that CANNOT BE TRUE simultaneously, assuming that there is at least one student in the class.
 A Statement 1 and Statement 3 B Statement 1 and Statement 2 C Statement 2 and Statement 4 D Statement 3 and Statement 4
Question 16 Explanation:
Explanation: Statement 1 & Statement 3 cannot be true simultaneously.
Statement 1: All students are inquisitive. Statement 3: No student is inquisitive Question 17
A palindrome is a word that reads the same forwards and backwards. In a game of words, a player has the following two plates painted with letters. From the additional plates given in the options, which one of the combinations of additional plates would allow the player to construct a five-letter palindrome. The player should use all the five plates exactly once. The plates can be rotated in their plane.
 A B C D Question 17 Explanation:
From the question we know that letters can rotate in the plane, now we will verify the options one by one. Option 1: A,D,D,D,J from these letters we can't form a palindrome.
Option 2: A,D,R,A,R from these we can form RADAR which is a palindrome.
Option 3: A,D,Z,E,D from these letters we can't form a palindrome.
Option 4: A,D,I,L,Y from these letters we can't form a palindrome.
 Question 18
Some people believe that “what gets measured, improves”.
Some others believe that “what gets measured, gets gamed”.
One possible reason for the difference in the beliefs is the work culture in organizations. In organizations with good work culture, metrics help improve outcomes.
However, the same metrics are counterproductive in organizations with poor work culture.
Which one of the following is the CORRECT logical inference based on the information in the above passage?
 A Metrics are useful in organizations with poor work culture B Metrics are useful in organizations with good work culture C Metrics are always counterproductive in organizations with good work culture D Metrics are never useful in organizations with good work culture
Question 18 Explanation:
From the given passage consider the sentence “In organizations with good work culture, metrics help improve outcomes” Option A: Metrics are useful in organizations with poor work culture. Which is incorrect based on the sentence.
Option B: Metrics are useful in organizations with good work culture. Which is correct and relevant to the considered sentence.
Option C: Metrics are always counterproductive in organizations with good work culture. Which is incorrect because we can't always guarantee good work culture.
Option D: Metrics are never useful in organizations with a good work culture. Incorrect metrics will help to make a good work culture.
 Question 19
In a recently conducted national entrance test, boys constituted 65% of those who appeared for the test. Girls constituted the remaining candidates and they accounted for 60% of the qualified candidates. Which one of the following is the correct logical inference based on the information provided in the above passage?
 A Equal number of boys and girls qualified B Equal number of boys and girls appeared for the test C The number of boys who appeared for the test is less than the number of girls who appeared D The number of boys who qualified the test is less than the number of girls who Qualified
Question 19 Explanation:
The girls accounted for 60% of the qualified candidates.
60% can be written as ⅗. Which means among 5 qualified persons three of them are girls.
So, The number of boys qualified is less than the number of girls qualified.
 Question 20
A box contains five balls of same size and shape. Three of them are green coloured balls and two of them are orange coloured balls. Balls are drawn from the box one at a time. If a green ball is drawn, it is not replaced. If an orange ball is drawn, it is replaced with another orange ball. First ball is drawn. What is the probability of getting an orange ball in the next draw?
 A 1/2 B 8/25 C 19/50 D 23/50
Question 20 Explanation:
Case 1: The first ball is Green
Probability of first ball is green = 3/5
If the first ball is green then we are not replacing so the probability of the second ball being orange is = 2/4.
Probability of first ball green and second ball orange is =3/5 * 2/4 = 6/20
Case 2: The first ball is orange
Probability of first ball is orange =2/5
If the first ball is orange then we are replacing it with another orange ball/ so the number balls will remain the same.
Probability of second ball is orange = 2/5
Probability of first ball is orange and second ball is also orange = 2/5*2/5 = 4/25
Therefore, Total probability = (6/20 + 4/25) = 23/50.
 Question 21
The corners and mid-points of the sides of a triangle are named using the distinct letters P, Q, R, S, T and U, but not necessarily in the same order. Consider the following statements:
The line joining P and R is parallel to the line joining Q and S.
P is placed on the side opposite to the corner T.
S and U cannot be placed on the same side.
Which one of the following statements is correct based on the above information?
 A P cannot be placed at a corner B S cannot be placed at a corner C U cannot be placed at a mid-point D R cannot be placed at a corner
Question 21 Explanation:
From the given information draw the image, such that The Image is satisfying all given conditions. From this we can say that S cannot be placed at the corner. By satisfying all given conditions we can draw in different ways but in all cases S cannot be placed at the corner.
 Question 22
A plot of land must be divided between four families. They want their individual plots to be similar in shape, not necessarily equal in area. The land has equally spaced poles, marked as dots in the below figure. Two ropes, R1 and R2, are already present and cannot be moved.
What is the least number of additional straight ropes needed to create the desired plots? A single rope can pass through three poles that are aligned in a straight line. A 2 B 4 C 5 D 3
Question 22 Explanation:
Given Conditions are Shape is to be similar, area need not to be same, A rope can pass through three poles. Therefore least number of additional ropes required = 3.
 Question 23

Out of a group of 21 persons, 9 eat vegetables, 10 eat fish and 7 eat eggs : 5 persons eat all three. How many persons eat at least two out the three dishes?

