S4 Flashcards by Eklavya Sarkar | Brainscape

Flashcards in S4 Deck (8)

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1

Let f, g : ℝ → ℝ be two functions.

If f is injective, then f is monotonic.

False

Take f(x) = x for x ≥ 0, f(x) = ¹/_x for x < 0.

2

Let f, g : ℝ → ℝ be two functions.

If f is bijective and increasing, then its inverse f⁻¹ is decreasing.

False:

Take f(x) = x.

3

Let f, g : ℝ → ℝ be two functions.

If f ◦ g is decreasing, then f and g are decreasing.

False:

Take f = x, g = −x.

4

The image of the circle |z| = 1 under the map f(z) = ¹/_z is a circle.

True:

The circle |z| = 1 is sent to itself by f.

5

z² +1 divides z⁶ +3z⁴ +z² −1.

True:

You can check that i, −i are both roots of the RHS.

6

Let z_{k, k=1,···,n} be the n roots of zⁿ +b_n−1zⁿ⁻¹ +···+b₀, then ∏z_i = (−1)ⁿb₀.

True:

You can check by writing the polynomial as show.

7

For all complex numbers z, |z| > ¹/_|z| .

False:

The inequality is false everytime |z| < 1

8

There are infinitely many complex numbers z such that |z| = 3.

True:

The locus described by |z| = 3 is a circle.

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