Kryteax Aptitude

Aptitude

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01. A bag contains an equal number of one rupee, 50 paisa and 25 paisa coins. If the total

value is ₨ 35 then the number of coins of each type are

(a) 10 (b) 40

(c) 20 (d) 18

02. ‘A’ lent Rs 5000 to ‘B’ for 2 years and Rs 3000 to ‘C’ for 4 years on simple interest at the

same rate of interest and received Rs 2200, in all from both them as interest. Then the rate of

interest per annum is

(a) 10% (b) 20%

(c) 8% (d) 4%

03. A Father’s age is three times the sum of the age of his two children, after 20 years his age

will be equal to the sum of their ages. Then the Father’s age (in years) is

(a) 30 (b) 40

(c) 35 (d) 45

04. 8 : 81 : : 64 : ______________

(a) 125 (b) 137

(c) 625 (d) 525

05. In a row of Boys, where A is th10 from left and B is

th9 from the right. On interchanging

their positions, A becomes th15 from the left. The number of Boys in the row are:

(a) 23 (b) 24

(c) 25 (d) 31

06. QIOK : MMKO :: YAWC : _______________

(a) UESG (b) USGA

(c) VUES (d) SUEG

07. n n7 4 is divisible by 11 for

(a) odd values of n (b) even values of n

(c) both odd and even values of n (d) None of these

Aptitude


08. From the diagram, the value of x is

16 28 29

13 12 16

14 10 15

15 30 x

(a) 60 (b) 30

(c) 20 (d) 45

09. The side of a square is 22 m. The radius (in m) of the circle, whose circumference is equal

to the perimeter of square, is

(a) 28 (b) 3.5

(c) 14 (d) 7

10. Two stations A and B are 110 km apart on a straight line. One train starts from A at 7:00

AM and travels towards B at 20 km/hr speed. Another train starts from B at 8:00 AM and

travels towards A at 25 km/hr speed. The time at which they will meet is

(a) 10:30 AM (b) 11:00 AM

(c) 10:00 AM (d) 11:30 AM

11. Four examiners can examine certain number of answer papers in 10 days by working 5

hours a day. The number of working hours in a day would 2 examiners take have to work in

order to examine twice the number of answer papers in 20 days is

(a) 8 hours (b) 10 hours

(c) 9 hours (d) 11 hours

12. Which of the following statements is incorrect?

(a) 10

log 10 1 (b) 10

log 1 0

(c) log 2 3 log 2 3 (d) log6 log1 log2 log3

Aptitude


13. A reduction of 20%, in the price of rice enables a person to buy 3.5 kg more rice for Rs

385. The original price (in Rs) of rice per kg is

(a) 20 (b) 22.50

(c) 25 (d) 27.50

14. The mean marks of 10 Boys in a class is 70%, whereas the mean marks of 15 girls is 60%.

The mean (in %) marks of all 25 students is

(a) 64 (b) 60

(c) 55 (d) 52

15. If N 1! 3! 5! 7!........................................99! then remainder obtained when N is divided

by 24 is

(a) 7 (b) 6

(c) 3 (d) 5

16. The unit’s digit in the product of

443 1076 4513 2194 773

197 296 273 2379 3125 is

(a) 7 (b) 6

(c) 3 (d) 0

17. Red light flashes 3 times per minute and Green light flashes 5 times in every 2 minutes. If

both light start flashing at the same time, then how many times do they flash together in

each hour?

(a) 30 (b) 24

(c) 20 (d) 60

18. From a beaker full of pure cola, Mohit withdraws 25% of cola and replaces it with water.

He again withdraws 25% of mixture of cola and again replaces with water and the process

was repeated two more times till he finds 567 ml of pure cola left in the mixture. Then the

actual initial quantity (in ml) of cola was

(a) 1792 (b) 1000

(c) 1296 (d) None of these

Aptitude


19. The numbers that can be made with digits 0, 7, 8 which are greater than 0 and less than a

million are

(a) 496 (b) 486

(c) 728 (d) 1084

20. 6n 6n7 6 , where ‘n’ is an integer greater than 0, is divisible by

(a) 13 (b) 127

(c) 559 (d) All of these

21. Two third of a consignment was sold at a profit of 5% and remainder was sold at a profit

of 2%. If total profit was Rs 800, then the value (in Rs) of the consignment is

(a) 20000 (b) 18000

(c) 16000 (d) 24000

22. If 1 2 3 1 2 3 1 2 3

N a a a a a a a a a ................................,N is a repeating nor terminating decimal

number, then the natural number by which N can be multiplied so that the resultant is a

natural number is

(a) 495 (b) 1001

(c) 891 (d) 4995

23. A and B decided to meet on 14th February, 2014 between 6 and 7 pm. The probability

that they will meet provided one cannot wait for the other for more than 12 minutes is

