Waraqaalee qormaataa

Qormaata galuumsa kutaa 12ffaa Herregaa, bara 2016 A.L.I (2024)

Gaaffilee dhugaa qormaata galuumsa yunivarsiitii kutaa 12 kan Herregaa, akkuma barattoonni bara 2016 A.L.I itti fudhatanitti. Tokkoon tokkoon gaaffii deebii sirrii fi ibsa qaba.

Bilisaan shaakaluu jalqabaa Waraqaa gaaffii

Damee saayinsii uumamaa
Eenyutu fudhata
2016 A.L.I (2024)
Bara qormaataa
Gaaffilee 52 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

Let $p$ and $q$ be propositions. Which one of the following propositions is equivalent to the proposition $\neg p \lor (p \land q)$ ?

A. $\neg p \lor \neg q$
B. $p \Rightarrow q$
C. $\neg p \land q$
D. $\neg p \Rightarrow q$

Deebii fi ibsa ilaalaa

Deebii: B $p \Rightarrow q$

Maaliif

Using the distributive law, we have

\neg p \lor (p \land q) \equiv (\neg p \lor p) \land (\neg p \lor q) \equiv \text{True} \land (\neg p \lor q) \equiv \neg p \lor q

, which is equivalent to the conditional statement

p \Rightarrow q

.

Gaaffii 2

Which one of the following is NOT true about the function $f(x) = \tan^{-1} x$ ?

A.The line $x = -\frac{\pi}{2}$ is a vertical asymptote to the graph of $f$ .
B.The graph of $f$ is the image of the graph of $y = \tan x$ after it is reflected with respect to the line $y = x$ .
C.The graph of $f$ intersects the $x$ -axis at $(0, 0)$ .
D.The line $y = \frac{\pi}{2}$ is a horizontal asymptote to the graph of $f$ .

Deebii fi ibsa ilaalaa

Deebii: A The line $x = -\frac{\pi}{2}$ is a vertical asymptote to the graph of $f$ .

Maaliif

The domain of

f(x) = \tan^{-1} x

is the entire set of real numbers

\mathbb{R}

, and it has horizontal asymptotes at

y = \pm \frac{\pi}{2}

but no vertical asymptotes. The line

x = -\frac{\pi}{2}

is actually a vertical asymptote for

y = \tan x

, not for its inverse.

Gaaffii 3

Which one of the following statements is true?

A.If $x$ and $y$ are integers and $x$ divides $y$ , then $y$ divides $x$ .
B.If $x$ is an even integer and $y$ is an odd integer, then $x^2 + y^2$ is odd.
C.If $x$ and $y$ are even integers, then $x + y$ is an odd integer.
D.If $x$ and $y$ are positive integers, then $\frac{x}{y}$ is a positive integer.

Deebii fi ibsa ilaalaa

Deebii: B If $x$ is an even integer and $y$ is an odd integer, then $x^2 + y^2$ is odd.

Maaliif

If

x

is even, then

x^2

is even; if

y

is odd, then

y^2

is odd. The sum of an even number and an odd number,

x^2 + y^2

, is always odd.

Gaaffii 4

Let $p$ , $q$ , and $r$ be propositions. If $p$ is True , $q$ is False , and $r$ is False , which one of the following propositions has truth value True ?

A. $(p \land \neg q) \Leftrightarrow (p \lor r)$
B. $(q \land r) \Leftrightarrow (p \land \neg r)$
C. $p \Rightarrow (q \lor r)$
D. $(p \lor q) \Rightarrow r$

Deebii fi ibsa ilaalaa

Deebii: A $(p \land \neg q) \Leftrightarrow (p \lor r)$

Maaliif

Substituting the truth values, we have

p \land \neg q \equiv \text{True} \land \text{True} \equiv \text{True}

, and

p \lor r \equiv \text{True} \lor \text{False} \equiv \text{True}

. Thus,

(p \land \neg q) \Leftrightarrow (p \lor r) \equiv \text{True} \Leftrightarrow \text{True}

, which is True.

