divelab/Sys2Bench huggingface dataset:
Question: Matt is touring a nation in which coins are issued in two amounts, 2¢ and 5¢, which are made of iron and copper, respectively. If Matt has twu iron coins and twu copper coins, how many different sums from 1¢ to 14¢ can he make with a combination of his coins?
Options:
A)8
B)11
C)9
D)4
E)12
Rationale: The total sum is 2*2 + 2*5 = 14¢. If you can make each sum from 1 to 14 (1¢, 2¢, 3¢, ..., 14¢), then the answer would be 14 (maximum possible).
Now, with 2¢ and 5¢ we cannot make 1¢ and 3¢. We also cannot make 13¢ and 11¢ (since total sum is 14¢ we cannot remove 1¢ or 3¢ to get 13¢ or 11¢).
Also discarded 6 and 8, for lack of even numbers after 4
So, out of 14 sums 6 are for sure not possible, so the answer must be 14 - 6 = 8 sums or less. Only A fits.
Answer: A.
Answer: A
Question: Which one of the following numbers is the greatest positive integer x such that 3^x is a factor of 81^5?
Options:
A)5
B)8
C)10
D)15
E)20
Rationale: 81 = 3*3*3*3= 3^4
81^5 =(3^4)^5
=3^20
we have to find the greatest positive integer value for x
when x=20, it divides exactly once and that is the least positive integer value of the quotient
therefore, the correct option is E
Answer: E
Question: A factory has three types of machines, each of which works at its own constant rate. If 7 Machine As and 11 Machine Bs can produce 261 widgets per hour, and if 8 Machine As and 22 Machine Cs can produce 600 widgets per hour, how many widgets could one machine A, one Machine B, and one Machine C produce in one 8-hour day?
Options:
A)408
B)475
C)550
D)625
E)700
Rationale: Let Machine A produce A widgets per hour. B produce B widgets per hour and C produce C widgets per hour.
7A+11B=261 ---(1)
8A+22C=600 ---(2)
Dividing (2) by 2
4A+11C=300.....(3)
Adding (1)(3)
11A+11B+11C = 561
A+B+C=51 per hour
So for eight hrs = 51*8 = 408 = Answer = A
Answer: A
Question: What is the least number should be added to 1156, so the sum of the number is completely divisible by 25?
Options:
A)19
B)2
C)5
D)6
E)8
Rationale: (1156 / 25) gives remainder 6
6 + 19 = 25, So we need to add 19
Answer : A
Answer: A
Question: The expression (5x-1)/(x+3) is equivalent to which of the following?
Options:
A)(5-2)/3
B)5 – (2/3)
C)5 – (x)/(x+3)
D)5 – (17)/(x+3)
E)5 - (16)/(x+3)
Rationale: Easiest way is to pick any number and put in (5x-1)/(x+3). Then check the options with the same number.
I took x=1 and got E as the answer.
Answer: E
Question: Find value of X: (323^23 + 73^44 + 413^30) - (317 × 91) = ?
Options:
A)34455
B)35546
C)52150
D)68542
E)56842
Rationale: It's speed and accuracy which decides the winner. You can use various methods including Vedic mathematics to do calculations quickly.
(32323+7344+41330)−(317×91)=80997−28847=52150
C
Answer: C
Question: In one can of mixed nuts, 30% is peanuts. In another can of mixed nuts that is one-half the size of the first one, 40% is peanuts. If both cans are emptied into the same bowl, what percentage of the mixed nuts in the bowl is peanuts?
Options:
A)16 2/3%
B)20%
C)25%
D)33 1/3%
E)35%
Rationale: Let number of mixed nuts in first can = 100
Number of peanuts in first can = (30/100)*100 = 30
Since second can is one- half of first , number of mixed nuts in second can = 50
Number of peanuts in second can = (40/100)*50 = 20
Total number of peanuts in both cans = 30+20 = 50
Total number of mixed nuts in both cans = 100+50 = 150
% of peanuts if both cans are mixed = (50/150) *100% = 33.33 %
Answer D
Answer: D
Question: Jim and Joe split the bill based on the amount of space they take up in their apartment. Jim pays for two-fifths of it and Joe pays the rest. If Jim pays 30 dollars, how much will Joe have to pay?
