RikoteMaster/aqua-rat-mcqa huggingface dataset:
Question: The two lines y=x and x=-9 intersect on the coordinate plane. What is the value of the area of the figure formed by the intersecting lines and the x-axis?
A.32.5
B.36.5
C.40.5
D.44.5
Answer: C.40.5
Explanation: The point of intersection is (-9,-9).
The triangle has a base of length 9 and a height of 9.
area = (1/2)*base*height = (1/2)*9*9 = 40.5
The answer is C.
Question: The average weight of 25 persons sitting in a boat had some value. A new person added to them whose weight was 46 kg only. Due to his arrival, the average weight of all the persons decreased by 5 kg. Find the average weight of first 25 persons?
A.55 kg
B.48 kg
C.52 kg
D.None
Answer: D.None
Explanation: Answer: Option D
25x + 46 = 26(x â€“ 5)
x = 176
Question: The charge for a single room at Hotel P is 70 percent less than the charge for a single room at Hotel R and 10 percent less than the charge for a single room at Hotel G. The charge for a single room at Hotel R is what percent greater than the charge for a single room at Hotel G?
A.100%
B.120%
C.150%
D.200%
Answer: D.200%
Explanation: P = 0.3R = 0.9G
R = 0.9G/0.3 = 3G
Thus R is 200% greater than G.
The answer is D.
Question: If q + | q | = 0, which of the following must be true?
A.q > 0
B.q≥0
C.q < 0
D.q≤0
Answer: D.q≤0
Explanation: Manipulate the equations:
q + | q | = 0
|q| = -y
-q > 0 OR -q = 0
This means q could be 0 or q is less than 0.
D. q≤0
E. q=0
Answer: D
Question: The average age of husband, wife and their child 3 years ago was 27 years and that of wife and the child 5 years ago was 20 years. The present age of the husband is
A.40
B.35
C.45
D.55
Answer: A.40
Explanation: Explanation:
Sum of the present ages of husband, wife and child = (27 * 3 + 3 * 3) years = 90 years.
Sum of the present ages of wife and child = (20 * 2 + 5 * 2) years = 50 years.
Husband's present age = (90 - 50) years = 40 years
Answer: Option A
Question: A guy was asked to specify his age in years. He said, “Take my age 3 years hence, multiply it by three and subtract thrice of my age 3 years ago and you will know my age.” What was the age of that guy?
A.18
B.15
C.13
D.19
Answer: A.18
Explanation: A
18
Current age of the guy = A years.
Then, 3 (A + 3) – 3 (A – 3) = A
(3A + 9) – (3A – 9) = A
A = 18
Question: 6, 11, 21, 36, 56, 81, ?
A.111
B.71
C.81
D.61
Answer: A.111
Explanation: The pattern is + 5, + 10, + 15, + 20, + 25, + 30, + 35, + 40, + 45, + 50,....
Answer : A.
Question: If two painters can complete four rooms in two hours, how many painters would it take to do 24 rooms in 8 hours?
A.7
B.8
C.4
D.6
Answer: C.4
Explanation: Explanation:
Two painters can complete four rooms in two hours.So 24 rooms can be painted in 8 hrs by 4 painters
Answer: C) 4 painters
Question: Two trains are running in opposite directions with the same speed. If the length of each train is 60 m and they cross each other in 12 sec, then the speed of each train is?
A.30
B.31
C.18
D.25
Answer: C.18
Explanation: Let the speed of each train be x m/sec.
Then, relative speed of the two trains = 2x m/sec.
So, 2x = (60 + 60)/12 => x = 5
Speed of each train = 5 m/sec.
= 5 * 18/5 =18 km/hr.
Answer: Option C
Question: Diana invested $61,293 in an account with a fixed annual percent of interest, compounding quarterly. At the end of five full years, she had $76,662.25 in principal plus interest. Approximately what was the annual percent rate of interest for this account?