(a) 5

9 (b)

7

16

(c) 9

25 (d)

11

36

24. If petrol becomes cheaper by 25%. The percentage by which Ashish can drive his bike

more, so that his budget for petrol remains unaltered, is

(a) 22.2% (b) 11.2%

(c) 33.3% (d) 10.2%

Aptitude


25. The maximum number of students, among whom 429 mangoes and 715 oranges can be

equally distributed, are

(a) 143 (b) 173

(c) 133 (d) 193

26. A man spends 1

3 of his income on food, of the rest

1

4on house rent and

1

5 on cloth. He

still has Rs 1760 left with him. Then his income (in Rs) is

(a) 4400 (b) 4800

(c) 5800 (d) 5400

27. Sectoral growth rate (in %) of major industry groups (2006-2007) are given as

Sector Weight 2006 2007

Mining 11.46 2.9

Manufacturing 77.11 8.7

Electricity 11.43 7.1

From table, the weighted average growth rate (in %) of all the industries in 2006-2007 was

(a) 7.85 (b) 6.85

(c) 4.85 (d) 9.85

28. The angle of elevation of the top of a tower 30 m high, from two points on the level

ground on its opposite sides are 45° and 60°. The distance (in m) between the two points is

(a) 20 3 3 (b) 10 3 3

(c) 15 3 3 (d) 30 3 3

29. Two coins are tossed 4 times then the probability that there will be equal number of

‘heads’ and ‘tails’ is

(a)35

64 (b)

7

1

2

(c) 35

128 (d)

8

1

2

Aptitude


30. A trader marked his goods at 20% above the cost price. He sold half the stock at the

marked price, one quarter at a discount of 20% on the marked price and the rest at a

discount of 40% on the marked price. Then his total gain (in %) is

(a) 4 (b) 6

(c) 8 (d) 2

31. 10 men and 15 women together can complete a work in 6 days. It takes 100 days for the

man alone to complete the same work. The number of days required for one woman to

complete the same work are

(a) 225 (b) 325

(c) 125 (d) 425

32. A man can row at a speed of 4.5 km/hr is still water to a certain upstream point and back

to the starting point in a river which flows at 1.5 km/hr. his average speed (in km/hr) for total

journey is

(a) 4 (b) 8

(c) 10 (d) 12

33. If 2x 1 a a ............. and 2y 1 b b ........... then 2 2 3 31 ab a b a b ........ is

given by (where a and b are less than 1)

(a) xy (b) xy

x y 1

(c) xy

x y 1 (d)

xy

x y 1

34. The probability that January of a leap year should have 5 Sundays is

(a)1

7 (b)

2

7

(c) 3

7 (d)

6

7

Aptitude


35. If 2 3 is a root of the equation 2x px 6 0 and the equation

2x px q has equal

roots, then value of Q is

(a)1

4 (b)

3

4

(c) 5

4 (d)

3

4

36. The term that comes next in the series YEB, WFD, UHG, SKI is

(a) QOL (b) QGL

(c) TOL (d) QNL

37. If a, b, c, d and e are positive real numbers, then the minimum value of

1 1 1 1 1

a b c d ea b c d e

(a) 25 (b) 5

(c) 125 (d) cannot be determined

38. A man can do a piece of work in 91 days. In order to complete the work faster, he starts

increasing his efficiency by 20% everyday (100% on day 1, 120% on day 2, 140% on day 3

and so on). How many days (minimum) would be required to complete the work now?

(a) 14 (b) 25

(c) 13 (d) 61

39. 10 men and their wives participate in a corporate mixed-doubles tennis championship.

What is the probability that no couple play in the same game?