Options:
A)45
B)50
C)55
D)40
E)60
Rationale: 30 x (5/2) = 150/2 = 75
150 - 30 = 45
Answer is A
Answer: A
Question: What is the units digit of 3333^(333)*3333^(222)?
Options:
A)0
B)2
C)4
D)6
E)8
Rationale: Each of the other explanations to this question has properly explained that you need to break down the calculation into pieces and figure out the repeatingpatternof the units digits.
Here's another way to organize the information.
We're given [(2222)^333][(3333)^222]
We can 'combine' some of the pieces and rewrite this product as....
([(2222)(3333)]^222) [(2222)^111]
(2222)(3333) = a big number that ends in a 6
Taking a number that ends in a 6 and raising it to a power creates a nice pattern:
6^1 = 6
6^2 = 36
6^3 = 216
Etc.
Thus, we know that ([(2222)(3333)]^222) will be a gigantic number that ends in a 6.
2^111 requires us to figure out thecycleof the units digit...
2^1 = 2
2^2 = 4
2^3 = 8
2^4 = 16
2^5 = 32
2^6 = 64
2^7 = 128
2^8 = 256
So, every 4powers, the pattern of the units digits repeats (2, 4, 8, 6.....2, 4, 8, 6....).
111 = 27 sets of 4 with a remainder of 3....
This means that 2^111 = a big number that ends in an 8
So we have to multiply a big number that ends in a 6 and a big number that ends in an 8.
(6)(8) = 48, so the final product will be a gigantic number that ends in an 6.
Final Answer:
D
Answer: D
Question: If H = {1, 7, 18, 20, 29, 33}, how much less is the mean of the numbers in H than the median of the numbers in H?
Options:
A) 1.0
B) 1.5
C) 2.0
D) 2.5
E) 3.0
Rationale: This is a good question to understand the difference between mean and median.
Mean: Average of all the numbers. (Sum of all the elements divided by the number of elements)
Median: Arrange the elements of the set in increasing order. If the number of terms is odd, the middle term is the median. If the number of terms is even, the average of middle two terms is the median
Coming to this question,
Mean = (1 + 7 + 18 + 20 + 29 + 33)/6 = 18
Median = (18 + 20)/2 = 19
Difference = 1
Option A
Answer: A
Question: A dog is tied to a tree by a long nylon cord. If the dog runs from the due North side of the tree to the due South side of the tree with the cord extended to its full length at all items, and the dog ran approximately 30 feet, what was the approximate length of the nylon cord L in feet?
Options:
A)30
B)25
C)15
D)10
E)5
Rationale: Because the cord was extended to its full length at all items, the dog ran along a semi-circular path, from north to south.
The circumference of a full circle is 2*pi*r, but since we only care about the length of half the circle, the semi-circle path is pi*r.
L=pi*r = 30. Round pi = 3, then r = 10.
Chord is about 10 feet long.D
Answer: D
Question: A man's speed with the current is 20 kmph and speed of the current is 6 kmph. The Man's speed
against the current will be
Options:
A)11 kmph
B)12 kmph
C)14 kmph
D)8 kmph
E)None of these
Rationale: Explanation:
Speed with current is 20,
speed of the man + It is speed of the current
Speed in sƟll water = 20 - 6 = 14
Now speed against the current will be
speed of the man - speed of the current
= 14 - 6 = 8 kmph
Answer: D
Answer: D
Question: In a group of ducks and cows, the total number of legs are 24 more than twice the number of heads. Find the total number of cows.