A.1.2%
B.4.5%
C.10%
D.18%
Answer: B.4.5%
Explanation: 61,293 (1+(r/4))^20 = 76,662.25
Interest Accrued within Given Time Period: 76,662.25 - 61,293 = (estimated) 15,000
So, Per Year: 15,000/5 = 3000
Next, in order to find the rate:
(estimated) 60,000 * Rate = 3,000
3,000 / 60000 = 5% .... Since 60000 is lower than the actual figure of 61293, the actual rate will be a little less than 5%.
The only answer close to that is Choice B.
Question: Seven children — A, B, C, D, E, F — are going to sit in seven chairs in a row. The children CFE have to sit next to each other, and the others can sit in any order in any remaining chairs. How many possible configurations are there for the children?
A.84
B.720
C.1440
D.4320
Answer: A.84
Explanation: for such questions , we can treat both as one then total person=4..
these 6 can be arranged in 4! ways.. but within these 4 , one consists of two people who can be arranged in 6 ways CFE or FC.. so ans =4!*6=84
ans A
Question: A train traveling at 108 kmph crosses a platform in 30 seconds and a man standing on the platform in 12 seconds. What is the length of the platform in meters?
A.240 meters
B.360 meters
C.420 meters
D.600 meters
Answer: B.360 meters
Explanation: Answer
Distance covered by the train when crossing a man and when crossing a platform
When a train crosses a man standing on a platform, the distance covered by the train is equal to the length of the train.
However, when the same train crosses a platform, the distance covered by the train is equal to the length of the train plus the length of the platform.
The extra time that the train takes when crossing the platform is on account of the extra distance that it has to cover. i.e., length of the platform.
Compute length of platform
Length of the platform = speed of train * extra time taken to cross the platform.
Length of platform = 108 kmph * 12 seconds
Convert 108 kmph into m/sec
1 kmph = 5/18 m/s (This can be easily derived. But if you can remember this conversion, it saves a good 30 seconds).
∴ 108 kmph = 5/18∗108 = 30 m/sec
Therefore, length of the platform = 30 m/s * 12 sec = 360 meters.
Choice B
Question: X Y
+Y X
________
The sum of the two digit numbers above is a three digit number PQ2, where each letter X, Y, P, and Q represents a different non zero digit. Which of the following can be the value of X?
I)	7
II)	8
III)	9
A.I only
B.II only
C.III only
D.I , II and III
Answer: D.I , II and III
Explanation: it can be a combination of 9,3 or 8,4 or 7,5
I choose D as an answer.
Question: 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.4%
B.4.02%
C.4.04%
D.5%
Answer: C.4.04%
Explanation: 100 cm is read as 102 cm.
A1 = (100 x 100) cm2 and A2 (102 x 102) cm2.
(A2 - A1) = [(102)2 - (100)2]
= (102 + 100) x (102 - 100)
= 404 cm2.
Percentage error = 404 x 100 % = 4.04%
100 x 100
C)
Question: To furnish a kitchen in a model home an interior decorator is to select 2 chairs and 2 tables from the collection of chairs and tables in a warehouse that are all different from each other. If there are 5 chairs in the warehouse and if 120 different combinations are possible, how many tables are there in the warehouse?
A.A)5
B.B)8
C.C)10
D.D)15
Answer: A.A)5
Explanation: Total number of ways to choose 2 chairs from 5= 5C2=10
Total number of combination=100
That means we need get 10 combinations from the selection of tables....
Screening through the answers.... 5C2=10..... 10*10=100
Answer is 6... A
Question: What will be the ratio of simple interest earned by certain amount at the same rate of interest for 7 years and that for 11 years?
A.8 : 11
B.9 : 11
C.7 : 11
D.Data inadequate
Answer: D.Data inadequate
Explanation: Let the principal be P and rate of interest be R%.