Gaaffii 5

Which one of the following is the equation of the tangent line to the graph of $f(x) = 2x^3 - 4x^2 + 5x + 3$ at the point $(0, 3)$ ?

A. $3x + 5y - 3 = 0$
B. $3x - 5y - 3 = 0$
C. $5x - y + 3 = 0$
D. $5x + y + 3 = 0$

Deebii fi ibsa ilaalaa

Deebii: C $5x - y + 3 = 0$

Maaliif

The derivative of

f(x)

is

f'(x) = 6x^2 - 8x + 5

, which gives a tangent slope of

f'(0) = 5

at

x = 0

. The equation of the tangent line is

y - 3 = 5(x - 0)

, which simplifies to

5x - y + 3 = 0

.

Gaaffii 6

Let $v = \vec{AB}$ and $w = \vec{CD}$ be vectors in the plane, where $A(1, -1)$ , $B(2, -2)$ , $C(-1, 3)$ , and $D(4, -3)$ . Which one of the following is equal to $v - w$ ?

A. $5i - 6j$
B. $i - 15j$
C. $-4i + 5j$
D. $4i - 5j$

Deebii fi ibsa ilaalaa

Deebii: C $-4i + 5j$

Maaliif

The vector

v = B - A = (1, -1) = i - j

and

w = D - C = (5, -6) = 5i - 6j

. Subtracting these gives

v - w = (i - j) - (5i - 6j) = -4i + 5j

.

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Kunneen gaaffilee dhugaa qormaata galuumsa Herregaa bara 2016 A.L.I ti?

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Qormaata galuumsa kutaa 12ffaa Herregaa, bara 2016 A.L.I (2024)

Gaaffilee qormaata kanaa yaalaa

Let ppp and qqq be propositions. Which one of the following propositions is equivalent to the proposition ¬p∨(p∧q)\neg p \lor (p \land q)¬p∨(p∧q)?

Which one of the following is NOT true about the function f(x)=tan⁡−1xf(x) = \tan^{-1} xf(x)=tan−1x?

Which one of the following statements is true?

Let ppp, qqq, and rrr be propositions. If ppp is True , qqq is False , and rrr is False , which one of the following propositions has truth value True ?

Which one of the following is the equation of the tangent line to the graph of f(x)=2x3−4x2+5x+3f(x) = 2x^3 - 4x^2 + 5x + 3f(x)=2x3−4x2+5x+3 at the point (0,3)(0, 3)(0,3)?

Let v=AB⃗v = \vec{AB}v=AB and w=CD⃗w = \vec{CD}w=CD be vectors in the plane, where A(1,−1)A(1, -1)A(1,−1), B(2,−2)B(2, -2)B(2,−2), C(−1,3)C(-1, 3)C(−1,3), and D(4,−3)D(4, -3)D(4,−3). Which one of the following is equal to v−wv - wv−w?

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Qormaatilee Herregaa kan waggoota biroo

Qormaata kana taa'uu dura Herregaa qayyabadhu

Let $p$ and $q$ be propositions. Which one of the following propositions is equivalent to the proposition $\neg p \lor (p \land q)$ ?

Which one of the following is NOT true about the function $f(x) = \tan^{-1} x$ ?

Let $p$ , $q$ , and $r$ be propositions. If $p$ is True , $q$ is False , and $r$ is False , which one of the following propositions has truth value True ?

Which one of the following is the equation of the tangent line to the graph of $f(x) = 2x^3 - 4x^2 + 5x + 3$ at the point $(0, 3)$ ?

Let $v = \vec{AB}$ and $w = \vec{CD}$ be vectors in the plane, where $A(1, -1)$ , $B(2, -2)$ , $C(-1, 3)$ , and $D(4, -3)$ . Which one of the following is equal to $v - w$ ?