Options:
A)12
B)14
C)16
D)18
E)20
Rationale: Let the number of ducks be d
and number of cows be c
Then, total number of legs = 2d + 4c = 2(d + 2c)
total number of heads = c + d
Given that total number of legs are 24 more than twice the number of heads
=> 2(d + 2c) = 24+ 2(c + d)
=> d + 2c = 12 + c + d
=> 2c = 12 + c
=> c = 12
i.e., total number of cows = 12
Answer is A.
Answer: A
Question: The cost price of a cow and a horse is Rs 3 lakhs. The cow is sold at 20% profit and the horse is sold at 10% loss. Overall gain is Rs 4200. What is the cost price of the cow?
Options:
A)29
B)88
C)271
D)273
E)111
Rationale: Answer:E
Answer: E
Question: I bought three toys for my triplet boys (one for each). I placed the toys in the dark store. One by one each boy went to the store and pick the toy. What is the probability that no boy will choose his own toy?
Options:
A)1/4
B)1/3
C)1/6
D)2/3
E)1/9
Rationale: B
1/3
Assuming T1 is the Toy for brother1, T2 is the toy for brother2 and T3 is the toy for brother3.
Following are the possible cases for toys distribution:
Boy1 Boy2 Boy3
T1 T2 T3
T1 T3 T2
T2 T1 T3
T2 T3 T1 .... (A)
T3 T1 T2 .... (B)
T3 T2 T1
In both steps (A) & (B), no one gets the correct toy.
Therefore probability that none brother can get the own toy is 2/6 = 1/3
Answer: B
Question: Present ages of X and Y are in the ratio 5:6 respectively. Seven years hence this ratio will become 6:7 respectively. What is X's present age in years?
Options:
A)40
B)35
C)28
D)49
E)70
Rationale: Let the present ages of X and Y be 5x and 6x years respectively.
Then, (5x + 7)/(6x + 7) = 6/7
7(5x + 7) = 6(6x + 7) => x = 7
X's present age = 5x = 35 years.
ANSWER B
Answer: B
Question: n a 1000m race, if A gives B a start of 40m, A wing by 19sec. But if A gives B start of 30sec, B wins by 40m.
Find the time taken by B to run 4000m race?
Options:
A)500
B)700
C)400
D)600
E)800
Rationale: then 960/B -1000/A = 19 => 24/B - 25/A = 19/40
and 1000/B - 960/A = 30 => 25/B - 24/A = 30/40
solving equation A speed =8 and B=20/3
for 1000m b will take 4000/20/3= 600 sec
D
Answer: D
Question: Out of 10 persons working on a project, 4 are graduates. If 3 are selected, what is the probability that there is at least one graduate among them?
Options:
A)5/5
B)5/8
C)5/1
D)5/6
E)5/2
Rationale: P(at least one graduate) = 1 - P(no graduates)
= 1 - 6C3/10C3
= 1 - (6 * 5 * 4)/(10 * 9 * 8)
= 5/6
Answer:D
Answer: D
Question: K and L start the business, with L investing the total capital of Rs. 40,000 on the condition that K pays L interest @ 8% per annum on his half of the capital. L is a working partner and receives Rs.1200 per month from the total profit and any profit reamaining is equally shared by both of them. At the end of the year, it was found that the income of L is twice that of K. Find the total profit for the year ?
Options:
A)46409
B)46400
C)46129
D)46411
E)41222
Rationale: Interest received by L from K = 8% of half of Rs.40,000
= Rs.1600
Amount received by L per annum for being a working partner = 1200\small \times12 = Rs.14,400
Let 'A' be the part of remaing profit that 'L' receives as his share.
Total income of 'K' = only his share from the reamaing profit
= 'A', as both share equally.
Given income of L = Twice the income of K
Rightarrow (1600 + 14400 + A ) = 2A
Rightarrow A= Rs.16000
Thus total profit = 2A + Rs.14,400= 2(16000) + 14400
= 32000 +14400 = Rs.46,400.
Answer: B
Answer: B
Question: A man whose speed is 7.5 kmph in still water rows to a certain upstream point and back to the starting point in a river which flows at 1.5 kmph, find his average speed for the total journey?