(a) 0.6222 (b) 0.3111

(c) 0.4285 (d) 0.2777

40. How many ways can 9! be expressed as a product of two co-primes?

(a) 6 (b) 8

(c) 10 (d) 12

Aptitude


41. Study the following graph and answer the question below

Average annual exports during the period 1994-1997 for company P is what percent of the

average annual exports for company Q? ________

(a) 113.33 (b) 123.25

(c) 145.65 (d) 135.61

42. If |x|+y+z=15, then find the number of possible integral solutions?

(a) 250 (b) 254

(c) 286 (d) 290

43. Study the following pie-diagrams and answer the question below.

If the total weight of the human body were to be shown in a single pie chart, the weight of

water present in hormones and enzymes should be shown as an arc of the circle subtended

at an angle of

(a) 54° (b) 126°

(c) 108° (d) 252°

Aptitude


44. Study the graph and answer the question that follows.

Income and expenditure (in crore Rs.) of four companies in the year 2014.

If the income of company Q in 2014 was 20% more than its income in 2013 and the

company had earned a profit of 10% in 2013, then its expenditure in 2013 (in crore Rs.) was

_________(2 decimal places)

(a) 25.61 (b) 22.77

(c) 15.65 (d) 35.68

45. If 4 16

2log log 0.5 2

x , then

(a) 4

4log x 7 (b) 4

2log x 7

(c) 16

2log x 7 (d) 16

log x 7

46. The sum of all possible two digit numbers formed from three different one digit natural

numbers when divided by sum of the original three numbers is

(a) 36 (b) 28

(c) 22 (d) 15

47. In a city, Star sports, ESPN and ZEE sports are three sports channel watched by 56%, 80%

and 45% of the population respectively. 40% watch exactly two of the three channels and

10% watch none. What percent of the people watch all three channels?

(a) 36.5 (b) 56.41

(c) 45.65 (d) 25.5

48. In an examination, there are 5 subjects and each has the same maximum. A boy’s marks

are in the ratio 3:4:5:6:7 and his aggregate is 3/5th of the full marks. In how many subjects did

he get more than 50% marks?

(a) 1 (b) 2

(c) 3 (d) 4

Aptitude


49. Two pipes A and B can separately fill a cistern in 15 and 20 minutes respectively and the

waste pipe C can carry off 10 litters per minute. If the three pipes are opened when the

cistern is full, it is emptied in 2 hours. How many litters does the cistern hold?

(a) 40 (b) 80

(c) 60 (d) 120

50. The three interior angles of a quadrilateral are in AP such that the difference between the

largest and the second largest angle is equal to the smallest angle considering the same

three angles. Also the fourth angle is equal to one of the three angles that are in AP. At least

one of the angles of the quadrilateral is 900. The least angle of the quadrilateral can be

(a) 40 (b) 45

(c) 51.3 (d) Data insufficient

Aptitude


Answer Key

1 2 3 4 5 6 7 8 9 10

c a a c a a a b c c

11 12 13 14 15 16 17 18 19 20

b c d a a d a a c d

21 22 23 24 25 26 27 28 29 30

a d c c a b a b c D

31 32 33 34 35 36 37 38 39 40

a a b c d a a d a b

41 42 43 44 45 46 47 48 49 50

a d c b a c d c b b

Aptitude


Solutions

01. Ans: (c)

Solution:

Let there are ‘X’ number of coins of each type

Total value 35 1 0.5 0.25 x

x 20

02. Ans: (a)

Solution:

Let the rate of interest be r%

Total interest Interest from B Interest from C

5000 r 2 3000 r 42200

100 100

2200 100r 120r

r 10%

03. Ans: (a)

Solution:

Let the current sum of ages of two children be ‘x’

Let the current Father’s age be ‘y’

y 3x ...(i)

After 20 years

y 20 x 20 20 (Because he has two children)

y x 20 ....(ii)

Equating value of ‘x’ from equation (ii) in equation (i)

yy 20

3

y 30years

04. Ans: (c)

Solution:

8 : 81 : : 64 : ______________ 3 4 3 42 :3 : : 4 :5

∴ Answer is 625

Aptitude


05. Ans: (a)

Solution:

A B| | | | | | | | | | | | | | | | | | | | |

Total boys A’s new position B’s old position 1 23

06. Ans: (a)

Solution:

Sum of the position of the first number and next two add upto 26

Q is 17, I is 9 and O is 15, K is 11

M is 13, M is 13 and K is 11, O is 15

Y is 25, A is 1 and W is 23, C is 3

So, term satisfying this relation is UESG.