∴ Required Ratio = [(P×R×7/100) /(P×R×11/100)]=1PR/11PR=7/11=7:11
Answer D
Question: A boat can travel with a speed of 42 km/hr in still water. If the speed of the stream is 10 km/hr, find the time taken by the boat to go 80 km downstream
A.1 hour 40 min
B.2 hour 40 min
C.1 hour 20 min
D.1 hour 30 min
Answer: D.1 hour 30 min
Explanation: Speed of the boat in still water = 42 km/hr
speed of the stream = 10 km/hr
Speed downstream = (42+10) = 52 km/hr
Distance travelled downstream = 80 km
Time taken =distance/speed
=80/52
= 1.54 hours
= 1 hour 30 min.
ANSWER:D
Question: A train m leaves meerut at 5 a.m. and reaches delhi at 9 a.m. another train leaves delhi at 7 a.m. and reaches meerut at 10.30 a.m. at what time do the two trains_lavel3 cross each other ?
A.7:56am
B.7:58am
C.7:46am
D.8:56am
Answer: A.7:56am
Explanation: Let the total distance be x
so the speed of 1st train is x/4 and 2nd train x/3.5
In 2 hours 1st train covers half of the total distance .
so remaining is only half of the total distance(ie x/2).
let t be the time taken
t*x/4 + t*x/3.5 = x/2
t=14/15 i.e. 56 min
ie
total time taken= 2 hrs + 56 min
time they cross each other is 7:56am
ANSWER:A
Question: How many boxes do we need if we have to carry 250 apples into boxes that each hold 25 apples?
A.A)9
B.B)5
C.C)7
D.D)10
Answer: D.D)10
Explanation: Sol.
Apples 250
Each carries 25
= 250/25
= 10
Answer : D
Question: A set S = {x, -8, -6, -5, 5, 7, 9, y} with elements arranged in increasing order. If the median and the mean of the set are the same, what is the value of |x|-|y|?
A.Cannot be determined
B.-1
C.1
D.2
Answer: D.2
Explanation: Median of the set = (-5+5)/2 = 0
As per statement, Mean of the set = 0
Mean of the set
|y|- |x| +21-19 = 0 (where x is negative n y is positive)
|y|- |x| = -2
So the absolute difference between two numbers is 2
Answer D
Question: When 242 is divided by a certain divisor the remainder obtained is 8. When 698 is divided by the same divisor the remainder obtained is 9. When the sum of the two numbers 242 and 698 is divided by the divisor, the remainder obtained is 4. What is the value of the divisor?
A.11
B.17
C.13
D.23
Answer: C.13
Explanation: Explanatory Answer
When 242 is divided by a certain divisor the remainder obtained is 8.
Let the divisor be d.
When 242 is divided by d, let the quotient be 'x'. The remainder is 8.
Therefore, 242 = xd + 8
When 698 is divided by the same divisor the remainder obtained is 9.
Let y be the quotient when 698 is divided by d.
Then, 698 = yd + 9.
When the sum of the two numbers, 242 and 698, is divided by the divisor, the remainder obtained is 4.
242 + 698 = 940 = xd + yd + 8 + 9
940 = xd + yd + 17
As xd and yd are divisible by d, the remainder when 940 is divided by d should have been 17.
However, because we know that the remainder is 4, it would be possible only when 17/d leaves a remainder of 4.
If the remainder obtained is 4 when 17 is divided by 'd', then 'd' has to be 13.
Choice C
Question: A grocer has a sale of Rs. 6400, Rs. 3545, Rs. 9500, Rs. 7500 for 4 consecutive months. How much sale must he have in the fifth month so that he gets an average sale of Rs. 5500?
A.Rs. 480
B.Rs. 555
C.Rs. 499
D.Rs. 490
Answer: B.Rs. 555
Explanation: Total sale for 4 months = Rs. (6400 +3545 +9500 + 7500 ) = Rs. 26945.
Required sale = Rs. [ (5500 x 5) - 26945 ]
= Rs. (27500 - 26945)
= Rs. 555.
B)
Question: the average distance between the sun and a certain planet is approximatly 4.7 10^8 inches. which of the following is the closest to the average distence between sun and the planet in kelometers? (1 km is approx 3.9 x 10^4 inches)
A.12.05 x(10)^3
B.22.05 x(10)^4
C.14.05 x(10)^6
D.13.05 x(10)^8
Answer: A.12.05 x(10)^3
Explanation: No need to solve entire problem.