Options:
A)8.2kmph
B)9.0kmph
C)7.0kmph
D)6.5kmph
E)7.2kmph
Rationale: M = 7.5
S = 1.5
DS = 9
US = 6
AS = (2 * 9 * 6) /15 = 7.2
Answer:E
Answer: E
Question: What is the median from the below series
90, 92, 93, 88, 95, 88, 97, 87, and 98
Options:
A)85
B)89
C)92
D)98
E)100
Rationale: Ordering the data from least to greatest, we get:
87, 88, 88, 90, 92, 93, 95, 96, 98
The median score is 92.
C
Answer: C
Question: A rectangular field is to be fenced on three sides leaving a side of 20 feet uncovered.If the area of the field is 600 sq.ft, how many feet of fencing will be required ?
Options:
A)80feet
B)70feet
C)60feet
D)50feet
E)20feet
Rationale: Explanation:
We are given with length and area, so we can find the breadth.
as Length * Breadth = Area
=> 20 * Breadth = 600
=> Breadth = 30 feet
Area to be fenced = 2B + L = 2*30 + 20
= 80feet
Answer: Option A
Answer: A
Question: The population of a town is 7600. It decreases annually at the rate of 10% p.a. What was its population 2 years ago?
Options:
A)9400
B)8000
C)8500
D)9500
E)10000
Rationale: Formula :
( After =100 denominator
Ago = 100 numerator)
7600 Ã— 100/90 Ã— 100/90 = 9382
A)
Answer: A
Question: A trader marks his books 30% more than the cost price. If he allows 30% discount, then find his gain or loss percent?
Options:
A)2% profit
B)2% loss
C)4% loss
D)No profit No loss
E)9% loss
Rationale: Let CP of an book = 100
MP of an book = 130
Discount 30 % on 130 = 39
SP of the book = 130 - 39 = 91
loss = 9%
Answer : E
Answer: E
Question: A boat can move upstream at 33 kmph and downstream at 35 kmph, then the speed of the current is?
Options:
A)1
B)2
C)7
D)8
E)9
Rationale: US = 33
DS = 35
M = (35 - 33)/2 = 1
Answer: A
Answer: A
Question: If a certain number is divided by 3, the quotient, dividend, and divisor, added together, will amount to 63. What is the number?
Options:
A)18
B)28
C)45
D)38
E)59
Rationale: Let x = the number sought.
Then x/3 + x + 3 = 63.
And x = 45.
C
Answer: C
Question: A contractor undertook to do a piece of work in 8 days. He employed certain number of laboures but 3 of them were absent from the very first day and the rest could finish the work in only 14 days. Find the number of men originally employed ?
Options:
A)6
B)7.5
C)5
D)6
E)7
Rationale: Let the number of men originally employed be x.
8x = 14(x â€“ 3)
or x = 7
Answer E
Answer: E
Question: nd the area of trapezium whose parallel sides are 20 cm and 15 cm long, and the distance between them is 14 cm?
Options:
A)230cm2
B)245cm2
C)255cm2
D)260cm2
E)280cm2
Rationale: Area of a trapezium = 1/2 (sum of parallel sides) * (perpendicular distance between them)
= 1/2 (20 + 15) * (14)
= 245 cm2
Answer:B
Answer: B
Question: A sum of money is to be divided among Ann, Bob and Chloe. First, Ann receives $4 plus one-half of what remains. Next, Bob receives $4 plus one-third of what remains. Finally, Chloe receives the remaining $32. How much money M did Bob receive?
Options:
A)20
B)22
C)24
D)26
E)52
Rationale: Notice that we need not consider Ann's portion in the solution. We can just let K = the money REMAINING after Ann has received her portion and go from there.
Our equation will use the fact that, once we remove Bob's portion, we have $32 for Chloe.
So, we getK - Bob's $ = 32
Bob received 4 dollars plus one-third of what remained
Once Bob receives $4, the amount remaining is K-4 dollars. So, Bob gets a 1/3 of that as well.
