# zad 1

# a)
(2 * 3^3)/(sin(pi/6)) + log(2401,7) # 112

# b)
abs(-9 + 5^-3) + (2^(1/3))^(1/5) # 10.03929

# c)
2^(1/3) * log((3^5 + 5^2 - 12),2) + (3 + (7^(4/3)))^(1/5) # 10.03929

# zad 2

a = c(1,-10,12,3,2,1,-8,4,5)
A = matrix(a, 3, 3)
b = c(1,3,-1,2,1,5)
B = matrix(b,3, 2)

# a)
det(A) # 572

# b)
solve(A)

#[,1]        [,2]       [,3]
#[1,]  0.01048951 -0.04020979 0.04895105
#[2,]  0.17132867  0.17657343 0.13286713
#[3,] -0.05944056  0.06118881 0.05594406


# c)
t(A)

#[,1] [,2] [,3]
#[1,]    1  -10   12
#[2,]    3    2    1
#[3,]   -8    4    5

# d)

A%*%B
#     [,1] [,2] [,3]
#[1,]    1  -10   12
#[2,]    3    2    1
#[3,]   -8    4    5


A%*%t(A)
#[,1] [,2] [,3]
#[1,]   74  -36  -25
#[2,]  -36  120  -98
#[3,]  -25  -98  170

A%*%A
#[,1] [,2] [,3]
#[1,] -125    1  -36
#[2,]   18  -22  108
#[3,]   62   43  -67

# e)
A[1,] * B[,2]
# [1]   2   3 -40

# zad 3 na kolokwium

c = c(1,2,5,-2,1,2,5,-3,-1,0,2,1,-2,3,5,2)
C = matrix(c, 4, 4)
D = c(-7,21,40,2)
solve(C,D)
# [1] 1 2 0 5
solve(C)[2,3]
# [1] -0.1578947

# zad 4 na kolokwium
w = (1:100)^2
summary.factor(w%%10)
# 0  1  4  5  6  9 
# 10 20 20 10 20 20 
summary.factor(w%%2 == 0)
# FALSE  TRUE 
# 50    50 
summary.factor(w%%3 == 0)
# FALSE  TRUE 
# 67    33 

summary.factor(w%%3 == 0 & w%%2 == 0)
# FALSE  TRUE 
# 84    16 

# zad 5 na kolokwium
w = seq(21,29903, by=7)^3
summary.factor(w %% 100 == 23)
#FALSE  TRUE 
# 4226    43 

# zad 6