Follow BODMAS rule to solve this question, as per the order given below, 

Step-1: Parts of an equation enclosed in 'Brackets' must be solved first, and in the bracket, 

Step-2: Any mathematical 'Of' or 'Exponent' must be solved next, 

Step-3: Next, the parts of the equation that contain 'Division' and 'Multiplication' are calculated, 

Step-4: Last but not least, the parts of the equation that contain 'Addition' and 'Subtraction' should be calculated

950 + 50 × 15 - 14 × 22 + √? = 113 + 92

⇒ 950 + 750 - 308 + √? = 1331 + 81

⇒ 1392 + √? = 1412

⇒ √? = 20

⇒ ? = 400

Follow the BODMAS rule to solve the question,

Step–1– Parts of an equation enclosed in 'Brackets' must be solved first, and in the bracket, the BODMAS rule must be followed,

74% of 159 – [36.5% of 142 + 25.4% of 203] = 13.5% of ? – 10.5% of 120

⇒ 74% of 159 – {[(36.5/100) × 142] + [(25.4/100) × 203]} = [(13.5/100) × ?] – [(10.5/100) × 120]

⇒ 74% of 159 – [51.83 + 51.56] = [0.135 × ?] – 12.6

⇒ 74% of 159 – 103.39 = 0.135 × ? – 12.6

Any mathematical 'Of' or 'Exponent' must be solved,

⇒ [(74/100) × 159] – 103.39 = 0.135 × ? – 12.6

⇒ 117.66 – 103.39 = 0.135 × ? – 12.6

⇒ 14.27 = 0.135 × ? – 12.6

⇒ 14.27 + 12.6 = 0.135 × ?

⇒ 26.87 = 0.135 × ?

17⁴ + √2400.5 + 50.67 + 17% of 400 + √528.9 = (?) + 44

Approximating the value to the nearest integer:

⇒ 17⁴ + √2401 + 51 + 17% of 400 + √529 = (?) + 44

⇒ 83521 + 49 + 51 + 68 + 23 = (?) + 44

⇒ 83521 + 191 – 44 = (?)

Follow BODMAS rule to solve this question, as per the order given below,

Step-1- Parts of an equation enclosed in 'Brackets' must be solved first, and in the bracket,

Step-2- Any mathematical 'Of' or 'Exponent' must be solved next,

Step-3- Next, the parts of the equation that contain 'Division' and 'Multiplication' are calculated,

Step-4- Last but not least, the parts of the equation that contain 'Addition' and 'Subtraction' should be calculated.

Given expression is,

\(\Rightarrow 9\frac{{10}}{2} \times \left( {\frac{3}{8} \times \frac{{16}}{9}} \right) - 6\frac{5}{3} = 5\frac{5}{2} + 4\frac{1}{2} - ?\)

⇒ (28/2) × [(3/8) × (16/9)] – 23/3 = 15/2 + 9/2 - ?

⇒ 14 × (2/3) – 23/3 = 15/2 + 9/2 - ?

⇒ 28/3 – 23/3 = 15/2 + 9/2 - ?

⇒ 28/3 – 23/3 = 24/2 - ?

⇒ 5/3 = 12 - ?

⇒ ? = 12 – 5/3

Using laws of Indices:

(a)^m × (a)ⁿ = a^{(m + n)}

(a)^m ÷ (a)ⁿ = a^{(m - n)}

(a^m)ⁿ = (a)^mn

(a)^(-m) = 1/a^m

(a)⁰ = 1

(a)^1/m = ^m√a

Given expression is:

⇒ 3⁸ × 3⁴ ÷ 243 = 9 × 3 × 81 × 3⁴ ÷ ?

⇒ 3⁸ × 3⁴ ÷ 3⁵ = 3² × 3 × 3⁴ × 3⁴ ÷ ?

⇒ 3^{(8 + 4 – 5)} = 3^{(2 + 1 + 4 + 4)} ÷ ?

⇒ 3⁷ = 3¹¹ ÷ ?

