Find the area of the figure. A = 4 m 11 m 13 m 5 m

Find the exact values of sin(2u), cos(2u), and tan(2u) given 2 COS u where д < u < π. 2

(1) Let R be a field of real numbers and X=R³, X is a vector space over R, let M={(a,b,c)/ a,b,cE R,a+b=3-c}, show that whether M is a hyperplane of X or not (not by definition). متکاری Xn-XKE 11Xn- Xmit (2) Show that every converge sequence in a normed space is Cauchy sequence but the converse need not to be true. EK 2x7 (3) Write the definition of continuous map between two normed spaces and write with prove the equivalent statement to definition. (4) Let be a subset of a normed space X over a field F, show that A is bounded set iff for any sequence in A and any sequence in F converge to zero the sequence converge to zero in F. އ

Establish the identity. 1 + cos u 1 - cos u 1 - cos u 1 + cos u = 4 cot u csc u

sin(cos Find the exact value of the expression sin cos -1 3 -15 + sin 5 13

Find the exact values of sin(u + v), cos(u – v), and tan(u + v) given sin u =— - 25 12 and COS V=- 4/5

Solve the equation 1 + cos 0 2 = 2sin 0 for 0 ≤ 0 < 2π.

Find all solutions of cos(x + 1) = cos(x − 1) = Π - - = 1 in the interval [0, 2π). 6

Solve the equation 2 cos 2x + √√3 = 0 for 0 ≤ 0 < 2π.