Alternate Logic: The difference between first letters of the 2 given ratios is -4, +4, -4, +4.

Q (17) – M (13) = +4

I (9) – M (13) = -4

O (15) – K (11) = +4

K (11) – O (15) = -4

Following the same pattern, the correct answer is UESG.

07. Ans: (a)

Solution:

n na b is divisible by a b , when n is odd.

You can also follow a hit and trial method. Substitute n= 0, 1, 2, 3 and see if the value is

divisible by 11. You’ll find that the value is only divisible by 11 when n= 1 and 3, which gives

you (a) as the correct answer.

08. Ans: (b)

Solution:

In a column, sum of top two numbers is equal to sum of bottom two numbers.

Column 1 16 13 14 15

Column 2 28 12 10 30

Column 3 29 16 15 x

x 30

Aptitude


09. Ans: (c)

Solution:

Let the radius be ‘r’ m

4 22 2 r r 14m

10. Ans: (c)

Solution:

A B

110 km

Let train B has travelled for ‘x’ has after its starting time till trains from A and B meet

Total distance 110

= speed of train from A× time taken by train from A + speed of train from B × time taken by

train from B

110 20 x 1 25 x

90 45x

x 2hrs

∴ at 10:00 hrs they will meet.

11. Ans: (b)

Solution:

Let ‘X’ number of answer paper are examined by four examiners in 10 days by workings 5

hours

∴ Paper examined by 1 examiner in 1 day by working 1 hour x

4 10 5

Let ‘H’ number of hours in a day are taken by 2 examiners to examine ‘2x’ answer papers in

20 days.

x2 20 H 2x

4 10 5

H 10hours

12. Ans: (c)

Solution:

log 2 3 log2 log3

Aptitude


13. Ans: (d)

Solution:

Let the original price be Rs x and the buys ‘y’ kg of rice with Rs 385

xy 385 on 20% reduction of price, person can buy 3.5 kg more rice

4

x y 3.5 3855

4 4

xy x 3.5 3855 5

4

385 2.8x 3855

385

2.8 x 775

x Rs27.5

14. Ans: (a)

Solution:

70% 10 60% 15Mean of 25 students 64%

25

15. Ans: (a)

Solution:

4! Onwards every factorial will be a multiple of 24. As 4! 24

5! 4! 5 6! 4! 5 6 and so on

So, remainder will be 1! 3! 7

16. Ans: (d)

Solution:

In the given number there is an even number (296) as well as a multiple of 5 i.e. (3125).

So, the product will surely end in a zero.

17. Ans: (a)

Solution:

Red light flashes after every 20 second and Green light flashes after every 24 second.

They will flash together after LCM (20, 24)=120 seconds or 2 minutes

Hence, they will flash together 30 times in an hour.

Aptitude


18. Ans: (a)

Solution:

Quantity of cola left after 4th operation 4

1initial quantity 1

4

4

3567 initial quantity

4

So,

4

4initial quantity 567 1792 ml

3

19. Ans: (c)

Solution:

Numbers less than a million will be 63 1 728 (as zero has to be excluded)

20. Ans: (d)

Solution:

n na b will always be divisible by a b

, if n is even

So, 6n 6n7 6 is divisible by

a b i.e. 7 6 13

2n 2n

6n 6n 3 37 6 7 6 will always be divisible by 3 37 6 and 3 37 6

(as power is even).

So, the number is divisible by 127 and 559 as well.

21. Ans: (a)

Solution:

Let ‘x’ be the value of consignment

2x 5 x 2

profit 8003 100 3 100

x Rs20000

22. Ans: (d)

Solution:

1 2 3N a a a

…(i) 3

1 2 3 1 2 310 N a a a a a a

…(ii)

(i)-(ii) 1 2 3

999N a a a

Aptitude


So, N have to be multiplied by 999 or any multiple of 999.

Hence, 999 5 4995 is the answer.