⇒ ? = 3¹¹ ÷ 3⁷

⇒ ? = 3^{(11 – 7)}

⇒ ? = 3⁴

\(\begin{array}{l}{\rm{I}}.\frac{{131{a^2}}}{7} + 29a + \frac{{240}}{7} = \frac{{89{a^2}}}{7} - \frac{5}{7}\\ \Rightarrow \frac{{131{a^2}}}{7} - \frac{{89{a^2}}}{7} + 29a + \frac{{240}}{7} + \frac{5}{7} = 0\end{array}\) 

⇒ 6a² + 29a + 35 = 0

⇒ 6a² + 15a + 14a + 35 = 0

⇒ 3a(2a + 5) + 7(2a + 5) = 0

⇒ (3a + 7)(2a + 5) = 0

Then, a = -7/3 = -2.33 or a = -5/2 = -2.5

\({\rm{II}}.2{b^2} + {\rm{\;}}9\left( {b + 1} \right) = \frac{{5b + 5}}{6}\) 

⇒ 12b² + 54b + 54 = 5b + 5

⇒ 12b² + 49b + 49 = 0

⇒ 12b² + 28b + 21b + 49 = 0

⇒ 4b(3b + 7) + 7(3b + 7) = 0

⇒ (4b + 7)(3b + 7) = 0

Then, b = -7/3 = -2.33 or b = -7/4 = -1.75

So, when a = -2.33, a = b for b = -2.33 and a < b for b = -1.75

And when a = -2.5, a < b for b = -2.33 and a < b for b = -1.75

I: x² - 2√3x + 1 = 0

Adding two to both the sides

⇒ x² - 2√3x + 3 = 2

⇒ (x - √3)² = 2 [∵ a² + 2ab + b² = (a + b)²]

⇒ x - √3 = ± √2

⇒ x = √3 + √2 or √3 - √2

II: y² + √3 (√6 – 1)y + 4 - √6 = 0

⇒ y² + (3√2 – √3)y + √2 (2√2 - √3) = 0

⇒ y² + √2y + (2√2 – √3)y + √2(2√2 - √3) = 0

⇒ y (y + √2) + (2√2 - √3) (y + √2) = 0

⇒ (y + √2) (y + 2√2 - √3) = 0

⇒ y = - √2 or √3 - 2√2

When x = √3 + √2, y = - √2 then x > y

When x = √3 + √2, y = √3 – 2√2 then x > y

When x = √3 - √2, y = - √2 then x > y

When x = √3 – √2, y = √3 – 2√2 then x > y

I. \(\frac{4}{{\sqrt {\rm{x}} }}\; + \;\frac{{23}}{{\sqrt {\rm{x}} }}\; = \;\sqrt {\rm{x}} \)

⇒ \(\frac{{27}}{{\sqrt {\rm{x}} }}\; = \;\sqrt {\rm{x}} \)

⇒ 27 = x

Then, x = 27

II. \({y^6} - \frac{{{{\left( { - 3} \right)}^{\frac{{39}}{2}}}}}{{\sqrt {\rm{y}} }}\; = \;0\)

We know, \(\sqrt {\rm{y}} {\rm{\;}} = {\rm{\;}}{\left( y \right)^{\frac{1}{2}}}\)

⇒ \({y^6}{\rm{\;}} = {\rm{\;}}\frac{{{{\left( { - 3} \right)}^{\frac{{39}}{2}}}}}{{\sqrt {\rm{y}} }}\)

⇒ \({y^6} \times {\left( y \right)^{\frac{1}{2}}}{\rm{\;}} = {\rm{\;}}{\left( { - 3} \right)^{\frac{{39}}{2}}}\)

As per the rule of exponents, \({a^{\rm{x}}} \times {a^{\rm{y}}}{\rm{\;}} = {\rm{\;}}{a^{{\rm{x\;}} + {\rm{\;y}}}}\)

⇒ \({\left( y \right)^{6{\rm{\;}} + {\rm{\;}}\frac{1}{2}}}{\rm{\;}} = {\rm{\;}}{\left( { - 27} \right)^{\frac{{13}}{2}}}\)

⇒ \({\left( y \right)^{\frac{{13}}{2}}}{\rm{\;}} = {\rm{\;}}{\left( { - 27} \right)^{\frac{{13}}{2}}}\)

Then, y = -27

So, when x = 27, x > y for y = -27

I. x² + 2x - 5.9696 = 0

⇒ x² - 1.64x + 3.64x - 5.9696 = 0

⇒ x (x – 1.64) + 3.64 (x – 1.64) = 0

⇒ (x – 1.64) (x + 3.64) = 0

Then, x = 1.64 or x = -3.64

II. 2y² + 4y + 1.1808 = 0

⇒ y² + 2y + 0.5904 = 0

⇒ y² + 0.36y + 1.64y + 0.5904 = 0

⇒ y(y + 0.36) + 1.64 (y + 0.36) = 0

⇒ (y + 0.36)(y + 1.64) = 0

Then, y = -0.36 or y = -1.64

So, when x = 1.64, x > y for y = -0.36 and x > y for y = -1.64

And when x = -3.64, x < y for y = -0.36 and x < y for y = -1.64

I. \(\frac{{{3^3} + \;{6^2}}}{7} = {x^2}\)

\(\Rightarrow \frac{{27 + 36}}{7} = {x^2}\\ \Rightarrow 63 = 7{x^2}\\ \Rightarrow {x^2} = 9\)

⇒ x = ±3

II. \(17{y^3} = \left( {15 \times 9} \right) + 12{y^3}\)

\(\Rightarrow {\rm{\;}}17{y^3} - 12{y^3} = \left( {15 \times 9} \right)\\ \Rightarrow 5{y^3} = \left( {15 \times 9} \right)\\ \Rightarrow {y^3} = \frac{{\left( {15 \times 9} \right)}}{5}\\ \Rightarrow {y^3} = {\left( 3 \right)^3}\)

Then, y = 3

So, when x = +3, x = y for y = +3

And when x = -3, x < y for y = +3

The pattern is as follows:

The logic used here is the multiplication of the number with 3 and adding and subtracting consecutive odd integers alternatively from it as shown below

⇒ 6 = (1 × 3) + 3

⇒ 13 = (6 × 3) – 5

⇒ 46 = (13 × 3) + 7

⇒ 130 ≠ (46 × 3) – 9 (not following the pattern of the series)

130 should be 129 to make a pattern in the series.

⇒ 398 = (129 × 3) + 11

⇒ 1181 = (398 × 3) – 13

The series follows the following pattern:

⇒ (1 × 2 × 3) – (1 + 2 + 3) = 0

⇒ (2 × 3 × 4) – (2 + 3 + 4) = 15

⇒ (3 × 4 × 5) – (3 + 4 + 5) = 48

⇒ (4 × 5 × 6) – (4 + 5 + 6) = 105

⇒ (5 × 6 × 7) – (5 + 6 + 7) = 192

⇒ (6 × 7 × 8) – (6 + 7 + 8) = 315 (Not 305)

66,

66 × 1 - 3 = 63

63 × 3 - 9 = 180

180 × 9 - 27 = 1593

1593 × 27 - 81 = 42930

The pattern of the given series is as follows:

11² × 2 = 242

12² × 2 = 288

13² × 2 = 338

14² × 2 = 392

15² × 2 = 450

The pattern of the given series is as follows:

5 × 1 × + 0.25 × 1 = 5.25

5.25 × 2 + 0.25 × 4 = 11.5

11.5 × 3 + 0.25 × 9 = 36.75

Now, the series on the same pattern starting from 3 would be,

⇒ 3

⇒ A = 3 × 1 + 0.25 × 1 = 3.25

⇒ B = 3.25 × 2 + 0.25 × 4 = 7.5

⇒ C = 7.5 × 3 + 0.25 × 9 = 24.75

⇒ D = 24.75 × 4 + 0.25 × 16 = 103

⇒ E = 103 × 5 + 0.25 × 25 = 521.25

∴ the number in place of C is 24.75