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According to question,

⇒ When we divide 1056 by 23,

⇒ The remainder is 2

⇒ When we add 2 to 1056 it becomes 1058 and it becomes divisible by 23

∴ The correct answer is option 3.

Given:

A. 0.1 × 0.1 × 0.1

B. 0.03/3

C. 0.01/2

D. 0.1 × 0.02 × 0.2

Calculations:

Considering option A,

 0.1 × 0.1 × 0.1

⇒ 0.001

Considering option B,

0.03/3

⇒ 0.01

Considering option C,

0.01/2

⇒ 0.005

Considering option D,

0.1 × 0.02 × 0.2

⇒ 0.0004

On comparing all values,

0.0004<0.001<0.005<0.01

∴ The smallest value is 0.0004.

Calculation:

2/5 = 0.4

8/25 = 0.32

7/15 = 0.47

3/10 = 0.3

∴ The smallest fraction is 3/10

Calculation:

5/8 = 0.625, 5/6 = 0.8333, 4/3 = 1.33, 3/5 = 0.60

As we can se 3/5 is the smallest fraction.

A prime number is a whole number greater than 1 that can only be divided by itself and 1. The smallest prime number is  2. The number 2 is the only even prime number.

Given :

(X/Y) × 100 = 90

To find :

(Y/X) × 100

Calculation :

\(\frac{X}{Y} × 100 = 90\)

\( \Rightarrow X = \frac{9}{{10}}\;Y\)      -----(1)

Now,

\(\frac{Y}{X} × 100\)

Putting the value of X from (1)

\( = \;\frac{Y}{{\frac{{9Y}}{{10}}}} × \;100\)

= (10/9) × 100

= 1000/9

= 111.11

∴ Y is 111.11% of X

Given :

17% of P = 13% of Q

Calculation :

17% of P = 13% of Q

⇒ (17/100) × P = (13/100) × Q

⇒ 17P = 13Q

⇒ Q/P = 17/13

∴ Q : P is 17 : 13.

Calculation:

'?' % OF 40 = 1.20

(?/100) × 40 = 1.20

(?/100) = 1.20/40

? = 1.20/40 × 100 = 3

∴ The value of '?' is 3. 

√(11)⁴ × (6)² × (5)² × (100000000)/(11)² × (2)² × (5)² × (100000000)

√(11)^{4 – 2} × (6/2)²

⇒ √(11)² × (3)²

⇒ 11 × 3

⇒ 33

Given:

Our given expression is \(9 \frac{67}{121}\)

Calculation:

Converting mixed fraction to the fraction,

Our given expression is \(9 \frac{67}{121}\)

⇒ (9 × 121 + 67)/121

⇒ 1156/121

Using prime factorization in numerator and denominator,

1156 = 2 × 2 × 17 × 17

And, 121 = 11 × 11

The square root of the given number = (1156/121)^1/2

⇒ {(2 × 2 × 17 × 17)/(11 × 11)}^1/2

⇒ 34/11

∴ The square root of the given expression is 34/11

Calculations :

Let x be the value which we got after dividing √6561 by square root of √6561 

⇒ 6561 = 3⁸

⇒ √6561 = √3⁸

⇒ √6561 = 3⁴ = 81 

⇒ √81 = 9 

⇒ 81 ÷ 9 = x 

⇒ x = 9

∴ The result we got will be 9

Given:

Our number is 20 – 6√11

Formula used:

(a – b)2 = a2 + b2 – 2ab

Calculation:

Our number is 20 – 6√11

⇒ {(√11)2 + 32 – 2 × 3 × √11}1/2

⇒ {(√11 – 3)2}1/2  

⇒ ±(√11 – 3)

∴ The required value of given number is ±(√11 – 3)

Calculation:

2100/50 = ?

⇒ 42 = ?

∴ The value of ? is 42

Calculation:

√0.000841 = 0.029

∴ The required answer is 0.029

Given:

The volume ratio of A, B is 6 ∶ 5

The volume of A is increased by 30%

Calculation:

Let the volume of A, B be 60, 50

New volume of A = 60(1 + 30/100) = 78

Difference between the volumes of A, B is 78 – 50 = 28

∴ Percentage Increase in volume = 100 × 28/50 = 56%

Given:

Raju ate \(\frac{3}{8}\) part of a pizza and Adam ate \(\frac{3}{10}\)part of the remaining pizza. Then Renu ate \(\frac{4}{7}\) part of the pizza that was left.

Concept Used:

Concept of fractional substraction

Calculation:

Let, the total pizza is 1 part

Raju ate \(\frac{3}{8}\) part

Remaining (1 - 3/8) part

⇒ 5/8 part

Adam ate \(\frac{3}{10}\)part of the remaining pizza

⇒ Adam ate (5/8 × 3/10) = 15/80 = 3/16 part of the total pizza

Total pizza is eaten (3/8 + 3/16) 

⇒ 9/16 part

Remaining pizza (1 - 9/16)

⇒ 7/16 part

Then Renu ate \(\frac{4}{7}\) part of the pizza

Renu ate (7/16 × 4/7) = 1/4 part of the pizza

Together they ate (3/8 + 3/16 + 1/4) part

⇒ 13/16 part

The fraction of the pizza till left (1 - 13/16)

⇒ 3/16 part

∴ 3/16 fraction of the pizza is still left.

Given:

One-sixth of one-half of three-eighth of a number = 20

Calculation:

Let the number be 'x'

(1/6)(1/2)(3/8)x = 20

⇒ x = 20 × 8 × 2 × 6/3 

⇒ x = 640

40% of that number = 40% of 640

⇒ 640 × 40/100 = 256

∴ The 40% of that number is 256

Given:

Kachita's attendance in her school for academic session 2018- 2019 was 216 days.

Kachita's attendance % = 90%

Formula:

x as a percentage of y = (x/y) × 100 

Calculation:

Let total working days of the school were x days, then

x × (90/100) = 216

⇒ x = 216 × (100/90)

⇒ x = 240 days.

∴ Total working days of the school were 240 days.

Given: 

\(\sqrt {30 + \sqrt {30 + \sqrt {30 + \sqrt {30 + ...\,\,...\,\,...\,\,...\,\,...\,\,.\,\infty } } } } \) 

Calculation: 

Let 'a' be the value of \(\sqrt {30 + \sqrt {30 + \sqrt {30 + \sqrt {30 + ...\,\,...\,\,...\,\,...\,\,...\,\,.\,\infty } } } } \)

⇒ a = \(\sqrt {30 + \sqrt {30 + \sqrt {30 + \sqrt {30 + ...\,\,...\,\,...\,\,...\,\,...\,\,.\,\infty } } } } \)

Squaring both sides, we get 

⇒ a² = 30 + \(\sqrt {30 + \sqrt {30 + \sqrt {30 + \sqrt {30 + ...\,\,...\,\,...\,\,...\,\,...\,\,.\,\infty } } } } \)

⇒ a² = 30 + a 

⇒ a² - a - 30 = 0 

⇒ a² - 6a + 5a - 30 = 0 

⇒ (a - 6)(a + 5) = 0 

⇒ a = 6, -5 

a = - 5 is not possible

∴ The value of \(\sqrt {30 + \sqrt {30 + \sqrt {30 + \sqrt {30 + ...\,\,...\,\,...\,\,...\,\,...\,\,.\,\infty } } } } \) is 6.