 A Theory Explanation.
 Question 24
Listening to music during exercise improves exercise performance and reduces discomfort. Scientists researched whether listening to music while studying can help students learn better and the results were inconclusive. Students who needed external stimulation for studying fared worse while students who did not need any external stimulation benefited from music. Which one of the following statements is the CORRECT inference of the above passage?
 A Listening to music has no effect on learning and a positive effect on physical exercise. B Listening to music has a clear positive effect on physical exercise. Music has a positive effect on learning only in some students. C Listening to music has a clear positive effect both on physical exercise and on learning. D Listening to music has a clear positive effect on learning in all students. Music has a positive effect only in some students who exercise.
Question 24 Explanation:

From the first statement “Listening to music during exercise improves exercise performance and reduces discomfort. “ It is clear that listening to music has a positive effect on physical exercise.

From the statement “Scientists researched whether listening to music while studying can help students learn better and the results were inconclusive. “   it is clear that only on some students music has a positive effect.

 Question 25 A transparent square sheet shown above is folded along the dotted line. The folded sheet will look like _______.
 A B C D Question 25 Explanation:
Fold the sheet along the dotted line and join the shapes, we get option 2.
 Question 26
Gauri said that she can play the keyboard _______ her sister.
 A as worse as B as well as C as better as D as nicest as
Question 26 Explanation:
‘worse’, ‘better’ are comparative words which need the word ‘than’ for comparison. Out of the remaining two options ‘as well as’ is the correct answer.
 Question 27
Pen : Write :: Knife : _______ Which one of the following options maintains a similar logical relation in the above?
 A Vegetables B Sharp C Blunt D Cut
Question 27 Explanation:
A pen is used to write. Similarly, a knife is used to cut.
 Question 28
The number of students in three classes is in the ratio 3:13:6. If 18 students are added to each class, the ratio changes to 15:35:21. The total number of students in all the three classes in the beginning was:
 A 66 B 22 C 88 D 110
Question 28 Explanation:

Given Initial ratio : 3:13:6

Total number of students initially = 3x+13x+6x = 22x

From the given question, we can write 3x+18 = 15y -----I

13x+18 = 35y-----II

6x+18 = 21y------III

I - II => -10X = -20Y

X=2Y------IV

Put IV in III => 6(2y) + 18 = 21y

12y + 18 = 21y

9y = 18

y = 2 -----V

Substitute V in I => 3x + 18 = 30

3x = 12

x = 4

Number of initial students = 22x = 22(4) = 88

 Question 29 The number of units of a product sold in three different years and the respective net profits are presented in the figure above. The cost/unit in Year 3 was ₹ 1, which was half the cost/unit in Year 2. The cost/unit in Year 3 was one-third of the cost/unit in Year 1. Taxes were paid on the selling price at 10%, 13% and 15% respectively for the three years. Net profit is calculated as the difference between the selling price and the sum of cost and taxes paid in that year.
The ratio of the selling price in Year 2 to the selling price in Year 3 is ______.
 A 1:1 B 3:4 C 1:2 D 4:3
Question 29 Explanation:

According to the given data:

Year     Cost/unit   Number of units     Cost Price

1           ₹ 3                    100                     300

2           ₹ 2                     200                    400

3           ₹ 1                     300                    300

Cost Price = Cost/unit x Number of units

Net Profit = Selling Price - (Cost Price + (Tax%  x selling price )

Year 2:

296 = SP - (400 + (13/100) x SP)

=>296 = 0.87SP - 400

=> 0.87SP = 696

=> SP = ₹ 800

Year 3:

210 = SP - (300 + (15/100) x SP)

=> 210 = 0.85SP - 300

=>  0.85SP = 510

=> SP = ₹ 600

The ratio of selling price in Year 2 to the selling price in year 3 = 800:600

= 4:3

 Question 30 A jigsaw puzzle has 2 pieces. One of the pieces is shown above. Which one of the given options for the missing piece when assembled will form a rectangle? The piece can be moved, rotated or flipped to assemble with the above piece.
 A B C D Question 30 Explanation:
Rotate 90 degrees to the left and flip the piece in Option 2.
 Question 31 A 2 B 4 C 8 D 6
Question 31 Explanation:

