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Qormaata galuumsa kutaa 12ffaa Herregaa, bara 2011 A.L.I (2019)

Gaaffilee dhugaa qormaata galuumsa yunivarsiitii kutaa 12 kan Herregaa, akkuma barattoonni bara 2011 A.L.I itti fudhatanitti. Tokkoon tokkoon gaaffii deebii sirrii fi ibsa qaba.

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  • Damee lamaanuu

    Eenyutu fudhata

  • 2011 A.L.I (2019)

    Bara qormaataa

  • Gaaffilee 64 deebii waliin

    Kuusaa kana keessa

Gaaffilee qormaata kanaa yaalaa

Deebii banuun dura tokkoon tokkoon gaaffii ofii keessaniin deebisaa. Ibsi gara kutaa kitaaba barnootaa sirriitti waan qajeelchuuf, yeroo gaaffii dogoggortan fuula kam irra deebi'uu akka qabdan ni beektu.

Gaaffii 1

Which one of the following is the set of critical numbers of f(x)=38x8/3−6x2/3f(x) = \frac{3}{8}x^{8/3} - 6x^{2/3}?

  1. A.{−2,2}\{-2, 2\}
  2. B.{−1,1}\{-1, 1\}
  3. C.{−1,0,1}\{-1, 0, 1\}
  4. D.{−2,0,2}\{-2, 0, 2\}
Deebii fi ibsa ilaalaa

Deebii: D {−2,0,2}\{-2, 0, 2\}

Maaliif

The critical numbers of a function are where its derivative is zero or undefined. By differentiating, we get f′(x)=x2−4x1/3f'(x) = \frac{x^2 - 4}{x^{1/3}}, which is zero at x=±2x = \pm 2 and undefined at x=0x = 0, all of which are in the domain of ff.

Gaaffii 2

What is the greatest lower bound of the sequence {(−1)n1n+1}n=0∞\left\{(-1)^n \frac{1}{n+1}\right\}_{n=0}^{\infty}?

  1. A.-1
  2. B.0
  3. C.−1/2-1/2
  4. D.1/21/2
Deebii fi ibsa ilaalaa

Deebii: C −1/2-1/2

Maaliif

Listing the first few terms of the sequence yields {1,−1/2,1/3,−1/4,1/5,−1/6,…}\{1, -1/2, 1/3, -1/4, 1/5, -1/6, \ldots\}. The negative terms are increasing towards 0, with the absolute minimum and greatest lower bound being the first negative term, which is −1/2-1/2.

Gaaffii 3

Which one of the following is true about the pair of lines:3x+9y−24=03x + 9y - 24 = 0 and 4x+12y+32=04x + 12y + 32 = 0?

  1. A.perpendicular lines
  2. B.intersecting lines
  3. C.parallel and distinct lines
  4. D.representing the same lines
Deebii fi ibsa ilaalaa

Deebii: C parallel and distinct lines

Maaliif

Both equations can be rewritten in slope-intercept form to find their slopes: y=−13x+83y = -\frac{1}{3}x + \frac{8}{3} and y=−13x−83y = -\frac{1}{3}x - \frac{8}{3}. Since they have the same slope but different yy-intercepts, they represent parallel and distinct lines.

Gaaffii 4

Which one of the following is equal to f(x)=(x+4)2f(x) = \sqrt{(x+4)^2}, for every x∈Rx \in \mathbb{R}?

  1. A.g(x)=x+4g(x) = x + 4
  2. B.g(x)=x+2g(x) = x + 2
  3. C.g(x)=∣x+4∣g(x) = |x+4|
  4. D.g(x)=∣x∣+4g(x) = |x|+4
Deebii fi ibsa ilaalaa

Deebii: C g(x)=∣x+4∣g(x) = |x+4|

Maaliif

By definition, for any real number uu, u2=∣u∣\sqrt{u^2} = |u|. Substituting u=x+4u = x+4 gives g(x)=∣x+4∣g(x) = |x+4|.

Gaaffii 5

The center of a circle on the line y=2xy = 2x and the line x=1x = 1 is tangent to the circle at (1,6)(1,6). How long is the radius of the circle?

  1. A.5
  2. B.4
  3. C.3
  4. D.2
Deebii fi ibsa ilaalaa

Deebii: D 2

Maaliif

The tangent line x=1x = 1 is vertical, meaning the radius connecting the center (h,2h)(h, 2h) to the tangent point (1,6)(1, 6) must be horizontal, so 2h=6  ⟹  h=32h = 6 \implies h = 3. The radius is the horizontal distance between (3,6)(3,6) and (1,6)(1,6), which is ∣3−1∣=2|3-1| = 2.

Gaaffii 6

What is the value of lim⁡n→∞(3n+2n6n)\lim_{n \to \infty} \left(\frac{3^n + 2^n}{6^n}\right)?

  1. A.1
  2. B.0
  3. C.∞\infty
  4. D.56\frac{5}{6}
Deebii fi ibsa ilaalaa

Deebii: B 0

Maaliif

We can split the expression into lim⁡n→∞[(1/2)n+(1/3)n]\lim_{n \to \infty} \left[ (1/2)^n + (1/3)^n \right]. Since both bases are between −1-1 and 11, their limits as n→∞n \to \infty are both 0, making the overall limit 0.

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Yaadannoon gabaabaan Herregaa kutaa 12 kaardii 33 qaba. Tokkoon tokkoon boqonnaa boqonnaadhaan kitaaba biyyaalessaa irraa qophaa'ee seericha, mallattoowwan isaa maal bakka akka bu'an fi dogoggora qabxii nama dhabsiisu ni ibsa.

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Kunneen gaaffilee dhugaa qormaata galuumsa Herregaa bara 2011 A.L.I ti?

Eeyyee. Qormaata galuumsa yunivarsiitii biyyaalessaa kan Herregaa barattoonni bara 2011 A.L.I (2019) fudhatan irraa fudhataman. Kuusaan keenya gaaffilee isaa 64 qaba; tokkoon tokkoon isaanii deebii fi ibsa gara kitaaba barnootaatti qajeelchu qabu.

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