1/3 of K-4 is (K-4)/3
So ALTOGETHER, Bob receives4 + (K-4)/3
So, our equation becomes:K -[4 + (K-4)/3 ]= 32
Simplify to get: K - 4 - (K-4)/3 = 32
Multiply both sides by 3 to get: 3K - 12 - K + 4 = 96
Simplify: 2K - 8 = 96
Solve: K = 52
Plug this K-value intoK - Bob's $ = 32to get: 52 - Bob's $ = 32
So, Bob's $ M= 20
Answer:
B
Answer: B
Question: A sum of money at simple interest amounts to Rs. 815 in 3 years and to Rs. 854 in 4 years. The sum is :
Options:
A)Rs. 700
B)Rs. 690
C)Rs. 650
D)Rs. 698
E)Rs. 765
Rationale: Explanation:
Simple Interest (SI) for 1 year = 854-815 = 39
Simple Interest (SI) for 3 years = 39 × 3 = 117
Principal = 815 - 117 = Rs.698
Answer: Option D
Answer: D
Question: A train ,140 meters long travels at a speed of 45 km/hr crosses a bridge in 30 seconds. The length of the bridge is
Options:
A)270 m
B)245 m
C)235 m
D)220 m
E)240 m
Rationale: Explanation:
Assume the length of the bridge = x meter
Total distance covered = 140+x meter
total time taken = 30s
speed = Total distance covered /total time taken = (140+x)/30 m/s
=> 45 Ã— (10/36) = (140+x)/30
=> 45 Ã— 10 Ã— 30 /36 = 140+x
=> 45 Ã— 10 Ã— 10 / 12 = 140+x
=> 15 Ã— 10 Ã— 10 / 4 = 140+x
=> 15 Ã— 25 = 140+x = 375
=> x = 375-140 =235
Answer: Option C
Answer: C
Question: Points A, B, and, C have xy-coordinates (2,0), (8,12), and (14,0), respectively. Points X, Y, and Z have xy-coordinates (6,0), (8,4), and (10,0), respectively. What fraction C of the area of triangle ABC is the area of triangle XYZ?
Options:
A) 1/9
B) 1/8
C) 1/6
D) 1/5
E) 1/3
Rationale: If you notice, both triangles ABC and XYZ have a side on X axis. we can take these sides as bases for each triangle, therefore
Area of ABC is 1/2*12*12 (Height of ABC is the y coordinate of the third point (8,12))
similarly Area of XYZ is 1/2*4*4
dividing area of XYZ with that of ABC gives C= 1/9.A
Answer: A
Question: In how many different number of ways 6 men and 2 women can sit on a shopa which can accommodate persons?
Options:
A)1250
B)1640
C)1240
D)1680
E)1270
Rationale: 8p4 = 8 x 7 x 6 × 5 = 1680
D)
Answer: D
Question: The length of a train and that of a platform are equal. If with a speed of 90 k/hr, the train crosses the platform in one minute, then the length of the train (in meters) is?
Options:
A)228
B)267
C)200
D)750
E)281
Rationale: Speed = [90 * 5/18] m/sec = 25 m/sec; Time = 1 min. = 60 sec.
Let the length of the train and that of the platform be x meters.
Then, 2x/60 = 25 è x = 25 * 60 / 2 = 750 .Answer: D
Answer: D
Question: If L = 775 × 778 × 781, what is the remainder when L is divided by 14?
Options:
A)6
B)7
C)8
D)9
E)10
Rationale: L 775/14 leaves a remainder 5
778/14 leaves a remainder 8
781/14 leaves a remainder 11
5*8*11 =440
So the remainder will be the remainder of 440/14 which is 6
Ans A
Answer: A
Question: Sum of two numbers prime to each other is 16 and their L.C.M. is 63. What are the numbers?
Options:
A)9 and 10
B)8 and 9
C)11 and 5
D)7 and 8
E)7 and 9
Rationale: As two numbers are prime, only options satisfy all
but option C will not make the product of numbers i.e 63
ANSWER:E
Answer: E
Question: Two persons A and B can complete a piece of work in 20 days and 40 days respectively. If they work together, what part of the work will be completed in 5 days?