Expand the given equation:

x2+1/4 - x -x2 -9/4 +3x = x+2

=> -2 + 2x = x + 2

X = 4

 Question 32
If θ is the angle, in degrees, between the longest diagonal of the cube and any one of the edges of the cube, then, cos θ =
 A B C D Question 32 Explanation: Question 33
Six students P, Q, R, S, T and U, with distinct heights, compare their heights and make the following observations.
Observation I: S is taller than R.
Observation II: Q is the shortest of all.
Observation III: U is taller than only one student.
Observation IV: T is taller than S but is not the tallest.
The number of students that are taller than R is the same as the number of students shorter than _______.
 A S B P C T D R
Question 33 Explanation:
The order will be PTSRUQ
Number of students taller than R = Number of students shorter than S = 3
 Question 34
Consider the following sentences:
(i) Everybody in the class is prepared for the exam.
(ii) Babu invited Danish to his home because he enjoys playing chess.
Which of the following is the CORRECT observation about the above two sentences?
 A (i) is grammatically incorrect and (ii) is ambiguous B (i) is grammatically correct and (ii) is ambiguous C (i) is grammatically correct and (ii) is unambiguous D (i) is grammatically incorrect and (ii) is unambiguous
Question 34 Explanation:
i) Everybody in the class is prepared for the exam This is grammatically correct statement
ii). Babu invited Danish to his home because he enjoys playing chess.
This statement is ambiguous as there is no guarantee that Danish knows playing chess.
 Question 35

 Items Cost (₹) Profit % Marked Price (₹) P 5,400 --- 5,860 Q --- 25 10,000
Details of prices of two items P and Q are presented in the above table. The ratio of cost of item P to cost of item Q is 3:4. Discount is calculated as the difference between the marked price and  the selling price. The profit percentage is calculated as the ratio of the difference between selling price and cost, to the cost Profit %=Selling price-CostCost100. The discount on item Q, as a percentage of its marked price, is ______.
 A 25 B 5 C 10 D 12.5
Question 35 Explanation: Question 36
The ratio of boys to girls in a class is 7 to 3. Among the options below, an acceptable value for the total number of students in the class is:
 A 37 B 73 C 21 D 50
Question 36 Explanation:
The Ratio of boys to girls is 7:3
Number of boys = 7x
Number of girls = 3x
Total number of students = 10x
So the possible value should multiple by 10.
 Question 37
A polygon is convex if, for every pair of points, P and Q belonging to the polygon, the line segment PQ lies completely inside or on the polygon.
Which one of the following is NOT a convex polygon?
 A B C D Question 37 Explanation: Question 38 A circular sheet of paper is folded along the lines in the directions shown. The paper, after being punched in the final folded state as shown and unfolded in the reverse order of folding, will look like _______.
 A B C D Question 38 Explanation: Question 39
We have 2 rectangular sheets of paper, M and N, of dimensions 6 cm × 1 cm each. Sheet M is rolled to form an open cylinder by bringing the short edges of the sheet together. Sheet N is cut into equal square patches and assembled to form the largest possible closed cube. Assuming the ends of the cylinder are closed, the ratio of the volume of the cylinder to that of the cube is ______.
 A B C D Question 39 Explanation: Question 40
______ is to surgery as writer is to ______
Which one of the following options maintains a similar logical relation in the above sentence?
 A Hospital, library B Medicine, grammar C Plan, outline D Doctor, book
Question 40 Explanation:
The given statement is to fill the relationship of the subject and the action done by the subject.
Doctor is to surgery as writer is to book.
 Question 41
There are five bags each containing identical sets of ten distinct chocolates. One chocolate is picked from each bag. The probability that at least two chocolates are identical is _____
 A 0.81225 B 0.4235 C 0.6976 D 0.3024
Question 41 Explanation:
Explanation :
There are 5 bags having identical sets of 10 distinct chocolates. one chocolate is drawn from each bag, probability to find at least 2 chocolate is identical
The total number of such selections possible = 10*10*10*10*10 = 10^5.
We need to select such that at least two of them are identical.
We can count the number of ways with no chocolate being repeated and subtract it from total possible ways to get the required probability.
The number of ways to select all different/distinct = 10*9*8*7*6
Required number of selections= 10^5-10*9*8*7*6 = 69760
Required probability = 69760/10^5 = 0.69760
 Question 42
Some people suggest anti-obesity measures (AOM) such as displaying calorie information in restaurant menus. Such measures sidestep addressing the core problems that cause obesity: poverty and income inequality.
Which one of the following statements summarizes the passage?
 A AOM are addressing the problem superficially. B The proposed AOM addresses the core problems that cause obesity. C AOM are addressing the core problems and are likely to succeed. D If obesity reduces, poverty will naturally reduce, since obesity causes poverty.
Question 42 Explanation:
The problems mentioned are poverty and income inequality. But AOMs are not addressing the main problems directly. But it is a kind of generic step taken to address obesity. So AOMs are only addressing the obesity problem superficially but not addressing the real problems.
 Question 43
Given below are two statements 1 and 2, and two conclusions I and II.
Statement I: All bacteria are microorganisms.
Statement II: All pathogens are microorganisms.
Conclusion I: Some pathogens are bacteria.
Conclusion II: All pathogens are not bacteria.
Based on the above statements and conclusions, which one of the following options is logically CORRECT?
 A Only conclusion I is correct B Only conclusion II is correct C Neither conclusion I nor II is correct D Either conclusion I or II is correct
 Question 44

