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Qormaata galuumsa kutaa 12ffaa Herregaa, bara 2017 A.L.I (2025)

Gaaffilee dhugaa qormaata galuumsa yunivarsiitii kutaa 12 kan Herregaa, akkuma barattoonni bara 2017 A.L.I itti fudhatanitti. Tokkoon tokkoon gaaffii deebii sirrii fi ibsa qaba.

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  • Damee saayinsii hawaasaa

    Eenyutu fudhata

  • 2017 A.L.I (2025)

    Bara qormaataa

  • Gaaffilee 59 deebii waliin

    Kuusaa kana keessa

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Gaaffii 2

Let f(x)f(x) be a polynomial of degree at least 1, g(x)g(x) be a non-zero polynomial and c∈Rc \in \mathbb{R}. Which of the following states the Factor Theorem?

  1. A.f(c)=0f(c) = 0 if and only if x−cx - c is a factor of f(x)f(x).
  2. B.For distinct real numbers aa and bb, if f(a)f(b)<0f(a)f(b) < 0, then f(x)f(x) has at least one zero between aa and bb.
  3. C.There exist polynomials q(x)q(x) and r(x)r(x) such that f(x)=g(x)q(x)+r(x)f(x) = g(x)q(x) + r(x).
  4. D.If f(x)f(x) is divided by x−cx - c, then the remainder is f(c)f(c).
Deebii fi ibsa ilaalaa

Deebii: A f(c)=0f(c) = 0 if and only if x−cx - c is a factor of f(x)f(x).

Maaliif

The Factor Theorem states that for any polynomial f(x)f(x) of degree at least 1, x−cx - c is a factor of f(x)f(x) if and only if f(c)=0f(c) = 0.

Gaaffii 3

Let A={1,2}A = \{1,2\} and B={1,3}B = \{1,3\} be sets. Then which one of the following Cartesian products is correct?

  1. A.B×A={(1,1),(3,1),(1,2),(3,2),(2,3)}B\times A = \{(1,1),(3,1),(1,2),(3,2),(2,3)\}.
  2. B.B×B={(1,1),(1,3),(3,3),(2,3)}B\times B = \{(1,1),(1,3),(3,3),(2,3)\}.
  3. C.A×B={(1,1),(1,3),(2,1),(2,3),(3,2)}A\times B = \{(1,1),(1,3),(2,1),(2,3),(3,2)\}.
  4. D.A×A={(1,1),(2,2),(2,1),(1,2)}A\times A = \{(1,1),(2,2),(2,1),(1,2)\}.
Deebii fi ibsa ilaalaa

Deebii: D A×A={(1,1),(2,2),(2,1),(1,2)}A\times A = \{(1,1),(2,2),(2,1),(1,2)\}.

Maaliif

For A={1,2}A = \{1,2\}, the Cartesian product A×AA \times A consists of all ordered pairs (x,y)(x, y) where both xx and yy are in AA, yielding {(1,1),(1,2),(2,1),(2,2)}\{(1,1), (1,2), (2,1), (2,2)\}. The set in option D contains exactly these elements.

Gaaffii 4

Consider the matrix (2−310−1π314)\begin{pmatrix} 2 & -3 & 1 \\ 0 & -1 & \pi \\ 3 & 1 & 4 \end{pmatrix}. What is the cofactor of entry π\pi?

  1. A.77
  2. B.−11-11
  3. C.−11π-11\pi
  4. D.−7π-7\pi
Deebii fi ibsa ilaalaa

Deebii: B −11-11

Maaliif

The entry π\pi is at row 2, column 3, so its cofactor is (−1)2+3det⁡(2−331)=−(2⋅1−(−3)⋅3)=−(2+9)=−11(-1)^{2+3} \det \begin{pmatrix} 2 & -3 \\ 3 & 1 \end{pmatrix} = -(2 \cdot 1 - (-3) \cdot 3) = -(2 + 9) = -11.

Gaaffii 5

When converting the recurring decimal 0.25‾0.\overline{25} into a fraction using convergent geometric series, then what are the first term and the common ratio respectively?

  1. A.0.250.25 and 0.010.01
  2. B.2525 and 0.010.01
  3. C.0.250.25 and 0.10.1
  4. D.2.52.5 and 0.10.1
Deebii fi ibsa ilaalaa

Deebii: A 0.250.25 and 0.010.01

Maaliif

The recurring decimal 0.25‾0.\overline{25} can be written as the infinite series 0.25+0.0025+0.000025+…0.25 + 0.0025 + 0.000025 + \dots, which is a geometric series with the first term a1=0.25a_1 = 0.25 and common ratio r=0.01r = 0.01.

Gaaffii 6

What are the maximum and minimum values respectively of z=3x+yz = 3x + y subject to constraints {3x+2y≤202x−y≥4x≥0,y≥0\begin{cases} 3x + 2y \leq 20 \\ 2x - y \geq 4 \\ x \geq 0, y \geq 0 \end{cases}?

  1. A.2020 and 66
  2. B.4040 and 66
  3. C.1616 and 66
  4. D.1818 and 88
Deebii fi ibsa ilaalaa

Deebii: A 2020 and 66

Maaliif

The vertices of the feasible region defined by the constraints are (2,0)(2, 0), (20/3,0)(20/3, 0), and (4,4)(4, 4). Evaluating z=3x+yz = 3x + y at these vertices gives 66, 2020, and 1616 respectively, making the maximum value 2020 and the minimum value 66.

Gaaffii 7

What is the domain of the function f(x)=3−x2f(x) = \sqrt{3 - x^2}?

  1. A.[3,∞)[\sqrt{3}, \infty)
  2. B.[−3,3][-\sqrt{3}, \sqrt{3}]
  3. C.[−3,3][-3, 3]
  4. D.(−∞,3]∪[3,∞)(-\infty, \sqrt{3}] \cup [\sqrt{3}, \infty)
Deebii fi ibsa ilaalaa

Deebii: B [−3,3][-\sqrt{3}, \sqrt{3}]

Maaliif

For the square root function f(x)=3−x2f(x) = \sqrt{3 - x^2} to be defined on real numbers, the radicand must be non-negative: 3−x2≥0  ⟹  x2≤33 - x^2 \ge 0 \implies x^2 \le 3, which gives the interval [−3,3][-\sqrt{3}, \sqrt{3}].

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