Options:
A)1/8
B)1/3
C)3/6
D)3/8
E)5/3
Rationale: A's one day's work = 1/20
B's one day's work = 1/40
(A + B)'s one day's work = 1/20 + 1/40 = 3/40
The part of the work completed in 5 days = 5 (3/40) = 3/8.
Answer:D
Answer: D
Question: Two friends A and B leave point A and point B simultaneously by bicycle and travel towards Point B and Point A on the same route at their respective constant speeds. They meet along the route and immediately proceed to their respective destinations in 32 minutes and 50 minutes respectively. How long will B take to cover the entire journey between Point B and point A?
Options:
A)65
B)70
C)75
D)80
E)90
Rationale: Let x per minute be the speed of A and y per minute be the speed of B.
After meeting at a point, A travels for 32 mins and B travels for 50 mins. So distance covered by each of them post point of crossing
A= 32x and B=50y
The distance covered by A and B before they cross each would be distance covered by B and A post crossing respectively.
Therefore distance covered by B before he meets A= 32x
Time taken by B cover 32x distance= 32x/y mins
Therefore total time taken by B= 32x/y + 50 mins ................. I
We need to find value of x in terms of y to arrive at final answer.
Total distance= 32x+50y
Combined speed of A and B= x+y
Therefore time taken before A and B meet en-route= (32x+50y)/(x+y)
Time taken by B reach destination after meeting A= 50 mins
Total travel time for B= [(32x+50y)/(x+y)]+50 mins ...................II
Equate I and II
32x/y+50= [(32x+50y)/(x+y)]+50
(32x+50y)/y=(82x+100y)/(x+y)
32x^2+50xy+32xy+50y^2=82xy+100y^2
32x^2+82xy-82xy+50y^2-100y^2=0
32x^2-50y^2=0
32x^2=50y^2
16x^2=25y^2
Taking square root.. (since x and y denote speed, square root can't be negative)
4x=5y
y=4x/5 ............ III
substitute in I
=32x/(4x/5) + 50
=32x*5/4x + 50
=40+50
=90 mins
E
Answer: E
Question: The division of a whole number N by 14 gives a quotient of 18 and a remainder of 2. Find N.
Options:
A)797
B)254
C)597
D)997
E)297
Rationale: According to the division process of whole numbers, N can be written, using multiplication, as follows
N = quotient *divisor + remainder = 18* 14 + 2 = 254 correc answer B
Answer: B
Question: If 2/3 of the air in a tank is removed with each stroke of a vacuum pump, how many strokes does it take before less than 1% of the original amount of air in the tank remains?
Options:
A) 2
B) 3
C) 4
D) 5
E) 6
Rationale: My approach is to find more or less good number and plug in. Lets take 90 as the total volume of an air, and the question asks after how many stokes there will be less than 0,9 air in a tank, if each stroke takes 2/3 of an air.
So after he1 strokethe volume will be 30 (90-60=30), and so on
2 stroke- 30-20=10
3 stroke- 10-6,7=3,3
4 stroke- 3,3-2,2=1,1
5 stroke- 1,1 - minus something that gets us less than 0,9 for sure. so the answer is: after 5 strokes there will be less than 1% of air left in a tank. OptionD.
Answer: D
Question: A man jumps into a river from a overbridge ,at the same time a hat drops in the river which flows with the stream of the water. That man travels 10 minute upstream and returns back where he was asked to catch the hat. He catches hat at a distance of 1000 yard down at a second bridge . Find the speed of the river.
Options:
A)50 yards per minute
B)40 yards per minute
C)60 yards per minute
D)55 yards per minute
E)45 yards per minute
Rationale: Suppose x is the distance travelled by the man in 10 min upstream.