23. Ans: (c)

Solution: If A arrives at time ‘x’ and B arrives at time ‘y’ then for them to meet the range of y

will be ‘x-12’ to ‘x+12’ so the favourable area is the area bounded between these two lines.

favourable area 48 48 9Required probability 1

total area 60 60 25

24. Ans: (c)

Solution:

If prices becomes cheaper by 25%,

then consumption can be increased by 25 1

100 100% 33.3%100 25 3

25. Ans: (a)

Solution:

H.C.F. of 429 and 715 is 143

26. Ans: (b)

Solution:

Let the income of man is Rs. ‘P’

P 1 P 1 PP P 1760 P

3 4 3 5 3

P P 2P

1760 P3 6 15

P P 2P

1760 P3 6 15

30 10 5 4 P1760

30

P Rs4800

27. Ans: (a)

Solution:

Weighted average growth 11.46 2.9 77.11 8.7 11.43 7.1

33.234 670.857 81.15311.46 77.11 11.43

7.85%

Aptitude


28. Ans: (b)

Solution:

Distance between two points 35 30AB BC 10 3 3 m

tan45 tan60

29. Ans: (c)

Solution:

If coin 1 gets 4 heads, then coin 2 must get 4 tails for required condition

Permutation for this case 4 4

4 4P P

14! 4!


Permutation for this case

4 4

3 3P P

163! 3!



2 2P P

362! 2!



1 1P P 16



0 0P P 1

Total possible cases 1 16 36 16 1 70

∴ 8

70 35Pr obability

1282

30. Ans: (d)

Solution:

Let the C.P be ‘x’

∴ Marked price of good is 6

x5

6 6 80 6 60x x x x x x51 1 15 100 5 100Total profit 100 100 100

2 x 4 x 4 x

1 6 1 20 1 140100 100 100 10 1 7 2%

2 5 4 500 4 500

31. Ans: (a)

Solution:

A man takes 100 days to complete the work

Aptitude


Work done by a single man in 1 day 1

100

Let the woman require ‘x’ days to complete the same work

Work done by a single woman in a day 1

x

1 1

10 15 6 1100 x

1 15 1

10 x 6

15 1 1 10 6 4 1

x 6 10 60 60 15

x 225 days

32. Ans: (a)

Solution:

Speed in still water 4.5 km / hr

Speed of river 1.5 km / hr w

While going upstream river will flow against the man

Hence, Upstream speed 3 per hour

While going downstream river will flow with the man

Hence, Downstream speed 6 per hour

Time taken while going upstream d

hr3

Time taken while going downstream d

hr6

Total dis tance 2dAverage speed 4 km / hr

d dTotal time

3 6

33. Ans: (b)

Solution: 2x 1 a a .............

G.P. with r a

1x

1 a

x 1a

x

Similarly, y 1

by

Now, a, b are less than 1 ab 1

Aptitude


∴ 2 11 ab ab ........

1 ab

(G.P. with r=ab)

xy xy1 1

1 ab xy xy x y 1 x y 1x 1 y 11

x y

34. Ans: (c)

Solution:

January has 31 days

So, apart from 4 full weeks we have 3 days left.

To have a 5th Sunday we have 3 possible chances

Out of the 7 total chances

3Pr obability

7

35. Ans: (d)

Solution:

2 3 is a root of 2x px 6 0

2

2 3 p 2 3 6 0

6p 3

2 3

For 2x px q 0

2

pq

2

2

3 3q

2 4

36. Ans: (a)

Solution:

22 1

32 2

22 3

32 4

Y E B

W F D

U H G

S K I

Q O L

Aptitude


37. Ans: (a)

Solution:

For positive real numbers1 2 3 n

a ,a ,a ...............a ,

2

1 2 3 n

1 2 3 n

1 1 1 1a a a ............... a .............. n

a a a a

Hence the minimum value of 1 1 1 1 1

a b c d ea b c d e

is 25 25

38. Ans: (d)

Solution:

In first day he would complete 1

91of the work

Second day (20% more) 1 1.2

120%of91 91

Third day 1.4

91and so on

1 1.2 1.4 x........... 1

91 91 91 91

1 1 x

1 1.2 1.4 .......... x 1 x 1 1 x x 1 182 x 1391 91 2

Here, x is the nth term, but we need to find ‘n’

13=1+ (n-1) (0.2) [ (an=a+(n-1)d)]

n 61

39. Ans: (a)

Solution:

A game can be played without selecting couples 10 8

2 2C C 2 ways

Total number of games that can be played 10 10

2 2C C 2 ways

The required probability= 10 8

2 2

10 10

2 2

C C 2 280.6222

45C C 2

Aptitude


40. Ans: (b)

Solution

7 4 1 19! 2 3 5 7

co-prime numbers are called coprime if only positive integer that evenly divides both of

them is 1

So, Factors can be selected in 42 16 ways; hence, 8 pairs are possible.