Distance in KM = 47*(10)^8/39*(10)^4 which is -470*(10)^7/39*(10)^4 = 12.05 x(10)^3
Answer : A
Question: A man spends 35% of his income on food, 25% on children education and 80% of the remaining on house rent. What percent of his income he is left with?
A.1%
B.8%
C.12%
D.5%
Answer: B.8%
Explanation: Let the total income be x
then, income left = (100-80)% of x - [100-(35+25)]% of x = 20% of 40% of x = 8% of x
Answer is B
Question: A and B invests Rs.3000 and Rs.8000 respectively in a business. If A doubles his capital after 6 months. In what ratio should A and B divide that year's profit?
A.9:6
B.9:8
C.9:16
D.9:9
Answer: C.9:16
Explanation: (3*6 + 6*6): (8*12)
54:96 => 9:16.Answer:C
Question: The sum of the present ages of a father and his son is 42 years. Six years ago, father's age was four times the age of the son. After 6 years, son's age will be:
A.14
B.16
C.18
D.20
Answer: C.18
Explanation: Let the present ages of son and father be x and (42 - x) years respectively.
Then, (42 - x) - 6 = 4(x - 6)
36 - x = 4x - 24
x = 12
The son's age after 6 years will be 18.
The answer is C.
Question: For integers x and y, if 91x = 8y, which of the following must be true?
I. x > y
II. y is not an integer
III. The cube root of x is an integer
A.I only
B.II only
C.III only
D.I and II
Answer: A.I only
Explanation: Statement 1: y>x
When y=x=0, equation holds but y is not greater than x
When x=-8 and y=-91, equation again holds but x>y
TRUE
Statement 2: y is not an integer
Since x and y are integers, 91x and 8y must also be integers.
It is given that 91x=8y
or 13*7*x = 8 y
or 13x = 8y/7
To balance the equation, y must be an integer
NOT TRUE
Statement 3: The cube root of x is an integer
x can be equal to 2*2*2*3 and for this value of x,y will be 13*7*3
So, x may or may not be a cube root.
NOT TRUE
Answer:-A
Question: In the game of Yahtzee, a "full house" occurs when a player rolls five fair dice and the result is two dice showing one number and the other three all showing another (three of a kind). What is the probability that a player will roll a full house on one roll of each of the three remaining dice if that player has already rolled a pair on the first two?
A.1/3
B.5/54
C.5/72
D.5/216
Answer: B.5/54
Explanation: We have that the player rolled XX on the first two dice. To get a "full house" the next three dice must show YYY or XYY in any combination.
The probability of YYY is 5/6*1/6*1/6 = 5/216.
The probability of XYY is 3*1/6*5/6*1/6 = 15/216 (multiply by 3 because XYY can occur in three ways: XYY, YXY, YYX).
The overall probability is 5/216 + 15/216 = 20/216 = 5/54.
Answer: B.
Question: If the volume of two cubes are in the ratio 64:1, the ratio of their edges is:
A.3:1
B.3:2
C.3:5
D.4:1
Answer: D.4:1
Explanation: Explanation:
Let the edges be a and b of two cubes, then
a3/b3=64/1
=>(a/b)3=(4/1)3
a/b=4/1
=>a:b=4:1
Option D
Question: If p and q are prime numbers, how many divisors does the product p^4 * q^9 have?
A.30
B.35
C.40
D.50
Answer: D.50
Explanation: when a number N=a^x*b^y, where a and b are prime numbers, and x, y are positive integers, the number of divisors of N=(x+1)(y+1)
Therefore, the answer is D. 5*10=50
Question: The smallest number which when divided by 20, 25, 35, 40 leaves the remainder 6 When divided by
14, 19, 23 and 34 respectively is the difference between divisor and The corresponding remainder is 6.