Then
x/(v-u) = 10 where v is man's speed and u is river speed)
total time taken to swim by the man upstream and down stream is equal to the time taken by the hat to travel 1000 yards with the speed of river
1000/u = 10 + (x+1000)/(u+v)
now substitute x = 10(v-u) and simplify
we will get
100v = 2uv
means
2u = 100 and u = 50 yards per minute
ANSWER:A
Answer: A
Question: Two cars cover the same distance at the speed of 48 and 64 kmps respectively. Find the distance traveled by them if the slower car takes 1 hour more than the faster car.
Options:
A)87
B)138
C)192
D)83
E)93
Rationale: Explanation:
48(x + 1) = 64x
X = 3
48 * 4 = 192 km
Answer: Option C
Answer: C
Question: total dinning bill of 5 people was $139.00 and 10% tip divided the bill evenly ?what is the bill amount each person shared .
Options:
A)32.84
B)22.84
C)30.58
D)24.84
E)30.84
Rationale: dinner bill of 5 person = 139 + 10% tip
so,
10% of 139 = (139*10)/100 = 13.9
So, the actual total amount = 139+13.9 = $ 152.9
so per head bill = 152.9/5 = $ 30.58
ANSWER:C
Answer: C
Question: The distance between Delhi and Mathura is 170 kms. A starts from Delhi with a speed of 20 kmph at 7 a.m. for Mathura and B starts from Mathura with a speed of 25 kmph at 8 p.m. from Delhi. When will they meet?
Options:
A)10.50 a.m.
B)11.20 a.m.
C)10.30 a.m.
D)11.40 a.m.
E)1.40 a.m.
Rationale: D = 170 â€“ 20 = 150
RS = 20 + 25 = 45
T = 150/45 = 3 1/3 hours
8 a.m. + 3 hrs 20 min = 11.20 a.m.
ANSWER:B
Answer: B
Question: If f(x)= 5x^3- 2x+8 and g(y)= 6y+4, then g(f(x)) =
Options:
A)30x^3-12x+52
B)11x^2-12x+44
C)8x^3-8x+32
D)30x^3+4x+4
E)30x^3-12x+44
Rationale: g(f(x)) = 6(f(x)) +4 = 6(5x^3- 2x+8) + 4 = 30x^3-12x+52 =>A
Answer: A
Question: A walks around a circular field at the rate of one round per hour while B runs around it at the rate of six rounds per hour. They start in the same direction from the same point at 7.30 a.m. They shall first cross each other at:
Options:
A)5.30 a.m
B)6.42 a.m
C)7.55 a.m
D)7.42 a.m
E)None
Rationale: Sol.
Since A and B move in the same direction along the circle, so they will first meet each other when there is a difference of one round between the two
Relative speed of A and B = (6-1) = 5 rounds per hour.
Time taken to complete one round at this speed = 1/5 hr = 12 min.
Answer D
Answer: D
Question: If 5a + 7b = t, where a and b are positive integers, what is the largest possible value of t for which exactly one pair of integers (a, b) makes the equation true?
Options:
A)35
B)48
C)69
D)70
E)74
Rationale: 5*a1 + 7*b1 = t
5*a2 + 7*b2 = t
5*(a1 - a2) = 7*(b2 - b1)
since we are dealing with integers we can assume that a1 - a2 = 7*q and b2 - b1 = 5*q where q is integer, so whenever we get a pair for (a;b) we can find another one by simply adding 7 to a and subtracting 5 from b or vice versa, subtracting 7 from a and adding 5 to b.
Lets check how it works for our numbers, starting from the largest:
E)74 = 5*12 + 7*2 (a1 = 12, b1 = 2), subtract 7 fromaand add 5 tobrespectively, so a2 = 5 and b2 = 7, second pair - bad
D)70 = 5*7 + 7*5 (a1 = 7, b1 = 5), if we add 7 toawe will have to subtract 5 from b but b can't be 0, so - no pair, if we subtract 7 froma, we'll get a = 0 which also isn't allowed - no pair, thus this is the only pair for (a;b) that works,good!, thus
D is the answer
Answer: D