To complete a work for 6 days, 8 men require

For complete a work for 1 day = 6*8 = 48 men

For complete a work in half a day(1/2) = 48*2 = 96 men.

Given rope of length = 40m (radius)

Area of semi circle =(pi*r^2)/2 = (1600 * pi)/2 = 800*pi ---(1)

Area of quarter circle = (1/2 * 800 * pi) = 400*pi ----- (2)

Area of sector is = r*r*(pi/2)/2 = 25*pi ----- (3)

By adding above eq. (1) (2) & (3)

Total area = 1225*pi

As per quesiton there are 2 goats

so area = 1225*pi*2 = 2450pi.

2ab = (a^2 + b^2) - (a - b)^2

= 29 - 9 = 20

ab = 10.

20*10 +15*8 +x*4/20+15+x = 8.5

x= 5 child.

Number of runs made by running = 110 - (3 * 4 + 8 * 6)

= 110 - (60) = 50.

Required percentage = ({50/110}*100)% = 45{5/11}%.

m = (2 - ?3)

=> 1/m = (2 + ?3)

(m^6+m^5+m^4+1)/m^3

= m^3 + m^2 + m + 1/m^3

= (m + 1/m)^3 - 3(m +1/m) + m(m+1)

= 4^3 - 3*4 + (2-?3)(3-?3)

= 64 - 12 + 9 - 5?3

= (61 - 5?3).

Let Swati's age = 2X and Varun's age = 5X

(2X + 8)/(5X + 8) = 1/2 => X = 8

Swati's age = 16 years

and Varun's age = 40 years

Difference of their ages = 24 years.

Given Last value int seq. = 54.

and in every Loop result multiplied By 3.

So By Backward Approach

54/3=18 => 18/3=6

=> 6/3=2.

So, The Starting Value Is 2.

A+B=132

B+C=130

C+D=102

B+D=116

A+2B+C=262

B+C+2D=218

A+B-2D=44

132-2D=44

D=44.

m^2+m^-2=23

=> (m + m^-1)^2 - 2 = 23

=> (m + m^-1) = 5

m^3+m^-3 = (m + m^-1)^3 - 3(m + m^-1) = 5^3 - 3*5 = 110.

X^{2}-X-6=0

(x-3)(X+2)=0

X=3,-2.

length covered by 80% of the pencils =80/100 * 14000000 * 5 = 56000000 inches

Minimum length covered by remaining 20% of the pencils =20/100 * 14000000 * 3 = 8400000 inches

Maximum length covered by remaining 20% of the pencils =20/100 * 14000000 * 6 = 16800000 inches

Hence, total length spanned by the pencils is (56000000 +8400000) to (56000000 +16800000 )

ie, 64400000 inches to 72800000 inches.

persons who can speak English = 1 - (1/5) = 4/5

persons who can speak Hindi = 1 - (3/7) = 4/7

persons who can speak any or both = 1 - 1/10 = 9/10

so total persons speak both = 4/5 + 4/7 - 9/10 = 33/70

persons speak any = 1 - 33/70 - 1/10 = 3/7

% = 300/7% = 42.86%.

The order of each letter in the dictionary is ABLORU.

Now, with A in the beginning, the remaining letters can be permuted in 5! ways.

Similarly, with B in the beginning, the remaining letters can be permuted in 5! ways.

With L in the beginning, the first word will be LABORU, the second will be LABOUR.

Hence, the rank of the word LABOUR is 5!+5!+2 =242.

Let's call the odd number 2a+1, where a is an integer.

Then (2a+1)^2 = 4a^2 + 4a + 1 = 4a(a+1) + 1.

Since a is odd, a+1 is even.

So, 4a(a+1) has to be a multiple of 8

Therefore 4a(a+1) + 1 must leave a remainder of 1 when divided by 8.

Lets assume son's age = x

So, father's age = x + 30

**After 5 years:**

Son's age = x + 5

Father's age = x + 35

In 5 years the father is thrice as old as his son so:

3(x + 5) = x + 35

x = 10

So, current father's age = 40 Years.

Since each one of the three numbers a, b, and c is divisible by 3, the numbers can be represented as 3p,3q, and 3r, respectively, where p, q, and r are integers.

The product of the three numbers is 3p*3q*3r =27(pqr).

Since p, q, and r are integers, pqr is an integer and therefore abc is divisible by 27.

If any one choose container A then prob = (1/2)*(4/10)=1/5

If any one choose container A then prob = (1/2)*(40/100)=1/5.

The volume of the cylinder is Pi x r^2 x h = Pi x 3^2 x 6 = 54Pi

Volume of a cylinder is 4/3 x Pi x r^3

Now 54Pi = 12 x 4/3 x Pi x r^3

54 = 16 x r^3

r^3 = 54/16 = 27/8

r = 3/2

Therefore diameter is 3 cms.

1*360, 2*180, 3*120, 4*90, 5*72, 6*60, 8*45, 9*40, 10*36, 12*30, 15*24, 18*20

SOL 2) 360 => 2^3 * 3^2 * 5^1 [i.e. a^x * b^y * c^z)

No. of ways can be written as a prod of 2 no's is

(x+1)*(y+1)(z+1)/2

=> (3+1)*(2+1)*(1+1)/2

=> 24/2 = 12.

Let their marks be (x + 9) and x.

Then, x + 9 = (56/100)(x + 9 + x)

25(x + 9) = 14(2x + 9)

3x = 99

x = 33

So, their marks are 42 and 33.

Let 10'splace digit is x and unit's place digit y

**First milestone :** 10x+y

**Second milestone :** 10y+x

**Third milestone:** 100x+y

Since the speed is uniform so

Distance covered in first Hr = Distance covered in Second Hr

(10y+x)-(10x+y) = (100x+y)-(10y+x)

By solving,

y=6x

but since x and y are digits so only possible combination is x=1 and y=6,

So average speed = 45 Kmph.

Two tallest boys can be arranged in 2! ways, rest 18 can be arranged in 18! ways.

Girls can be arranged in 6! ways.

Total number of ways in which all the students can be arranged = 2! * 18! * 6! = 18! *1440.

Total no. of families who take newspaper=80

Hindu newspaper = 60% = (80*60/100) = 48

Regional newspaper = 80% = (80*80/100) = 64

TOI newspaper = 70% = (70*80/100) = 56

Find common families take H.C.F (48,64,56) = 8

Therefore,(8/80)*100=10%

So, common families who take all newspaper = 10%.

log 0.317=0.3332 and log 0.318=0.3364

then

log 0.319=log0.318+(log0.318-log0.317)

=0.3396.

Given A works trice as much as B

So the ratio between A and B = 3:1

so 3x-x=60

and x = 30

so 1/90+1/30=221/2 days.