Given:

3x+y = 81 and 81x-y = 3

Calculation:

According to the question

3x+y = 81

⇒ 3x+y = (3)⁴

So, x + y = 4 .....(1)

Again, 81x-y = 3

⇒ (3)^{4x - 4y} = (3)¹

⇒ 4x - 4y = 1 ....(2)

Now, multiplying equation (1) by 4, we get

⇒ 4x + 4y = 16 ....(3)

Adding equation (2) and (3), we get 

⇒ 8x = 17

Calculation:

(ax² + bx + c) is always positive, if there exists no real root of the equation & a is positive.

The condition for real root is (b²- 4ac) <0

If we apply the strict inequality we have (b2- 4ac) <0, we can conclude 'a' is positive.

∴ The sign of the quadratic polynomial ax2 + bx + c is always positive if a is positive and b2 - 4ac ≤ 0.

Given:

The value of k that (2x - 1) may be a factor of 4x4 - (k - 1)x3 + kx2 - 6x + 1

Calculation:

On putting 2x = 1 in the expression 4x4 - (k - 1)x3 + kx2 - 6x + 1

Given:

The roots of the equation x2 - 2ax + a2 + a - 3 = 0 are real and less than 3

Concept used:

We have a formula for solving quadratic equation ax² + bx + c = 0 is

Calculation:

x2 - 2ax + a2 + a - 3 = 0 ....(1)

⇒ (x - a)² = 3 - a

Since, (x - a)² ≥ 0

⇒ 3 - a ≥ 0

⇒ a ≤ 3

Now, according to the above concept

The roots of the equation (1) is

⇒ x = a ± √(3 - a)

Since, roots are less than 3

⇒ a + √(3 - a) < 3

⇒ √(3 - a) < 3 - a

⇒ 3 - a < 9 + a² - 6a

⇒ a² - 5a + 6 > 0

⇒ (a - 2)(a - 3) > 0

⇒ a < 2 or a > 3

∴ The required answer is a < 2.

Multiply the whole equation with 'abc'

⇒ bcx² + acx + ab = 0  

Let roots are α and 1/α 

Now,

Given:

A shopkeeper earns a profit of Rs. 10,000 by selling 20 units at the transport charge of Rs. 400.

He also earns a profit of Rs: 12,000 by selling 25 units at the transport charge of Rs. 600.

Calculation:

Let the linear expression of transportation charge (t) and the quantity of the commodity (q) is as follows:

Earned profit = x.t + y.q

Then, according to the question

⇒ 10,000 = 400t + 20q

⇒ 500 = 20t + q ....(1)

Again, 

⇒ 12,000 = 600t + 25q

⇒ 480 = 24t + q ....(2)

Now, solving equation (1) and (2), we get

⇒ 20 = -4t

⇒ t = -5

Now, putting the value of t in equation (1), we get

⇒ 500 = -100 + q

⇒ q = 600

So, the linear expression in t and q = 600q - 5t.

∴ The required linear expression is 600q - 5t.

Given :

If (x + y) : (y + z) : (z + x) = 9 : 6 : 11

and x + y + z = 26

Calculation:

 (x + y) = 9k , then y + z = 6k and z + x = 11k

So, (x + y) + (y + z) + (z + x) = 9k + 6k +11k

⇒ 2( x + y + z) = 26k

⇒ ( x + y + z) = 13k

So, ( x + y + z) -  (y + z) = 13k - 6k

⇒ x = 7k , similarly y = 2k and z = 4k

As per the question

13k = 26

⇒ k = 2

So x = 7 × 2 

⇒ 14 

similarly,  y = 4 and z = 8

So,

(x + y2 + z ) = 14 + 4² + 8

⇒ 14 + 16 + 8 = 38 

So value  of (x + y2 + z ) is 38

Given:

If x4 + x-4 = 47, x > 0,

Concept used:

(a + b)² = a² + b² + 2ab

Calculations:

According to the question,

Subtracting 2 from both the sides, we get,