A.1396
B.1394
C.1494
D.1384
Answer: B.1394
Explanation: The smallest number which when divided by 20, 25, 35, 40 leaves the remainder 6 When divided by
14, 19, 23 and 34 respectively is the difference between divisor and The corresponding remainder is 6.
= 1400-6 = 1394
Answer is B
Question: A certain sum of money at simple interest amounted Rs.680 in 8 years at 3% per annum, find the sum?
A.548
B.675
C.588
D.678
Answer: C.588
Explanation: 680 = P [1 + (8*3)/100]
P = 548
Answer: C
Question: Find the odd man out
1, 2, 5, 14, 41, 123
A.2
B.14
C.124
D.123
Answer: D.123
Explanation: 1*3-1 =2
2*3-1 = 5
5*3-1 = 14
14*3-1 = 41
41*3 -1 = 122
ANSWER:D
Question: What is the value of (P + Q)/(P - Q) if P/Q is 7?
A.4/3
B.2/3
C.2/6
D.7/8
Answer: A.4/3
Explanation: (P + Q)/(P - Q) = [(P/Q) + 1]/[(P/Q) - 1]
= (7 + 1)/(7 - 1)
= 8/6
= 4/3
ANSWER:A
Question: A car traveling at a certain constant speed takes 5 seconds longer to travel 1 km than it would take to travel 1 km at 75 km/hour. At what speed, in km/hr, is the car traveling?
A.70
B.72
C.74
D.75
Answer: A.70
Explanation: Time to cover 1 kilometer at 80 kilometers per hour is 1/75 hours = 3,600/75 seconds = 48 seconds;
Time to cover 1 kilometer at regular speed is 48 + 5 = 53 seconds = 53/3,600 hours = 1/70 hours;
So, we get that to cover 1 kilometer 1/70 hours is needed --> regular speed 70 kilometers per hour (rate is a reciprocal of time or rate=distance/time).
Answer: A
Question: A train 120 m long is running with a speed of 62 kmph. In what time will it pass a man who is running at 8 kmph in the same direction in which the train is going
A.5 sec
B.6 sec
C.7 sec
D.8 sec
Answer: D.8 sec
Explanation: Explanation:
Speed of the train relative to man = (62 - 8) kmph
= (54× 5/18) m/sec = 15 m/sec
Time taken by the train to cross the man
= Time taken by It to cover 120 m at 15m / sec
= 120 ×1/15 sec = 8sec
Answer: Option D
Question: A mixture contains milk and water in the ratio 6:3. On adding 10 litters of water, the ratio of milk to water becomes 6:5. The quantity of milk in the original mixture is?
A.90
B.45
C.110
D.40
Answer: B.45
Explanation: Explanation:
milk:water = 6:3
6x : 3x + 10 = 6 : 5
5[6x] = 6[3x + 10]
30x = 18x + 60
30x - 18x = 60
12x = 60
x = 5
The quantity of milk in the original mixture is = 6 : 3 = 6 + 3 = 9
9x = 45
Short cut method - 2 : for Only milk problems
milk : water
6 : 3
6 : 5
milk ratio same but water ratio 2 part incress per 10 liters
2 part of ratio -------> 10 liters
9part of ratio -------> 45 liters
Answer: Option B
Question: The average of 9 numbers is 7 and the average of 7 other numbers is 10. What is the average of all 16 numbers?
A.9
B.8 5/16
C.7 7⁄8
D.7 1⁄2
Answer: B.8 5/16
Explanation: The average of 9 numbers is 7....
The sum of those numbers is (9)(7) = 63.
The average of 7 other numbers is 10....
The sum of those numbers is (7)(10) = 70
The sum of ALL 16 numbers is 63+70 = 133. The average of those numbers is 133/16 = 8 5/16.
Final Answer:
[Reveal]Spoiler:
B
Question: A, B and C started a business with capitals of Rs. 8000, Rs. 10000 and Rs. 12000 respectively. At the end of the year, the profit share of B is Rs. 1500. The difference between the profit shares of A and C is?