Alternate Solution:

If p qN a b .........., where a, b, - - - - are prime numbers, then N can be written as a product

of two co primes in 2n-1 ways, where n is the number of prime factors to N. Here, n=4 (Since

the prime factors are 2, 3, 5, 7)

Hence the number of ways 9! Can be expressed as a product of two prime factors =24-1=8

41. Ans: (a)

Solution:

Average annual exports of P (during 1994-97) = 30 20 36 50

34crores4

Average annual exports of Q (during 1994-97) = 20 30 30 40

30crores4

Percentage of P’s exports with respect to that of Q=34

100 113.33%30

42. Ans: (d)

Solution:

Given x y z 15

The following three cases exist.

Case (1): x 0 y z 15 y,z 0

Number of solutions= 15 2 1 16

2 1 1C C 16

Case (2): x 15 y z 0

Number of solutions 12 2 x 15

Case (3): x y z 15 x,y,z 0

15 3 1 17

r 1 2x y z 15 C C

Number of solutions 17 1

2C 2 272

Hence, 272+16+2=290

Aptitude


43. Ans: (c)

Solution:

Let total weight of body be x

Weight of hormones and enzymes 1 1 1 18 2x

x x x x3 6 10 30 5

Weight of water present in hormones and enzymes=75 2 30

x x100 5 100

Angle of subtended arc= 0

30x

100 360 108x

44. Ans: (b)

Solution:

Profit%=Income Expenditure

100Expenditure

Let income of company Q in 2013 = Rs. x crore

Then income of Q in 2014=120

crore100

12030 x

100

x 25

Income of Q in 2013 = Rs.25 crore

Let expenditure of Q in 2013 be Rs.E crore

Then, 25 E

10 100E

2510 1 100

E

10E 25 22.77

11

Expenditure of Q in 2013 = Rs.22.77 crore

Aptitude


45. Ans: (a)

Solution:

4 16

2

16 16

2

16

16 2

2

16 16

2

16 16

16

4

2log log 0.5 2

x

2 1log log 2

x 2

2 1log 2

x 2

2log 2

x

log 2 log x 2

1 1log x 2 log 2

4 4

72log x

4

7log x

4

46. Ans: (c)

Solution:

Let the one digit numbers be x, y, z

Sum of all possible two digit number

10x y 10x z 10y x 10y z 10z x 10z y 22 x y z

Therefore sum of all possible two digit numbers when divided by sum of one digit numbers

gives 22.

47. Ans: (d)

Solution:

Aptitude


As 10% do not watch any of the channels, so 90% watch at least one of the channels

56 a b x 80 a c x 45 b c x a b c x 90

90 56 80 45 a b c 3x x

2x 51 a b c 40

x 25.5%

48. Ans: (c)

Solution:

If the maximum of each paper is 100x, total marks = 500x

The boy’s aggregate =3

500x 300x5 , which when divided in the given ratio gives marks:

36x, 48x, 60x, 72x and 84x

He scored more than 50 % in three subjects

49. Ans: (b)

Solution:

Let V be the volume

In one minute, A and B can fill V

15 and

V

20of the cistern respectively.

Also, C can carry off 10 litters in one minute.

We have, V V

V 120 10 015 20

V 8V 6V 1200 0

15V 1200

V 80litters

50. Ans: (b)

Solution:

Let the three angles (in A.P) be x, x a, x 2a

Given that x 2a x a x , i.e. 2a a x a x

Hence the angles are in the form x, 2x, 3x. Let the 4th angle be P

Three cases exist when P is equal to one of x, 2x and 3x respectively.

Case1: P=x

Aptitude


x 2x 3x x 360 7x 360

x 51.4

Angles are 51.4,51.4; 51.4 2; 51.4 3

Since no angle is 90; we reject this possibility

Case 2: P=3x

x 2x 3x 3x 360 9x

x 40

Angles are 40;80;120;120 (Rejected)

Case 3: P=2x

0

x 2x 3x 2x 8x 360

x 45

Angles are 0 0 0 045 ,90 ,135 ,90 .

Thus the smallest angle= 045 .

Documents

Kryteax Aptitude