To find the value of k for which the given system of equations has a solution, we need to solve the system of equations and examine the conditions for consistency. Let's solve the system step by step.

Given system of equations:
1) x + y + z = 1
2) 2x + y + 4z = k
3) 4x + y + 10z = k^2

Step 1: Eliminate y from equations 2 and 3
To eliminate y, let's subtract equation 1 from equation 2 and equation 1 from equation 3:

2x + y + 4z - (x + y + z) = k - 1
4x + y + 10z - (x + y + z) = k^2 - 1

Simplifying these equations, we get:
x + 3z = k - 1 ...(4)
3x + 9z = k^2 - 1 ...(5)

Step 2: Eliminate x from equations 4 and 5
To eliminate x, let's multiply equation 4 by 3 and subtract it from equation 5:

3( x + 3z ) = 3(k - 1)
3x + 9z - (3x + 9z) = k^2 - 1 - 3(k - 1)

Simplifying this equation, we get:
0 = k^2 - 3k + 2

Step 3: Solve the quadratic equation
The equation obtained in step 2 is a quadratic equation. To find the value of k, we need to solve this equation:

k^2 - 3k + 2 = 0

Factoring the equation, we get:
(k - 2)(k - 1) = 0

So, the possible values of k are:
k - 2 = 0 => k = 2
k - 1 = 0 => k = 1

Case 1: k = 2
Let's substitute k = 2 in equations 4 and 5 to find the values of x and z:

x + 3z = 2 - 1
3x + 9z = 2^2 - 1

Simplifying these equations, we get:
x + 3z = 1 ...(6)
3x + 9z = 3 ...(7)

From equation 6, we can express x in terms of z:
x = 1 - 3z

Substituting this in equation 7, we get:
3(1 - 3z) + 9z = 3
3 - 9z + 9z = 3
3 = 3

As the equation is satisfied for any value of z, the system of equations is consistent and has infinitely many solutions for k = 2.

Case 2: k = 1
Let's substitute k = 1 in equations 4 and 5 to find the values of x and z:

x + 3z = 1 - 1
3x + 9z = 1^2 - 1

Simplifying these equations,

Ans