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Given:

 If 4 students sit on each bench, then 3 benches are left unoccupied

3 students sit on each bench, 3 students are left standing

Calculation:

Let the total number of students be x.

According to the question, (x/4) = Number of benches needed for all the students

When 4 students sit on one bench, number of benches left = 3

Then, according to the first condition, the total number of benches = (x/4) + 3      ----(1)

 According to the question, (x/3) = number of benches needed for all - 1

When 3 students sit on one bench, number of benches needed extra = 1

Then, according to the second condition, the total number of benches available = (x/3) - 1      ----(2)

Equating (1) and (2),

(x/3) - 1 = (x/4) + 3

⇒ (x/12) = 4 

⇒ x = 48

∴ The total number of students present in the class is 48.

Given:

a = 3 + 2√2

Concept Used:

a² – b² = (a – b)(a + b)

a³ + b³ = (a + b)³ – 3ab(a + b)

Calculation:

a = 3 + 2√2

1/a = 1/(3 + 2√2)

⇒ 1/a = (3 – 2√2)/{(3 + 2√2) × (3 – 2√2)}

⇒ 1/a = (3 – 2√2)/{3² – (2√2)²}

⇒ 1/a = (3 – 2√2)/(9 – 8)

⇒ 1/a = (3 – 2√2)

Now,

a + 1/a = 3 + 2√2 + 3 – 2√2

⇒ a + 1/a = 6

(a⁶ – a⁴ – a² + 1)/a³

⇒ a³ – a – 1/a + 1/a³

⇒ (a³ + 1/a³) – (a + 1/a)

⇒ {(a + 1/a)³ – 3(a + 1/a)} – (a + 1/a)

⇒ (6³ – 3 × 6) – 6

⇒ 216 – 18 – 6

⇒ 192

∴ The required value of (a⁶ – a⁴ – a² + 1)/a³ is 192

Given

Smaller number = 31.855

Difference of numbers = 47.364

Calculation

Let the numbers be 'x' and 'y'

Then, y = 31.855

And x - y = 47.364

⇒ x - 31.855 = 47.364

⇒ x = 47.364 + 31.855 = 79.219

∴ The bigger number is 79.219

Given:

 x + y + z = 24 and xy + yz + zx = 36

Formula used:

(x + y + z)² = x² + y² + z² + 2xy + 2yz + 2zx

Calculations:

(x + y + z)2 = x2 + y2 + z2 + 2xy + 2yz + 2zx

⇒ (24)² = x2 + y2 + z2 + 2 × 36

⇒ 576 = x2 + y2 + z2 + 72

⇒ x2 + y2 + z² = 576 – 72

⇒ x² + y² + z² = 504

Now, (x – y)2 + (y – z)2 + (z – x)² + 3

⇒ x² + y² – 2xy + y² + z² – 2yz + z² + x² – 2zx + 3 

⇒ 2x2 + 2y2 + 2z2 – 2xy – 2yz – 2zx + 3

⇒ 2(x2 + y2 + z2 – xy – yz – zx) + 3

⇒ 2(504 – 36) + 3

⇒ 2 × 468 + 3

⇒ 936 + 3 = 939

∴ The value of (x – y)2 + (y – z)2 + (z – x)2 + 3 is 939.