A.8867
B.299
C.279
D.600
Answer: D.600
Explanation: Ratio of investments of A, B and C is 8000 : 10000 : 12000 = 4 : 5 : 6
And also given that, profit share of B is Rs. 1500
=> 5 parts out of 15 parts is Rs. 1500
Now, required difference is 6 - 4 = 2 parts
Required difference = 2/5 (1500) = Rs. 600.Answer: D
Question: The average salary of a person for the months of January, February, March and April is Rs.8000 and that for the months February, March, April and May is Rs.8300. If his salary for the month of May is Rs.6500, find his salary for the month of January?
A.2177
B.2876
C.4500
D.5300
Answer: D.5300
Explanation: Sum of the salaries of the person for the months of January, February, March and April = 4 * 8000 = 32000 ----(1)
Sum of the salaries of the person for the months of February, March, April and May = 4 * 8300 = 33200 ----(2)
(2)-(1) i.e. May - Jan = 1200
Salary of May is Rs.6500
Salary of January = Rs.5300.Answer: D
Question: What is the product of all the prime factors of 24?
A.36
B.24
C.48
D.12
Answer: B.24
Explanation: 24 , factors are 2,3,4,8,6,12. Now again prime Numbers are Only Two i.e 2 and 3 , this time we would have 2*2*2*3=24
Answer : B
Question: Let p and q be the root of the quadratic equation x2 - (∝ - 2)x - ∝ -1 =0. what is the minimum possible value of p2 + q2 ?
A.0
B.3
C.4
D.5
Answer: D.5
Explanation: Explanation :
p + q = α –2 and pq = –α – 1
(p + q)2 = p2 + q2 + 2pq,
Thus (α –2)2 = p2 + q2 + 2(–α – 1)
p2 + q2 = α2 – 4α + 4 + 2α + 2
p2 + q2 = α2 – 2α + 6
p2 + q2 = α2 – 2α + 1 + 5
p2 + q2 = (α – 1)2 + 5
Thus, minimum value of p2 + q2 is 5.
Answer : D
Question: what is the Max number of rectangular boxes, each measuring 5 inches by 2 inches by 7 inches, that can be packed into a rectangular packing box measuring 15 inches by 20 inches by 35 inches, if all boxes are aligned in the same direction?
A.200
B.350
C.100
D.150
Answer: D.150
Explanation: The 5 inch side should be aligned to the 15 inch side ( 3 layer)
2 inch side should be aligned to the 20 inch side. (10 layer)
7 inch side should be aligned to the 35 inch side.(5 layer)
Maximum number of rectangles =3*10*5=150
Answer is D
Question: When a number is divided by 6 &then multiply by 12 the answer is 10 what is the no.?
A.4.5
B.5
C.5.6
D.5.7
Answer: B.5
Explanation: if $x$ is the number, x/6 * 12 = 10
=> 2x = 10
=> x = 5
B
Question: In a group of 80 children and 10 youngsters, each child got sweets that are 20% of the total number of children and each youngster got sweets that are 25% of the total number of children. How many sweets were there?
A.1140
B.1160
C.1280
D.1200
Answer: C.1280
Explanation: Number of sweets each child got = 20% of 80 = 20/100 * 80 = 16.
Number of sweets 80 children got = 80 * 16 = 1080.
Number of sweets each youngster got = 25% of 80 = 25/100 * 80 = 20.
Number of sweets 10 youngsters got = 10 * 20 = 200.
Total number of sweets = 1080 + 200 = 1280.
C)
Question: The average marks of a class of 22 students is 40 and that of another class of 28 students is 60. Find the average marks of all the students?
A.51.2
B.59.5
C.52.8
D.52.5
Answer: A.51.2
Explanation: Sum of the marks for the class of 22 students = 22 * 40 = 880
Sum of the marks for the class of 28 students = 28 * 60 = 1680
Sum of the marks for the class of 50 students =
880 + 1680 = 2560
Average marks of all the students = 2560/50
= 51.2
Answer:A
Question: If the letters of the word PLACE are arranged taken all at a time, find how many do not start with AE?