Based on the given statements, select the most appropriate option to solve the given question, What will be the total weight of 10 poles each of same weight?

Statements:
(I) One fourth of the weight of a pole is 5 kg
(II) The total weight of these poles is 160 kg more than the total weight of two poles.
 A Statement I alone is not sufficient B Statement II alone is not sufficient C Statement II alone is not sufficient D Both statements I and II together are not sufficient.
Question 44 Explanation:
Statement-I:
One fourth of the weight of a pole is 5Kg. ⇒ Weight of pole is 4×5 = 20Kg
Weight of 10 poles each of same weight = 10×20 = 200 Kg
∴Statement I alone is sufficient.
Statement-II:
Let, Weight of each pole = W Kg
Given,
10W = 2W + 160
⇒ 8W = 160
W = 20Kg
∴ Weight of each pole = 20 Kg
∴ Weight of 10 poles = 10×20 Kg = 200 Kg
∴ Statement II alone is sufficient.
Either I or II alone is sufficient.
 Question 45

Consider a function f(x) = 1 - |x| on -1 ≤ x ≤ 1. The value of x at which the function attains a maximum and the maximum value of the function are

 A 0, -1 B -1, 0 C 0, 1 D -1, 2
Question 45 Explanation:
f(x) = 1 - |x| on -1 ≤ x ≤ 1
In the given function, it is given as
-|x| ⇒ To obtain the maximum of this function we have to minimize the value -|x| and the minimum value is 0.
∴ Maximum value of f(x) is at f(0) and i.e., f(x) = 1
Maximum value is 1 at x=0.
 Question 46

A generic term that include various items of clothing such as a skirt, a pair of trousers and a shirt is

 A fabric B textile C fibre D apparel
Question 46 Explanation:
Apparel- clothing, especially outerwear; garments; attire; raiment
 Question 47

Choose the statement where underlined word is used correctly.

 A The industrialist load a personnel jet. B I write my experience in my personnel diary. C All personnel are being given the day off. D Being religious is a personnel aspect.
Question 47 Explanation:
personnel- people employed in an organization or engaged in an organized undertaking such as military service.
 Question 48

We __________________ our friend’s birthday and we ______________ how to make it up to him.

 A Completely forgot - - - don’t just know B Forgot completely - - - don’t just know C Completely forgot - - - just don’t know D Forgot completely - - - just don’t know
Question 48 Explanation:
We completely forgot our friend's birthday and we just don't know how to make it up to him.
 Question 49

In a triangle PQR, PS is the angle bisector of ∠QPR and ∠QPS = 60º. What is the length of PS? A (q+r)/qr B qr/(q+r) C √(q2 + r2) D (q+r)2 /qr
Question 49 Explanation:
Given, ∠QPS = 60º
and PS is angle bisector of ∠QPR
⇒ ∠QPS = ∠SPR = 60º
and ∠QPR = 120º
Let, PS = s
Area of ΔPQR = Area of ΔPQS + Area of ΔSPR
1/2 qr Sin∠QPR = 1/2 rs Sin∠QPS + 1/2 sq∠SPR
1/2 qr Sin 120º = 1/2 rs Sin 60º + 1/2 sq 60º
1/2 qr Sin × √3/2 = 1/2 rs × √3/2 + 1/2 × √3/2
qr = rs + sq
∴ s = qr/q+r
PS = qr/q+r
 Question 50

Out of the following four sentences, select the most suitable sentence with respect to grammar and usage.

 A Since the report lacked needed information, it was of no use to them. B The report was useless to them because there were no needed information in it. C Since the report did not contain the needed information, it was not real useful to them D Since the report lacked needed information, it would not have been useful to them.
Question 50 Explanation:
(B) there was no needed information
(C) not really useful
(D) would not have been
There are 50 questions to complete.

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