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Qormaata galuumsa kutaa 12ffaa Herregaa, bara 2015 A.L.I (2023)

Gaaffilee dhugaa qormaata galuumsa yunivarsiitii kutaa 12 kan Herregaa, akkuma barattoonni bara 2015 A.L.I itti fudhatanitti. Tokkoon tokkoon gaaffii deebii sirrii fi ibsa qaba.

Bilisaan shaakaluu jalqabaa Waraqaa gaaffii

Damee lamaanuu
Eenyutu fudhata
2015 A.L.I (2023)
Bara qormaataa
Gaaffilee 57 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

If $f(x) = \frac{\sqrt{x + 2}}{x + 2}$ and $g(x) = \frac{1}{x} - 2$ then what is the value of $f(g(x))$ for all $x > 0$ ?

A.√x - 2
B.√x + 2
C.√x
D.1/√x

Deebii fi ibsa ilaalaa

Deebii: C √x

Maaliif

Substituting

g(x) = \frac{1}{x} - 2

into

f(x)

gives

f(g(x)) = \frac{\sqrt{(1/x - 2) + 2}}{(1/x - 2) + 2} = \frac{\sqrt{1/x}}{1/x} = \frac{x}{\sqrt{x}} = \sqrt{x}

for all

x > 0

.

Gaaffii 2

Which of the following is Not the property of $f(x) = x^{2/5}$ ?

A.f(xy) = f(x)f(y)
B.decreasing on (-∞,0]
C.the range is real number
D.the graph is symmetric with respect to the y-axis

Deebii fi ibsa ilaalaa

Deebii: C the range is real number

Maaliif

Since

f(x) = x^{2/5} = (x^{1/5})^2

is always non-negative for all real numbers

x

, its range is

[0, \infty)

, not the set of all real numbers.

Gaaffii 4

Which one of the following function is one to one correspondence function from A = [0,1] to B = [2,3]?

A.f(x)=x+2
B.f(x)=2x+2
C.f(x)=x+3
D.f(x)=2x+1

Deebii fi ibsa ilaalaa

Deebii: A f(x)=x+2

Maaliif

For a function to be a bijection from

[0, 1]

to

[2, 3]

, it must map the endpoints to the endpoints. For

f(x) = x + 2

, we have

f(0) = 2

and

f(1) = 3

, and since it is strictly increasing and linear, it is a one-to-one correspondence.

Gaaffii 6

Suppose $f(x) = \frac{Q(x)}{x(x^2 - 1)}$ where $Q(x)$ is a quadratic function. Which one of the following is necessarily true about the graph $f$ ?

A.x=0, x=1 and x=-1 are vertical asymptote of the graph f
B.The graph of f does not intersect with its horizontal asymptote.
C.The vertical asymptote of the graph of f is only x=-1, if Q(x)=x^2-x
D.The vertical asymptote of the graph of f is only x=1, if Q(x)=2x^2

Deebii fi ibsa ilaalaa

Deebii: C The vertical asymptote of the graph of f is only x=-1, if Q(x)=x^2-x

Maaliif

If

Q(x) = x^2 - x = x(x-1)

, the common factors

x

and

x-1

in the numerator and denominator cancel out, leaving

f(x) = \frac{1}{x+1}

(with holes at

x=0

and

x=1

). Thus, the only vertical asymptote is

x = -1

.

Gaaffii 7

What is the solution set of $\frac{2}{x} = \frac{x - 2}{x^2 - 2x} = 1 - \frac{2x - 2}{3x - 2}$ ? (Note: The equation has two equals signs, likely a typo; probably it is $\frac{2}{x} = \frac{x - 2}{x^2 - 2x}$ and also equals the other expression. We solve as two equations.)

A.{-2,1}
B.{1,2}
C.{-1}
D.{1}

Deebii fi ibsa ilaalaa

Deebii: D {1}

Maaliif

By correcting the typo where the first minus sign was written as an equals sign, the equation

\frac{2}{x} - \frac{x-2}{x^2-2x} = 1 - \frac{2x-2}{3x-2}

simplifies to

\frac{1}{x} = \frac{x}{3x-2}

, which yields

x = 1

and

x = 2

. Since

x = 2

is excluded from the domain of the original expression, the only valid solution is

x = 1

.

Gaaffii 8

Given line $4y = 3x + 24$ and another line $\ell$ parallel to this line at a distance of 4 units from it. A possible equation for $\ell$ is

A.y = (3/4)x
B.y = (3/4)x + 1
C.y = (3/4)x + 6
D.y = (3/4)x - 6

Deebii fi ibsa ilaalaa

Deebii: B y = (3/4)x + 1

Maaliif

The distance between parallel lines

3x - 4y + 24 = 0

and

3x - 4y + C = 0

is given by

\frac{|C - 24|}{\sqrt{3^2 + (-4)^2}} = 4

, which simplifies to

|C - 24| = 20

. This gives

C = 4

, resulting in the line

3x - 4y + 4 = 0

, or

y = \frac{3}{4}x + 1

.

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Gaaffilee 57 kuusaa kana keessa jiran hunda akka waraqaa qormaataa maxxanfameetti, furtuu deebii ibsa 57 waliin immoo akka faayilii lammaffaatti. Maxxansaa yookiin daataa malee dubbisaa.

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Qormaata galuumsa kutaa 12ffaa Herregaa, bara 2015 A.L.I (2023)

Gaaffilee qormaata kanaa yaalaa

If f(x)=x+2x+2f(x) = \frac{\sqrt{x + 2}}{x + 2}f(x)=x+2x+2​​ and g(x)=1x−2g(x) = \frac{1}{x} - 2g(x)=x1​−2 then what is the value of f(g(x))f(g(x))f(g(x)) for all x>0x > 0x>0 ?

Which of the following is Not the property of f(x)=x2/5f(x) = x^{2/5}f(x)=x2/5?

Which one of the following function is one to one correspondence function from A = [0,1] to B = [2,3]?

Suppose f(x)=Q(x)x(x2−1)f(x) = \frac{Q(x)}{x(x^2 - 1)}f(x)=x(x2−1)Q(x)​ where Q(x)Q(x)Q(x) is a quadratic function. Which one of the following is necessarily true about the graph fff ?

Given line 4y=3x+244y = 3x + 244y=3x+24 and another line ℓ\ellℓ parallel to this line at a distance of 4 units from it. A possible equation for ℓ\ellℓ is

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Qormaatilee Herregaa kan waggoota biroo

Qormaata kana taa'uu dura Herregaa qayyabadhu

If $f(x) = \frac{\sqrt{x + 2}}{x + 2}$ and $g(x) = \frac{1}{x} - 2$ then what is the value of $f(g(x))$ for all $x > 0$ ?

Which of the following is Not the property of $f(x) = x^{2/5}$ ?

Suppose $f(x) = \frac{Q(x)}{x(x^2 - 1)}$ where $Q(x)$ is a quadratic function. Which one of the following is necessarily true about the graph $f$ ?

Given line $4y = 3x + 24$ and another line $\ell$ parallel to this line at a distance of 4 units from it. A possible equation for $\ell$ is