A.94
B.113
C.114
D.121
Answer: C.114
Explanation: Total no'of arrangements 5P5 = 5! = 120
no'of arrangements start with AE = 1 × 6 = 6
no'of arrangements which do not start with AE = 120 - 6 = 114
C)
Question: 1600 men have provisions for 28 days in the temple. If after 4 days, 400 men leave the temple, how long will the food last now?
A.18
B.77
C.32 days
D.16
Answer: C.32 days
Explanation: 1600 ---- 28 days
1600 ---- 24
1200 ---- ?
1600*24 = 1200*x
x = 32 days
Answer: C
Question: Running 3/4th of his usual rate, a man is 12min late. Find his usual time in hours ?
A.6/4 hrs
B.3/5 hrs
C.9/4 hrs
D.3/4 hrs
Answer: B.3/5 hrs
Explanation: Explanation:
Walking at 3/4th of usual rate implies that time taken would be 4/3th of the usual time. In other words, the time taken is 1/3rd more than his usual time
so 1/3rd of the usual time = 12min
or usual time = 3 x 12 = 36min = 36/60 hrs = 3/5 hrs.
Answer: B
Question: A sum of Rs. 125000 amounts to Rs. 15500 in 4 years at the rate of simple interest. What is the rate of interest?
A.7%
B.9%
C.5%
D.6%
Answer: D.6%
Explanation: S.I. = (15500 - 12500) = Rs. 3000\
Rate = (100 * 3000) / (12500 * 4)
= 6%
Answer: D
Question: The difference between the squares of two consecutive odd integers
is a square<1000 whose root equals the combined sum of the digits
of the two integers. What is the sum of the digits M of the larger integer?
A.M=2
B.M=5
C.M=8
D.10
Answer: A.M=2
Explanation: Really hard. But let's start with few concepts:
1. Square of an odd integer will be odd.
2. Difference of odd and odd will be even.
3. Any odd integer can be expressed as difference of two squares.
4. An even integer can be expressed as difference of two squares only if that even integer is a multiple of 4.
Assume two odd integers to be (2x-1) and (2x+1).
Difference of their squares = (2x+1)^2 - (2x-1)^2
= 4x^2 + 1 + 4x - (4x^2 + 1 - 4x)
= 4x^2 + 1 + 4x - 4x^2 - 1 + 4x
= 8x
Now root of 8x needs to be an integer such that it is equal to the sum of the digits of the two odd integers.
8 = 2^3, so x needs to be such that itcompletesa perfect square.
If we find x, we can find the value of 2x+1 (larger integer) and then sum of its digits.
Let's check the options, starting with c.
For sum to be 8, few possibilities are: 17, 35, 53, 71
If we take 17, the pair is 15 and 17, meaning x = 8.
8x = 64
root of 64 = 4 but 4 is not equal to 1+5+1+7.
Reject.
If we take 35, the pair is 33 and 35, meaning x = 17.
8x = 8*17
Reject since perfect square is not possible.
If we take 53, the pair is 51 and 53, meaning x = 26.
8x = 8*26
Reject since perfect square is not possible.
If we take 71, the pair is 69 and 71, meaning x = 35.
8x = 8*35
Reject since perfect square is not possible.
I tried each option and the possibilities and then got lost.
Then, it occurred to me that I was only checking 2-digit integers. What about 3-digits?
Starting with option a, only 2-digit odd integer whose sum is 2 is 101.
If we take 101, the pair is 99 and 101, meaning x = 50.
8x = 8*50 = 400
root of 400 = 20 which is indeed equal to 9+9+1+1.
Answer (A).
Question: One side of a rectangular field is 15 m and one of its diagonals is 17 m. Find the area of the field in meter square
A.100
B.110
C.120
D.130
Answer: C.120
Explanation: Other side = ((17) 2- (15)2)(1/2)
= (289- 225)(1/2)
= (64)(1/2)
= 8 m.
Area = (15 x 8) m2 = 120 m2.
Option C