1st question 
import numpy as np

A = np.matrix("8 -8 -2; 4 -3 -2; 3 -4 1")

print("A :", A)

ev1, ev2 = np.linalg.eig(A)

print("Eigen Values :", ev1)
print("Eigen Vector :", ev2)

3rd question 

import math

eqn = input("Enter the function in terms of x (e.g., x**3 - x - 2): ")
a = float(input("Enter interval start (a): "))
b = float(input("Enter interval end (b): "))
tol = float(input("Enter tolerance (e.g., 0.0001): "))

fa = eval(eqn, {"x": a, "math": math})
fb = eval(eqn, {"x": b, "math": math})

if fa * fb >= 0:
    print("Root does not lie in this interval.")
else:
    while (b - a) / 2 > tol:
        mid = (a + b) / 2
        f_mid = eval(eqn, {"x": mid, "math": math})

        if f_mid == 0:
            break
        elif eval(eqn, {"x": a, "math": math}) * f_mid < 0:
            b = mid
        else:
            a = mid

    print(f"\nApproximate Root: {((a + b) / 2):.5f}")

5th question 



x, y, z, w = 0.0, 0.0, 0.0, 0.0

n = 15

print(f"{'k':<4} {'x':<12} {'y':<12} {'z':<12} {'w':<12}")

for k in range(1, n+1):

    x_new = (6 + y - 2*z) / 10
    y_new = (25 + x + z - 3*w) / 11
    z_new = (-11 - 2*x + y + w) / 10
    w_new = (15 - 3*y + z) / 8

    x, y, z, w = x_new, y_new, z_new, w_new

    print(f"{k:<4} {x_new:<12.4f}{y_new:<12.4f}{z_new:<12.4f}{w_new:<12.4f}")

7th question 

print("Choose M(x,y):")
print("1. M = x*y")
print("2. M = x + y")

m_choice = int(input("Enter your choice for M: "))

print("\nChoose N(x,y):")
print("1. N = x**2/2")
print("2. N = x - y")

n_choice = int(input("Enter your choice for N: "))


def dM_dy(x):
    if m_choice == 1:
        return x
    elif m_choice == 2:
        return 1


def dN_dx(x):
    if n_choice == 1:
        return x
    elif n_choice == 2:
        return 1


x = 2

if dM_dy(x) == dN_dx(x):
    print("\nThe differential equation is Exact")
else:
    print("\nThe differential equation is Not Exact")

9th question 


from sympy import symbols, Function, dsolve, Eq

x = symbols('x')
y = Function('y')

eq = Eq(y(x).diff(x, 2) - 8*y(x).diff(x) + 15*y(x), 0)

solution = dsolve(eq)

print("y =", solution.rhs)