Let F₁ be the set of parallelograms, F₂ be the set of rectangles, F₃ be the set of rhombus and F₄ be the set of squares. Then, show that F₁ is equal to the union of all sets.

2

(a)

Two finite sets have m and n elements. The number of subsets of the first set is 112 more than that of the second set.Find the values of m and n.

2

(a)

Prove that \(\cos\theta\cos2\theta\cos2^{ 2 }\theta ...{ \cos2 }^{ n-1 }\theta =\frac { { \sin2 }^{ n }\theta }{ { 2 }^{ n }\sin\theta } for\ all\ n\in N.\)

2

(a)

For every positive integer n, prove that 7ⁿ-2ⁿ is divisible by 5.

2

(a)

Let \(T=[x:\frac{x+5}{x-7}-5=\frac{4x-40}{13x-x}]\) Is T an empty set? Justify your answer.

2

(a)