True or False with reasons. Decide whether or not the following statements are true or false. Include a short reason for your answer in the space provided below. (i) If R and S are isomorphic commutative rings, then each ring homomorphism f : R → S is an isomorphism. TRUE or FALSE. (ii) If I and J are ideals in a commutative ring R, then I S J is also an ideal. TRUE or FALSE. (iii) If R is an integral domain, then every unit in R[x] has degree zero. TRUE or FALSE. (iv) If f : R → S is a homomorphism of commutative rings, then S has a subring that is isomorphic to R/ ker f. TRUE or FALSE. (v) If I is a proper ideal in a commutative ring R, then R has a subring that is isomorphic to R/I. TRUE or FALSE

p ÷ r = tan(x)

step-by-step explanation:

given that there is a figure below shows a right triangle: a right triangle is shown with hypotenuse equal to q units and the length of the legs equal to p units and r units.

graph is missing so i will create the graph as per given information.

the angle between the legs having lengths p units and q units is y degrees. the angle between the legs having lengths r units and q units is x degrees.

now we need to find about what is p ÷ r equal to.

so we can use trigonometric ratio tan(x)=[altitude]/[base] to find the value of p ÷ r, because p is the altitude and r is the base.

p ÷ r=tan(x)

hence final answer is p ÷ r=tan(x).

center (1,3) with radius 6

step-by-step explanation:

we must complete the square to find the center and radius of the circle.

first make sure the x and y squared terms have 1 as their coefficients. we also make sure x and y terms together.

we now create space between the x and y terms with parenthesis.

we complete the square by taking the middle terms -2x and the -6y - divide each and square them.

we add the squares to both sides.

simplify.

and write the quadratics in factored form.

the center is (h,k) or (1,3). the radius is the square root of 36 which is 6.

1 and 0

step-by-step explanation: